# Palindrome Partitions and the Calkin-Wilf Tree

David J. Hemmer  Department of Mathematical Sciences
Michigan Technological University
Houghton, MI 49931<EMAIL_ADDRESS>and  Karlee J. Westrem  Department of
Mathematical Sciences
Michigan Technological University
Houghton, MI 49931<EMAIL_ADDRESS>
(Date: February, 2024)

###### Abstract.

There is a well-known bijection between finite binary sequences and integer
partitions. Sequences of length $r$ correspond to partitions of perimeter
$r+1$. Motivated by work on rational numbers in the Calkin-Wilf tree, we
classify partitions whose corresponding binary sequence is a palindrome. We
give a generating function that counts these partitions, and describe how to
efficiently generate all of them. Atypically for partition generating
functions, we find an unusual significance to prime degrees. Specifically, we
prove there are nontrivial palindrome partitions of $n$ except when $n=3$ or
$n+1$ is prime. We find an interesting new “branching diagram” for partitions,
similar to Young’s lattice, with an action of the Klein four group
corresponding to natural operations on the binary sequences.

##  1\. Introduction

Recall that $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{r})$ is a
partition of $n$, denoted $\lambda\vdash n$, if
$\lambda_{1}\geq\lambda_{2}\cdots\geq\lambda_{r}>0$ are positive integers with
$\sum_{i=1}^{r}\lambda_{i}=n$. To $\lambda$ we associate its _Young diagram_
$[\lambda]$. It is well-known that we can encode partitions by infinite binary
sequences, unbounded in each direction, which begin with infinitely many zeros
and end with infinitely many ones, see for example [StanleyEC2, Exercise 7.59]
or [ArmstrongResultsandconjectures]. This encoding can be used, for example,
to prove the important fact in representation theory of the symmetric group
that the $p$-core of a partition is unique. To encode a partition $\lambda$,
draw the Young diagram of $\lambda$ with the left hand vertical edge and upper
horizontal edge extended to infinity. Label the southeast boundary of the
diagram, including the horizontal and vertical rays, with a zero next to each
vertical edge and a one next to each horizontal edge. To obtain the
corresponding binary sequence $C_{\lambda}$, simply read the numbers moving
from southwest to northeast. For example in Figure 1.1 we see that if
$\lambda=(5,5,3,3,1)$ then $C_{\lambda}=\cdots 00101100110011\cdots.$

11$\cdots$0001100110100$\vdots$ Figure 1.1. The coding $C_{\lambda}$ of the
partition $\lambda=(5,5,3,3,1)$

After the infinitely many zeros, $C_{\lambda}$ has an initial one,
corresponding to the first rightward move, and a final zero, corresponding to
the last upwards move, before the infinite sequence of ones.

######  Definition 1.1.

Define $B(\lambda)$ to be the finite binary sequence between this initial one
and final zero. So in Figure 1.1, we see that $B(5,5,3,1)=01100110$, labelled
in red.

Of course one can easily recover $\lambda$ from $B(\lambda)$. Given a binary
sequence $C$ we denote the corresponding partition by $P(C)$, so
$\lambda=P(B(\lambda))$. Notice that the partition $\tau=(1)$ corresponds to
the empty sequence $B(\tau)=\emptyset$, so $P(\emptyset)=(1)$.

Suppose we have a sequence $C$ with $A$ zeros and $B$ ones. It is clear from
Figure 1.1, that $B+1$ is the first part of $P(C)$ and $A+1$ is the number of
nonzero parts of $P(C)$.

There are two natural operations on these binary sequences. The _reverse_
sequence just reverses the order in the sequence, and we denote it by $C^{r}$.
The _inverse_ of a sequence is obtained by switching zeros and ones, and is
denoted $\overline{C}$.

######  Remark 1.2.

There is an equivalent way to define $B(\lambda)$ in terms of operations on
the Young diagram. Start with the Young diagram of $\lambda$. If $[\lambda]$
has a single box in the last row (i.e. if the last part is equal to 1), then
remove it and record a “1”. If not, then remove the entire first column from
the Young diagram (i.e. subtract one from every part), and record a “0”.
Continue until only one box remains. The sequence you obtain will be
$B(\lambda)$.

Reversing $B(\lambda)$ has a nice geometric description in terms of Young
diagrams:

######  Proposition 1.3.

Let $\lambda$, $B(\lambda)$, $A$ and $B$ be as above. Consider the boxes not
in the first row or column as a partition $\tilde{\lambda}$ sitting inside an
$A\times B$ rectangle. Replace $\tilde{\lambda}$ with the partition obtained
by taking its complement inside the $A\times B$ rectangle and rotating it 180
degrees, while preserving the first row and column of $\lambda$. What remains
will be the Young diagram for $P(B(\lambda)^{r})$.

###### Proof.

This is clear from Figure 1.2. When we rotate the rectangle by 180 degrees,
vertical edges remain vertical, horizontal edges remain horizontal. So we can
simply rotate all the labels (in red) and the new partition will be correctly
labelled with binary sequence equal to $B(\lambda)^{r}$. ∎

######  Example 1.4.

Consider the sequence 101011. Check that for $\lambda=(3,3,3,2,1)$ we have
$B(\lambda)=101011$. Following the algorithm in Prop. 1.3, we see that the
partition $(3,3,2,1,1)$ corresponds to the reverse sequence 101011. This is
illustrated in Figure 1.2. Notice that $n$ is not typically preserved by this
operation.

00010101$B(3,3,3,2,1)={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}010100}$00101001$B(3,3,2,1,1)={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}001010}$
Figure 1.2. Calculating the partition corresponding to the reverse sequence

##  2\. The Calkin-Wilf Tree

We are going to be studying properties of the sequence $B(\lambda)$, and in
particular when it is a palindrome. This question was motivated by work of
[KenyonCalkinWilfThesis] on binary sequences and the Calkin-Wilf tree. This is
an infinite perfect binary tree with each vertex labelled by a rational
number. One can also label this tree by binary sequences in a natural way, and
also by Young diagrams of partitions. We give a quick description in this
section as motivation, but it is not necessary to understand our results.

The Calkin-Wilf tree, $T_{\mathbb{Q}}$, and corresponding Calkin-Wilf
sequence, is defined in [CalkinWilfRecountingTheRationals], although it has
appeared also much earlier. It is an infinite binary tree with each vertex
labelled by a positive rational number written in lowest terms. It is defined
inductively as follows. The root is $\frac{1}{1}$. The left child of
$\frac{p}{q}$ is $\frac{p}{p+q}$ and the right child is $\frac{p+q}{q}$. Part
of the tree is shown on the right side of Figure 2.1. It is not difficult to
prove that every positive rational number appears exactly once, and that if
$\frac{p}{q}$ is a reduced fraction, then so are both of its children. To
obtain the Calkin-Wilf sequence, read left to right top to bottom, obtaining
$\\{1,\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{2},\frac{2}{3},\frac{3}{1},\ldots\\}$.
Let $l(n)$ be the $n$-th term in the Calkin-Wilkin sequence. In
[GraverListingthepositiverationals], the author gives a method to calculate
$l(n)$ without generating the entire tree above it. In the opposite direction,
in [GoblerListingTheRationals] the author gives a method to start with a
positive rational number and locate its position in the sequence, and hence on
the tree. One first computes the simple continued fraction expansion of the
rational number, and then uses it to read off the binary sequence, using the
tree on the left of Figure 2.1 to locate the fraction on $T_{\mathbb{Q}}$. In
Figure 2.1 we give, on the left, the infinite binary tree labeled by binary
sequences starting with 1. So the root is labelled 1, the left child
corresponds to appending a zero and the right child to appending a one to the
end of the sequence.

$1/2$$1/3$$1/4$$4/3$$3/2$$3/5$$5/2$$2/1$$2/3$$2/5$$5/3$$3/1$$3/4$$4/1$

Figure 2.1. The Binary Tree and Calkin-Wilf tree

Ignoring the initial 1 on each label, we have a bijection between binary
sequences of length $n$ and entries in the tree at level $n+1$. For example
the binary sequence 001 corresponds to the fraction $4/3$, as labelled in red.

In [KenyonCalkinWilfThesis], Kenyon studies the correspondence between binary
sequences and rational numbers illustrated in Figure 2.1. She calls the
sequences “paths”, as they encode the path from the root vertex to the vertex
in question, with zero representing left and one representing right. Again
note that she does not consider the initial one as part of the path, the
rational number $\frac{1}{1}$ corresponds to an empty path.

It is an easy exercise that the inverse paths (sequences) corresponds to
reciprocal fractions. It is not at all clear what reversing the path does to
the corresponding rational numbers.

Kenyon gathers data on paths that are equal to their reverse, called
_palindromic paths_ , and those whose reverse path is equal to the inverse
path, called _antipalindromic paths_. So, for example, $0110$ is a palindromic
path and corresponds to the rational number $5/7$. The path $0101$ is an
antipalindromic path, corresponding to the rational number $8/5$.

We emphasize that the tree on the left of Figure 2.1 is labelled in a way to
match the previous literature, but the reader should keep in mind when we
discuss paths, palindromic paths, etc. that the initial one is not included.

###  2.1. Partitions in the Calkin-Wilf Tree

From Remark 1.2 we see there is a natural way to place Young diagrams at the
nodes of a binary tree. Start with the partition $\lambda=(1)$ at the root.
For any partition $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{t}),$ let its left child
be $(\mu_{1},\mu_{2},\ldots,\mu_{t},1)$ and its right child be
$(\mu_{1}+1,\mu_{2}+1,\ldots,\mu_{t}+1)$. Then following the unique path from
a diagram upward to the root will correspond to the operation described in
Remark 1.2. This is illustrated in Figure 2.2

centertableaux 1centertableaux1,1centertableaux 1,1,1centertableaux
1,1,1,1centertableaux 2,2,200011centertableaux 2,2centertableaux
2,2,1centertableaux 3,3centertableaux2centertableaux 2,1centertableaux
2,1,1centertableaux 3,2centertableaux 3centertableaux 3,1centertableaux 4
Figure 2.2. Partitions on the Calkin-Wilf Tree

######  Remark 2.1.

For the Young diagrams in Figure 2.2, the diagram $[\lambda]$ occupies the
same spot as the binary sequence which is $B(\lambda)^{r}$. This is because we
defined $B(\lambda)$ reading from southwest to northeast to match the existing
literature. For example if $\lambda=(2,2,2)$ we see that $B(\lambda)=100$ but
$\lambda$ occupies the same position as $001$ in the tree. The choice of zero
or one for vertical/horizontal moves is arbitrary, as is the choice of which
direction to read the sequence off, but the results are all comparable
independent of these choices.

For a partition $\lambda\vdash n$, recall the conjugate or transpose partition
$\lambda^{\prime}\vdash n$ can be obtained by swapping rows and columns in the
Young diagram. For a box $(i,j)$ in the Young diagram, the $(i,j)$ hook length
is defined as $\lambda_{i}-i+\lambda^{\prime}_{j}-j+1$.

######  Proposition 2.2.

  1. (1)

Let $\lambda\vdash n$. Then $B(\lambda^{\prime})=\overline{B(\lambda)^{r}}$ is
the inverse of the reverse of $B(\lambda)$. Thus, antipalindromic paths
correspond to self-conjugate partitions $\lambda=\lambda^{\prime}$.

  2. (2)

Given $\lambda$ and $B(\lambda)$, to find the partition $P(B(\lambda)^{r})$
corresponding to the reverse sequence of $B(\lambda)$ one can use the
algorithm described in Prop. 1.3.

  3. (3)

Row $t$ of Figure 2.2 contains all partitions with first column hook length
$h_{11}(\lambda)=t$. This value is also sometimes called the perimeter of the
partition.

######  Remark 2.3.

From Proposition 2.2 we see that we have an action of the Klein four group on
the diagram in Figure 2.2. The operations come from the the operations of
inverse (I) and reversal (R) on the corresponding sequences. We have seen that
$R$ fixes the first row and column while replacing the rest with its
complement inside a rectangle, whereas $R\circ I=I\circ R$ takes the transpose
of the partition. Thus $I$ does both operations, taking the transpose and then
performing the complement operation.

##  3\. Palindrome Partitions

Say a partition $\lambda\vdash n$ is a _palindrome partition_ if the sequence
$B(\lambda)$ is a palindrome, i.e. $B(\lambda)=B(\lambda)^{r}$. For example in
Figure 1.1, we see that $B(55331)=01100110$, so $(5,5,3,1)$ is a palindrome
partition. Define $PP(n)$ to be the number of palindrome partitions of $n$.
Observe that the partitions $(n)$ and $(1^{n})$ are palindromes, corresponding
to the sequences of all ones and all zeros respectively, so $PP(n)\geq 2$.
Table 3.1 gives the sequence $PP(n)$ up to $n=40$. This sequence does not yet
appear in the Online Encyclopedia of Integer Sequences at oeis.org. The values
of $n+1$ where $PP(n)=2$ already suggests something interesting is happening!

Table 3.1. Count of Palindrome Partitions $n$ |  $PP(n)$ | $n$ |  $PP(n)$ | $n$ |  $PP(n)$ | $n$ | $PP(n)$
---|---|---|---|---|---|---|---
1 | 1 | 11 | 10 | 21 | 12 | 31 | 38
2 | 2 | 12 | 2 | 22 | 2 | 32 | 34
3 | 2 | 13 | 8 | 23 | 36 | 33 | 18
4 | 2 | 14 | 10 | 24 | 12 | 34 | 46
5 | 4 | 15 | 10 | 25 | 14 | 35 | 104
6 | 2 | 16 | 2 | 26 | 24 | 36 | 2
7 | 4 | 17 | 18 | 27 | 36 | 37 | 20
8 | 4 | 18 | 2 | 28 | 2 | 38 | 46
9 | 6 | 19 | 20 | 29 | 60 | 39 | 108
10 | 2 | 20 | 16 | 30 | 2 | 40 | 2

We will use the description in Proposition 1.3 to compute a generating
function $\sum_{n=0}^{\infty}PP(n)q^{n}$ for $PP(n)$. We will sum over all
possible choices for $\lambda_{1}$ and $\lambda^{\prime}_{1}$. Let “Case 1” be
when both both $\lambda_{1}$ and $\lambda^{\prime}_{1}$ are odd, so $A$ and
$B$ are both even. Let $A=2k$ and $B=2l$. We must count partitions $\mu$ in
the $2k\times 2l$ rectangle so that $\mu$ is equal to its complement inside
the rectangle. Dividing the large rectangle into 4 rectangles of size $k\times
l$, it is clear that the upper left rectangle must be included in $\mu$ and
the lower right rectangle must be empty. We can select an arbitrary partition
$\tau$ for lower left rectangle, and then its complement in the $k\times l$
rectangle for the upper right, as shown on the left side of Figure 3.1. We
have $\mu$ occupying $2kl$ boxes. It is well know that there are
$\binom{k+l}{k}$ choices for $\tau$ fitting inside a $k\times l$ rectangle. We
also have $2k+2l+1$ boxes in the first row and column. Summing over all
possible $k$ and $l$ we have a contribution to the generating function of:

(3.1) $\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\binom{k+l}{k}q^{2kl+2k+2l+1}.$

Now suppose $\lambda_{1}$ is odd and $\lambda^{\prime}_{1}$ is even, call this
“Case 2a.” So we are considering $\mu$ inside a $2k+1\times 2l$ box as shown
on the right side of Figure 3.1. It is clear that the shaded $1\times l$
rectangle must be entirely included in $\mu$. If it were not there would be a
shaded box in the $1\times l$ rectangle to its right with an unshaded box to
its left, and $\mu$ would not be a valid palindrome partition. So in this case
we have $\mu$ filling up $(2k+1)l$ boxes, $\binom{k+l}{k}$ choices for $\tau$
and $2k+2l+2$ boxes in the first row and column. Summing over all possible $k$
and $l$ we have a contribution to the generating function of:

(3.2) $\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\binom{k+l}{k}q^{2kl+2k+3l+2}.$

Now “Case 2b” is when $\lambda_{1}$ is even and $\lambda^{\prime}_{1}$ is odd.
This would simply give the transpose of the diagram on the right of Figure
3.1, and gives another contribution equal to (3.2), as the sum is symmetric in
$k$ and $l$.

There is no “Case 3” where both $\lambda_{1}$ and $\lambda^{\prime}_{1}$ are
even, as there can be no palindrome partitions as $\mu$ would need to fill
half of the boxes in an odd by odd rectangle. The three cases are summarized
in Table 3.2.

Table 3.2. Possible parities for $\lambda_{1}$ and $\lambda^{\prime}_{1}$ Case |  $A=\lambda^{\prime}_{1}-1=\\#$ 0’s |  $B=\lambda_{1}-1=\\#$ 1’s | n | 2(n+1)
---|---|---|---|---
1 | 2k | 2l | 2k+2l+2kl+1 | 4(k+1)(l+1)
2a. | 2k+1 | 2l | 2k+3l+2kl+2 | (2l+2)(2k+3)
2b. | 2k | 2l+1 | 3k+2l+2kl+2 | (2k+2)(2l+3)

Thus we have proven:

######  Theorem 3.1.

We have the following generating function:

$\sum_{n=0}^{\infty}PP(n)q^{n}=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\binom{k+l}{k}q^{2kl+2k+2l+1}+2\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\binom{k+l}{k}q^{2kl+2k+3l+2}.$

kkll$\mu=$kk$\tau$$\tau$$\tau$$\tau$ll1 Figure 3.1. Counting Palindrome
Partitions

In Table 3.2 we added a fourth column showing the factorization of $n+1$. We
can now use this to explain why $PP(n)=2$ occurs only when $n=3$ or $n+1$ is
prime:

######  Theorem 3.2.

The number $PP(n)=2$ if and only if $n=3$ or $n+1$ is prime.

###### Proof.

Consider the factorizations of $n+1$ in Table 3.2. If $n+1$ is prime, we are
either in Case 2a with $l=0$ or Case 2b with $k=0$, corresponding to the
partitions $(1^{n})$ and $(n)$ respectively. In this case there are no other
palindrome partitions. Now suppose $n+1$ is not prime, so let $n+1=xy$ with
$x,y>1$. We will explain how to find additional palindrome partitions of $n$.

If $n+1$ is odd, then so are $x$ and $y$. We can find $\lambda$ in Case 2a
with $l=x-1\geq 2$ and $k=(y-3)/2\geq 0$. Then any of the $\binom{k+l+1}{k}$
choices for $\tau$ will produce a palindrome partition $\lambda\vdash n$ not
equal to $(n)$ or $(1^{n})$. A corresponding choice in Case 2b will give the
conjugate $\lambda^{\prime}$. In the smallest example where $x=y=3$, we can
choose $k=0$, $l=2$ or $k=2,l=0$ and we obtain $\lambda=(5,3)$ and its
conjugate $(2,2,2,1,1)$.

Finally suppose $n+1=2x$. If $x>2$ is even, we can choose $k=1$, $l=x/2>1$ in
Case 1, to obtain a palindrome partition $\lambda\vdash n$ not equal to $(n)$
or $(1^{n})$. If $x\geq 3$ is odd, we can choose $l=1$ and $k=(x-3)/2$ in Case
2a to obtain a palindrome partition $\lambda\vdash n$ not equal to $(n)$ or
$(1^{n})$.

This leaves only the special case where $n+1=4$, where we see that $n+1$ is
not prime but the only palindrome partitions of $n=3$ are $(3)$ and $(1,1,1)$.
∎

######  Remark 3.3.

The sequence of integers $n$ with $PP(n)=2$, given by
$\\{2,3,4,6,10,12,16,18,22,\ldots\\}$, is A068499 at oeis.org : “Numbers $n$
such that $m!$ reduced modulo $(n+1)$ is not zero.” Of course this differs in
only one term from the sequence of $n$ with $n+1$ prime.

##  4\. Generating Palindrome Partitions of $n$

We have seen that, given $n$, there are always two palindrome partitions of
$n$ corresponding to sequences $B(\lambda)$ of length $n-1$, namely the all
zero sequence corresponding to $(1^{n})$ and the all one sequence
corresponding to $(n)$. If we want to generate all palindrome partitions of
$n$, we need to know what other sequence lengths can occur, and for each
length, how many zeros and ones are included. This will specify the first part
$\lambda_{1}$ and the first column $\lambda^{\prime}_{1}$, and values for $k$
and $l$. We simply choose all possible $\mu$ as in the proof of Theorem 3.1,
to obtain a list of all palindrome partitions of $n$.

Table 4.1. Palindrome partitions of $n=11$ $\lambda$ | 11 | 7,4 | 5,5,1 | 5,4,2 | 5,3,3 | $3^{3},1^{2}$ | $3^{2},2^{2},1$ | $3,2^{4}$ | $2^{4},1^{3}$ | $1^{11}$
---|---|---|---|---|---|---|---|---|---|---
$B(\lambda)$ | $1^{10}$ | $1^{3}01^{3}$ | $01^{4}0$ | 101101 | 110011 | 001100 | 010010 | $10^{4}1$ | $0^{3}10^{3}$ | $0^{10}$
$l(B(\lambda))$ | 10 | 7 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 10

######  Definition 4.1.

Let $PL(n)$ be the number of lengths among all sequences $B(\lambda)$ as
$\lambda$ runs over all palindrome partitions of $n$. This is also the number
of distinct perimeters among all palindrome partitions of $n$.

So in Table 4.1, we see that $PL(11)=3$ corresponding to the lengths 6,7, 10.
For a given length $m$, we have $m=A+B$, where the sequence has $A$ zeros and
$B$ ones. Recall that swapping zeros and ones gives a sequence also
corresponding to a palindrome partition of $n$. We now classify precisely
which lengths occur and prove that, for each possible length, the split into
zeros and ones is unique (up to swapping zeros and ones).

######  Theorem 4.2.

$PL(n)$ is the number of factorizations $xy=2(n+1)$ where $0<x\leq y\leq n$.
This is the sequence https://oeis.org/A211270 shifted by one. Moreover,
suppose there is a palindrome partition $\lambda\vdash n$ with $B(\lambda)$
having length $m=A+B$ with $A$ zeros and $B$ ones. Then any other palindrome
partition $\mu\vdash n$ with $B(\mu)$ of length $m$ must have $A$ zeros and
$B$ ones or $B$ zeros and $A$ ones.

###### Proof.

Suppose we have a palindrome partition $\lambda$ of $n$ where $B(\lambda)$ has
length $m=A+B$ as in Table 3.2. Thus $k$ and $l$ are determined and we obtain
a factorization as desired from the final column.

Conversely, suppose we have such a factorization $xy=2(n+1)$. We prove there
is a unique corresponding length $m$ and decomposition $m=A+B$. It is the case
that we have one odd factor or no odd factor. If one of the factors is odd,
say $x$, then $x=2l+3$ and $y=2k+2$, as in case $2$. Then it follows that
$A=2k=y-2$ and $B=2l=x-3$. If neither factor is odd, then $x=2l+2$ and
$y=2k+2$ which is case $1$. So $A=x-2$ and $B=y-2$, and we have a palindrome
partition with $B(\lambda)$ of length $A+B$. Suppose that for a factorization
of $2(n+1)$ we have that both $x$ and $y$ are even where $m=2k+2l$ and also
$m=2\tilde{k}+2\tilde{l}$ for some $k,l,\tilde{k},\tilde{l}$ where
$k\neq\tilde{k}$ and $l\neq\tilde{l}$. It could be the case that $k=\tilde{l}$
and $l=\tilde{k}$, which would be interchanging the zeros and ones. Consider a
nontrivial solution where $k+l=\tilde{k}+\tilde{l}$. Then
$2(n+1)=(2k+2)(2l+2)=(2\tilde{k}+2)(2\tilde{l}+2)$ which simplifies to
$kl=\tilde{k}\tilde{l}$. Then

$\displaystyle kl$ $\displaystyle=\tilde{k}\tilde{l}$ $\displaystyle kl+l^{2}$
$\displaystyle=\tilde{k}\tilde{l}+l^{2}$ $\displaystyle l(k+l)$
$\displaystyle=\tilde{k}\tilde{l}+l^{2}$ $\displaystyle
l(\tilde{k}+\tilde{l})$ $\displaystyle=\tilde{k}\tilde{l}+l^{2}$
$\displaystyle(l-\tilde{l})(\tilde{k}-l)$ $\displaystyle=0.$

Thus $l=\tilde{l}$ or $\tilde{k}=l$. Now suppose for a factorization of
$2(n+1)$ that $x$ is odd without loss of generality, which would mean
$m=2k+2l+1$ and $m=2\tilde{k}+2\tilde{l}+1$ for some $k,l,\tilde{k},\tilde{l}$
where $k\neq\tilde{k}$ and $l\neq\tilde{l}$. Again, the case that
$k=\tilde{l}$ and $l=\tilde{k}$, which would be interchanging the zeros and
ones. Suppose we have a nontrivial solution where $k+l=\tilde{k}+\tilde{l}$.
Then $2(n+1)=(2k+2)(2l+3)=(2\tilde{k}+2)(2\tilde{l}+3)$ which simplifies to
$2kl+k=2\tilde{k}\tilde{l}+\tilde{k}$. Then

$\displaystyle 2kl+k$ $\displaystyle=2\tilde{k}\tilde{l}+\tilde{k}$
$\displaystyle 2kl+k-k+2k^{2}$
$\displaystyle=2\tilde{k}\tilde{l}+\tilde{k}-k+2k^{2}$ $\displaystyle 2k(l+k)$
$\displaystyle=2\tilde{k}\tilde{l}+\tilde{k}-k+2k^{2}$ $\displaystyle
2k(\tilde{l}+\tilde{k})$
$\displaystyle=2\tilde{k}\tilde{l}+\tilde{k}-k+2k^{2}$
$\displaystyle(\tilde{k}-k)(2k-1-2\tilde{l})$ $\displaystyle=0.$

Thus $k=\tilde{k}$ or $2k=2\tilde{l}+1$. Therefore, it’s the case that for
each word length of $m$, there is a unique number of zeros and ones, up to
swapping zeros and ones. ∎

######  Example 4.3.

We demonstrate how to use the results above to efficiently generate all
$PP(29)=60$, palindrome partitions of $n=29$. There are five relevant
factorizations of $2(n+1)=30$. For each, we can compute $k,l,A,B$ as in the
proof above, using Table 3.2. Once we have these, the first row $\lambda_{1}$
and column $\lambda^{\prime}_{1}$ are fixed. Then we have $\binom{k+l}{l}$
choices for the partition $\tau$ as in Figure 3.1. Table 4.2 shows the
results, where the entires in the penultimate column sum to 60 as expected.

Table 4.2. Generating Palindrome Partitions of 29 $60=x\cdot y$ | l | k | # of zeros | # of ones | # of $\lambda$ | Example
---|---|---|---|---|---|---
$2\cdot 30$ | 0 | 14 | 0 | 28 | 1 | $(29)$
$2\cdot 30$ | 14 | 0 | 28 | 0 | 1 | $(1^{29})$
$3\cdot 20$ | 9 | 0 | 1 | 18 | 1 | $(19,10)$
$3\cdot 20$ | 0 | 9 | 18 | 1 | 1 | $(2^{10},1^{9})$
$4\cdot 15$ | 6 | 1 | 2 | 13 | $\binom{7}{1}$ | $(14,14,1)$
$4\cdot 15$ | 1 | 6 | 13 | 2 | $\binom{7}{1}$ | $(3,2^{13})$
$5\cdot 12$ | 5 | 1 | 3 | 10 | $\binom{6}{1}$ | $(11,11,6,1)$
$5\cdot 12$ | 1 | 5 | 10 | 3 | $\binom{6}{1}$ | $(4,3^{5},2^{5})$
$6\cdot 10$ | 4 | 2 | 4 | 8 | $\binom{6}{2}$ | $(9^{3},1,1)$
$6\cdot 10$ | 2 | 4 | 8 | 4 | $\binom{6}{2}$ | $(5^{5},1^{4})$

##  5\. Partitions with weight fixed by reversal

As we mentioned in Remark 2.3, sequence reversal gives an action of order two
on all partitions given by $\lambda\rightarrow P(B(\lambda)^{r})$. The
palindrome partitions are fixed by this operation. In this section, we
consider partitions where $\lambda$ and $P(B(\lambda)^{r})$ are both
partitions of the same integer. In other words, these partitions remain in the
same row of Figure 2.2 under this action. Let $R(n)$ be the number of
partitions $\lambda\vdash n$ where $P(B(\lambda)^{r})\vdash n$.

Computing $R(n)$ is similiar to counting $PP(n)$, but in Figure 3.1 we no
longer have the requirement that the partition be equal to the rotation of its
complement. Instead we require that it fill exactly half of the boxes in the
$2k\times 2l$ or $(2k+1)\times 2l$ rectangle. (As with the palindrome case, if
the rectangle has an odd number of boxes then there are no partitions to
count).

Recall that the q-binomial coefficient ${n\brack k}_{q}$ is the generating
function for partitions that fit inside an $(n-k)\times k$ rectangle, and is
called a Gaussian polynomial. The coefficient of $q^{m}$ counts partitions of
$m$ fitting inside a $(n-k)\times k$ rectangle, so it has leading term
$q^{k(n-k)}$. In place of the binomial coefficients in Theorem 3.1, we will
need the coefficient of the middle degree term in this polynomial. There is a
nice description of this coefficient in the case either $k$ or $n-k$ is even,
which we prove here.

######  Definition 5.1.

Let $T(n,k)$ be the number of nondecreasing sequences of length n, with
integer entries in $[-k,k]$, summing to zero.

For example if $n=5$ and $k=4$ a possible sequence would be
$\\{-4,-2,1,1,4\\}$. The array $T(n,k)$ is given as sequence A183917 in the
OEIS.

######  Proposition 5.2.

The coefficient of $q^{kl}$ in the Gaussian polynomial ${2k+l\brack l}_{q}$ is
$T(l,k)$.

###### Proof.

The coefficient in question counts all partitions of $kl$ which fit inside an
$l\times 2k$ rectangle, i.e. have at most $l$ parts, each less than or equal
to $2k$. We give a bijection to the sequences counted by $T(l,k)$. Given such
a partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{l})\vdash kl$,
the corresponding sequence is
$\\{\lambda_{l}-k,\lambda_{l-1}-k,\lambda_{l-2}-k,\ldots,\lambda_{1}-k\\}$.
For example the sequence above corresponds to the partition $(8,5,5,2,0)$
inside the $5\times 8$ rectangle. ∎

We can now write the generating function for $R(n)$ as in Theorem 3.1. The
first term is from the $2k\times 2l$ rectangle so we get $T(2l,k)$. The second
term is the $(2k+1)\times 2l$ rectangle, which has the same count as a
$2l\times(2k+1)$ rectangle, namely $T(k+1,l)$. Thus:

######  Theorem 5.3.

We have the following generating function:

$\sum_{n=0}^{\infty}R(n)q^{n}=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}T(2l,k)q^{2kl+2k+2l+1}+2\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}T(k+1,l)q^{2kl+2k+3l+2}.$

Table 5.1. Partitions with weight fixed by reversal $n$ |  $R(n)$ | $n$ |  $R(n)$ | $n$ |  $R(n)$ | $n$ | $R(n)$
---|---|---|---|---|---|---|---
1 | 1 | 11 | 10 | 21 | 12 | 31 | 76
2 | 2 | 12 | 2 | 22 | 2 | 32 | 90
3 | 2 | 13 | 8 | 23 | 52 | 33 | 18
4 | 2 | 14 | 14 | 24 | 28 | 34 | 198
5 | 4 | 15 | 10 | 25 | 14 | 35 | 320
6 | 2 | 16 | 2 | 26 | 52 | 36 | 2
7 | 4 | 17 | 20 | 27 | 80 | 37 | 20
8 | 4 | 18 | 2 | 28 | 2 | 38 | 142
9 | 6 | 19 | 28 | 29 | 120 | 39 | 388
10 | 2 | 20 | 28 | 30 | 2 | 40 | 2

Of course $PP(n)\leq R(n)$. Comparing Tables 3.1 and 5.1, we see the smallest
$n$ with $PP(n)<R(n)$ is $n=14$, where $PP(14)=10$ and $R(14)=14$. The four
extra partitions which are not palindromes are
$\lambda=(44222),R(\lambda)=(43331)$ and $\mu=(5522),R(\mu)=(5441)$.

######  Remark 5.4.

The proof of Theorem 3.2 applies to the more general setting of computing
$R(n)$, i.e. $R(n)=2$ precisely when $P(n)=2$.

##  6\. Problems

We suggest some problems for future research.

######  Problem 6.1.

Can we write the generating functions in Theorem 3.1 and 5.3 in a more compact
form $\sum f(n)q^{n}$? Can we see clearly from the generating function why
$f(n)=2$ when $n+1$ is prime?

######  Problem 6.2.

The traditional Young’s lattice has all partitions of $n$ in row $n$ with
edges corresponding to removing and/or adding a single box. Classically this
describes the branching of irreducible representations of the symmetric group.
Does Figure 2.2 have any representation- theoretic interpretation?

######  Problem 6.3.

Proposition 1.3 gives a quick way to identify palindrome partitions $\lambda$
without calculating $B(\lambda)$. Can one do the same for fractions in the
Calkin-Wilf tree corresponding to palindrome sequences, i.e. determine if a
fraction corresponds to a palindrome without doing the continued fraction
expansion? Can we describe what the reversal operation does to fractions?

##  7\. Acknowledgements

The authors would like to thank Matthew Just and Robert Schneider for
suggesting looking at this operation on partitions and William Keith for help
with generating functions.

## References


aainstitutetext: Physics Department and Center for Education and Research in
Cosmology and Astrophysics (CERCA), Case Western Reserve University,
Cleveland, OH 44106, USAbbinstitutetext: Walter Burke Institute for
Theoretical Physics, California Institute of Technology, Pasadena, CA 91125

# Baryonic Higgs and Dark Matter

Pavel Fileviez Pérez b Clara Murgui a and Alexis D. Plascencia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss the correlation between dark matter and Higgs decays in gauge
theories where the dark matter is predicted from anomaly cancellation. In
these theories, the Higgs responsible for the breaking of the gauge symmetry
generates the mass for the dark matter candidate. We investigate the Higgs
decays in the minimal gauge theory for Baryon number. After imposing the dark
matter density and direct detection constraints, we find that the new Higgs
can have a large branching ratio into two photons or into dark matter.
Furthermore, we discuss the production channels and the unique signatures at
the Large Hadron Collider.

##  1 Introduction

The discovery of the Standard Model (SM) Higgs boson with a mass of 125 GeV at
the Large Hadron Collider (LHC) has opened a possible new portal to dark
matter Silveira:1985rk ; McDonald:1993ex ; Patt:2006fw . If dark matter
acquires mass through spontaneous symmetry breaking, then it can be expected
that dark matter will be coupled to the SM Higgs. The ATLAS and CMS
collaborations have throughly search for dark matter by looking for the
invisible decays of the SM Higgs boson. The most recent upper bound reported
by the ATLAS collaboration from their Run II analysis of data with
$\sqrt{s}=13$ TeV and $139\text{ fb}^{-1}$ luminosity is given by

$\text{Br}(h\to\text{invisible})<0.13\text{ at }95\%\text{
CL}\text{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{ATLAS-
CONF-2020-008}{\@@citephrase{(}}{\@@citephrase{)}}}.}$ (1)

Moreover, a statistical combination of the results from both Run I and II by
ATLAS gives $\text{Br}(h\to\text{invisible})<0.11\,(95\%\text{ CL})$
ATLAS:2020kdi , while a combined analysis by the CMS collaboration finds that
$\text{Br}(h\to\text{invisible})<0.19\,(95\%\text{ CL})$ Sirunyan:2018owy .

Nevertheless, the discovery of the SM Higgs also motivates the question of
whether there are other fundamental scalars to be discovered at the LHC. This
bring us to question the origin of the dark matter mass. Among the different
possibilities that could give rise to the mass of the dark matter candidate,
the idea of generating the dark matter mass through the spontaneous breaking
of a given gauge symmetry is specially attractive from a theoretical point of
view. In that context, the dark matter mass would be connected to the energy
scale of the force acting on it.

In theories where an anomalous symmetry is promoted to a local symmetry, a
dark matter candidate can be predicted from the cancellation of gauge
anomalies; for example when gauging baryon or lepton number Perez:2014qfa ;
FileviezPerez:2011pt ; Duerr:2013dza . In this case, the dark matter $\chi$
will interact through a new Abelian gauge symmetry $\,{\rm U}(1)_{X}$ with
charge $n_{X}$. Then, through the following Yukawa interaction

$-{\cal L}\supset y_{\chi}\chi\chi h_{X}+\text{h.c.},$ (2)

the dark matter can get a mass once the new Higgs $h_{X}$ acquires a vacuum
expectation value (vev) that spontaneously breaks the $\,{\rm U}(1)_{X}$ and
gives mass to the corresponding gauge mediator that determines the scale
$\Lambda_{X}\sim M_{Z_{X}}$ where the new symmetry is broken. Hence, the dark
matter mass is linked to such scale as follows:

$M_{\chi}=\frac{\sqrt{2}\,y_{\chi}}{n_{X}g_{X}}M_{Z_{X}}.$ (3)

Figure 1: Collider experiments can search for dark matter through the SM Higgs
boson decays to invisible. However, in theories where the dynamical origin of
the dark matter mass can be understood, those decays can be strongly
suppressed by the scalar mixing angle. Consequently, it might be more
convenient to look for signals arising from the invisible decays of a the new
Higgs, $h_{X}$, responsible for the dark matter mass and the breaking of the
new symmetry $\,{\rm U}(1)_{X}$.

As we will discuss in the next section, it is possible to define simple
anomaly-free theories where the dark matter candidate is predicted by the
condition of anomaly cancellation. In this context, one would expect that the
new Higgs $h_{X}$ mixes with the SM Higgs through the scalar potential as
illustrated in Fig. 1. This mixing can be parametrized by an angle
$\theta_{X}$, which is already constrained by collider experiments to be at
most $\sin\theta_{X}<0.3$ Ilnicka:2018def ; Adhikari:2020vqo . However, as we
will learn along this manuscript, direct detection bounds on dark matter can
place a more stringent bound on the mixing angle between the two scalar
bosons.

A dark matter scenario such as the one just described could be seen at the
collider searches for invisible decays of the SM Higgs boson, but such signals
will always suffer from the mixing suppression of $\sin^{2}\theta_{X}$. Then,
the mixing portal might not be the best option to detect dark matter. In this
paper we suggest to look for dark matter in alternative portals that will of
course depend on the new force we are considering. In the theories we
consider, where the dark matter candidate gets mass through a new Higgs
mechanism, we will propose ways to study the correlations between the physics
of the new Higgs at colliders and the dark matter phenomenology.

In this work we study the motivated scenario in which the new gauge symmetry
is identified to be baryon number $\,{\rm U}(1)_{B}$ Perez:2014qfa ;
FileviezPerez:2011pt ; Duerr:2013dza . Our study focuses on the theory with
the least number of new representations to cancel the gauge anomalies
presented in Ref. Perez:2014qfa . As we will show, it is crucial to consider a
complete theory in order to predict the decays of the new Higgs (to which we
will refer as Baryonic Higgs) since the loop-induced decays with the anomaly-
canceling fermions running in the loop can be the dominant ones due to the
fact that they do not suffer from mixing suppression. We show that one can
predict the branching ratios for the Baryonic Higgs in agreement with the
cosmological constraints on the dark matter density. Our main predictions can
be tested in the near future at the LHC and dark matter experiments.

The paper is structured as follows: In Section 2 we study the dark matter
phenomenology and characterize the properties of the second Higgs in the
theory. In Section 3 we discuss the production of this new Higgs at the LHC
and the signatures to look for it. We present our conclusions in Section 4. In
Appendix A we present all the Feynman rules that were used in our
calculations. Finally, Appendix B contains the full analytical results for all
the loop-induced and tree-level decays of the Baryonic Higgs.

##  2 Higgs Decays and DM from Anomaly Cancellation

In the previous section we discussed a class of theories for dark matter where
the dark matter mass is generated by the spontaneous symmetry breaking of the
gauge symmetry. In this context, the dark matter candidate is charged under
the new force and, therefore, it is part of the anomalons, i.e. fermions
needed to satisfy the anomaly cancellation conditions that otherwise would
spoil the gauge invariance of the theory. One of the most attractive examples
of this type of theories is local baryon number since

  * (a)

it is one of the simplest extensions of the SM where it is possible to
understand the spontaneous breaking of baryon number without giving issues
regarding the stability of the proton, which in these theories is predicted to
be stable to any order in perturbation theory,

  * (b)

it predicts the existence of dark matter from anomaly cancellation,

  * (c)

the cosmological bound on the relic density measured by the Planck satellite
$\Omega h^{2}\leq 0.12$ translates into an upper bound of $M_{Z_{B}}\lesssim
28$ TeV FileviezPerez:2019jju  and since all new particles acquire their mass
from the symmetry breaking scale, this implies an upper bound on the full
theory,

  * (d)

this theory can live at the low scale in agreement with all experimental
constraints with a gauge coupling of order $g_{B}\sim 0.1$. Therefore, one can
hope to test this theory at the LHC.

In this work we will discuss the minimal theory for local baryon number
Perez:2014qfa , where only four extra fermionic representations are added to
define an anomaly-free theory. These new fermionic fields are given by

where the numbers in parenthesis correspond to the quantum numbers under the
gauge groups $\,{\rm SU}(3)_{c}$, $\,{\rm SU}(2)_{L}$, $\,{\rm U}(1)_{Y}$ and
$\,{\rm U}(1)_{B}$, respectively. The Yukawa interactions in this theory are
given by

$\displaystyle-\mathcal{L}\supset$ $\displaystyle\hskip
2.84526pty_{1}\overline{\Psi}_{R}H\chi_{L}+y_{2}H^{\dagger}\Psi_{L}\chi_{L}+y_{3}H^{\dagger}\Sigma_{L}\Psi_{L}+y_{4}\overline{\Psi}_{R}\Sigma_{L}H$
$\displaystyle+y_{\Psi}\overline{\Psi}_{R}\Psi_{L}S^{*}_{B}+\frac{y_{\chi}}{\sqrt{2}}\chi_{L}\chi_{L}S_{B}+y_{\Sigma}\text{Tr}\Sigma^{2}_{L}S_{B}+{\rm
h.c.},$ (5)

where $H\sim(\mathbf{1},\mathbf{2},1/2,0)$ is the SM Higgs and the new Higgs,
whose quantum numbers are indirectly fixed by anomaly cancellation,

$S_{B}\sim(\mathbf{1},\mathbf{1},0,3),$ (6)

is responsible for the spontaneous breaking of $\,{\rm U}(1)_{B}$ and
generates the masses for the new fermions in the theory, including the dark
matter candidate.

The scalar potential of this theory reads as

$\displaystyle
V=-\mu_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}-\mu_{B}^{2}S^{*}_{B}S_{B}+\lambda_{B}(S^{*}_{B}S_{B})^{2}+\lambda_{HB}(H^{\dagger}H)(S^{*}_{B}S_{B}),$
(7)

and the Higgses can be written as

$\displaystyle H=\matrixquantity(h^{+}\\\
\frac{1}{\sqrt{2}}(h_{0}+ia_{0})),\quad\text{ and }\quad
S_{B}=\frac{1}{\sqrt{2}}(s_{B}+ia_{B}).$ (8)

Spontaneous symmetry breaking of baryon number is achieved once $S_{B}$
acquires the vev, $\langle S_{B}\rangle=v_{B}/\sqrt{2}$, while the electroweak
spontaneous symmetry is triggered by the vev of the SM Higgs, $\langle
h^{0}\rangle=v_{0}/\sqrt{2}$. Both Higgses mix through the scalar potential in
Eq. (7). In the broken phase, the physical Higgses are defined as follows

$\begin{split}h&=h_{0}\cos{\theta_{B}}-s_{B}\sin{\theta_{B}},\\\
h_{B}&=s_{B}\cos{\theta_{B}}+h_{0}\sin{\theta_{B}},\end{split}$ (9)

where the mixing angle $\theta_{B}$ that diagonalizes the mass matrix for the
Higgses is given by

$\displaystyle\tan{2\theta_{B}}=\frac{v_{0}v_{B}\lambda_{HB}}{v_{B}^{2}\lambda_{B}-v_{0}^{2}\lambda_{H}}.$
(10)

This theory predicts the existence of a new gauge boson $Z_{B}$, which does
not interact with leptons at tree-level, and a Baryonic Higgs $h_{B}$. For the
physical fermion fields we have two charged $F^{\pm}_{i}\,\,(i=1,2)$ and four
neutral: $F^{0}_{j}\,\,(j=1,2,3)$ and the dark matter candidate $\chi$.

After spontaneous symmetry breaking, the local $\,{\rm U}(1)_{B}$ is broken to
a ${\mathcal{Z}}_{2}$ symmetry which protects the dark matter candidate from
decaying. This symmetry acts only in the new sector as follows:

$\\{\Psi_{L}\to-\Psi_{L},\ \Psi_{R}\to-\Psi_{R},\ \Sigma_{L}\to-\Sigma_{L},\
\chi_{L}^{0}\to-\chi_{L}^{0}\\}.$

In order to have a consistent scenario for cosmology, we then require that

$M_{\chi}<M_{F_{i}^{0}},M_{F^{\pm}_{i}},$

and the lightest neutral state can be a good dark matter candidate because it
is automatically stable and neutral. In this article we will investigate the
dark matter properties when $\chi\sim\chi_{L}+(\chi_{L})^{C}$ since in this
case the Baryonic Higgs can have a large branching ratio to dark matter even
when its mass is not far from the electroweak scale. For simplicity, we will
take the limit where the Yukawa couplings with the SM Higgs $y_{i}$, with
$i=1,...,4$, are negligible. In this context, the Feynman rules of the theory
are listed in Appendix A.

The most relevant parameters for our dark matter studies are the new gauge
coupling, $g_{B}$, the mixing angle between the two Higgses, $\theta_{B}$, the
dark matter mass, $M_{\chi}$, the mass of the Baryonic Higgs, $M_{h_{B}}$, and
the mass of the new gauge boson, $M_{Z_{B}}$. The masses of the anomaly-
canceling fermions, $M_{F_{i}^{0}}\text{ and }M_{F^{\pm}_{i}}$, become
relevant when we study the Baryonic Higgs decays. For phenomenological studies
of this type of theory see Refs. Ohmer:2015lxa ; Duerr:2017whl ; Duerr:2017uap
; Duerr:2014wra ; FileviezPerez:2020mtk ; FileviezPerez:2019jju ;
FileviezPerez:2018jmr ; FileviezPerez:2020gfb .

###  2.1 Dark Matter: Relic Density and Direct Detection

In order to investigate the predictions for the Higgs decays we first need to
discuss the dark matter relic density and direct detection constraints. Since
we are mainly interested in the scenarios with a light Baryonic Higgs $h_{B}$,
so that it can be produced at the LHC, we focus on the simplest scenario where
$\chi$ is singlet-like under the SM. In the context of the minimal theory for
baryon number described in the previous section, our dark matter candidate is
a Majorana fermion, $\chi=\chi_{L}+(\chi_{L})^{C}$, and has the following
annihilation channels,

$\chi\chi\to\bar{q}q,\,Z_{B}Z_{B},\,Z_{B}h,\,Z_{B}h_{B},\,hh,\,hh_{B},\,h_{B}h_{B},\,WW,\,ZZ.$

Notice that the channels $Z_{B}h,\,hh,\,hh_{B},\,WW$ and $ZZ$ are suppressed
by the mixing angle $\theta_{B}$, which from collider bounds has to be
$\sin\theta_{B}\leq 0.3$ Ilnicka:2018def ; Adhikari:2020vqo . The channels
$\bar{q}q,\,Z_{B}Z_{B},Z_{B}h_{B}$ and $h_{B}h_{B}$, if allowed, do not suffer
from mixing suppression and also define the most interesting regions that
satisfy the relic density constraints, as we show in this section.

Figure 2: Contour plot for the dark matter relic density in the $M_{\chi}$
versus $M_{Z_{B}}$ plane. In the left (right) panel the mass of the second
Higgs has been fixed to $M_{h_{B}}=200$ GeV (500 GeV). The solid black line
gives the measured relic abundance by the Planck satellite
$\Omega_{\chi}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx . The colored region
satisfies the constraint $\Omega_{\chi}h^{2}\leq 0.12$ and the different
colors correspond to different dominant annihilation channels as indiciated in
the legend. The solid green line shows the region ruled out by Xenon-1T
Aprile:2018dbl , the dashed black line shows the projected sensitivity for
Xenon-nT Aprile:2015uzo , both for zero mixing angle. For small mixing angles,
the dark matter relic density and the direct detection are almost independent
of the scalar mixing angle as explained in the text.

In Fig. 2 we present our results for the calculation of the dark matter relic
density which has been computed numerically using MicrOMEGAs 5.0.6
Belanger:2018ccd . In both panels we show, in solid black, the contour lines
that give the relic abundance measured by the Planck satellite
$\Omega_{\chi}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx . The colored regions
show the parameter space that gives a relic density that agrees with the later
bound but lies below it. On the left panel, the mass of the Baryonic Higgs is
fixed to $M_{h_{B}}=200$ GeV, while on the right panel we show the parameter
space in agreement with the relic density bound for a heavier Baryonic Higgs
of $M_{h_{B}}=500$ GeV. For these plots, as well as for the rest of the
results presented here, we will assume a gauge coupling $g_{B}=0.1$ in order
to be in agreement with the collider bounds coming from dijet searches
regardless of the $Z_{B}$ mass considered. The monojet bounds become relevant
ony for smaller masses below the range we consider ATLAS:2020wzf . For a
detailed study of the dark matter phenomenology for larger values of the gauge
coupling see Ref. FileviezPerez:2020mtk .

Label | Dominant annihilation channel | Properties
---|---|---
Scenario I |  $\chi\chi\to\bar{q}q$ | $\\!M_{\chi}\simeq M_{Z_{B}}/2$
Scenario II |  $\chi\chi\to Z_{B}h_{B}$ | $\quad\quad\quad 2M_{\chi}>M_{h_{B}}+M_{Z_{B}}$
Scenario III |  $\chi\chi\to Z_{B}Z_{B}$ | $\\!\\!\\!\\!M_{\chi}>M_{Z_{B}},\,\,\,\,\,M_{\chi}\simeq M_{h_{B}}/2$
Table 1: Classification of the three different scenarios for which the relic
density bound $\Omega_{\chi}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx  is
satisfied in the minimal theory for local baryon number.

The parameter space of the theory can be classified depending on which
annihilation channel gives the largest contribution to the relic density, as
illustrated in Table 1, what gives rise to four possible scenarios:

  * •

Scenario I ($\chi\chi\to\bar{q}q$): The annihilation into a quark antiquark
pair is the dominant one only at the resonance $M_{\chi}\approx M_{Z_{B}}/2$.
This region is shown by the thin diagonal in Fig. 2. This region satisfies the
relic density bound independently of the value of $M_{h_{B}}$.

  * •

Scenario II ($\chi\chi\to Z_{B}h_{B}$): As shown in both plots in Fig. 2 the
$\chi\chi\to Z_{B}h_{B}$ annihilation can be dominant in a large region of the
parameter space and it does not rely on a resonance.

  * •

Scenario III ($\chi\chi\to Z_{B}Z_{B}$): In the scenario with $M_{h_{B}}\gg
M_{\chi}$ we have to rely on the $Z_{B}Z_{B}$ annihilation channel to achieve
the measured relic abundance. As shown on the left plot in Fig. 2, this
annihilation can be dominant in the resonance $M_{\chi}\approx M_{h_{B}}/2$.
However, it does not always have to be close to the resonance, as shown in the
right plot of Fig. 2. This scenario, while disfavoured by direct detection
searches for light $h_{B}$ masses, will be fully probed by Xenon-nT in the
near future for $M_{h_{B}}\lesssim 1\text{ TeV}$.

  * •

Scenario IV ($\chi\chi\to h_{B}h_{B})$: This annihilation is dominant in the
limit when $M_{h_{B}}\ll M_{\chi}\ll M_{Z_{B}}$ and a large Yukawa coupling
$y_{\chi}\approx M_{\chi}/v_{B}$. These conditions imply a large gauge
coupling above the perturbativity limit so we do not discuss this scenario any
further.

In Fig. 2, we color each region depending on which annihilation channel gives
the dominant contribution to the relic abundance: the red area is dominated by
$\chi\chi\to\bar{q}q$, the orange area is dominated by $\chi\chi\to
Z_{B}Z_{B}$, while in the blue area the annihilation channel $\chi\chi\to
Z_{B}h_{B}$ dominates.

Regarding direct detection of the dark matter, the cross-section mediated by
the $Z_{B}$ is velocity suppressed,

$\displaystyle\sigma_{\chi N}^{\text{SI}}(Z_{B})$ $\displaystyle=$
$\displaystyle\frac{27}{8\pi}\frac{g_{B}^{4}M_{N}^{2}}{M_{Z_{B}}^{4}}v^{2},$
(15)

where $M_{N}$ is the nucleon mass and $v$ the velocity of the non-relativistic
dark matter. There is also the channel mediated by Higgs mixing:

$\sigma_{\chi N}^{\text{SI}}(h_{i})=\frac{72}{8\pi
v_{0}^{2}}\sin^{2}\theta_{B}\cos^{2}\theta_{B}M_{N}^{4}\frac{g_{B}^{2}M_{\chi}^{2}}{M_{Z_{B}}^{2}}\left(\frac{1}{M_{h}^{2}}-\frac{1}{M_{h_{B}}^{2}}\right)^{2}f_{N}^{2},$
(16)

which is suppressed by the mixing angle. The parameter $f_{N}$ corresponds to
the effective Higgs-nucleon-nucleon coupling that we take $f_{N}=0.3$
Hoferichter:2017olk .

In Fig. 2 we show in solid green the parameter space ruled out by Xenon-1T
Aprile:2017iyp ; Aprile:2018dbl , while the dashed black line shows the region
that will be probed by Xenon-nT Aprile:2015uzo  in the case of zero mixing
angle, these constraints become stronger for larger mixing angles. In Fig. 3
we present our predictions for the spin-independent cross-section as a
function of the dark matter mass with all points satisfying the measured relic
abundance. As can be seen, the projected sensitivity for Xenon-nT will probe
the zero mixing angle case for $M_{\chi}<390$ GeV. Strikingly, this result
tells us that one could test this theory if our dark matter candidate is not
too heavy.

Figure 3: Predictions for the direct-detection spin-independent cross-section
as a function of the dark matter mass. The purple, red, green and orange lines
correspond to $\sin\theta_{B}=0.1$, $0.05$, $0.01$ and $0.001$, respectively.
The brown line corresponds to the maximal mixing scenario,
$\sin\theta_{B}=0.3$, while the blue dashed line represents the minimal one,
$\theta_{B}=0$. All points give the measured relic abundance by the Planck
satellite $\Omega_{\chi}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx  and satisfy
constraints from the LHC. The solid black lines show current experimental
bounds from Xenon-1T Aprile:2017iyp ; Aprile:2018dbl , the dashed black line
shows the projected sensitivity for Xenon-nT Aprile:2015uzo  and the dashed
gray line shows the coherent neutrino scattering limit Billard:2013qya .

###  2.2 Baryonic Higgs Decays

The new Higgs present in the theory can have the following decays:

$h_{B}\to\gamma\gamma,\,\,gg,\,\,\gamma Z,\,\,\gamma
Z_{B},\,\,ZZ,\,\,ZZ_{B},\,\,Z_{B}Z_{B},\,\,WW,\,\,{\chi}\chi,\,\,\overline{F_{i}}F_{i},\,\,hh,\,\,\overline{f_{i}}f_{i},$

where $\chi$, $F_{i}$ and $f_{i}$ correspond to dark matter, anomaly-canceling
fermions (also referred to as anomalons throughout the text) and SM fermions,
respectively. All the Feynman diagrams and the full analytical results for the
tree-level and loop-induced decays of the Baryonic Higgs are given in Appendix
B. Among them, the decays $h_{B}\to Z_{B}Z_{B},\,\,\chi\chi,\,\,\bar{F}F$
occur at tree-level, with no suppression if allowed by kinematics. Therefore,
when any of these decays is open, it dominates the branching ratios. The
decays $h_{B}\to\gamma\gamma,\,\,Z\gamma$ are loop suppressed but can be
dominant whenever the tree-level decays are kinematically closed. The decays
$h_{B}\to\,\,WW,\,\,ZZ$ have a tree-level and one-loop contribution, the
former is the dominant for large mixing angles and the latter dominates for
small mixing angles. The rest of decays, $h_{B}\to
Z_{B}\gamma,\,\,gg,\,\,ZZ_{B}$, on top of being loop-induced are suppressed by
the mixing angle $\theta_{B}$ as well. In the limit of very heavy anomaly-
canceling fermions these decays were studied in Ref. Ohmer:2015lxa .

As we discussed in the previous sections, having a complete theory that
includes the anomaly-canceling fermions is crucial to make the correct
predictions for the different decays of the Baryonic Higgs. In the previous
section we demonstrated that the scalar mixing angle $\theta_{B}$ is strongly
constrained by direct detection bounds, which has the following implications
for the Baryonic Higgs decays in the three different scenarios that can be in
agreement with the cosmological bound on the relic abundance:

  * •

Scenario I ($\chi\chi\to\bar{q}q$): In this scenario, the decays
$h_{B}\to\chi\chi$ and $h_{B}\to Z_{B}Z_{B}$ can be open or closed depending
on the value of $M_{h_{B}}$. The left panel in Fig. 4 shows the results for
$M_{h_{B}}<2M_{\chi}<2M_{Z_{B}}$, where the invisible decays and the decay to
a pair of new gauge bosons are kinematically closed, while in the right panel
we show the results for $2M_{\chi}<M_{h_{B}}<2M_{Z_{B}}$, and hence, $h_{B}$
decays predominantly into dark matter.

Figure 4: Branching ratios of the Baryonic Higgs $h_{B}$ as a function of the
scalar mixing angle $\theta_{B}$ in the context of Scenario I, where
$\chi\chi\to\bar{q}q$ is the dominant annihilation channel. The relevant
parameters have been fixed as indicated in the plot to satisfy the dark matter
relic density $\Omega h^{2}=0.12$. The different colors correspond to
different decay channels as shown in the plot. The area shaded in red is ruled
out by Xenon-1T direct detection constraints, while in yellow we show the
region that will be probed by Xenon-nT.

  * •

Scenario II ($\chi\chi\to Z_{B}h_{B}$): This scenario requires that
$2M_{\chi}>M_{h_{B}}+M_{Z_{B}}$, and therefore, the invisible decay is closed.
Furthermore, $M_{Z_{B}}$ has to be around $M_{\chi}$ for low masses of the
dark matter in order to achieve the correct relic density, so that in this
case the $h_{B}\to Z_{B}Z_{B}$ decay is also closed. This implies that for
small mixing angles the loop-induced decays into $\gamma\gamma$ and $\gamma Z$
can be significant, as it is shown on the left panel of Fig. 5. However, as we
show in the right panel of that figure, for larger dark matter masses
$M_{Z_{B}}\lesssim M_{\chi}$ and hence the $h_{B}\to Z_{B}Z_{B}$ can dominate
for small mixing angles.

Figure 5: Branching ratios of the Baryonic Higgs $h_{B}$ as a function of the
scalar mixing angle $\theta_{B}$ in the context of Scenario II, where
$\chi\chi\to Z_{B}h_{B}$ is the dominant annihilation channel. The relevant
parameters have been fixed as indicated in the plot to satisfy the dark matter
relic density $\Omega h^{2}=0.12$. The different colors correspond to
different decay channels as shown in the plot. The area shaded in red is ruled
out by Xenon-1T direct detection constraints, while in yellow we show the
region that will be probed by Xenon-nT.

  * •

Scenario III ($\chi\chi\to Z_{B}Z_{B}$): For this annihilation channel to be
open, $M_{Z_{B}}\lesssim M_{\chi}$ is required. Furthermore, as shown in Fig.
2, the condition $M_{h_{B}}\gtrsim 2M_{Z_{B}}$ is always satisfied. This
implies that the decay $h_{B}\to Z_{B}Z_{B}$ is open and it is the dominant
decay. On the other hand, whenever $M_{h_{B}}\gtrsim 2M_{\chi}$ the Higgs will
also have a branching ratio into invisible. The branching ratios for the
second Higgs in this scenario are shown in Fig. 6.

Figure 6: Branching ratios of the Baryonic Higgs $h_{B}$ as a function of the
scalar mixing angle $\theta_{B}$ in the context of Scenario III, where
$\chi\chi\to Z_{B}Z_{B}$ is the dominant annihilation channel. The relevant
parameters have been fixed as indicated in the plot to satisfy the dark matter
relic density $\Omega h^{2}=0.12$. The different colors correspond to
different decay channels as shown in the plot. The area shaded in red is ruled
out by Xenon-1T.

We would like to mention in passing that for $M_{Z_{B}}<125$ GeV the SM Higgs
can also decay into $h\to\gamma Z_{B}$ with the top quark in the loop.
However, we find that ${\rm BR}(h\to\gamma Z_{B})<10^{-5}$ which means that
this exotic Higgs decay will be hard to detect in the near future.

From the above scenarios, we would also like to highlight the interesting
framework of Scenario II where it is possible to have large branching ratios
of the Baryonic Higgs to the electroweak gauge bosons and the photon,
$\text{BR}(h_{B}\to\gamma\gamma)\gtrsim 1\%$. In this case, one can expect to
detect this Baryonic Higgs at colliders through the number of events predicted
for those clean channels.

##  3 Signatures at the LHC

In this section we study the main signatures of the minimal theory for local
baryon number at the LHC. We first focus on the associated production of the
Baryonic Higgs and use the fact that its decay to photons or missing energy
can be relevant for certain regions of the parameter space. We also highlight
that this theory predicts unequivocally the branching ratios of the
leptophobic gauge boson $Z_{B}$.

###  3.1 Associated production of $h_{B}$ and $Z_{B}$

We have shown in the previous section that the dark matter direct detection
experimental bounds set a strong limit on the mixing between the SM Higgs and
the Baryonic Higgs, $\sin\theta_{B}<0.1$, which becomes stronger for light
dark matter masses, being able to rule out any scalar mixing for
$M_{\chi}\lesssim 180$ GeV for $g_{B}=0.1$ as Fig. 3 shows. Therefore, all the
SM-like Higgs production mechanisms for the Baryonic Higgs are typically
suppressed.

In these theories we have the possibility to use the associated production
mechanism $pp\to Z_{B}^{*}\to h_{B}Z_{B}$, that is not suppressed by the
mixing between the two Higgses. The cross-section for $pp\to Z_{B}^{*}\to
Z_{B}h_{B}$ is given by

$\sigma(pp\to h_{B}Z_{B})(s)=\int_{\tau_{0}}^{1}d\tau\frac{d{\cal
L}^{pp}_{q\bar{q}}}{d\tau}\sigma(q\bar{q}\to Z_{B}h_{B})(\hat{s}),$ (17)

which weights the contribution of the partonic cross-section of the process
represented in Fig. 7,

$\sigma=\frac{\cos^{2}\theta_{B}g_{B}^{4}}{144\pi
s^{2}}\frac{(M_{h_{B}}^{4}\\!\\!-\\!2M_{h_{B}}^{2}(M_{Z_{B}}^{2}\\!\\!+\\!s)+M_{Z_{B}}^{4}\\!+\\!10M_{Z_{B}}^{2}s\\!+\\!s^{2})\sqrt{(s\\!+\\!M_{Z_{B}}^{2}\\!\\!-\\!M_{h_{B}}^{2})^{2}\\!-\\!4sM_{Z_{B}}^{2}}}{(s\\!-\\!M_{Z_{B}}^{2})^{2}\\!+\\!\Gamma_{Z_{B}}^{2}M_{Z_{B}}^{2}},$
(18)

by the contribution of each quark to the proton, the latter being parametrized
by the corresponding parton distribution function as follows,

$\frac{d{\cal
L}_{q\bar{q}}^{pp}}{d\tau}=\int_{\tau}^{1}\frac{dx}{x}\left[f_{q/p}(x,\mu)\,f_{\bar{q}/p}\left(\frac{\tau}{x},\mu\right)+f_{q/p}\left(\frac{\tau}{x},\mu\right)f_{\bar{q}/p}(x,\mu)\right],$
(19)

where $\tau=\hat{s}/s$, being $\hat{s}$ the partonic center-of-mass energy
squared, $s$ is the center-of-mass energy squared at the hadronic level,
$\tau_{0}=(M_{h_{B}}+M_{Z_{B}})^{2}/s$ is the production threshold, and $\mu$
refers to the factorization scale, which we take it to be
$\mu=(M_{h_{B}}+M_{Z_{B}})/2$ in our calculations.

Figure 7: Feynman graph for the process $\bar{q}q\to Z_{B}^{*}\to Z_{B}h_{B}$.

Figure 8: Contour lines for the production cross-sections for the process
$pp\to Z_{B}^{*}\to Z_{B}h_{B}$ at the LHC with center-of-mass energy of 14
TeV. These results have a very weak dependence on the mixing angle for
$\sin\theta_{B}<10^{-2}$. In the upper (lower) panel the scalar mixing angle
is fixed to $\sin\theta_{B}=10^{-3}$ (with identical results for
$\sin\theta_{B}=10^{-2}$). The region shaded in gray is excluded by direct
searches for the $Z_{B}$ at the LHC.

The cross-sections are obtained using MadGraph5aMC@NLO - v2.7.0 Alwall:2014hca
, we cross-checked our results in a Mathematica notebook and the use of the
MSTW2008 Martin:2009iq  set of parton distribution functions. In Fig. 8 we
present the results for the cross-sections of the process $pp\to Z_{B}^{*}\to
Z_{B}h_{B}$ in the $g_{B}$ vs $M_{Z_{B}}$ plane. For the upper (lower) panel
we have fixed the mass of the second Higgs to 200 GeV (500 GeV), in agreement
with the scenarios considered in Section 2. The gray regions in the plot are
excluded from direct searches of the $Z_{B}$ gauge boson by the ATLAS and CMS
collaborations, for a discussion of these bounds see Ref.
FileviezPerez:2020mtk . Since this cross-section scales as
$\cos^{2}\theta_{B}$, the results have a very mild dependence on the mixing
angle for small values of $\theta_{B}$.

Figure 9: Contour lines for the production cross-sections for the process
$pp\to Z_{B}^{*}\to Z_{B}h_{B}$ at the LHC with center-of-mass energy of 14
TeV. We show the results in the $M_{Z_{B}}$ vs $M_{h_{B}}$ plane. These
results have almost no dependence on the mixing angle for
$\sin\theta_{B}<10^{-2}$.

In Fig. 9 we present the result for the cross-section in the $M_{Z_{B}}$ vs
$M_{h_{B}}$ plane. The gauge coupling has been fixed to $g_{B}=0.1$ in order
to avoid the dijet constraints. As this plot shows for masses of the $Z_{B}$
around 100 GeV the production cross-section can be as large as 10 pb, while
for masses close to a TeV the cross-section drops down to 0.005 pb.

The expected number of events at the LHC is given by,

$N_{\rm events}(x\bar{x}y\bar{y})={\mathcal{L}}\times\sigma(pp\to Z_{B}^{*}\to
Z_{B}h_{B})\times{\rm BR}(Z_{B}\to x\bar{x})\times{\rm BR}(h_{B}\to
y\bar{y}).$ (20)

In Fig. 10 we show our results for the expected number of events at the LHC
with center-of-mass energy of 14 TeV and integrated luminosity of 3000 ${\rm
fb}^{-1}$ for different combination of final states. The branching ratios for
$Z_{B}$ and $h_{B}$ are computed at each point on the plots.

Figure 10: Contour lines for the expected number of events at the LHC with
center-of-mass energy of 14 TeV assuming an integrated luminosity of
$\mathcal{L}=3000\,{\rm fb}^{-1}$. We show the most relevant final states: For
the upper panels we show $b\bar{b}\gamma\gamma$ and $t\bar{t}\gamma\gamma$
while for the lower panels we show $b\bar{b}E_{T}^{\rm miss}$ and
$t\bar{t}E_{T}^{\rm miss}$. In order to calculate the branching ratios of
$h_{B}$ the masses of the new fermions have been scaled as the label shows on
the plots. The gray region is excluded by the direct searches for a $Z_{B}$ at
the LHC. The scalar mixing angle is fixed to $\sin\theta_{B}=10^{-3}$
($10^{-2}$) for the upper (lower) panels.

For the two plots in the upper panels of Fig. 10 we consider Scenario II
($\chi\chi\to Z_{B}h_{B}$), and hence, we have that $2M_{\chi}\gtrsim
M_{h_{B}}+M_{Z_{B}}$. As we have discussed in Section 2, in this context, if
the Higgs is light enough such that the $Z_{B}Z_{B}$ channel remains closed,
the branching ratio to two photons can be relevant. In order to compute the
${\rm BR}(h_{B}\to\gamma\gamma)$, we scale the masses of the anomaly-canceling
fermions as $M_{\Psi}=3\,M_{Z_{B}}$ and $M_{\Sigma}=3.5\,M_{Z_{B}}$. The dark
matter mass remains a free parameter and can be fixed to the value that
satisfies the correct dark matter relic density. In most of the parameter
space in the plots we find that ${\rm BR}(h_{B}\to\gamma\gamma)\simeq
10^{-2}$.

For the two plots in the lower panels of Fig. 10 we consider Scenario I
($\chi\chi\to q\bar{q}$) that is at the resonance $M_{\chi}\simeq
M_{Z_{B}}/2$. Since we want the $h_{B}$ decay into missing energy to be the
dominant one we only focus on the range for
$M_{Z_{B}}=[M_{h_{B}}/2,M_{h_{B}}]$. The masses of the anomaly-canceling
fermions are scaled as $M_{\Psi}=1.3\,M_{Z_{B}}$ and
$M_{\Sigma}=1.5\,M_{Z_{B}}$ in this context. In a large region of the
parameter space considered we find that ${\rm BR}(h_{B}\to\chi\chi)\simeq
0.9$. For the left panel we consider the $b\bar{b}E_{T}^{\rm miss}$ final
state for which the ATLAS collaboration ATLAS:2019ivx  has a recent analysis,
albeit for smaller Higgs masses that the ones we consider. The right panel
shows our results for the $t\bar{t}E_{T}^{\rm miss}$ final state. In summary,
in this theory it is possible to predict a large number of events in agreement
with the collider bounds where the Baryonic Higgs has large branching ratios
into two photons or into dark matter.

###  3.2 The leptophobic gauge boson decays

The new gauge boson in the theory $Z_{B}$ is coupled to the quarks and the
anomaly-canceling fermions, and hence, $Z_{B}$ could decay into all of these
particles including dark matter. However, in order to prevent overclosing the
Universe we find that, as Fig. 2 explicitly shows,

$M_{Z_{B}}\lesssim 2M_{\chi}.$ (21)

In consequence, the invisible decays of the $Z_{B}$ are kinematically
forbidden and the gauge boson can decay only into the SM quarks. In Fig. 11 we
show the predictions of the branching ratios of the $Z_{B}$. We note also that
these predictions do not depend on $g_{B}$ since this parameter factorizes out
in the branching ratios.

Figure 11: Branching ratios of the $Z_{B}$ boson. In the parameter space where
the relic density bound is satisfied, $\Omega h^{2}\leq 0.12$, the $Z_{B}$ can
only decay to quarks, and hence, the branching ratios into different quarks is
completely determined as a function of the $Z_{B}$ mass.

We would like to emphasize that this simple prediction is a consequence of
studying the correlation between the dark matter cosmological constraints and
the predictions for collider physics.

##  4 Summary

In this work, we have discussed the possibility to search for the decays of a
new Higgs in theories for dark matter where a Majorana dark matter is
predicted from the cancellation of gauge anomalies. We have considered the
minimal gauge theory for baryon number, in which four extra fermion
representations are needed for anomaly cancellation, including a dark matter
candidate. Taking into account these new fermions is crucial to compute the
predictions for the decay of the new Higgs, since the loop-induced decays with
the anomalons running in the loop can have large branching ratios in some
regions of the parameter space.

We showed that there are three different scenarios consistent with the relic
density constraint; each one determined by the annihilation channel that gives
the dominant contribution to the dark matter relic density. In the scenario at
the resonance $\chi\chi\to Z_{B}^{*}\to q\bar{q}$ it is possible for the
Baryonic Higgs to decay mostly into a pair of dark matter particles so the
invisible branching ratio can be very large $\text{BR}(h_{B}\to\chi\chi)\simeq
90\%$.

In the second scenario, the annihilation channel $\chi\chi\to Z_{B}h_{B}$
dominates. This scenario is appealing since it does not rely on any resonance
to achieve the correct dark matter relic abundance, and hence, a large region
in the parameter space corresponds to this scenario. Furthermore, the
branching ratio of the new Baryonic Higgs into photons can be much larger than
the SM Higgs branching ratio into photons, i.e.
$\text{BR}(h_{B}\to\gamma\gamma)\gtrsim 1\%$.

For the third scenario, the $\chi\chi\to Z_{B}Z_{B}$ is the dominant
annihilation channel and the condition $M_{h_{B}}\lesssim 2M_{Z_{B}}$ is
satisfied, thus, the second Higgs can decays into a pair of the new gauge
bosons. In fact, we find that the $h_{B}\to Z_{B}Z_{B}$ decay dominates,
followed by the invisible decay $h_{B}\to\chi\chi$ which, if kinematically
open, usually has a branching ratio $\text{BR}(h_{B}\to\chi\chi)\simeq 10\%$.

We also demonstrated that the dark matter direct detection experimental bounds
set a strong limit on the mixing between the SM Higgs and the Baryonic Higgs.
Consequently, all the SM-like Higgs production mechanisms are typically
suppressed. Motivated by this, we studied the cross section for the associated
production $pp\to Z_{B}^{*}\to h_{B}Z_{B}$ and showed that it can be large in
agreement with all experimental constraints. Therefore, it is possible to
expect a large number of events for the exotic signatures
$t\bar{t}E_{T}^{miss}$ and $b\bar{b}E_{T}^{miss}$ in the scenarios where the
invisible branching ratio can be large, and $t\bar{t}\gamma\gamma$ and
$b\bar{b}\gamma\gamma$ in the scenario where the Baryonic Higgs can decay into
two photons with a large branching ratio. These results further motivate new
studies in the search for a new Higgs that decays into dark matter at ATLAS
and CMS. In summary, our results show the importance of the correlation
between the cosmological constraints and the predictions for Higgs decays in a
theory predicting the existence of dark matter from anomaly cancellation.

Acknowledgments: The work of P.F.P. has been supported by the U.S. Department
of Energy, Office of Science, Office of High Energy Physics, under Award
Number de-sc0020443. This material is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of High Energy Physics, under
Award Number DE-SC0011632. C.M. thanks the support provided by the Walter
Burke Institute for Theoretical Physics. We would like to thank E. Golias for
several discussions.

##  Appendix A Feynman Rules

In this appendix we list the Feynman rules that have been used in this work.
In the following expressions, as well as in Appendix B, we define the dark
matter field and the Majorana anomalons as:

$\chi=\chi^{0}_{L}+(\chi^{0}_{L})^{C},\quad\quad\text{ and
}\quad\quad\Sigma^{0}=\Sigma^{0}_{L}+(\Sigma^{0}_{L})^{C},$ (22)

while the Dirac anomalons are given by

$\Psi^{+}=(\Psi_{1}^{+})_{L}+(\Psi_{2}^{+})_{R},\quad\quad\text{ and
}\quad\quad\Sigma^{+}=\Sigma^{+}_{L}+(\Sigma_{L}^{-})^{C}.$ (23)

Taking the above definitions into account and working in the context where we
neglect the Yukawa couplings between the new fermions and the SM Higgs,
$y_{i}$ with $i=1,...,4$ from Eq. (5), the simplified Feynman rules read as:

$\begin{aligned}
&\vbox{\hbox{\includegraphics[height=51.6665pt]{psi_psi_A.pdf}}}:\quad-
ie\gamma^{\mu},\\\\[8.61108pt]
&\vbox{\hbox{\includegraphics[height=51.6665pt]{psi_psi_h.pdf}}}:\quad-\frac{i}{\sqrt{2}}\cos\theta_{B}y_{\Psi},\end{aligned}\qquad\qquad\begin{aligned}
&\vbox{\hbox{\includegraphics[height=51.6665pt]{sigma_sigma_A.pdf}}}:\quad-
ie\gamma^{\mu},\\\\[8.61108pt]
&\vbox{\hbox{\includegraphics[height=51.6665pt]{sigma_sigma_h.pdf}}}:\quad-i\sqrt{2}\cos\theta_{B}y_{\Sigma},\\\
\end{aligned}$

$\begin{aligned}
&\vbox{\hbox{\includegraphics[height=51.6665pt]{psi_psi_W.pdf}}}:\quad-\frac{ig}{\sqrt{2}}\gamma^{\mu},\\\\[8.61108pt]
&\vbox{\hbox{\includegraphics[height=51.6665pt]{psi_psi_Z.pdf}}}:\quad-
ig\frac{s_{W}}{t_{2W}}\gamma^{\mu},\\\\[8.61108pt]
&\vbox{\hbox{\includegraphics[height=51.6665pt]{psi0_psi0_Z.pdf}}}:\quad\frac{ig}{2c_{W}}\gamma^{\mu},\end{aligned}\qquad\qquad\begin{aligned}
&\vbox{\hbox{\includegraphics[height=51.6665pt]{sigma_sigma_W.pdf}}}:\quad
ig\gamma^{\mu},\\\\[8.61108pt]
&\vbox{\hbox{\includegraphics[height=51.6665pt]{sigma_sigma_Z.pdf}}}:\quad-
ig\frac{s_{W}}{t_{W}}\gamma^{\mu},\\\
&\vbox{\hbox{\includegraphics[height=51.6665pt]{psi0_psi0_h.pdf}}}:\quad-\frac{i}{\sqrt{2}}\cos\theta_{B}y_{\Psi}.\end{aligned}$

##  Appendix B Higgs Decays

In this Appendix, we present the Higgs decays including the loop-induced
channels and the full analytic expressions. To compute the loop functions we
make use of the Package-X Patel:2015tea  Mathematica package. In the Feynman
diagrams below, the orange dot symbolizes that the vertex can only occur by
the mixing of the Baryonic Higgs with the SM Higgs.

  * •

${\boldmath h_{B}\to\gamma\gamma}$:

$\displaystyle=$
$\displaystyle\vbox{\hbox{\includegraphics[height=73.19421pt]{hBgammagamma_F.pdf}}}+\vbox{\hbox{\includegraphics[height=73.19421pt]{hBgammagamma_fSM.pdf}}}+\vbox{\hbox{\includegraphics[height=73.19421pt]{hBgammagamma_W.pdf}}}$
$\displaystyle\Gamma(h_{B}\to\gamma\gamma)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{64\pi^{3}M_{h_{B}}^{5}}\left|\cos\theta_{B}\sum_{F^{+}}\frac{3g_{B}M_{F^{+}}^{2}}{M_{Z_{B}}}F_{F^{+}}+\frac{\sin\theta_{B}}{v_{0}}\left(\sum_{f^{+}}N_{c}^{f}Q_{f}^{2}m_{f^{+}}^{2}F_{f^{+}}-F_{W}\right)\right|^{2}.\quad\quad\quad$

  * •

$\boldmath h_{B}\to Z_{B}\gamma$:

$\displaystyle=$
$\displaystyle\underbrace{\vbox{\hbox{\includegraphics[height=73.19421pt]{hBZBgamma_F.pdf}}}}_{=0}+\vbox{\hbox{\includegraphics[height=73.19421pt]{hBZBgamma_q.pdf}}}$
$\displaystyle\Gamma(h_{B}\to Z_{B}\gamma)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{B}\,g_{B}^{2}\,\frac{\alpha}{288\,\pi^{4}}\,\frac{M_{h_{B}}^{2}-M_{Z_{B}}^{2}}{v_{0}^{2}M_{h_{B}}^{3}}\left|\sum_{q}Q_{q}m_{q}^{2}A_{q}\right|^{2}.$

  * •

$h_{B}\to Z\gamma$:

$\displaystyle=$
$\displaystyle\vbox{\hbox{\includegraphics[height=73.19421pt]{hBZgamma_F.pdf}}}\\!\\!\\!\\!\\!+\vbox{\hbox{\includegraphics[height=73.19421pt]{hBZgamma_fSM.pdf}}}\\!\\!\\!\\!\\!+\vbox{\hbox{\includegraphics[height=73.19421pt]{hBZgamma_W.pdf}}}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\Gamma(h_{B}\to Z\gamma)$
$\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{8\pi^{3}}\frac{M_{h_{B}}^{2}\\!-\\!M_{Z}^{2}}{M_{h_{B}}^{3}}\left|\frac{\sin\theta_{B}}{v_{0}\,s2\theta_{W}}\left(\sum_{u}m_{u}^{2}\left(1\\!-\\!\frac{8}{3}\sin^{2}\theta_{W}\right)A_{u}+\sum_{d}\frac{m_{d}^{2}}{2}\left(1\\!-\\!\frac{4}{3}\sin^{2}\theta_{W}\right)A_{d}\right.\right.\quad\quad\quad$
$\displaystyle+$
$\displaystyle\left.\sum_{e}\frac{m_{e}^{2}}{2}\left(1\\!-\\!4\sin^{2}\theta_{W}\right)A_{e}-\frac{\cos^{2}\theta_{W}}{2}A_{W}\right)+\left.\frac{3\,g_{B}\cos\theta_{B}}{M_{Z_{B}}}\left(\frac{M_{\Sigma^{+}}^{2}}{t\theta_{W}}A_{\Sigma^{+}}+\frac{M_{\Psi^{+}}^{2}}{t2\theta_{W}}A_{\Psi^{+}}\right)\right|^{2}.$

  * •

$h_{B}\to gg$:

$\displaystyle\vbox{\hbox{\includegraphics[height=64.58313pt]{hBgg.pdf}}}\qquad\Gamma(h_{B}\to
gg)=\sin^{2}\theta_{B}\frac{\alpha_{s}^{2}M_{h_{B}}^{3}}{128\,v_{0}^{2}\,\pi^{3}}\left|\sum_{q}\frac{2m_{q}^{2}}{M_{h_{B}}^{4}}F_{q}\right|^{2}.$

  * •

$h_{B}\to Z_{B}Z_{B}$:

$\displaystyle\vbox{\hbox{\includegraphics[height=51.6665pt]{hBZBZB.pdf}}}\qquad\Gamma(h_{B}\to
Z_{B}Z_{B})=\frac{9g_{B}^{2}}{32\pi}\cos^{2}\theta_{B}\sqrt{1-\frac{4M_{Z_{B}}^{2}}{M_{h_{B}}^{2}}}\,\frac{M_{h_{B}}^{4}-4M_{h_{B}}^{2}M_{Z_{B}}^{2}+12M_{Z_{B}}^{4}}{M_{h_{B}}M_{Z_{B}}^{2}}.\quad\quad\quad$

  * •

$h_{B}\to WW$:

$\displaystyle=$
$\displaystyle\vbox{\hbox{\includegraphics[height=64.58313pt]{hBtoWW_C1.pdf}}}+\vbox{\hbox{\includegraphics[height=64.58313pt]{hBtoWW_C2.pdf}}}+\vbox{\hbox{\includegraphics[height=51.6665pt]{hBWW_mixing.pdf}}}$
$\displaystyle\Gamma(h_{B}\to WW)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{M_{h_{B}}^{2}-4M_{W}^{2}}}{16\pi
M_{h_{B}}^{2}M_{W}^{4}}\left(\cos^{2}\theta_{B}\frac{9g_{B}^{2}}{M_{Z_{B}}^{2}}\left|\sum_{F}g_{WF}^{2}B_{F}[W]\right|^{2}+2(M_{h_{B}}^{2}-2M_{W}^{2})\times\right.$
$\displaystyle\text{Re}\left\\{\cos\theta_{B}\frac{3g_{B}}{M_{Z_{B}}}\sum_{F}g_{WF}^{2}B_{F}[W]\left(\cos\theta_{B}\frac{3g_{B}}{M_{Z_{B}}}\sum_{F}g_{WF}^{2}C_{F}^{*}[W]+\frac{\sin\theta_{B}}{v_{0}}M_{W}^{2}\right)\right\\}$
$\displaystyle\left.+\left|\cos\theta_{B}\frac{3g_{B}}{M_{Z_{B}}}\sum_{F}g_{WF}^{2}C_{F}[W]+\frac{\sin\theta_{B}}{v_{0}}M_{W}^{2}\right|^{2}(M_{h_{B}}^{4}-4M_{h_{B}}^{2}M_{W}^{2}+12M_{W}^{4})\right).$

  * •

$h_{B}\to ZZ$:

$\displaystyle=$
$\displaystyle\vbox{\hbox{\includegraphics[height=64.58313pt]{hBtoZZ_F.pdf}}}+\vbox{\hbox{\includegraphics[height=51.6665pt]{hBZZ_mixing.pdf}}}$
$\displaystyle\Gamma(h_{B}\to ZZ)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{M_{h_{B}}^{2}-4M_{Z}^{2}}}{128\pi
M_{h_{B}}^{2}M_{Z}^{4}}\left(\cos^{2}\theta_{B}\frac{9g_{B}^{2}}{M_{Z_{B}}^{2}}\left|\sum_{F}g_{ZF}^{2}B_{F}[Z]\right|^{2}+2(M_{h_{B}}^{2}-2M_{Z}^{2})\times\right.$
$\displaystyle\text{Re}\left\\{\cos\theta_{B}\frac{3g_{B}}{M_{Z_{B}}}\sum_{F}g_{ZF}^{2}B_{F}[Z]\left(\cos\theta_{B}\frac{3g_{B}}{M_{Z_{B}}}\sum_{F}g_{ZF}^{2}C_{F}^{*}[Z]+2\frac{\sin\theta_{B}}{v_{0}}M_{Z}^{2}\right)\right\\}$
$\displaystyle\left.+\left|\cos\theta_{B}\frac{3g_{B}}{M_{Z_{B}}}\sum_{F}g_{ZF}^{2}C_{F}[Z]+2\frac{\sin\theta_{B}}{v_{0}}M_{Z}^{2}\right|^{2}(M_{h_{B}}^{4}-4M_{h_{B}}^{2}M_{Z}^{2}+12M_{Z}^{4})\right).$

  * •

$h_{B}\to ZZ_{B}$:

$\displaystyle=$
$\displaystyle\underbrace{\vbox{\hbox{\includegraphics[height=64.58313pt]{hBZZB_F.pdf}}}}_{=\,0}+\vbox{\hbox{\includegraphics[height=64.58313pt]{hBZBZ_q.pdf}}}$
$\displaystyle\Gamma(h_{B}\to ZZ_{B})$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{B}\,g_{B}^{2}\,\alpha\,\frac{\lambda^{1/2}(M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2})}{64\,v_{0}^{2}\,\sin^{2}2\theta_{W}M_{h_{B}}^{3}M_{Z}^{2}M_{Z_{B}}^{2}}$
$\displaystyle\left(\left|\tilde{\Gamma}_{1}^{ZZ_{B}}\right|^{2}-2\,\text{Re}\left\\{\tilde{\Gamma}_{1}^{ZZ_{B}}(\tilde{\Gamma}_{2}^{ZZ_{B}})^{*}\right\\}(M_{h_{B}}^{2}-M_{Z}^{2}-M_{Z_{B}}^{2})\right.$
$\displaystyle\left.+\left|\tilde{\Gamma}_{2}^{ZZ_{B}}\right|^{2}(M_{h_{B}}^{4}-2M_{h_{B}}^{2}(M_{Z}^{2}+M_{Z_{B}}^{2})+M_{Z}^{4}+10\,M_{Z}^{2}M_{Z_{B}}^{2}+M_{Z_{B}}^{4})\right),$

where

$\tilde{\Gamma}_{1}^{ZZ_{B}}\equiv\sum_{q}v_{Z}^{q}m_{q}^{2}D_{q},\quad\text{
and }\quad\tilde{\Gamma}_{2}^{ZZ_{B}}\equiv\sum_{q}v_{Z}^{q}m_{q}^{2}E_{q}.$

Below we write explicitly the loop functions as a function of the fermion mass
$m_{f}$ or gauge boson mass $M_{V}$ running inside the loop and the Passarino-
Veltman functions defined in Package-X:

  * •

Loop functions entering in the $h_{B}$ decay to two massless gauge fields:

$\displaystyle F_{f}$ $\displaystyle=$ $\displaystyle
4M_{h_{B}}^{2}+(4M_{f}^{2}-M_{h_{B}}^{2})\log^{2}\left[1-\frac{M_{h_{B}}^{2}}{2M_{f}^{2}}\left(1+\sqrt{1-\frac{4M_{f}^{2}}{M_{h_{B}}^{2}\,}}\right)\right],$
$\displaystyle F_{W}$ $\displaystyle=$ $\displaystyle
M_{h_{B}}^{4}+6M_{h_{B}}^{2}M_{W}^{2}-3(M_{h_{B}}^{2}-2M_{W}^{2})M_{W}^{2}\log^{2}\left[1-\frac{M_{h_{B}}^{2}}{2M_{W}^{2}}\left(1+\sqrt{1-\frac{4M_{W}^{2}}{M_{h_{B}}^{2}}}\,\right)\right].$

  * •

Loop functions entering in the $h_{B}$ decay to a massive and a massless gauge
fields:

$\displaystyle A_{f}$ $\displaystyle=$
$\displaystyle(4m_{f}^{2}-M_{h_{B}}^{2}+M_{V}^{2})C_{0}[0,M_{h_{B}}^{2},M_{V}^{2};m_{f}]+2\left(\frac{M_{V}^{2}(\Lambda[M_{h_{B}}^{2}]-\Lambda[M_{V}^{2}])}{M_{h_{B}}^{2}-M^{2}_{V}}+1\right),$
$\displaystyle A_{W}$ $\displaystyle=$ $\displaystyle
2\,M_{W}^{2}\,C_{0}[0,M_{h_{B}}^{2},M_{V}^{2};M_{W}]\left((M_{h_{B}}^{2}-2M_{W}^{2})(\tan^{2}\theta_{W}-5)-2M_{V}^{2}(\tan^{2}\theta_{W}-3)\right)$
$\displaystyle+\left(\frac{M_{V}^{2}}{M_{h_{B}}^{2}-M_{V}^{2}}(\Lambda[M_{h_{B}}^{2}]-\Lambda[M_{V}^{2}])+1\right)\left(M_{h_{B}}^{2}(1-\tan^{2}\theta_{W})+2M_{W}^{2}(5-\tan^{2}\theta_{W})\right).$

  * •

Loop functions entering in the $h_{B}$ decay to a two equal massive gauge
bosons:

$\displaystyle B_{F}[V]$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{4M_{F}^{2}}{M_{h_{B}}^{2}-4M_{V}^{2}}\times$
$\displaystyle\left[(M_{h_{B}}^{2}-2M_{V}^{2})\left(M_{h_{B}}^{4}-4M_{F}^{2}(M_{h_{B}}^{2}-4M_{V}^{2})-6M_{h_{B}}^{2}M_{V}^{2}-4M_{V}^{4}\right)C_{0}[M_{h_{B}}^{2},M_{V}^{2},M_{V}^{2};M_{F}]\right.$
$\displaystyle\left.+\,4M_{V}^{2}(M_{h_{B}}^{2}+2M_{V}^{2})(\Lambda[M_{V}^{2}]-\Lambda[M_{h_{B}}^{2}])-2(M_{h_{B}}^{4}-6M_{h_{B}}^{2}M_{V}^{2}+8M_{V}^{4})\right),$
$\displaystyle C_{F}[V]$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{4M_{F}^{2}}{M_{h_{B}}^{2}-4M_{V}^{2}}\left[2(M_{h_{B}}^{2}+2M_{V}^{2}(\Lambda[M_{h_{B}}^{2}]-\Lambda[M_{V}^{2}])-4M_{V}^{2})\right.$
$\displaystyle\left.-(M_{h_{B}}^{4}-6M_{h_{B}}^{2}M_{V}^{2}-4M_{F}^{2}(M_{h_{B}}^{2}-4M_{V}^{2})+4M_{V}^{4})C_{0}[M_{h_{B}}^{2},M_{V}^{2},M_{V}^{2};M_{F}]\right].$

  * •

Loop functions entering in the decay to two different massive gauge boson:

$\displaystyle D_{q}$ $\displaystyle=$
$\displaystyle\frac{(M_{h_{B}}^{2}-(M_{Z}-M_{Z_{B}})^{2})(M_{h_{B}}^{2}-(M_{Z}+M_{Z_{B}})^{2})}{\pi^{2}(4M_{Z}^{2}M_{Z_{B}}^{2}-(M_{h_{B}}^{2}-M_{Z}^{2}-M_{Z_{B}}^{2})^{2})^{2}}\times$
$\displaystyle\left[((M_{h_{B}}^{2}-M_{Z}^{2}-M_{Z_{B}}^{2})\left(M_{h_{B}}^{6}-4m_{q}^{2}(M_{h_{B}}^{4}-2M_{h_{B}}^{2}(M_{Z}^{2}+M_{Z_{B}}^{2})+(M_{Z}^{2}-M_{Z_{B}}^{2})^{2}\right)\right.$
$\displaystyle-3M_{h_{B}}^{4}(M_{Z}^{2}+M_{Z_{B}}^{2})+M_{h_{B}}^{2}(3M_{Z}^{4}-10M_{Z}^{2}M_{Z_{B}}^{2}+3M_{Z_{B}}^{4})$
$\displaystyle-(M_{Z}^{2}-M_{Z_{B}}^{2})^{2}(M_{Z}^{2}+M_{Z_{B}}^{2}))C_{0}[M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2};m_{q}]/2$
$\displaystyle+M_{Z}^{2}(4M_{Z_{B}}^{2}(M_{h_{B}}^{2}+M_{Z}^{2})+(M_{h_{B}}^{2}-M_{Z}^{2})^{2}-5M_{Z_{B}}^{4})\Lambda[M_{Z}^{2}]$
$\displaystyle-\left(M_{h_{B}}^{4}(M_{Z}^{2}+M_{Z_{B}}^{2})-2M_{h_{B}}^{2}(M_{Z}^{4}-4M_{Z}^{2}M_{Z_{B}}^{2}+M_{Z_{B}}^{4})+(M_{Z}^{2}-M_{Z_{B}}^{2})^{2}(M_{Z}^{2}+M_{Z_{B}}^{2}\right)\Lambda[M_{h_{B}}^{2}]$
$\displaystyle+M_{Z_{B}}^{2}\left(M_{h_{B}}^{4}+M_{h_{B}}^{2}(4M_{Z}^{2}-2M_{Z_{B}}^{2})-5M_{Z}^{4}+4M_{Z}^{2}M_{Z_{B}}^{2}+M_{Z_{B}}^{4}\right)\Lambda[M_{Z_{B}}^{2}]$
$\displaystyle\left.+(M_{h_{B}}^{2}-M_{Z}^{2}-M_{Z_{B}}^{2})\left(4M_{Z}^{2}M_{Z_{B}}^{2}-(M_{h_{B}}^{2}-M_{Z}^{2}-M_{Z_{B}}^{2})^{2}\right)\right],$
$\displaystyle E_{q}$ $\displaystyle=$
$\displaystyle\displaystyle\left[\phantom{\frac{A}{B}}\\!\\!\\!\\!\\!\\!\\!\\!\left(4M_{h_{B}}^{2}M_{Z}^{2}M_{Z_{B}}^{2}+(4m_{q}^{2}-M_{h_{B}}^{2}+M_{Z}^{2}+M_{Z_{B}}^{2})\lambda[M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2}]\right)C_{0}[M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2};m_{q}]\right.$
$\displaystyle-2M_{Z}^{2}(M_{h_{B}}^{2}-M_{Z}^{2}+M_{Z_{B}}^{2})\Lambda[M_{Z}^{2}]+2\left(M_{h_{B}}^{2}(M_{Z}^{2}+M_{Z_{B}}^{2})-(M_{Z}^{2}-M_{Z_{B}}^{2})^{2}\right)\Lambda[M_{h_{B}}]$
$\displaystyle\left.\phantom{\frac{A}{B}}\\!\\!\\!\\!\\!\\!\\!\\!\\!-2M_{Z_{B}}^{2}(M_{h_{B}}^{2}+M_{Z}^{2}-M_{Z_{B}}^{2})\Lambda[M_{Z_{B}}^{2}]+2\lambda[M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2}]\right]\big{/}\left(4\pi^{2}\lambda[M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2}]\right),$

where $v_{Z}^{q}=T_{3}^{q}(1-4|Q_{q}|\sin^{2}\theta_{W})$ is the vector
coupling of the quarks with the $Z$ boson and $T_{3}^{q}$ is the weak isospin,
and the Källen function is defined as follows:

$\lambda[M_{h_{B}}^{2},M_{Z}^{2},M_{Z_{B}}^{2}]=M_{h_{B}}^{4}-2M_{h_{B}}^{2}M_{Z}^{2}-2M_{h_{B}}^{2}M_{Z_{B}}^{2}+M_{Z}^{4}-2M_{Z}^{2}M_{Z_{B}}^{2}+M_{Z_{B}}^{4}.$
(24)

Finally we list the baryonic Higgs decays to SM fermions, to the anomalons and
to a couple of SM Higgs bosons:

$\displaystyle\Gamma(h_{B}\to\bar{f}f)$ $\displaystyle=$ $\displaystyle
N_{c}^{f}\,\sin^{2}\theta_{B}\frac{m_{f}^{2}}{8\pi\,v^{2}}M_{h_{B}}\left(1-4\frac{m_{f}^{2}}{M_{h_{B}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma(h_{B}\to\bar{\Psi}^{0}\Psi^{0},\bar{\Psi}^{+}\Psi^{+})$
$\displaystyle=$
$\displaystyle\frac{9}{8\pi}g_{B}^{2}\,\cos^{2}\theta_{B}\,\frac{M_{\Psi}^{2}}{M_{Z_{B}}^{2}}M_{h_{B}}\left(1-4\frac{M_{\Psi}^{2}}{M_{h_{B}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma(h_{B}\to\bar{\Sigma}^{+}\Sigma^{+})$ $\displaystyle=$
$\displaystyle\frac{9}{8\pi}g_{B}^{2}\,\cos^{2}\theta_{B}\frac{M_{\Sigma^{+}}^{2}}{M_{Z_{B}}^{2}}M_{h_{B}}\left(1-4\frac{M_{\Sigma^{+}}^{2}}{M_{h_{B}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma(h_{B}\to\Sigma^{0}\Sigma^{0})$ $\displaystyle=$
$\displaystyle\frac{9}{16\pi}g_{B}^{2}\,\cos^{2}\theta_{B}\frac{M_{\Sigma^{0}}^{2}}{M_{Z_{B}}^{2}}M_{h_{B}}\left(1-4\frac{M_{\Sigma^{0}}^{2}}{M_{h_{B}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma(h_{B}\to\chi\chi)$ $\displaystyle=$
$\displaystyle\frac{9}{16\pi}g_{B}^{2}\,\cos^{2}\theta_{B}\,\frac{M_{\chi}^{2}}{M_{Z_{B}}^{2}}M_{h_{B}}\left(1-4\frac{M_{\chi}^{2}}{M_{h_{B}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma(h_{B}\to h\,h)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{M_{h_{B}}^{2}-4M_{h}^{2}}}{32\pi
M_{h_{B}}^{2}}|c_{112}|^{2}.$

The couplings between the gauge bosons and the new anomalons are given by (see
Feynman rules in Appendix A),

$g_{W\Sigma}=-\frac{e}{\sin\theta_{\text{W}}}=-\sqrt{2}\,g_{W\Psi},\quad
g_{Z\Psi_{0}}=-\frac{e}{\sin 2\theta_{W}},\quad g_{Z\Psi^{+}}=\frac{e}{\tan
2\theta_{W}},\quad g_{Z\Sigma^{+}}=\frac{e}{\tan\theta_{W}},$

whereas the trilinear interaction of Higgses in the scalar potential is
parametrized by

$\displaystyle c_{112}$ $\displaystyle=$
$\displaystyle-6\lambda_{h}v_{0}\cos^{2}\theta_{B}\sin\theta_{B}+6\lambda_{B}v_{B}\cos\theta_{B}\sin^{2}\theta_{B}$
$\displaystyle+\lambda_{HB}(v_{B}\cos^{3}\theta_{B}+2v_{0}\cos^{2}\theta_{B}\sin\theta_{B}-2v_{B}\cos\theta_{B}\sin^{2}\theta_{B}-v_{0}\sin^{3}\theta_{B}),$

and the couplings from the scalar potential can be expressed as a function of
the Higgs masses, vacuum expectation values and the mixing angle as follows:

$\displaystyle\lambda_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2v_{0}^{2}}(M_{h}^{2}\cos^{2}\theta_{B}+M_{h_{B}}^{2}\sin^{2}\theta_{B}),$
(25) $\displaystyle\lambda_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2v_{B}^{2}}(M_{h}^{2}\sin^{2}\theta_{B}+M_{h_{B}}^{2}\cos^{2}\theta_{B}),$
(26) $\displaystyle\lambda_{HB}$ $\displaystyle=$
$\displaystyle\frac{1}{v_{0}v_{B}}(M_{h_{B}}^{2}-M_{h}^{2})\sin\theta_{B}\cos\theta_{B}.$
(27)

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