Balancing polynomials, Fibonacci numbers and some new series for $\pi$

Robert Frontczak111Corresponding author. Statements and conclusions made in
this paper by R. F. are entirely those of the author. They do not necessarily
reflect the views of LBBW.

Landesbank Baden-Württemberg
Am Hauptbahnhof 2, 70173 Stuttgart
Germany
<EMAIL_ADDRESS>

Kalika Prasad
Central University of Jharkhand,
Ranchi, 835205, India
<EMAIL_ADDRESS>

###### Abstract

We evaluate some types of infinite series with balancing and Lucas-balancing
polynomials in closed form. These evaluations will lead to some new curious
series for $\pi$ involving Fibonacci and Lucas numbers. Our findings
complement those of Castellanos from 1986 and 1989.

2010 Mathematics Subject Classification: Primary 11B37, 11B39; Secondary
15A15.

_Keywords:_ balancing polynomial, Lucas-balancing polynomial, Fibonacci
number, Lucas number, infinite series for $\pi$.

##  1 Motivation and Preliminaries

Castellanos [1, 2] has found, among other things, the following curious series
for $\pi$:

$\frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{(2n+1)(2m+1)}}{5^{n}}\Big{(}\frac{2}{F_{2m+1}+\sqrt{F_{2m+1}^{2}+4/5}}\Big{)}^{2n+1},\quad
m\geq 0.$ (1)

When $m=0$ the expression simplifies to ([1, Eq. (46)] and [2, Eq. (3.4)])

$\frac{\pi}{4}=\sqrt{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}F_{2n+1}\Big{(}\frac{2}{3+\sqrt{5}}\Big{)}^{2n+1},$

which has the equivalent form

$\frac{\pi}{4}=\sqrt{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2n+1}}{\alpha^{4n+2}}.$
(2)

Another series involving the squares of $F_{2n+1}$ is ([2, Eqs. (3.2) and
(3.3)])

$\frac{\pi}{20}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{(2n+1)(2m+1)}^{2}}{(t+\sqrt{t^{2}+1})^{2n+1}},\quad
m\geq 0,$ (3)

with

$t=\frac{5F_{2m+1}^{2}}{4}\Big{(}1+\sqrt{1+\Big{(}\frac{16}{25}\Big{(}\frac{5}{2}F_{2m+1}^{2}-1\Big{)}/F_{2m+1}^{4}\Big{)}},$

which, for $m=0$, simplifies to

$\frac{\pi}{20}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2n+1}^{2}}{(3+\sqrt{10})^{2n+1}}.$
(4)

In the identities above, $F_{n}$ (respectively $L_{n}$) are the famous
Fibonacci (Lucas) numbers, defined for $n\geq 0$ by the recursion
$F_{n+2}=F_{n+1}+F_{n}$ ($L_{n+2}=L_{n+1}+L_{n}$) with initial conditions
$F_{0}=0,F_{1}=1$ ($L_{0}=2$ and $L_{1}=1$), and $\alpha=\frac{1+\sqrt{5}}{2}$
is the golden ratio. Castellanos’ series pose the question, whether more
series of these types exist? Here, we give a positive answer to this question.
Working with balancing and Lucas-balancing polynomials we derive more series
for $\pi$ involving Fibonacci and Lucas numbers and exhibiting such a
structure.

Recall that, for any integer $n\geq 0$ and $x\in\mathbb{C}$, balancing
polynomials $\big{(}B_{n}(x)\big{)}_{n\geq 0}$ and Lucas-balancing polynomials
$\big{(}C_{n}(x)\big{)}_{n\geq 0}$ are defined by the second-order homogeneous
linear recurrence [4]

$u_{n}(x)=6xu_{n-1}(x)-u_{n-2}(x),$ (5)

but with different initial terms. Balancing polynomials start with
$B_{0}(x)=0$ and $B_{1}(x)=1$, while for Lucas-balancing polynomials we set
$C_{0}(x)=1$, $C_{1}(x)=3x$. These polynomials have been introduced as a
natural extension of the popular balancing and Lucas-balancing numbers $B_{n}$
and $C_{n}$, respectively, and must be seen as a special member of the Horadam
sequence. Obviously, $B_{n}=B_{n}(1)$ and $C_{n}=C_{n}(1)$. The first few
polynomials are

$\displaystyle
B_{0}(x)=0,\quad\,\,B_{1}(x)=1,\quad\,\,B_{2}(x)=6x,\quad\,\,B_{3}(x)=36x^{2}-1,$
$\displaystyle B_{4}(x)=216x^{3}-12x,\,\,\quad B_{5}(x)=1296x^{4}-108x^{2}+1,$

and

$\displaystyle C_{0}(x)=1,\,\,\quad C_{1}(x)=3x,\,\,\quad
C_{2}(x)=18x^{2}-1,\,\,\quad C_{3}(x)=108x^{3}-9x,$ $\displaystyle
C_{4}(x)=648x^{4}-72x^{2}+1,\,\,\quad C_{5}(x)=3888x^{5}-540x^{3}+15x.$

The closed forms, known as Binet’s formulas, for balancing and Lucas-balancing
polynomials are given by

$B_{n}(x)=\frac{\lambda_{1}^{n}(x)-\lambda_{2}^{n}(x)}{\lambda_{1}(x)-\lambda_{2}(x)},\qquad
C_{n}(x)=\frac{\lambda_{1}^{n}(x)+\lambda_{2}^{n}(x)}{2},$ (6)

where $\lambda_{1}(x)=3x+\sqrt{9x^{2}-1}$ and
$\lambda_{2}(x)=3x-\sqrt{9x^{2}-1}$. We note the following properties:
$\lambda_{1}(x)\cdot\lambda_{2}(x)=1,\lambda_{1}(x)-\lambda_{2}(x)=2\sqrt{9x^{2}-1},\lambda_{1}(x)+\lambda_{2}(x)=6x$
and $0<\lambda_{2}(x)<\lambda_{1}(x)$ for $x>1/3$. The polynomials possess a
simple connection to Chebyshev polynomials [4] via

$B_{n}(x)=U_{n-1}(3x)\qquad\mbox{and}\qquad C_{n}(x)=T_{n}(3x),$

where $U_{n}(x)$ and $T_{n}(x)$ are Chebyshev polynomials of the first and
second kind, respectively. Some other interesting properties have been
discovered in the articles [5, 6, 7, 8, 9]. Balancing and Lucas-balancing
polynomials are also linked in various ways to Fibonacci and Lucas numbers.
Two such connections, which will be exploited in the text, are

$B_{n}\Big{(}\frac{L_{2m}}{6}\Big{)}=\frac{F_{2mn}}{F_{2m}}\,\,,\qquad
C_{n}\Big{(}\frac{L_{2m}}{6}\Big{)}=\frac{L_{2mn}}{2},$ (7)

and

$B_{2n+1}\Big{(}\frac{\sqrt{5}}{6}F_{2m+1}\Big{)}=\frac{L_{(2m+1)(2n+1)}}{L_{2m+1}}\,\,,\qquad
C_{2n+1}\Big{(}\frac{\sqrt{5}}{6}F_{2m+1}\Big{)}=\frac{\sqrt{5}}{2}F_{(2m+1)(2n+1)}.$
(8)

These relations can be deduced from the corresponding relations between
Fibonacci numbers and Chebyshev polynomials (see [2] and [4]).

In the present article, we study special infinite series involving $B_{n}(x)$
and $C_{n}(x)$, respectively. We evaluate these series in closed form. Based
on these results, new infinite series for $\pi$ with Fibonacci and Lucas
numbers will be stated as an immediate consequence. These new and curious
series must be seen as complements of Castellanos’ results from 1986 and 1989.

##  2 Complementing the work of Castellanos (Part 1)

The next theorem is our starting point.

######  Theorem 1.

For each real $x$ with $x>1/3$ and $z\in\mathbb{C}$ with
$|z|>|\lambda_{1}(x)|$, we have

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{B_{2n+1}(x)}{z^{2n+1}}=\frac{1}{2\sqrt{9x^{2}-1}}\arctan\Big{(}\frac{2\sqrt{9x^{2}-1}z}{z^{2}+1}\Big{)}$
(9)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{C_{2n+1}(x)}{z^{2n+1}}=\frac{1}{2}\arctan\Big{(}\frac{6xz}{z^{2}-1}\Big{)}.$
(10)

###### Proof.

Recall the Taylor series for the arctangent function

$\arctan(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}z^{2n+1},\quad|z|\leq 1.$
(11)

Hence, for all $x$ and $z$ with $|z|>|\lambda_{1}(x)|$,

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{B_{2n+1}(x)}{z^{2n+1}}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda_{1}(x)-\lambda_{2}(x)}\Big{(}\arctan\Big{(}\frac{\lambda_{1}(x)}{z}\Big{)}-\arctan\Big{(}\frac{\lambda_{2}(x)}{z}\Big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{9x^{2}-1}}\arctan\Big{(}\frac{2\sqrt{9x^{2}-1}z}{z^{2}+1}\Big{)},$

where in the last step we applied the identity

$\arctan(a)\pm\arctan(b)=\arctan\Big{(}\frac{a\pm b}{1\mp ab}\Big{)}.$ (12)

This proves the first identity. The proof of the second identity is very
similar and omitted. ∎

######  Corollary 2.

The following series representation for $\pi$ holds

$\frac{\pi}{8}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{C_{2n+1}}{(3+\sqrt{10})^{2n+1}}.$
(13)

######  Remark 3.

Observe the similarity of equation (13) to Castellanos’ series (4). To be
particular, the two series provide an interesting example for series whose
members’ denominators and the sum are the same but the numerators are
different. Namely,

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2n+1}^{2}}{(3+\sqrt{10})^{2n+1}}=\frac{\pi}{20}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{\frac{2}{5}C_{2n+1}}{(3+\sqrt{10})^{2n+1}}.$

Such series seem to be rare. Another example given by Mező [10] (and involving
Lucas numbers) are the series

$\sum_{n=1}^{\infty}\frac{L_{n}}{n2^{n}}=2\cdot\ln(2)=\sum_{n=1}^{\infty}\frac{2}{n2^{n}}.$

###  2.1 Series with even Fibonacci (Lucas) coefficients

Setting $x=L_{2m}/6,m\geq 0$, we first note that

$\lambda_{1}\Big{(}\frac{L_{2m}}{6}\Big{)}=\frac{L_{2m}+\sqrt{L_{2m}^{2}-4}}{2}=\alpha^{2m},$

with $\alpha=(1+\sqrt{5})/2$ and where we have used that
$L_{n}^{2}=5F_{n}^{2}+(-1)^{n}4$. We can now use (7) to obtain the relations,
valid for all $|z|>\alpha^{2m}$,

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2m(2n+1)}}{z^{2n+1}}=\frac{1}{\sqrt{5}}\arctan\Big{(}\sqrt{5}\frac{F_{2m}z}{z^{2}+1}\Big{)},$
(14)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{2m(2n+1)}}{z^{2n+1}}=\arctan\Big{(}\frac{L_{2m}z}{z^{2}-1}\Big{)}.$
(15)

Especially, for $z=L_{2m}$,

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2m(2n+1)}}{L_{2m}^{2n+1}}=\frac{1}{\sqrt{5}}\arctan\Big{(}\sqrt{5}\frac{F_{4m}}{L_{2m}^{2}+1}\Big{)},$
(16)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{2m(2n+1)}}{L_{2m}^{2n+1}}=\arctan\Big{(}\frac{L_{2m}^{2}}{L_{2m}^{2}-1}\Big{)}.$
(17)

Next, from $\alpha^{n}=\alpha F_{n}+F_{n-1},n\geq 1,$ we see that

$\alpha^{2m}<2F_{2m}+F_{2m-1}=F_{2m+2}.$

Hence, inserting $z=F_{2m+2}$, we also get the relations

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2m(2n+1)}}{F_{2m+2}^{2n+1}}=\frac{1}{\sqrt{5}}\arctan\Big{(}\sqrt{5}\frac{F_{2m}F_{2m+2}}{F_{2m+1}F_{2m+3}}\Big{)},$
(18)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{2m(2n+1)}}{F_{2m+2}^{2n+1}}=\arctan\Big{(}\frac{L_{2m}F_{2m+2}}{F_{2m}F_{2m+4}}\Big{)},$
(19)

where we have used the Catalan identities

$F_{2m+2}^{2}+1=F_{2m+3}F_{2m+1}\quad\mbox{and}\quad
F_{2m+2}^{2}-1=F_{2m+4}F_{2m}.$

To derive Castellanos-like expressions for $\pi$ with Fibonacci (Lucas)
coefficients, we use (14) and (15), and relate them to special arctan
arguments. We start with

$\frac{\sqrt{5}F_{2m}z}{z^{2}+1}=1\qquad\mbox{and}\qquad\frac{L_{2m}z}{z^{2}-1}=1.$

Solving $z^{2}-\sqrt{5}F_{2m}z+1=0,m\geq 1,$ gives

$z_{1/2}=\frac{\sqrt{5}F_{2m}\pm\sqrt{5F_{2m}^{2}-4}}{2}.$

But it can be checked easily that for all $m\geq 1$

$0<z_{2}=\frac{\sqrt{5}F_{2m}-\sqrt{5F_{2m}^{2}-4}}{2}<z_{1}=\frac{\sqrt{5}F_{2m}+\sqrt{5F_{2m}^{2}-4}}{2}<\alpha^{2m}$

with $\lim_{m\rightarrow\infty}z_{2}=0$ and $z_{1}\sim\alpha^{2m}$ as
$m\rightarrow\infty$. Similarly, solving the equation $z^{2}-L_{2m}z-1=0,m\geq
0$ gives

$z_{1/2}=\frac{L_{2m}\pm\sqrt{L_{2m}^{2}+4}}{2}$

and

$z_{2}=\frac{L_{2m}-\sqrt{L_{2m}^{2}+4}}{2}<0<\alpha^{2m}<z_{1}=\frac{L_{2m}+\sqrt{L_{2m}^{2}+4}}{2}.$

Note also that $\lim_{m\rightarrow\infty}z_{2}=0$ while $z_{1}\sim\alpha^{2m}$
as $m\rightarrow\infty$. Hence, inserting $z_{1}$ in (15) and simplifying
proves the next result.

######  Theorem 4.

For each integer $m\geq 0$ the following expression for $\pi$ is valid

$\frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}L_{2m(2n+1)}\Big{(}\frac{2}{L_{2m}+\sqrt{L_{2m}^{2}+4}}\Big{)}^{2n+1}.$
(20)

Equation (20) is the first Lucas number counterpart of (1). It is worth to
note, that the case $m=0$ yields

$\frac{\pi}{8}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\Big{(}\frac{1}{1+\sqrt{2}}\Big{)}^{2n+1},$

which can be obtained directly from (11) with $z=1/(1+\sqrt{2})$ and using the
arctan identity

$2\arctan(x)=\arctan\Big{(}\frac{2x}{1-x^{2}}\Big{)},\qquad x^{2}<1.$

When $m=1$, then

$\frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}L_{4n+2}\Big{(}\frac{2}{3+\sqrt{13}}\Big{)}^{2n+1}.$

In a similar manner, working with other arguments of the arctan function, we
can derive the following presumably new series for $\pi$ involving even Lucas
numbers.

######  Theorem 5.

For each integer $m\geq 0$ the following expressions for $\pi$ are valid

$\frac{\pi}{6}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}L_{2m(2n+1)}\Big{(}\frac{2}{\sqrt{3}L_{2m}+\sqrt{3L_{2m}^{2}+4}}\Big{)}^{2n+1},$
(21)
$\frac{\pi}{12}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}L_{2m(2n+1)}\Big{(}\frac{2(2-\sqrt{3})}{L_{2m}+\sqrt{L_{2m}^{2}+4(2-\sqrt{3})^{2}}}\Big{)}^{2n+1},$
(22)

and

$\frac{\pi}{5}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}L_{2m(2n+1)}\Big{(}\frac{2\sqrt{5-2\sqrt{5}}}{L_{2m}+\sqrt{L_{2m}^{2}+4(5-2\sqrt{5})}}\Big{)}^{2n+1}.$
(23)

###  2.2 Series with odd Fibonacci (Lucas) coefficients

Setting $x=\sqrt{5}F_{2m+1}/6,m\geq 0$, we observe that

$\lambda_{1}\Big{(}\frac{\sqrt{5}F_{2m+1}}{6}\Big{)}=\frac{\sqrt{5}F_{2m+1}+L_{2m+1}}{2}=\alpha^{2m+1},$

where again we have used $L_{n}^{2}=5F_{n}^{2}+(-1)^{n}4$. Proceeding as
before, inserting the value of $x$ into (9) and (10) we obtain the following
expressions via (8), valid for all $|z|>\alpha^{2m+1}$,

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{(2m+1)(2n+1)}}{z^{2n+1}}=\frac{1}{\sqrt{5}}\arctan\Big{(}\sqrt{5}F_{2m+1}\frac{z}{z^{2}-1}\Big{)},$
(24)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{(2m+1)(2n+1)}}{z^{2n+1}}=\arctan\Big{(}L_{2m+1}\frac{z}{z^{2}+1}\Big{)}.$
(25)

Especially, for $z=L_{2m+1}$,

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{(2m+1)(2n+1)}}{L_{2m+1}^{2n+1}}=\frac{1}{\sqrt{5}}\arctan\Big{(}\sqrt{5}\frac{F_{4m+2}}{L_{2m+1}^{2}+1}\Big{)},$
(26)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{(2m+1)(2n+1)}}{L_{2m+1}^{2n+1}}=\arctan\Big{(}\frac{L_{2m+1}^{2}}{L_{2m+1}^{2}-1}\Big{)}.$
(27)

Once more, deriving Castellanos-like expressions for $\pi$ is fairly simple.
We start with

$\frac{\sqrt{5}F_{2m+1}z}{z^{2}-1}=1\qquad\mbox{and}\qquad\frac{L_{2m+1}z}{z^{2}+1}=1.$

Solving $z^{2}-\sqrt{5}F_{2m+1}z-1=0,m\geq 1,$ gives

$z_{1/2}=\frac{\sqrt{5}F_{2m+1}\pm\sqrt{5F_{2m+1}^{2}+4}}{2}.$

But it can be checked easily that $z_{2}<\alpha^{2m+1}<z_{1}$ for all $m\geq
0$. Inserting the value $z_{1}$ in (24) gives Castellanos’ series (1).
Similarly, solving the equation $z^{2}-L_{2m+1}z+1=0,m\geq 0,$ gives

$z_{1/2}=\frac{L_{2m+1}\pm\sqrt{L_{2m+1}^{2}-4}}{2}$

We note that, for $m=0$ the zeros $z_{1/2}$ are complex numbers with
$|z_{1}|=|z_{2}|=1<\alpha$. For $m\geq 1$, the zeros are real and
$z_{2}<z_{1}<\alpha^{2m+1}$. Hence, no such series for $\pi$ will exist.

In the same manner other arctan arguments can be analyzed. Interestingly, the
careful analysis shows that no series involving odd Lucas numbers will come to
appearance. The next theorem contains additional odd-indexed Fibonacci series
for $\pi$ that we found.

######  Theorem 6.

For each integer $m\geq 0$ the following expressions for $\pi$ are valid

$\frac{\pi}{12}=\sqrt{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}F_{(2m+1)(2n+1)}\Big{(}\frac{2(2-\sqrt{3})}{\sqrt{5}F_{2m+1}+\sqrt{5F_{2m+1}^{2}+4(2-\sqrt{3})^{2}}}\Big{)}^{2n+1},$
(28)
$\frac{\pi}{6}=\sqrt{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}F_{(2m+1)(2n+1)}\Big{(}\frac{2}{\sqrt{15}F_{2m+1}+\sqrt{15F_{2m+1}^{2}+4}}\Big{)}^{2n+1},$
(29)

and

$\frac{\pi}{5}=\sqrt{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}F_{(2m+1)(2n+1)}\Big{(}\frac{2(\sqrt{5-2\sqrt{5}})}{\sqrt{5}F_{2m+1}+\sqrt{5F_{2m+1}^{2}+4(5-2\sqrt{5})}}\Big{)}^{2n+1}.$
(30)

The special case of (28) when $m=0$ appears in one of Castellanos’ papers [1,
Eq. (48)]. The generalization and the other two series seem to be new.

##  3 Complementing the work of Castellanos (Part 2)

Replacing $z$ in (9) by $z/\lambda_{1}(x)$ and $z/\lambda_{2}(x)$,
respectively, and combining the terms according to the Binet form gives

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{B_{2n+1}^{2}(x)}{z^{2n+1}}=\frac{1}{4(9x^{2}-1)}\Big{(}\arctan\Big{(}\frac{2\sqrt{9x^{2}-1}\lambda_{1}(x)z}{z^{2}+\lambda_{1}^{2}(x)}\Big{)}-\arctan\Big{(}\frac{2\sqrt{9x^{2}-1}\lambda_{2}(x)z}{z^{2}+\lambda_{2}^{2}(x)}\Big{)}\Big{)}$

Now, we can apply (12) and simplify to end with the following identity
involving squares of balancing polynomials

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{B_{2n+1}^{2}(x)}{z^{2n+1}}=\frac{1}{4(9x^{2}-1)}\arctan\Big{(}\frac{4(9x^{2}-1)z(z^{2}-1)}{z^{4}+6(12x^{2}-1)z^{2}+1}\Big{)},\quad
z>\lambda_{1}^{2}(x).$ (31)

Similarly, the same replacement in (10) gives

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{C_{2n+1}^{2}(x)}{z^{2n+1}}=\frac{1}{4}\arctan\Big{(}\frac{36x^{2}z(z^{2}-1)}{z^{4}-2(36x^{2}-1)z^{2}+1}\Big{)},\quad
z>\lambda_{1}^{2}(x).$ (32)

To get a series for, say, $\pi/128$ we set $x=1$ in (31) and are left with the
quadric

$z^{4}-32z^{3}+66z^{2}+32z+1=0.$ (33)

The roots are

$\displaystyle 8-\sqrt{47}-4\sqrt{7-\sqrt{47}},\qquad
8-\sqrt{47}+4\sqrt{7-\sqrt{47}},$ $\displaystyle
8+\sqrt{47}-4\sqrt{7+\sqrt{47}},\qquad 8+\sqrt{47}+4\sqrt{7+\sqrt{47}}.$

None of the roots satisfies the condition for convergence and therefore we can
conclude that no such series with $B_{2n+1}^{2}$ exists. Setting $x=1$ in the
above Lucas-balancing identity we obtain

$\frac{\pi}{16}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{C_{2n+1}^{2}}{z^{*2n+1}},$

where $z^{*}$ is a positive root of the polynomial equation

$z^{4}-36z^{3}-70z^{2}+36z+1=0.$ (34)

Here, also all four roots of equation (34) are real

$\displaystyle 9-7\sqrt{2}-3\sqrt{20-14\sqrt{2}},\qquad
9-7\sqrt{2}+3\sqrt{20-14\sqrt{2}},$ $\displaystyle
9+7\sqrt{2}-3\sqrt{20+14\sqrt{2}},\qquad 9+7\sqrt{2}+3\sqrt{20+14\sqrt{2}}$

and the only (biggest) root that satisfies the condition
$z>\lambda_{1}^{2}(1)$ is

$z^{*}=9+7\sqrt{2}+3\sqrt{20+14\sqrt{2}}=37,8254\ldots.$

Hence,

$\frac{\pi}{16}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{C_{2n+1}^{2}}{(9+7\sqrt{2}+3\sqrt{20+14\sqrt{2}})^{2n+1}}.$
(35)

From (31) and (32), with $x=L_{2m}/6$, we arrive at

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{F_{2m(2n+1)}^{2}}{z^{2n+1}}=\frac{1}{5}\arctan\Big{(}\frac{5F_{2m}^{2}z(z^{2}-1)}{z^{4}+2(5F_{2m}^{2}+1)z^{2}+1}\Big{)}$
(36)

and

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{2m(2n+1)}^{2}}{z^{2n+1}}=\arctan\Big{(}\frac{L_{2m}^{2}z(z^{2}-1)}{z^{4}-2(L_{2m}^{2}-1)z^{2}+1}\Big{)}.$
(37)

To get a new Castellanos’ type series for $\pi$ it is necessary to study the
equations

$z^{4}-L_{2m}^{2}z^{3}-2(L_{2m}^{2}-1)z^{2}+L_{2m}^{2}z+1=0$ (38)

and

$z^{4}-5F_{2m}^{2}z^{3}+2(5F_{2m}^{2}+1)z^{2}+5F_{2m}^{2}z+1=0.$ (39)

Note that, if in (38) we set $a=-L_{2m}^{2}$ then the quartic takes the form

$z^{4}+az^{3}+2(a+1)z^{2}-az+1=0.$ (40)

The four roots are given by

$z_{1/2}=-\frac{a}{4}-\frac{1}{4}\sqrt{a^{2}-8a-16}\pm\frac{1}{2}\sqrt{\frac{a^{2}-4a+a\sqrt{a^{2}-8a-16}}{2}}$
(41)

and

$z_{3/4}=-\frac{a}{4}+\frac{1}{4}\sqrt{a^{2}-8a-16}\pm\frac{1}{2}\sqrt{\frac{a^{2}-4a-a\sqrt{a^{2}-8a-16}}{2}}.$
(42)

Finally, if we set $\tilde{t}=-(a+\sqrt{a^{2}-8a-16})/4$ and
$t=-(a-\sqrt{a^{2}-8a-16})/4$ then

$z_{1/2}=\tilde{t}\pm\sqrt{\tilde{t}^{2}+1}\qquad\mbox{and}\qquad
z_{3/4}=t\pm\sqrt{t^{2}+1}.$

Among the four roots we choose the biggest root that satisfies the condition
for convergence and arrive at the main theorem.

######  Theorem 7.

For each $m\geq 0$ we have the following series for $\pi$ involving squared
even-indexed Lucas numbers

$\frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{2m(2n+1)}^{2}}{(t+\sqrt{t^{2}+1})^{2n+1}}$
(43)

with

$t=t(m)=\frac{L_{2m}^{2}+\sqrt{L_{2m}^{4}+8L_{2m}^{2}-16}}{4}.$ (44)

Theorem 7 is a wonderful Lucas analogue of Castellanos’ equation (3). For
$m=0$, $t(0)=1+\sqrt{2}$ and

$\frac{\pi}{16}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{1}{(1+\sqrt{2}+\sqrt{4+2\sqrt{2}})^{2n+1}}.$
(45)

For $m=1$, $t(1)=(9+\sqrt{137})/4$ and

$\frac{\pi}{16}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\frac{L_{4n+2}^{2}4^{2n}}{(9+\sqrt{137}+3\sqrt{26+2\sqrt{137}})^{2n+1}}.$
(46)

The analysis of the quartic (39) is very similar. With $b=-5F_{2m}^{2}$ the
relevant equation becomes

$z^{4}+bz^{3}+2(1-b)z^{2}-bz+1=0.$ (47)

The four roots are given by

$z_{1/2}=-\frac{b}{4}-\frac{1}{4}\sqrt{b^{2}+8b-16}\pm\frac{1}{2}\sqrt{\frac{b^{2}+4b+b\sqrt{b^{2}+8b-16}}{2}}$

and

$z_{3/4}=-\frac{b}{4}+\frac{1}{4}\sqrt{b^{2}+8b-16}\pm\frac{1}{2}\sqrt{\frac{b^{2}+4b-b\sqrt{b^{2}+8b-16}}{2}}.$

These roots can be expressed as

$z_{1/2}=\tilde{s}\pm\sqrt{\tilde{s}^{2}+1}\qquad\mbox{and}\qquad
z_{3/4}=s\pm\sqrt{s^{2}+1}$

with $\tilde{s}=\tilde{s}(m)=-(b+\sqrt{b^{2}+8b-16})/4$ and
$s=s(m)=-(b-\sqrt{b^{2}+8b-16})/4$. For $m=0$ and $m=1$ we have only complex
roots: $\tilde{s}(0)=-i,s(0)=i$ and
$\tilde{s}(1)=(5-\sqrt{31}i)/4,s(1)=(5+\sqrt{31}i)/4$ with $i=\sqrt{-1}$. For
$m\geq 2$, we get the real roots but no root satisfies the condition
$z>\lambda_{1}^{2}(L_{2m}/6)=\alpha^{4m}$.

In a similar manner other arctan arguments can be treated, yielding (if they
exist) other series for $\pi$ with squares of Fibonacci and Lucas numbers.

##  4 Similar series via subseries

Before closing the investigation, we present some additional results that are
closely related to those from Section 2. The next theorem will be useful to
prove some subseries analogues of these results.

######  Theorem 8.

For each real $x$ with $x>1/3$ and $z\in\mathbb{C}$ with
$|z|>|\lambda_{1}(x)|$, we have the following identity

$\displaystyle\sum_{n=0}^{\infty}\frac{1}{4n+3}\frac{B_{4n+3}(x)}{z^{4n+3}}$
$\displaystyle=$
$\displaystyle\frac{1}{8\sqrt{9x^{2}-1}}\ln\Big{(}\frac{z^{2}+2\sqrt{9x^{2}-1}z-1}{z^{2}-2\sqrt{9x^{2}-1}z-1}\Big{)}$
(48)
$\displaystyle\quad\quad-\frac{1}{4\sqrt{9x^{2}-1}}\arctan\Big{(}\frac{2\sqrt{9x^{2}-1}z}{z^{2}+1}\Big{)}$

and

$\sum_{n=0}^{\infty}\frac{1}{4n+3}\frac{C_{4n+3}(x)}{z^{4n+3}}=\frac{1}{8}\ln\Big{(}\frac{z^{2}+6xz+1}{z^{2}-6xz+1}\Big{)}-\frac{1}{4}\arctan\Big{(}\frac{6xz}{z^{2}-1}\Big{)}.$
(49)

###### Proof.

From Chen’s paper [3] we have the relation

$\sum_{n=0}^{\infty}\frac{z^{4n+3}}{4n+3}=\frac{1}{4}\ln\Big{(}\frac{1+z}{1-z}\Big{)}-\frac{1}{2}\arctan(z),\quad|z|<1.$
(50)

The remainder of the proof is the same as in the proof of Theorem 1. ∎

In the applications we can proceed as in the previous sections. First, from
(48) and (49) with $x=1$ it is evident that we can obtain some series for
$\pi$ with balancing and Lucas-balancing coefficients by appropriately
choosing the argument of the arctan function on the right. The structure of
the equations also shows that all series will start with a constant
logarithmic term. These arguments yield to the following exotic
representations:

$\frac{\pi}{3}=\ln\Big{(}\frac{46+4\sqrt{138}+8\sqrt{12}+4\sqrt{46}}{46+4\sqrt{138}-8\sqrt{12}-4\sqrt{46}}\Big{)}-16\sqrt{2}\sum_{n=0}^{\infty}\frac{1}{4n+3}\frac{B_{4n+3}}{(2\sqrt{6}+\sqrt{23})^{4n+3}},$
(51)
$\frac{\pi}{3}=\ln\Big{(}\frac{56+12\sqrt{21}+18\sqrt{3}+12\sqrt{7}}{56+12\sqrt{21}-18\sqrt{3}-12\sqrt{7}}\Big{)}-8\sum_{n=0}^{\infty}\frac{1}{4n+3}\frac{C_{4n+3}}{(3\sqrt{3}+2\sqrt{7})^{4n+3}},$
(52) $\displaystyle\frac{\pi}{6}$ $\displaystyle=$
$\displaystyle\ln\Big{(}\frac{2+8\sqrt{3}+4\sqrt{2+8\sqrt{3}}+4(2-\sqrt{3})(4+\sqrt{2+8\sqrt{3}})}{2+8\sqrt{3}+4\sqrt{2+8\sqrt{3}}-4(2-\sqrt{3})(4+\sqrt{2+8\sqrt{3}})}\Big{)}$
(53)
$\displaystyle\qquad-16\sqrt{2}\sum_{n=0}^{\infty}\frac{1}{4n+3}B_{4n+3}\Big{(}\frac{2-\sqrt{3}}{2\sqrt{2}+\sqrt{1+4\sqrt{3}}}\Big{)}^{4n+3},$

and

$\displaystyle\frac{\pi}{6}$ $\displaystyle=$
$\displaystyle\ln\Big{(}\frac{32-8\sqrt{3}+12\sqrt{4-\sqrt{3}}+6(2-\sqrt{3})(3+2\sqrt{4-\sqrt{3}})}{32-8\sqrt{3}+12\sqrt{4-\sqrt{3}}-6(2-\sqrt{3})(3+2\sqrt{4-\sqrt{3}})}\Big{)}$
(54)
$\displaystyle\qquad-8\sum_{n=0}^{\infty}\frac{1}{4n+3}C_{4n+3}\Big{(}\frac{2-\sqrt{3}}{3+2\sqrt{4-\sqrt{3}}}\Big{)}^{4n+3}.$

Working with Fibonacci and Lucas numbers we can derive the next series, where
each series is stated as a separate theorem without proof.

######  Theorem 9.

For each integer $m\geq 0$ the following series identity holds

$\displaystyle\frac{\pi}{2}$ $\displaystyle=$
$\displaystyle\ln\Big{(}\frac{1}{2}\Big{(}5F_{2m+1}^{2}+\sqrt{5}F_{2m+1}\sqrt{5F_{2m+1}^{2}+4}+2\Big{)}\Big{)}$
(55)
$\displaystyle\qquad-4\cdot\sqrt{5}\sum_{n=0}^{\infty}\frac{1}{4n+3}F_{(2m+1)(4n+3)}\Big{(}\frac{2}{\sqrt{5}F_{2m+1}+\sqrt{5F_{2m+1}^{2}+4}}\Big{)}^{4n+3}.$

The series in Theorem 9 is the Fibonacci subseries analogue of Castellanos’
series (1).

When $m=0$ the expression reads as

$\frac{\pi}{2}=\ln\Big{(}\frac{7+3\sqrt{5}}{2}\Big{)}-4\sqrt{5}\sum_{n=0}^{\infty}\frac{1}{4n+3}F_{4n+3}\Big{(}\frac{2}{3+\sqrt{5}}\Big{)}^{4n+3},$

which has the equivalent form

$\frac{\pi}{8}=\ln(\alpha)-\sqrt{5}\sum_{n=0}^{\infty}\frac{1}{4n+3}\frac{F_{4n+3}}{\alpha^{8n+6}}.$
(56)

We conclude with three other examples exhibiting this structure and leave the
derivations to the interested reader.

######  Theorem 10.

For each integer $m\geq 0$ the following expression for $\pi$ is valid

$\frac{\pi}{2}=\ln\Big{(}\frac{1}{2}\Big{(}L_{2m}^{2}+L_{2m}\sqrt{L_{2m}^{2}+4}+2\Big{)}\Big{)}-4\sum_{n=0}^{\infty}\frac{1}{4n+3}L_{2m(4n+3)}\Big{(}\frac{2}{L_{2m}+\sqrt{L_{2m}^{2}+4}}\Big{)}^{4n+3}.$
(57)

######  Theorem 11.

For each integer $m\geq 0$, the following identity for $\pi$ is valid

$\displaystyle\frac{\pi}{6}$ $\displaystyle=$
$\displaystyle\ln\left(\frac{(9+5\sqrt{3})L_{2m}^{2}+(3+\sqrt{3})L_{2m}\sqrt{(2+\sqrt{3})^{2}L_{2m}^{2}+4}+4}{(5+3\sqrt{3})L_{2m}^{2}+(1+\sqrt{3})L_{2m}\sqrt{(2+\sqrt{3})^{2}L_{2m}^{2}+4}+4}\right)$
(58)
$\displaystyle-4\sum_{n=0}^{\infty}\frac{1}{4n+3}L_{2m(4n+3)}\Big{(}\frac{2}{(2+\sqrt{3})L_{2m}+\sqrt{(2+\sqrt{3})^{2}L_{2m}^{2}+4}}\Big{)}^{4n+3}.$

######  Theorem 12.

For each integer $m\geq 0$, we have the following expression for $\pi$

$\displaystyle\frac{\pi}{6}$ $\displaystyle=$
$\displaystyle\ln\left(\frac{(9+5\sqrt{3})\sqrt{5}\Big{(}\sqrt{5}F_{2m+1}^{2}+F_{2m+1}\sqrt{5F_{2m+1}^{2}+4(2-\sqrt{3})^{2}}\Big{)}+4}{(5+3\sqrt{3})\sqrt{5}\Big{(}\sqrt{5}F_{2m+1}^{2}+F_{2m+1}\sqrt{5F_{2m+1}^{2}+4(2-\sqrt{3})^{2}}\Big{)}+4}\right)$
(59)
$\displaystyle-4\sqrt{5}\sum_{n=0}^{\infty}\frac{1}{4n+3}F_{(2m+1)(4n+3)}\Big{(}\frac{2(2-\sqrt{3})}{\sqrt{5}F_{2m+1}+\sqrt{5F_{2m+1}^{2}+4(2-\sqrt{3})^{2}}}\Big{)}^{4n+3}.$

##  5 Conclusion

Castellanos has derived curious series for $\pi$ involving odd Fibonacci and
odd squared Fibonacci numbers. The goal of the paper was to provide
complements of his work. This was achieved by studying some infinite series
involving balancing and Lucas-balancing polynomials. We have also indicated
how more series representations of this nature can be established.

## References

  * [1] D. Castellanos, Rapidly converging expansions with Fibonacci coefficients, _Fibonacci Quart._ 24 (1986), 70–82.
  * [2] D. Castellanos, A generalization of Binet’s formula and some of its consequences, _Fibonacci Quart._ 27 (1989), 424–438.
  * [3] H. Chen, On the summation of subseries in closed form, _Int. J. Math. Edu. Sci. Tech._ 41 (2010), 538–547.
  * [4] R. Frontczak, On balancing polynomials, _Appl. Math. Sci._ 13 (2019), 57–66.
  * [5] R. Frontczak, Powers of balancing polynomials and some consequences for Fibonacci sums, _Int. J. Math. Anal._ 13 (2019), 109–115.
  * [6] R. Frontczak, Relating Fibonacci Numbers to Bernoulli Numbers via Balancing Polynomials, _J. Integer Seq._ 22 (2019), Article 19.5.3.
  * [7] R. Frontczak and T. Goy, Combinatorial sums associated with balancing and Lucas-balancing polynomials, _Ann. Math. Inform._ 52 (2020), 97–105.
  * [8] R. Frontczak and T. Goy, Additional close links between balancing and Lucas balancing polynomials, _Adv. Stud. Cont. Math._ , 31 (2021), 287–300.
  * [9] D. S. Kim and T. Kim, On sums of finite products of balancing polynomials, _J. Comput. Appl. Math._ 377 (2020), 112913.
  * [10] I. Mező, Several generating functions for second-order recurrence sequences, J. Integer Seq. 12 (2009), Article 09.3.7.
  * [11] N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, 2020. Available at https://oeis.org.

##  6 Appendix: Numerical experiments

All series that have been derived in this paper have been verified numerically
for different values of the parameters $m$ and $n$ using MATLAB. Here, we
present some of the numerical results we obtained, i.e., our results
concerning the behavior of the partial sums, paying particular attention to
Sections 2 and 3. The reference value for $\pi$ is

$3.14159265358979323846264338327950288419716939937510582097494459230781...$

We begin with a comparison of the series (4) and (13), which is summarized in
the first table.

n | Series (4) (correct digits of $\pi$) | Series (13) (correct digits for $\pi$)
---|---|---
10 | 3.1415926540096031579541983575759 (8) | 3.1681253658683134338813758290598 (1)
25 | 3.1415926535897931755199500028705 (15) | 3.1394749981715568271090077701047 (1)
50 | 3.1415926535897931755212222064311 (15) | 3.1416593510252682593586123771523 (3)
75 | 3.1415926535897931755212222064311 (15) | 3.1415898900113144709178285437189 (4)
100 | 3.1415926535897931755212222064311 (15) | 3.1415927819755543664446705866538 (6)
Table 1: Comparison of series (4) and series (13)

Series (4) converges much faster to $\pi$ than (13). For instance, for $n=25$
the partial sum equals 3.1415926535897931755199500028705 and gives 15 correct
decimal places of $\pi$, whereas series (13) only gives one correct decimal
place in this case. Even for the big input $n=100$ (13) is able to get only 6
correct decimal places for $\pi$.

The series (20)-(23) and their Fibonacci counterparts from (28)-(30) exhibit
much better convergence properties as is seen from the next tables. Finally,
we state the numerical results for the series containing squared arguments in
the numerator.

$n$ | Series | Partial sum and correct decimal places | Partial sum and correct decimal places
---|---|---|---
|  | of $\pi$, when $m=0$ | of $\pi$, when $m=1$
$n=10$ | (20) | 3.1415926540617729889571081190938 (8) | 3.1421196570445914842979046044391 (2)
| (21) | 3.14159265358982738004655360471 (12) | 3.1415926671660013427841119922951 (7)
| (22) | 3.1415926535897947339924377936013 (14) | 3.1415926535897956856265976351507 (14)
| (23) | 3.1415926535921310968341963746407 (10) | 3.1415939762233089574900159429531 (5)
$n=25$ | (20) | 3.1415926535897935109299308733563 (15) | 3.1415924425521257791274315531176 (6)
| (21) | 3.141592653589793111803909553252 (15) | 3.1415926535897930747899824982812 (15)
| (22) | 3.1415926535897947339867034604622 (14) | 3.1415926535897944192246093796406 (14)
| (23) | 3.141592653589792871679833893134 (14) | 3.1415926535896632627309996371646 (12)
$n=50$ | (20) | 3.1415926535897935109305996238453 (15) | 3.1415926535907651359119064125043 (10)
| (21) | 3.1415926535897931118039095532521 (15) | 3.1415926535897930771752670402775 (15)
| (22) | 3.1415926535897947339867034604622 (14) | 3.1415926535897944192246093796406 (14)
| (23) | 3.1415926535897928716798338954132 (14) | 3.1415926535897928493289793465367 (14)
$n=75$ | (20) | 3.1415926535897935109305996238453 (15) | 3.1415926535897932445405743444239 (16)
| (21) | 3.1415926535897931118039095532521 (15) | 3.1415926535897930771752670402775 (15)
| (22) | 3.1415926535897947339867034604622 (14) | 3.1415926535897944192246093796406 (14)
| (23) | 3.1415926535897928716798338954132 (14) | 3.1415926535897928493289787796672 (14)
$n=100$ | (20) | 3.1415926535897935109305996238453 (15) | 3.1415926535897932504225822917772 (16)
| (21) | 3.1415926535897931118039095532521 (15) | 3.1415926535897930771752670402775 (15)
| (22) | 3.1415926535897947339867034604622 (14) | 3.1415926535897944192246093796406 (14)
| (23) | 3.1415926535897928716798338954132 (14) | 3.1415926535897928493289787796672 (14)

Table 2: Series with even indexed Lucas coefficients

$n$ | Series | Partial sum and correct decimal places | Partial sum and correct decimal places
---|---|---|---
|  | of $\pi$, when $m=0$ | of $\pi$, when $m=1$
$n=10$ | (28) | 3.1415926535897945091605064246388 (14) | 3.1415926535898033991093222132522 (12)
| (29) | 3.1415926536976031512950065859936 (9) | 3.1415927863118601169382604268352 (6)
| (30) | 3.1415926617777441429048832649656 (7) | 3.1416085221242421932891054056989 (3)
| (1) | 3.1415946628008058070270975343262 (5) | 3.1513648928833562151579600460063 (1)
$n=25$ | (28) | 3.1415926535897944941490876249881 (14) | 3.1415926535897942690609285922555 (14)
| (29) | 3.1415926535897929748407090095711 (14) | 3.1415926535897928788785708458934 (14)
| (30) | 3.1415926535897927600555556635439 (14) | 3.1415926535469317102875642213194 (10)
| (1) | 3.1415926535893304765153112605509 (12) | 3.1413911371820930200840194675568 (3)
$n=50$ | (28) | 3.1415926535897944941490876249881 (14) | 3.1415926535897942690609285922555 (14)
| (29) | 3.141592653589792974840740857073 (14) | 3.1415926535897933582722721433178 (15)
| (30) | 3.1415926535897927609931468741211 (14) | 3.1415926535897929816708906429346 (14)
| (1) | 3.1415926535897932529555146219012 (16) | 3.1415933178255789377527632793825 (5)
$n=75$ | (28) | 3.1415926535897944941490876249881 (14) | 3.1415926535897942690609285922555 (14)
| (29) | 3.141592653589792974840740857073 (14) | 3.1415926535897933582722721433093 (15)
| (30) | 3.1415926535897927609931468741211 (14) | 3.1415926535897929816235883662303 (14)
| (1) | 3.1415926535897932529555062020827 (16) | 3.1415926507102507935365145416425 (8)
$n=100$ | (28) | 3.1415926535897944941490876249881 (14) | 3.1415926535897942690609285922555 (14)
| (29) | 3.141592653589792974840740857073 (14) | 3.1415926535897933582722721433093 (15)
| (30) | 3.1415926535897927609931468741211 (14) | 3.1415926535897929816235883662988 (14)
| (1) | 3.1415926535897932529555062020827 (16) | 3.1415926536037885968968215682418 (9)

Table 3: Series with odd indexed Fibonacci coefficients including Castellaons series (1) n | Series (35) (correct digits of $\pi$)
---|---
10 | 3.1500288764501782022564579932578 (1)
25 | 3.1414500748004593281290444672926 (3)
50 | 3.1415929908436770508682631082098 (6)
75 | 3.1415926525406591485619208477698 (8)
100 | 3.1415926535934520603658154364136 (10)
Table 4: Series with odd squared Lucas-balancing coefficients n | Series (43) when $m=0$, | Series (43) when $m=1$
---|---|---
| (correct digits of $\pi$) | (correct digits of $\pi$)
10 | 3.141592653589793277636375358286 (16) | 3.1416003226810113749580541993994 (3)
25 | 3.1415926535897932279669723561782 (16) | 3.1415926535792334445640011923928 (10)
50 | 3.1415926535897932279669723561782 (16) | 3.1415926535897932072655992250783 (16)
75 | 3.1415926535897932279669723561782 (16) | 3.1415926535897932072618102929481 (16)
100 | 3.1415926535897932279669723561782 (16) | 3.1415926535897932072618102929499 (16)
Table 5: Comparison of series (45) and series (46)


# Electroweak baryogenesis, dark matter, and dark CP-symmetry

Jinghong Ma, Jie Wang, Lei Wang∗††footnotetext: *) Corresponding author. Email
address<EMAIL_ADDRESS>Department of Physics, Yantai University, Yantai
264005, P. R. China

###### Abstract

With the motivation of explaining the dark matter and achieving the
electroweak baryogenesis via the spontaneous CP-violation at high temperature,
we propose a complex singlet scalar ($S=\frac{\chi+i\eta_{s}}{\sqrt{2}}$)
extension of the two-Higgs-doublet model respecting a discrete dark CP-
symmetry: $S\to-S^{*}$. The dark CP-symmetry guarantees $\chi$ to be a dark
matter candidate on one hand and on the other hand allows $\eta_{s}$ to have
mixings with the pseudoscalars of the Higgs doublet fields, which play key
roles in generating the CP-violation sources needed by the electroweak
baryogenesis at high temperature. The universe undergoes multi-step phase
transitions, including a strongly first-order electroweak phase transition
during which the baryon number is produced. At the present temperature, the
observed vacuum is produced and the CP-symmetry is restored so that the
stringent electric dipole moment experimental bounds are satisfied.
Considering relevant constraints, we study the simple scenario of $m_{\chi}$
around the Higgs resonance region, and find that the dark matter relic
abundance and the baryon asymmetry can be simultaneously explained. Finally,
we briefly discuss the gravitational wave signatures at future space–based
detectors and the LHC signatures.

Introduction. The baryon asymmetry of the universe (BAU) presents one of the
major quests for particle cosmology. By the observation based on the Big-Bang
Nucleosynthesis, the BAU is pdg2020

$Y_{B}\equiv\rho_{B}/s=(8.2-9.2)\times 10^{-11},$ (1)

where $\rho_{B}$ is the baryon number density and $s$ is the entropy density.
Three necessary Sakharov conditions have to be fulfilled for a dynamical
generation of BAU: baryon number changing interactions, non-conservation of C
and CP, and departure from thermal equilibrium Sakharov . The electroweak
baryogenesis (EWBG) ewbg1 ; ewbg2  provides a promising and attractive
mechanism of explaining the BAU since it is testable at the energy frontier by
the LHC and at the precision frontier by the electric dipole moment (EDM)
experiments. To realize the EWBG, one needs extend the SM to produce
sufficient large CP violation and a strongly first-order electroweak phase
transition (EWPT), such as the singlet extension of SM (see e.g. bgs-1 ; bgs-3
; bgs-4 ; bgs-5 ; bgs-6 ; Beniwal:2018hyi ; Huang:2018aja ; Ghorbani:2017jls ;
cao ; huang ; Xie:2020wzn ; Ellis:2022lft ; Lewicki:2021pgr ; Idegawa:2023bkh
; Harigaya:2022ptp ) and the two-Higgs-doublet model (2HDM) (see e.g. bg2h-1 ;
bg2h-2 ; bg2h-3 ; bg2h-5 ; bg2h-4 ; bg2h-6 ; bg2h-7 ; bg2h-8 ; bg2h-9 ;
bg2h-11 ; Basler:2020nrq ; bg2h-13 ; bg2h-12 ; 2111.13079 ; 2207.00060 ;
Goncalves:2023svb ).

The negative results in the EDM searches for electrons impose stringent
constraints on the explicit CP-violation interactions in the scalar couplings
and Yukawa couplings edm-e . There are some cancellation mechanisms of CP-
violation effects in the EDM, which can relax the tension between the EWBG and
the EDM data 1411.6695 ; 2004.03943 ; 2111.13079 ; 2207.00060 . Alternatively,
a finite temperature spontaneous CP-violation mechanism is naturally
compatible with the EDM data, where the CP symmetry is spontaneously broken at
the high temperature and it is recovered at the present temperature. The novel
mechanism was achieved in the singlet scalar extension of the SM cao ; huang
in which a high dimension effective operator or extra fermions need to be
added, and the singlet pseudoscalar extension of 2HDM Huber:2022ndk ;
Liu:2023sey .

In addition to the BAU, the dark matter (DM) is one of the longstanding
questions of particle physics and cosmology. In this paper, we propose a
complex singlet scalar extension of the 2HDM respecting a discrete dark CP-
symmetry, and explain the observed DM relic density and the BAU
simultaneously. The dark CP-symmetry allows the imaginary component of singlet
scalar to have mixings with the pseudoscalars of scalar doublet fields, which
play key roles in generating the CP-violation sources needed by the EWBG at
high temperature. On the other hand, the dark CP-symmetry guarantees the real
component to be a DM candidate.

2HDM + $S$ respecting a dark CP-symmetry. Imposing a discrete dark CP-
symmetry, we extend the SM by a second Higgs doublet $\Phi_{2}$ and a complex
singlet $S$,

$\displaystyle\Phi_{1}=\left(\begin{array}[]{c}\phi_{1}^{+}\\\
\frac{(v_{1}+\rho_{1}+i\eta_{1})}{\sqrt{2}}\end{array}\right)\,,\Phi_{2}=\left(\begin{array}[]{c}\phi_{2}^{+}\\\
\frac{(v_{2}+\rho_{2}+i\eta_{2})}{\sqrt{2}}\end{array}\right),$ (6)
$\displaystyle S=\frac{(\chi+i\eta_{s})}{\sqrt{2}},$ (7)

with $v_{1}$ and $v_{2}$ being the vacuum expectation values (VEVs),
$v=\sqrt{v^{2}_{1}+v^{2}_{2}}=(246~{}\rm GeV)^{2}$ and $\tan\beta\equiv
v_{2}/v_{1}$. The singlet field $S$ has no VEV. Under the dark CP-symmetry,
$S\to-~{}S^{*}$ ($\chi\to-\chi$ ,$\eta_{s}\to\eta_{s}$ in the real
parametrization), and while all the other fields are not affected.

The full scalar potential is given as

$\displaystyle\mathrm{V}=m_{11}^{2}(\Phi_{1}^{\dagger}\Phi_{1})+m_{22}^{2}(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\left[\frac{\lambda_{5}}{2}(\Phi_{1}^{\dagger}\Phi_{2})^{2}\right.$
$\displaystyle+\left.\lambda_{6}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2})+\lambda_{7}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{2})+\rm
h.c.\right]$ $\displaystyle+m^{2}_{S}SS^{*}+\left[\frac{m^{\prime
2}_{S}}{2}SS+\rm
h.c.\right]+\left[\frac{\lambda^{\prime\prime}_{1}}{24}S^{4}+\rm h.c.\right]$
$\displaystyle+\left[\frac{\lambda^{\prime\prime}_{2}}{6}S^{2}SS^{*}+\rm
h.c.\right]+SS^{*}\left[\lambda^{\prime}_{1}\Phi_{1}^{\dagger}\Phi_{1}+\lambda^{\prime}_{2}\Phi_{2}^{\dagger}\Phi_{2}\right]$
$\displaystyle+\frac{\lambda^{\prime\prime}_{3}}{4}(SS^{*})^{2}+\left[S^{2}(\lambda^{\prime}_{4}\Phi_{1}^{\dagger}\Phi_{1}+\lambda^{\prime}_{5}\Phi_{2}^{\dagger}\Phi_{2})+\rm
h.c.\right]$
$\displaystyle+\left[\lambda^{\prime}_{6}SS^{*}\Phi_{2}^{\dagger}\Phi_{1}+\lambda^{\prime}_{7}(SS+S^{*}S^{*})\Phi_{2}^{\dagger}\Phi_{1}+\rm
h.c.\right]$
$\displaystyle+\left[-m_{12}^{2}\Phi_{2}^{\dagger}\Phi_{1}+\frac{\mu}{2}(S-S^{*})\Phi_{2}^{\dagger}\Phi_{1}+\rm
h.c.\right],$ (8)

where all the coupling coefficients and mass are real, and thus the scalar
potential sector is CP-conserved at zero temperature. The last term leads to
the mixings of $\eta_{s}$ and the pseudoscalars of Higgs doublet fields, and
$\chi$ is allowed to remain stable. For simplicity, we take
$\lambda_{6}=\lambda_{7}=\lambda^{\prime}_{6}=\lambda^{\prime}_{7}=\lambda^{\prime\prime}_{2}=0$
in the following discussions.

The stationary conditions give

$\displaystyle\quad
m_{11}^{2}=m_{12}^{2}t_{\beta}-\frac{1}{2}v^{2}\left(\lambda_{1}c_{\beta}^{2}+\lambda_{345}s_{\beta}^{2}\right)\,,$
$\displaystyle\quad
m_{22}^{2}=m_{12}^{2}/t_{\beta}-\frac{1}{2}v^{2}\left(\lambda_{2}s_{\beta}^{2}+\lambda_{345}c_{\beta}^{2}\right),$
(9)

where $t_{\beta}\equiv\tan\beta$, $s_{\beta}\equiv\sin\beta$,
$c_{\beta}\equiv\cos\beta$, and
$\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$.

In addition to the 125 GeV Higgs $h$, the physical scalar spectrum contains a
CP-even states $H$, a DM candidate $\chi$, two neutral pseudoscalars $A$ and
$X$, and a charged scalar $H^{\pm}$. The mass eigenstates $h$, $H$ and
$H^{\pm}$ and their masses are the same as those of the pure 2HDM. The
$\eta_{1}$, $\eta_{2}$ and $\eta_{s}$ are rotated into the $A$, $X$ and $G$ by
the two mixing angles $\theta$ and $\beta$, where $G$ is a Goldstone boson.
The parameters $\mu$, $m_{S}^{2}$, and $m_{S}^{\prime 2}$ are given as

$\displaystyle
m_{S}^{2}=\frac{1}{2}\left(m_{\chi}^{2}+m_{A}^{2}s_{\theta}^{2}+m_{X}^{2}c_{\theta}^{2}-\lambda^{\prime}_{1}v^{2}c_{\beta}^{2}-\lambda^{\prime}_{2}v^{2}s_{\beta}^{2}\right),$
$\displaystyle m^{\prime
2}_{S}=\frac{1}{2}\left(m_{\chi}^{2}-m_{A}^{2}s_{\theta}^{2}-m_{X}^{2}c_{\theta}^{2}-2\lambda^{\prime}_{4}v^{2}c_{\beta}^{2}-2\lambda^{\prime}_{5}v^{2}s_{\beta}^{2}\right),$
$\displaystyle\mu=\frac{\sqrt{2}(m_{X}^{2}-m_{A}^{2})}{v}s_{\theta}c_{\theta},$
(10)

where $s_{\theta}\equiv\sin\theta$ and $c_{\theta}\equiv\cos\theta$.

The general Yukawa interaction is given by

$\displaystyle-{\cal
L}=Y_{u2}\,\overline{Q}_{L}\,\tilde{{\Phi}}_{2}\,u_{R}+\,Y_{d2}\,\overline{Q}_{L}\,{\Phi}_{2}\,d_{R}\,+\,Y_{\ell
2}\,\overline{L}_{L}\,{\Phi}_{2}\,e_{R}\,$
$\displaystyle+Y_{u1}\,\overline{Q}_{L}\,\tilde{{\Phi}}_{1}\,u_{R}+\,Y_{d1}\,\overline{Q}_{L}\,{\Phi}_{1}\,d_{R}\,+\,Y_{\ell
1}\,\overline{L}_{L}\,{\Phi}_{1}\,e_{R}+\,\mbox{h.c.},$

where $Q_{L}^{T}=(u_{L}\,,d_{L})$, $L_{L}^{T}=(\nu_{L}\,,l_{L})$,
$\widetilde{\Phi}_{1,2}=i\tau_{2}\Phi_{1,2}^{*}$, and $Y_{u1,2}$, $Y_{d1,2}$
and $Y_{\ell 1,2}$ are $3\times 3$ matrices in family space. The Yukawa
coupling matrices are taken to be aligned to avoid the tree-level flavour
changing neutral current aligned2h ,

$\displaystyle(Y_{u1})_{ii}=\frac{\sqrt{2}m_{ui}}{v}\rho_{1u},~{}~{}~{}(Y_{u2})_{ii}=\frac{\sqrt{2}m_{ui}}{v}\rho_{2u},$
$\displaystyle(Y_{\ell 1})_{ii}=\frac{\sqrt{2}m_{\ell
i}}{v}\rho_{1\ell},~{}~{}~{}~{}(Y_{\ell 2})_{ii}=\frac{\sqrt{2}m_{\ell
i}}{v}\rho_{2\ell},$
$\displaystyle(X_{d1})_{ii}=\frac{\sqrt{2}m_{di}}{v}\rho_{1d},~{}~{}~{}(X_{d2})_{ii}=\frac{\sqrt{2}m_{di}}{v}\rho_{2d},$
(12)

where $X_{d1,2}=V_{CKM}^{\dagger}Y_{d1,2}V_{CKM}$,
$\rho_{1f}=(c_{\beta}-s_{\beta}\kappa_{f})$ and
$\rho_{2f}=(s_{\beta}+c_{\beta}\kappa_{f})$ with $f=u,d,\ell$. All the off-
diagonal elements are zero. The couplings of the neutral Higgs bosons
normalized to the SM are given by

$\displaystyle
y^{h}_{V}=\sin(\beta-\alpha),~{}y_{f}^{h}=\left[\sin(\beta-\alpha)+\cos(\beta-\alpha)\kappa_{f}\right],$
$\displaystyle
y^{H}_{V}=\cos(\beta-\alpha),~{}y_{f}^{H}=\left[\cos(\beta-\alpha)-\sin(\beta-\alpha)\kappa_{f}\right],$
$\displaystyle
y^{A}_{V}=0,~{}y_{A}^{f}=-i\kappa_{f}~{}{\rm(for}~{}u)c_{\theta},~{}y_{f}^{A}=i\kappa_{f}c_{\theta}~{}{\rm(for}~{}d,~{}\ell),$
$\displaystyle
y^{X}_{V}=0,~{}y_{X}^{f}=-i\kappa_{f}~{}{\rm(for}~{}u)s_{\theta},~{}y_{f}^{X}=i\kappa_{f}s_{\theta}~{}{\rm(for}~{}d,~{}\ell),$

where $\alpha$ is the mixing angle of the two CP-even Higgs bosons, and $V$
denotes $Z$ or $W$. In the following discussions, we choose
$\sin(\beta-\alpha)=1$ to satisfy the bound of the 125 GeV Higgs signal data.
The $\kappa_{u}$, $\kappa_{d}$, $\kappa_{\ell}$ are assumed to be small enough
so that the extra Higgses ($H$, $H^{\pm}$, $A$, $X$) can avoid the the
constraints of flavor observables and searches for additional Higgs bosons at
the collider.

Dark matter. The two neutral CP-even Higgs can mediate the interactions of DM,
$\lambda_{h}v\chi^{2}h/2$ and $\lambda_{H}v\chi^{2}H/2$ with

$\displaystyle\lambda_{h}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{2}^{\prime}+\lambda_{5}^{\prime})vs_{\beta}c_{\alpha}-(\lambda_{1}^{\prime}+\lambda_{4}^{\prime})vc_{\beta}s_{\alpha},$
$\displaystyle\lambda_{H}$ $\displaystyle\equiv$
$\displaystyle(\lambda_{2}^{\prime}+\lambda_{5}^{\prime})vs_{\beta}s_{\alpha}+(\lambda_{1}^{\prime}+\lambda_{4}^{\prime})vc_{\beta}c_{\alpha}.$
(14)

We consider a light DM whose freeze-out temperature is much lower than that of
EWPT, and thus the effect of EWPT on the current DM relic density can be
ignored. We take the new scalars to be much heavier than the DM so that the DM
pair-annihilation to these new scalars are kinematically forbidden. In the
parameter space chosen previously, the couplings of $H$ to the SM particles
can be ignored. Therefore, the DM relic density hardly constrains the
$\lambda_{H}$, and $\lambda_{1,2,4,5}^{\prime}$ are allowed to have room
enough to produce the pattern of EWPT need by the EWBG. The annihilation
processes mediated by $h$ are responsible for the relic density, and the DM
mass is favored in the 125 GeV Higgs resonance region. In Fig. 1, we show the
$\lambda_{h}$ versus $m_{\chi}$ allowed by the invisible decay of the 125 GeV
Higgs invisible , the DM relic density planck , the direct and indirect
searches experiments XENON:2018voc ; Fermi-LAT:2015att .

Figure 1: $\lambda_{h}$ consistent with the relic data versus $m_{\chi}$. The
dark thick line is allowed by the direct and indirect searches for DM, and the
light thick line is excluded by the indirect searches.

EWPT and Baryogenesis. We first consider the effective scalar potential at the
finite temperature. The neutral elements of $\Phi_{1}$ and $\Phi_{2}$ are
shifted by $\frac{h_{1}}{\sqrt{2}}$ and $\frac{h_{2}+ih_{3}}{\sqrt{2}}$. It is
plausible to take the imaginary part of the neutral elements of $\Phi_{1}$ to
be zero since the effective potential of Eq. (Electroweak baryogenesis, dark
matter, and dark CP-symmetry) only depends on the relative phase of the
neutral elements of $\Phi_{1}$ and $\Phi_{2}$ .

The effective potential is given by

$\displaystyle
V_{eff}=\frac{1}{2}(m_{11}^{2}+\Pi_{h_{1}})h_{1}^{2}+\frac{1}{2}(m_{22}^{2}+\Pi_{h_{2}})(h_{2}^{2}+h_{3}^{2})$
$\displaystyle+\frac{1}{2}(m_{S}^{2}+{m^{\prime}_{S}}^{2}+\Pi_{\chi})\chi^{2}+\frac{1}{2}(m_{S}^{2}-{{m^{\prime}_{S}}^{2}}+\Pi_{\eta_{s}})\eta_{s}^{2}$
$\displaystyle+\frac{1}{8}(\lambda_{1}h_{1}^{4}+\lambda_{2}h_{2}^{4}+\lambda_{2}h_{3}^{4})+\frac{1}{4}\lambda_{345}h_{1}^{2}h_{2}^{2}+\frac{1}{4}\bar{\lambda}_{345}h_{1}^{2}h_{3}^{2}$
$\displaystyle+\frac{\lambda_{2}}{4}h_{3}^{2}h_{2}^{2}-m_{12}^{2}h_{1}h_{2}-\frac{\mu}{\sqrt{2}}h_{3}\eta_{s}h_{1}$
$\displaystyle+\frac{\lambda^{\prime}_{1}}{4}(\chi^{2}+\eta_{s}^{2})h_{1}^{2}+\frac{\lambda^{\prime}_{2}}{4}(\chi^{2}+\eta_{s}^{2})(h_{2}^{2}+h_{3}^{2})$
$\displaystyle+\frac{\lambda^{\prime}_{4}}{2}(\chi^{2}-\eta_{s}^{2})h_{1}^{2}+\frac{\lambda^{\prime}_{5}}{2}(\chi^{2}-\eta_{s}^{2})(h_{3}^{2}+h_{2}^{2})$
$\displaystyle+(\frac{\lambda^{{}^{\prime\prime}}_{1}}{48}+\frac{\lambda^{\prime\prime}_{3}}{16})(\chi^{4}+\eta_{s}^{4})+\frac{1}{8}(\lambda^{\prime\prime}_{3}-\lambda^{\prime\prime}_{1})\chi^{2}\eta_{s}^{2},$
$\displaystyle\Pi_{h_{1}}=\left[{9g^{2}\over 2}+{3g^{\prime 2}\over
2}+6\lambda_{1}+4\lambda_{3}+2\lambda_{4}+2\lambda^{\prime}_{1}+6y_{t}^{2}c_{\beta}^{2}\right]{T^{2}\over
24},$ $\displaystyle\Pi_{h_{2}}=\left[{9g^{2}\over 2}+{3g^{\prime 2}\over
2}+6\lambda_{2}+4\lambda_{3}+2\lambda_{4}+2\lambda^{\prime}_{2}+{6y_{t}^{2}s_{\beta}^{2}}\right]{T^{2}\over
24},$ $\displaystyle\Pi_{h_{3}}=\Pi_{h_{2}},$
$\displaystyle\Pi_{\chi}=\left[4\lambda^{\prime}_{1}+4\lambda^{\prime}_{2}+8\lambda^{\prime}_{4}+8\lambda^{\prime}_{5}+2\lambda^{\prime\prime}_{3}\right]{T^{2}\over
24},$
$\displaystyle\Pi_{\eta_{s}}=\left[4\lambda^{\prime}_{1}+4\lambda^{\prime}_{2}-8\lambda^{\prime}_{4}-8\lambda^{\prime}_{5}+2\lambda^{\prime\prime}_{3}\right]{T^{2}\over
24},$ (15)

where $\bar{\lambda}_{345}=\lambda_{3}+\lambda_{4}-\lambda_{5}$ and
$y_{t}={\sqrt{2}m_{t}\over v}$.

Because baryogenesis is driven by diffusion processes in front of the bubble
wall, one needs to compute the $T_{n}$ at which bubble nucleation actually
starts. This can be calculated straightforwardly from the nucleation rate per
unit volume by bubble-0 ; bubble-1 ; bubble-2 , $\Gamma\approx
A(T)e^{-S_{3}/T}$, where $A(T)\sim T^{4}$ is a prefactor and $S_{3}$ is a
three-dimensional Euclidian action. The nucleation temperature $T_{n}$ is
obtained by $\Gamma/H^{4}$= 1, where $H$ is the Hubble parameter. It is
roughly estimated by $\frac{S_{3}(T)}{T}|_{T=T_{n}}=140$. The bubble wall VEV
profiles can be determined by the bounce equations of fields.

The WKB approximation method is used to evaluate the CP-violating source terms
and chemical potentials transport equations of particle species in the wall
frame with a radial coordinate $z$ bg2h-3 ; 0006119 ; 0604159 . A top quark
penetrating the bubble wall acquires a complex mass as a function of $z$,

$\displaystyle
m_{t}(z)=\frac{y_{t}}{\sqrt{2}}\sqrt{(c_{\beta}h_{1}(z)+s_{\beta}h_{2}(z))^{2}+s_{\beta}^{2}h^{2}_{3}(z)}~{}e^{i\Theta_{t}(z)},$
$\displaystyle{\rm
with~{}}\Theta_{t}(z)=\varphi_{Z}(z)+\arctan\frac{s_{\beta}h_{3}(z)}{c_{\beta}h_{1}(z)+s_{\beta}h_{2}(z)},$
$\displaystyle\partial_{z}\varphi_{Z}(z)=-\frac{h^{2}_{2}(z)+h^{2}_{3}(z)}{h^{2}_{1}(z)+h^{2}_{2}(z)+h^{2}_{3}(z)}\partial_{z}\varphi_{(}z),$
$\displaystyle\varphi_{(}z)=\arctan\frac{h_{3}(z)}{h_{2}(z)}.$ (16)

In our calculation, the imaginary part of the neutral element of $\Phi_{1}$ is
taken to be zero. As a result, the nonvanishing $Z_{\mu}$ field induces an
additional CP-violating force acting on the top quark, which is removed a
local axial transformation of top quark, reintroducing an additional overall
phase $\varphi_{Z}(z)$ into $m_{t}$ bg2h-4 .

We solve the transport equations derived by the complex mass of the top quark
bg2h-3 ; 0006119 ; 0604159 ; 9311367 ; 9506477 , and obtain the chemical
potential of the left-handed quarks $\mu_{B_{L}}$. The left-handed quark
number is converted into a baryon asymmetry by the weak sphalerons, which is
calculated as

$Y_{B}=\frac{405\Gamma_{ws}}{4\pi^{2}v_{w}g_{*}T_{n}}\int_{0}^{\infty}dz\mu_{B_{L}}(z)f_{sph}(z)\exp\left(-\frac{45\Gamma_{ws}z}{4v_{w}}\right)\,,$
(17)

where $\Gamma_{ws}\simeq 10^{-6}T_{n}$ is the weak sphaleron rate inside
bubble sphaleron-ws  and the wall velocity $v_{w}$ is taken as 0.1. The
function
$f_{sph}(z)=min(1,2.4\frac{T_{n}}{\Gamma_{ws}}e^{-40\xi_{n}(z)/T_{n}})$ with
$\xi_{n}(z)=\sqrt{\langle h_{1}\rangle^{2}+\langle h_{2}\rangle^{2}+\langle
h_{3}\rangle^{2}}$ is used to smoothly interpolate between the sphaleron rates
in the broken and unbroken phases bg2h-4 .

$t_{\beta}$ |  $m_{h}$(GeV) |  $m_{H}=m_{H^{\pm}}$(GeV) |  $m_{\chi}$(GeV) |  $m_{A}$(GeV) |  $m_{X}$(GeV) |  $m_{12}^{2}$(GeV$)^{2}$ | $\sin\theta$ | $\lambda^{\prime}_{1}$ | $\lambda^{\prime}_{2}$ | $\lambda^{\prime}_{4}$ | $\lambda^{\prime}_{5}$ | $\lambda^{\prime\prime}_{1}=\lambda^{\prime\prime}_{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1.0 | 125.0 | 509.43 | 55.86 | 74.15 | 382.17 | 3187.21 | 0.278 | 2.902 | 1.935 | -1.451 | -0.963 | 2.579
Table 1: Input parameters for the BP1, and other parameters are given above.

We focus on the following three-step PTs achieving the spontaneous CP-
violation at high temperature. At the first-step PT, the $\eta_{S}$ field
firstly acquires a nonzero VEV while $h_{1,2,3}$ still remains zero. The
second-step PT is a strongly first-order EWPT converting the origin phase of
$h_{1,2,3}$ into an electroweak symmetry broken phase, where $h_{3}$ is
required to be nonzero. In order to prevent the electroweak sphalerons to wash
out the produced BAU inside the bubbles of broken phase, the PT strength is
impose an bound pt-stren , $\frac{\xi_{n}}{T_{n}}>1.0$ in the broken phase.
After the third-step PT, the observed vacuum is produced and the CP-symmetry
is restored while the BAU is not changed. We employ the package
CosmoTransitions to analyze the PTs cosmopt . Some parameter space achieving
the three-step PTs are shown in Fig. 2, where we consider the constraints of
the vacuum stability, oblique parameter pdg2020 , dark matter observables, and
the 125 GeV Higgs signal data, and the data of BAU is not included. From Fig.
2, we find that the three-step PTs satisfying our requirements favor an
appropriate value of $\mu$ since the $\mu$ term of Eq. (Electroweak
baryogenesis, dark matter, and dark CP-symmetry) can lead to a close
correlation between $\langle h_{3}\rangle$ and $\langle\eta_{s}\rangle$ of the
potential minimum. As a result, according to Eq. (10), $m_{A}$ and $m_{X}$ is
required to have a large mass splitting.

Figure 2: The scattering plots achieving the three-step PTs with the
characteristics mentioned in the text, where we take $t_{\beta}=1$,
$m_{H}=m_{H^{\pm}}$, and $0.1<s_{\theta}<0.7$. Figure 3: The phase histories
of the BP1, where $\langle\chi\rangle$ is always 0. Figure 4: The radial
nucleation bubble wall VEV profiles for the first-order EWPT. Figure 5: The
$\mu_{i}$ and $u_{i}$ from the transport equations as functions of the
position of the bubble wall.

We pick out a benchmark point BP1 to discuss the EWPT and baryogenesis
detailedly, and the key input parameters are shown in Table 1. The phase
histories for the BP1 are shown in Fig. 3. Because the contributions of the
thermal mass terms to the effective potential are proportional to $T^{2}$, the
minimum of the potential is at the origin at a very high temperature. As the
Universe cools, at T=77.81 GeV, a second-order PT takes place during which
$\eta_{s}$ acquires a nonzero VEV and the other four fields remain zero. At
T=64.47 GeV, a strongly first-order EWPT starts which breaks electroweak
symmetry, (0, 0, 0, 0, 63.46) GeV $\to$ (64.02, 42.45, 50.02, 0.0, 28.06) GeV
for ($\langle h_{1}\rangle$, $\langle h_{2}\rangle$, $\langle h_{3}\rangle$,
$\langle\chi\rangle$, $\langle\eta_{s}\rangle$). The PT strength is 1.42, and
the BAU is produced via the EWBG mechanism. At T=53.15 GeV, another second-
order PT happens, and then CP-symmetry is recovered. The vacuum evolves along
the final phase, and ultimately ends in the observed values at T = 0 GeV.
Meanwhile, $\xi_{n}>1$ is always kept so that the BAU is not washed out by the
sphaleron processes. The freeze-out temperature of $\chi$ with a mass of 56.86
GeV is around 2.8 GeV, which is much lower than the PT temperatures.

The calculation of BAU depends on the bubble wall profiles, and we use the
FindBounce findbounce  to obtain the bubble wall VEV profiles for the first-
order EWPT of BP1, which is given in Fig. 4. The WKB method of calculating
transport equations needs the condition of $L_{W}T_{n}\gg 1$, where $L_{W}$ is
the width of bubble wall. The $L_{W}T_{n}$ of BP1 is approximately estimated
to be 3.0.

We solve the transport equations with the boundary conditions $\mu_{i}$
$(z=\pm\infty)=0$ ($i=t,~{}t^{c},~{}b,~{}h$), and obtain the chemical
potentials $\mu_{i}$ and velocity perturbations $u_{i}$ of each particle
species. We show the $\mu_{i}$ and $u_{i}$ for the BP1 in Fig. 5, which give
rise to the BAU, $Y_{B}\simeq 8.3\times 10^{-11}$.

The effective potential $V_{eff}$ presents an unbroken $Z_{2}$ symmetry under
which $h_{3}\to-h_{3}$ and $\eta_{s}\to-\eta_{s}$. When CP is spontaneously
broken, there is no bias between transitions to $+\langle\eta_{s}\rangle$ and
$-\langle\eta_{s}\rangle$. The baryon asymmetry produced in the regions with
$+\langle\eta_{s}\rangle$ would be cancelled out by that produced in the
regions with $-\langle\eta_{s}\rangle$. The issue can be solved by a soft
$Z_{2}$ symmetry breaking and CP-symmetry breaking term,
$-i\mu_{3}(S-S^{*})^{3}$. The regions with $-\langle\eta_{s}\rangle$ can
disappear before the EWPT for $\Delta V/T^{4}>10^{-16}$ with $\Delta V$ being
the potential difference between the vacua with $\pm\langle\eta_{s}\rangle$
1110.2876 ; McDonald . The needed $\mu_{3}$ is roughly estimated to be ${\cal
O}(10^{-15})$ GeV.

Comment on gravitational wave signatures, dark matter, and the LHC signatures.
The first-order EWPT needed by the EWBG can produce the gravitational wave. We
find that the gravitational wave signatures from the three-step PTs mentioned
above can easily exceed the sensitivity curve of the U-DECIGO detector udecigo
, such as those of BP1. A full exploration of the parameter space will
potentially find promising regions for detectable gravitational wave signal at
the BBO bbodecigo . The extra Higgses ($H$, $H^{\pm}$, $A$, $X$) couplings to
the SM fermions are significantly suppressed for $\kappa_{u,d,\ell}\to 0$.
Therefore, these extra Higgses are dominantly produced at the LHC via the
electroweak processes mediated by $Z,W^{\pm}$, and $\gamma$, and the main
decay modes include $H\to AZ$, $H^{\pm}\to AW^{\pm}$. The $A$ decay modes
depend on specific values of $\kappa_{u,d,\ell}$. For a heavy DM whose freeze-
out temperature is higher than the EWPT temperature, the EWPT can give
significant effects on the DM relic density. The studies of the LHC signatures
and the heavy DM will be carried out in the future.

Conclusion. We proposed a complex singlet scalar extension of the 2HDM
respecting a discrete dark CP-symmetry. The dark CP-symmetry guarantees $\chi$
to be a DM candidate on one hand and on the other hand allows $\eta_{s}$ to
have mixings with the pseudoscalars of the scalar doublet fields, which plays
key roles in producing the CP-violation sources needed by the EWBG at high
temperature. Imposing relevant theoretical and experimental constraints, we
studied the scenario of $m_{\chi}$ around the SM Higgs resonance region, and
found that the dark matter relic abundance and the BAU can be simultaneously
explained.

This work was supported by the National Natural Science Foundation of China
under grant 11975013.

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