# Observations on cycles in a variant of the Collatz Graph

Q Le  King’s College London School of Mathematics  Edward Smith  King’s
College London School of Mathematics

###### Abstract

It is well known that the Collatz Conjecture can be reinterpreted as the
Collatz Graph with root vertex 1, asking whether all positive integers are
within the tree generated. It is further known that any cycle in the Collatz
Graph can be represented as a tuple, given that inputting them into a function
outputs an odd positive integer; yet, it is an open question as to whether
there exist any tuples not of the form $(2,2,...,2)$, thus disproving the
Collatz Conjecture. In this paper, we explore a variant of the Collatz Graph,
which allows the 3x+1 operation to be applied to both even and odd integers.
We prove an analogous function for this variant, called the Loosened Collatz
Function (LCF), and observe various properties of the LCF in relation to
tuples and outputs. We prove a certain underlying unique factorisation monoid
structure for tuples to the LCF and provide a geometric interpretation of
satisfying tuples in higher dimensions. Research into this variant of the
Collatz Graph may provide reason as to why there exist no cycles in the
Collatz Graph.

###### keywords:

Collatz Conjecture, Graph Theory, 3x+1 Problem, Unsolved Problem, Cycle,
Unique Factorisation Monoids, Experimental Mathematics, General Mathematics

##  1 Introduction

The Collatz function is defined as

$c(n)=\begin{Bmatrix}\frac{n}{2}&\text{if}&n\equiv 0\text{ (mod 2)}\\\
3n+1&\text{if}&n\equiv 1\text{ (mod 2)}\end{Bmatrix}$
for $n\in\mathbb{N}_{1}$. Let $(x,c(x),c^{2}(x),...)$ be the trajectory of $x$
under the Collatz function. The Collatz Conjecture asks whether, for all
$x\in\mathbb{N}_{1}$, the trajectory of $x$ under the Collatz function yields
$1$. Recent empirical evidence (Barina,) has tested the first $2^{68}$
starting values and no counterexample has been found as of yet. It is well
known that this conjecture can be reformulated in graph-theoretic terms as the
Collatz graph, a graph that is defined by the inverse relation

$C(n)=\begin{Bmatrix}\\{2n\\}&\text{if}&n\equiv 0,1,2,3,5\text{ (mod 6)}\\\
\\{2n,\frac{n-1}{3}\\}&\text{if}&n\equiv 4\text{ (mod 6)}\end{Bmatrix}$
for $n\in\mathbb{N}_{1}$, where the vertices of the graph are positive
integers. Thus, the Collatz Conjecture is reframed to the question of whether
the Collatz graph is a tree containing all positive integers with root vertex
1.

We define $f,g:\mathbb{N}_{1}\to\mathbb{N}_{1}$ to be the two mappings in the
Collatz function that connect one vertex to another, $f:n\to\frac{n}{2}$ and
$g:n\to 3n+1$. We define cycles to be simple directed circuits - we will
specify when we refer to the more general directed circuit. We also say that
$f$ or $g$ is applied in a cycle if $f$ or $g$ is used on one vertex to
connect with another. If the Collatz graph contains cycles aside from the
trivial
$\dots\xrightarrow{f}1\xrightarrow{g}4\xrightarrow{f}2\xrightarrow{f}\dots$,
then the Collatz Conjecture is false, since there then exists a trajectory
that never yields 1.

It was proven (Böhm and Sontacchi,) that, for a cycle to exist in the Collatz
graph, there must exist a unique tuple $Y=(y_{1},\dots,y_{n})$, where each
entry represents the number of times you apply $f$ after applying $g$ in the
cycle. Furthermore, for such a tuple $Y$, it represents a cycle in the Collatz
graph if and only if the following function, which we call the Collatz
Function, outputs an odd number that has $g$ applied to it in the cycle
represented:

$F(Y)=\frac{\sum_{i=1}^{n}3^{n-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n}y_{i}}-3^{n}}$
where $y_{0}=0$. It is important to note that:

  1. 1.

each entry of $Y$ must be greater than 0, since having 0 means you applied $g$
twice consecutively, once on an odd integer, once on an even integer, which is
not permitted by the Collatz function.

  2. 2.

$y_{0}=0$ is required for the function to work and should not be seen as an
entry for the tuple, since $y_{0}$ always equals as $0$ regardless of the
inputted tuple.

We say that a tuple is an $n$-tuple if it has $n$ entries and a cycle is an
$n$-cycle if, within the cycle, $g$ is applied $n$ times. For example, the
trivial cycle
$\dots\xrightarrow{f}1\xrightarrow{g}4\xrightarrow{f}2\xrightarrow{f}\dots$
can be represented by the 1-tuple (2) since the output of the function is

$F((2))=\frac{1}{2^{2}-3}=1$.
and $g$ is applied one time in that cycle. In fact, the $n$-tuple (2,2,…,2)
represents the trivial cycle, repeated $n$ times. This can be seen by
investigating what happens when all entries of the tuple $(a,\dots,a)$ are
equal.

Many variants of the Collatz function has been researched, from extending the
domain to include integer, rational, real, and complex inputs to generalising
the mappings in the Collatz function. In this paper, we investigate a modified
Collatz graph called the Loosened Collatz graph, which is defined by the
relation

$L(n)=\begin{Bmatrix}\\{2n\\}&\text{if}&n\equiv 0,2\text{ (mod 3)}\\\
\\{2n,\frac{n-1}{3}\\}&\text{if}&n\equiv 1\text{ (mod 3)}\end{Bmatrix}$
for $n\in\mathbb{N}_{1}$, with the exception that $L(1)=2$. This is equivalent
to modifying the Collatz function into the graph generating relation

$l(n)=\begin{Bmatrix}\\{\frac{n}{2},3n+1\\}&\text{if}&n\equiv 0\text{ (mod
2)}\\\ \\{3n+1\\}&\text{if}&n\equiv 1\text{ (mod 2)}\end{Bmatrix}$
for $n\in\mathbb{N}_{1}$. This can be observed by noting that, for all
$x\equiv 1\text{ (mod 3)}$, $\frac{x-1}{3}$ is either even or odd and for all
$y\in\mathbb{N}_{1}$, one can easily construct $3y+1\equiv 1\text{ (mod 3)}$.

From this, we can see that the Collatz Function changes too, by extending the
domain of the entries of the tuple to include 0. Thus, for a cycle to exist in
the Loosened Collatz Graph, there must exist a unique tuple
$Y=(y_{1},\dots,y_{n})$, where each entry of the tuple represents the number
of times you apply $f$ after applying $g$ in the cycle, which may now be 0.
Further, for such a tuple $Y$, it represents a cycle in the Collatz graph if
and only if the following function, which we call the Loosened Collatz
Function (LCF), outputs an integer, even or odd, that has $g$ applied to it in
the cycle represented:

$F_{L}(Y)=\frac{\sum_{i=1}^{n}3^{n-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n}y_{i}}-3^{n}}$
where $y_{0}=0$. From this, we say that a tuple satisfies the LCF if:

  1. 1.

Each entry of the tuple is a non-negative integer.

  2. 2.

Inputting the tuple into the LCF outputs a positive integer.

Unlike the original Collatz graph, the Loosened Collatz graph has many cycles
that satisfy the LCF. A trivial case is the circuit

$\dots\xrightarrow{f}4\xrightarrow{g}13\xrightarrow{g}40\xrightarrow{f}20\xrightarrow{f}10\xrightarrow{f}5\xrightarrow{g}16\xrightarrow{f}8\xrightarrow{f}4\dots$
which can be represented by the tuple $(0,3,2)$,$(2,0,3)$ and $(3,2,0)$.

This paper investigates the Loosened Collatz Graph and makes several
observations and conjectures regarding it. Initially, we present a derivation
of the LCF from the Loosened Collatz Graph relation. Then, we prove the
satisfaction of the LCF of bitwise rotations of a tuple, given that a tuple
satisfies the LCF, and vice versa. We then look at the underlying monoid
structure of satisfying tuples and interpret the tuples as coordinates in
higher dimensional spaces, with the hopes that they are useful for other
researchers. We conclude with a summary of our research and further research
questions to explore.

##  2 Cycles in the Loosened Collatz Graph

The Loosened Collatz graph is the graph defined by the relation

$L(n)=\begin{Bmatrix}\\{2n\\}&\text{if}&n\equiv 0,2\text{ (mod 3)}\\\
\\{2n,\frac{n-1}{3}\\}&\text{if}&n\equiv 1\text{ (mod 3)}\end{Bmatrix}$
for $n\in\mathbb{N}_{1}$, with the exception that $L(1)=2$ to avoid having $0$
as a vertex of the graph. We define $f,g$ as in the introduction. By inversing
the relation, we have the graph generating relation

$l(n)=\begin{Bmatrix}\\{\frac{n}{2},3n+1\\}&\text{if}&n\equiv 0\text{(mod
2)}\\\ \\{3n+1\\}&\text{if}&n\equiv 1\text{(mod 2)}\end{Bmatrix}$
for $n\in\mathbb{N}_{1}$. From this, we can obtain the Loosened Collatz
Function (LCF).

###  2.1 Proof of The Loosened Collatz Function

######  Theorem 1.

For a cycle to exist in the Loosened Collatz graph, its tuple must satisfy the
function

$F_{L}(Y)=\frac{\sum_{i=1}^{n}3^{n-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n}y_{i}}-3^{n}}$
where $y_{0}=0$, each entry of the tuple $(y_{1},\dots,y_{n})$ corresponds to
the number of times you apply $f$ after applying $g$ in the cycle, including
0, and the output is one of the positive integers in that cycle, which has $g$
applied to it.

Although an analogous theorem has been proven true for the original Collatz
graph, it will be useful to prove it in this context, since some constructions
within the proof will be used later in the paper.

###### Proof.

Assume there exists an $n$-cycle, in which $g$ is applied $n$-times. Let
$x_{1}$ denote one of the numbers in the cycle to which $g$ is applied. After
$g$ is applied onto $x_{1}$, there exist $y_{1}$ applications of $f$ before
applying $g$ again. Let $x_{2}=f^{y_{1}}(g(x_{1}))$ be the number before
applying $g$ again. After $g$ is applied onto $x_{2}$, there exist $y_{2}$
applications of $f$ before applying $g$ again. Let $x_{3}=f^{y_{2}}(g(x_{2}))$
be the number before applying $g$ again. We may repeat this process until we
obtain $x_{1}=f^{y_{n}}(g(x_{n}))$. It is clear to see that
$y_{1},\dots,y_{n}$ are positive integers. So, we obtain the family of
equations

$x_{2}=f^{y_{1}}(g(x_{1}))$

$x_{3}=f^{y_{2}}(g(x_{2}))$

$\vdots$

$x_{n}=f^{y_{n-1}}(g(x_{n-1})$

$x_{1}=f^{y_{n}}(g(x_{n}))$
Notice that,

$x_{2}=\frac{3x_{1}+1}{2^{y_{1}}}$

$x_{3}=\frac{3^{2}x_{1}+2^{y_{1}}+3}{2^{y_{1}+y_{2}}}$

$x_{4}=\frac{3^{3}x_{1}+3\cdot
2^{y_{1}}+2^{y_{1}+y_{2}}+3^{2}}{2^{y_{1}+y_{2}+y_{3}}}$
To generalise the values in the numerator and denominator, we shall prove the
following lemma:

######  Lemma 1.

for $1\leq k\leq n$, $x_{k}=\frac{3^{k-1}x_{1}+\sum_{i=1}^{k-1}3^{k-i-1}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{k-1}y_{i}}}$, where $y_{0}=0$.

###### Proof.

We prove this statement by induction. For base cases 2 and 3, the claim is
true. Assume that the claim is true for $k=z$, for $1\leq z<n$. So,

$x_{z}=\frac{3^{z-1}x_{1}+\sum_{i=1}^{z-1}3^{z-i-1}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{z-1}y_{i}}}$.
We may consider the case for $k=z+1$. Since $x_{z+1}=f^{z}(g(x_{z}))$, by
performing $g$ and $f$, we obtain

$x_{z+1}=\frac{3(\frac{3^{z-1}x_{1}+\sum_{i=1}^{z-1}3^{z-i-1}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{z-1}y_{i}}})+1}{2^{y_{z}}}$.
Simplifying the fraction, we get

$x_{z+1}=\frac{3^{z}x_{1}+\sum_{i=1}^{z-1}3^{z-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}+2^{\sum_{i=1}^{z-1}y_{i}}}{2^{\sum_{i=1}^{z-1}y_{i}+y_{z}}}$.
We may allow $3^{z-z}$ to multiply $2^{\sum_{i=1}^{z-1}y_{i}}$, since this
does not change the overall result.

$x_{z+1}=\frac{3^{z}x_{1}+\sum_{i=1}^{z-1}3^{z-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}+3^{z-z}\cdot
2^{\sum_{i=1}^{z-1}y_{i}}}{2^{\sum_{i=1}^{z-1}y_{i}+y_{z}}}$.
Observe that $3^{z-z}\cdot 2^{\sum_{i=1}^{z-1}y_{i}}$ would be the term
generated by the sum $\sum_{i=1}^{z}3^{z-i}\cdot 2^{\sum_{j=0}^{i-1}y_{j}}$
for $i=z$. From this, we may write

$x_{z+1}=\frac{3^{z}x_{1}+\sum_{i=1}^{z}3^{z-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{z}y_{i}}}$,
which makes the claim true for $k=z+1$. By the principle of induction, we have
proven the claim true for all $k\in\mathbb{N}_{1}$. ∎

So, by Lemma 1, we have proven that

$x_{n}=\frac{3^{n-1}x_{1}+\sum_{i=1}^{n-1}3^{n-i-1}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n-1}y_{i}}}$,
and, if $x_{1}=f^{y_{n}}(g(x_{n}))$, then
$x_{1}=\frac{3^{n}x_{1}+\sum_{i=1}^{n}3^{n-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n}y_{i}}}$. Rearranging for $x_{1}$,
we get

$x_{1}=\frac{\sum_{i=1}^{n}3^{n-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n}y_{i}}-3^{n}}$.
Since this is true for any number in a cycle on which $g$ is applied, we may
allow the equation to become the function
$F_{L}(Y)=\frac{\sum_{i=1}^{n}3^{n-i}\cdot
2^{\sum_{j=0}^{i-1}y_{j}}}{2^{\sum_{i=1}^{n}y_{i}}-3^{n}}$, where $y_{0}=0$
regardless of the tuple $Y$, each entry of the tuple $(y_{1},\dots,y_{n})$
corresponds to the number of times you apply $f$ after applying $g$, including
0, and the output is one of the positive integers in that cycle, which has $g$
applied to it. ∎

Theorem 1 shows that we have extended the original Collatz Function to allow
0s as entries of the tuple when considering the Loosened Collatz graph.

A natural question to ask is whether there exists an algorithm to find tuples
that satisfy the LCF. We provide a partial answer to this - there do exist two
methods for finding cycles in the graph, which lend themselves to finding
satisfying tuples. However, there is no guarantee that they will always find
cycles.

The first method is to start from 1, 2, 8, or 16 and iterate $g$ on it a
number of times before applying the normal Collatz Function to the number
generated. Assuming the Collatz Conjecture is true, this will always result in
a cycle since all trajectories must yield 1, 2, 8, and 16, which can be seen
by observing the original Collatz graph. The second method is to
systematically search through every possible tuple combination of every
possible length.

Tuples found by the first method are called trivial tuples since all cycles
that these tuples represent found must include 1, 2, 8, or 16 and the method
used is relatively simple. Applying the same method to other numbers does not
always yield the same result. So, tuples found only by the second method are
called non-trivial tuples.

Now, we may state the following conjecture:

######  Conjecture 1.

For all $n\neq 0\text{ (mod 3)}\in\mathbb{N}_{1}$, $n$ is a vertex of a cycle.

Clearly, multiples of 3 cannot be in cycles by observing that, when applying
$g$, all integers $x$ go to $3x+1$, which is another integer.

###  2.2 Proof of Bitwise Satisfaction

Define
$R:(a_{1},a_{2},\dots,a_{n-1},a_{n})\to(a_{2},a_{3}\dots,a_{n},a_{1}),\mathbb{N}_{0}^{n}\to\mathbb{N}_{0}^{n}$
to be a left bit-wise rotation on an $n$-tuple.

######  Theorem 2.

An n-tuple $X=(x_{1},\dots,x_{n})$ satisfies the LCF if and only if all
bitwise rotations on that tuple also satisfy the LCF.

Recall that, for $X$ to satisfy the LCF, $F_{L}(X)$ must be a positive integer
that has $g$ applied to it in the cycle. Furthermore, recall that, by Theorem
1, for an $n$-cycle, $y_{k}$ is defined as the number of applications of $f$
on a number $x_{k}$ before applying $g$ again to produce a new number
$x_{k+1}$. Since the sequence $x_{1},\dots,x_{n},x_{1},\dots$ is periodic, so
too is the sequence $y_{1},\dots,y_{n},y_{1},\dots$. We may leverage these two
facts to prove this claim.

###### Proof.

Assume the claim is false. Then, there exists a cycle, in which both integer
and non-integer vertices have $g$ applied to them.

So, there exists a pair of consecutive vertices in the cycle in the Loosened
Collatz Graph, such that the first is an integer $x$ and the second is a non-
integer. Since, for all $x\in\mathbb{N}_{1}$, $g(x)\in\mathbb{N}_{1}$, then
the non-integer must be $f(x)$ or $\frac{x}{2}$.

Let $a,b\in\mathbb{N}_{1}$ such that $\frac{a}{2^{b}}\not\in\mathbb{N}_{1}$.
Applying $g$ produces $\frac{3a+2^{b}}{2^{b}}$, and since $2^{b}\nmid a$,
$2^{b}\nmid 3a+2^{b}$ either. Thus, $\frac{3a+2^{b}}{2^{b}}$ is a non-integer
of the form $\frac{c}{2^{d}}$, where $c,d\in\mathbb{N}_{1}$ such that
$2^{d}\nmid c$. Likewise, applying $f$ produces $\frac{a}{2^{b+1}}$, another
non-integer of the form $\frac{c}{2^{d}}$, where $c,d\in\mathbb{N}_{1}$ such
that $2^{d}\nmid c$. Thus, all non-integer vertices in the cycle must be of
the form $\frac{m}{2^{n}}$, for some $m,n\in\mathbb{N}_{1}$ such that
$2^{n}\nmid m$.

However, since applying $g$ or $f$ on any non-integer of the form expressed
above results in another non-integer of that form, it follows that there
cannot exist integer vertices in the cycle after the initial non-integer. But
this contradicts our initial assumption that there exists a cycle with both
integer and non-integer vertices.

Thus, if $X$ satisfies the LCF, then so too do all bitwise rotations of it.
Likewise, if all bitwise rotations of $X$ satisfy the LCF, then so too must
$X$, since all outputs must all be either positive integers or positive non-
integers.

∎

Theorem 2 shows that, for any $n$-tuple that we test with the LCF, we are
effectively testing at most $n$ different tuples, which are the bitwise
rotations of the $n$-tuple. So, Theorem 2 provides an easy way for processing
which tuples have been tested, helping us with the later search for non-
trivial tuples.

From this, we may state the following conjecture:

######  Conjecture 2.

Excluding the tuple $(2,2,...)$, a tuple satisfying the Loosened Collatz
Function must include $0$ in its entry.

If this conjecture is true, then there exist no cycles in the original Collatz
graph aside from the trivial cycle
$\dots\xrightarrow{f}1\xrightarrow{g}4\xrightarrow{f}2\xrightarrow{f}\dots$.

###  2.3 Proof of Unique Cycle Representation

For later sections, the following theorem and its argument will prove useful:

######  Theorem 3.

If $A=(a_{1},\dots,a_{n})$ and $B=(b_{1},\dots,b_{n})$ are $n$-tuples that
satisfy the LCF such that $F_{L}(R^{k}(A))=F_{L}(R^{k}(B))$ for all
$k\in\mathbb{N}_{1}$, then $A=B$.

###### Proof.

Recall that, in proving Theorem 1, we constructed a family of equations which
represented a cycle and, from there, constructed a tuple, which has the number
of applications of $f$ before applying $g$ as entries. If
$F_{L}(R^{k}(A))=F_{L}(R^{k}(B))$ for all $k\in\mathbb{N}$, then the families
of equations the tuples represent are:

$x_{2}=f^{a_{1}}(g(x_{1}))=f^{b_{1}}(g(x_{1}))$

$x_{3}=f^{a_{2}}(g(x_{2}))=f^{b_{2}}(g(x_{2}))$

$\vdots$

$x_{n}=f^{a_{n-1}}(g(x_{n-1})=f^{b_{n-1}}(g(x_{n-1}))$

$x_{1}=f^{a_{n}}(g(x_{n}))=f^{b_{n}}(g(x_{n}))$
Thus:

$\frac{3x_{1}+1}{2^{a_{1}}}=\frac{3x_{1}+1}{2^{b_{1}}}$

$\frac{3x_{2}+1}{2^{a_{2}}}=\frac{3x_{2}+1}{2^{b_{2}}}$

$\vdots$

$\frac{3x_{n-1}+1}{2^{a_{n-1}}}=\frac{3x_{n-1}+1}{2^{b_{n-1}}}$

$\frac{3x_{n}+1}{2^{a_{n}}}=\frac{3x_{n}+1}{2^{b_{n}}}$
It is easy to see that $a_{k}=b_{k}$ for all $1\leq k\leq n$.

∎

Theorem 3 shows that, for every cycle in the Loosened Collatz Graph, it can be
represented by a unique tuple, along with its bitwise rotations. It may be
possible to discover a normal form for such satisfying tuples, though we won’t
do so here.

##  3 The Monoid Structure of Satisfying Tuples

It may be useful to explore directed circuits more generally, instead of
cycles as we have defined them. This can be done by investigating tuples
instead of cycles in the Loosened Collatz Graph, which leads us to asking
whether there exist any algebraic properties of tuples. Further, we ask
whether it is possible to construct further tuples that satisfy the LCF. With
guidance from an article introducing monoids (Chapman,), we investigate how
cycles may be interpreted as such.

Define $S_{k}$ to be the union of the set of all tuples $X$ with non-negative
integer entries, such that $F_{L}(X)=k$ for some positive integer $k$, and the
empty set. So, $S_{k}=\\{X=(x_{1},\dots,x_{n}),x_{i}\in\mathbb{N}_{0}\text{
}|\text{ }F_{L}(X)=k\\}\cup\\{\emptyset\\}$, for some $k\in\mathbb{N}_{1}$.
Denote the binary operation of concatenation between two tuples
$X=(x_{1},\dots,x_{p})$ and $Y=(y_{1},\dots,y_{q})$ with $\cdot$, such that
$X\cdot Y=(x_{1},\dots,x_{p},y_{1},\dots,y_{q})$. It will be also useful to
denote the number of entries of a tuple $Z=(z_{1},\dots,z_{r})$ as $[Z]=r$ for
$r\in\mathbb{N}_{0}$.

######  Lemma 1.

If $F_{L}(X)=F_{L}(Y)=k$ for some $X,Y\in S_{k}$, then $F_{L}(X\cdot
Y)=F_{L}(Y\cdot X)=k$. Thus, $S_{k}$ is closed under concatenation.

###### Proof.

Recall the family of equations constructed in the proofs for Theorem 1 and
Theorem 3. If $F_{L}(X)=F_{L}(Y)=k$, where $X=(x_{1},\dots,x_{p})$ and
$Y=(y_{1},\dots,y_{q})$, then

$a_{2}=f^{x_{1}}(g(k))$

$a_{3}=f^{x_{2}}(g(a_{2}))$

$\vdots$

$a_{p}=f^{x_{p-1}}(g(a_{p-1}))$

$k=f^{x_{p}}(g(a_{p}))$
is the family of equations for the tuple $X$, where
$a_{2},\dots,a_{p}\in\mathbb{N}_{1}$ and $k=a_{1}$, and

$b_{2}=f^{y_{1}}(g(k))$

$b_{3}=f^{y_{2}}(g(b_{2}))$

$\vdots$

$b_{q}=f^{y_{q-1}}(g(b_{q-1}))$

$k=f^{y_{q}}(g(b_{q}))$
is the family of equations for the tuple $Y$, where
$b_{2},\dots,b_{q}\in\mathbb{N}_{1}$ and $k=b_{2}$. From this, let us observe
the following family of equations:

$c_{2}=f^{x_{1}}(g(c_{1}))$

$c_{3}=f^{x_{2}}(g(c_{2}))$

$\vdots$

$c_{p}=f^{x_{p-1}}(g(c_{p-1}))$

$c_{p+1}=f^{x_{p}}(g(c_{p}))$

$c_{p+2}=f^{y_{1}}(g(c_{p+1}))$

$c_{p+3}=f^{y_{2}}(g(c_{p+2}))$

$\vdots$

$c_{p+q}=f^{y_{q-1}}(g(c_{p+q-1}))$

$c_{p+q+1}=f^{y_{q}}(g(c_{p+q}))$
where $c_{1}=k$. With reference to the previous two families, we see that
$c_{i}=a_{i}$, for $1\leq i\leq p$, and so $c_{p+1}=k$. Likewise,
$c_{p+j}=b_{j}$, for $1\leq j\leq q$, and so $c_{p+q+1}=k$. Therefore, we
observe that $F_{L}((x_{1},\dots,x_{p},y_{1},\dots,y_{q}))=k$. Since
$(x_{1},\dots,x_{p},y_{1},\dots,y_{q})=(x_{1},\dots,x_{p})\cdot(y_{1},\dots,y_{q})=X\cdot
Y$, it follows that, if $F_{L}(X)=F_{L}(Y)=k$, then $F_{L}(X\cdot Y)=k$. A
similar argument can be used to show that $F_{L}(Y\cdot X)=k$ also holds. ∎

Thus, the set $S_{k}$ is closed under concatenation. It is worth remarking
that $S_{k}$ equipped with concatenation is not commutative under equality,
since, for $X,Y\in S_{k}\setminus\\{\emptyset\\}$, $X\cdot Y\neq Y\cdot X$,
even though the circuits they represent in the Loosened Collatz Graph pass
through the same edges and vertices the same number of times. So, we denote
$A\simeq B$ for tuples $A$ and $B$ $\in S_{k}$ if, when observing the circuits
of the graph, the unordered sets of edges and vertices in the circuits
represented by $A$ and $B$, as well as the number of times each edge and
vertex has been met, are equal. We can say that $A$ and $B$ are equivalent,
since the binary relation $\simeq$ is an equivalence relation. So, $X\cdot
Y\simeq Y\cdot X$, since both circuits have the same set of edges and vertices
and each edge and vertex has been passed through the same number of times in
both $X\cdot Y$ and $Y\cdot X$. It is also clear to see that $A\simeq B$
requires $A$ and $B$ to have the same number of entries, since the cycles they
represent must have $g$ applied an equal number of times in both.

From this, we can derive the following theorem:

######  Theorem 4.

The set $S_{k}$ equipped with concatenation is a commutative, cancellative
monoid under equivalence.

###### Proof.

Note that a set with its binary operator is a monoid if it follows the
associative law and contains an identity element. Let $X,Y,Z\in S_{k}$ be
tuples $X=(x_{1},\dots,x_{p}),Y=(y_{1},\dots,y_{q})$ and
$Z=(z_{1},\dots,z_{r})$. Observe that:

$\begin{split}(X\cdot Y)\cdot
Z&\simeq(x_{1},\dots,x_{p},y_{1},\dots,y_{q})\cdot(z_{1},\dots,z_{r})\\\
&\simeq(x_{1},\dots,x_{p},y_{1},\dots,y_{q},z_{1},\dots,z_{r})\\\
&\simeq(x_{1},\dots,x_{p})\cdot(y_{1},\dots,y_{q},z_{1},\dots,z_{r})\\\
&\simeq X\cdot(Y\cdot Z)\end{split}$ (1)

So, the associative law holds. It is clear to see that, for all $X\in S_{k}$,

$X\cdot\emptyset\simeq\emptyset\cdot X\simeq X$.
So, there exists an identity element, which proves that $S_{k}$ equipped with
concatenation is a monoid under equivalence. If $S_{k}$ is cancellative, then
$X\cdot Y\simeq X\cdot Z$ implies that $Y\simeq Z$. Following from above, if
$X\cdot Y\simeq X\cdot Z$, then

$(x_{1},\dots,x_{p},y_{1},\dots,y_{q})\simeq(x_{1},\dots,x_{p},z_{1},\dots,z_{r})$.
From this, observe that the $[X\cdot Y]$ and $[X\cdot Z]$ must be equal. So,
$p+q=p+r$, which leads to $q=r$. Further, by equivalence, $X\cdot Y$ and
$X\cdot Z$ have the same unordered set of edges and vertices encountered in
their circuits as well as the number of times each edge and vertex has been
met. Using the family of equations argument from Theorem 3, $Y$ and $Z$ must
be equivalent to keep the unordered set of edges and vertices equal after
removing the circuit represented by $X$. This leads to the implication that
$Y\simeq Z$. ∎

One could also prove that $S_{k}$ is a non-commutative, cancellative monoid
under equality as opposed to equivalence, though this will not be as useful
later. Since $S_{k}$ is a commutative, cancellative monoid under equivalence,
we can make a few key definitions and observations about elements in $S_{k}$,
adapted from a paper on Unique Factorisation in Abstract Algebra (Petersen,
(2014)):

  * •

$X|Y$ in $S_{k}$ if $X\cdot Z\simeq Y$ for some $Z\in S_{k}$.

  * •

Define $U\in S_{k}$ to be a unit in $S_{k}$ if there exists an $V\in S_{k}$
such that $U\cdot V\simeq V\cdot U\simeq\emptyset$, since $\emptyset$ is the
identity. Clearly, the only unit is $\emptyset$, since this is the only
element that has an ’inverse’, itself.

  * •

Define $S_{k}^{*}$ to be the set of elements of $S_{k}$ that are not units,
which is equal to $S_{k}\setminus\\{\emptyset\\}$.

  * •

If $X\in S_{k}^{*}$ and $Y,Z\in S_{k}$, then we say that:

    * –

$X$ is prime if, whenever $X|Y\cdot Z$, $X|Y$ or $X|Z$.

    * –

$X$ is irreducible or an atom if, whenever $X\simeq Y\cdot Z$, then $Y$ or $Z$
is a unit in $S_{k}$.

  * •

Define $A(S_{k})$ to be the set of tuples that are irreducible in $S_{k}$. By
using the family of equations construction for any tuple in $S_{k}$, we can
see that $A(S_{k})$ contains the set of all tuples representing circuits that
apply $g$ to $k$ once, which do not necessarily have to be cycles. Thus, for
all $X\in A(S_{k})$, $F_{L}(R^{l}(X))\neq k$ for $1\leq l<[X]$

  * •

$A,B\in S_{k}$ are associates if $A\simeq B\cdot U$ for some unit $U$. Since
the only unit is $\emptyset$, an associate of any element is simply itself.

From here, we can make connection between all primes and all atoms.

######  Corollary 1.

All primes are atoms and all atoms are primes in $S_{k}$.

###### Proof.

Supposing that $P\in S_{k}$ is prime, if $P\simeq A\cdot B$ for some $A,B\in
S_{k}$, then $P|A$ or $P|B$. If $P|A$, then $A\simeq P\cdot C$ for some $C\in
S_{k}$. Thus, $P\simeq A\cdot B\simeq P\cdot C\cdot B$ and, since $S_{k}$ is
cancellative, $P\simeq P\cdot C\cdot B$ implies $\emptyset\simeq C\cdot B$ and
so $C$ and $B$ is a unit. Likewise, we may apply the same logic if $P|B$ to
deduce that $A$ is a unit. Since $P$ is neither a unit nor can be expressed as
non-units, $P$ is atomic.

Supposing $X\in S_{k}$ is atomic, if $X\simeq Y\cdot Z$, then either $Y$ or
$Z$ is a unit of $S_{k}$. If $Y$ is a unit, then $Y=\emptyset$ since that is
the only unit of $S_{k}$, which leads to $X\simeq Z$, by the identity law. If
$X\simeq Z$, then $X|Z$, so $X$ is also prime. We may apply the same logic if
$Z$ is a unit instead of $Y$ to deduce that $X|Y$. This shows that all primes
are atomic and all atoms are primes. ∎

We shall use them interchangeably now.

######  Corollary 2.

$S_{k}$ is a unique factorisation monoid (UFM).

###### Proof.

For $S_{k}$ to be a UFM, the following conditions must hold for any element
$X\in S_{k}^{*}$:

  1. 1.

There exist atoms $\alpha_{1},\dots,\alpha_{k}\in S_{k}$ such that
$X\simeq\alpha_{1}\cdot\dots\cdot\alpha_{k}$.

  2. 2.

If $\alpha_{1},\dots,\alpha_{k}$ and $\beta_{1},\dots,\beta_{j}$ are atoms of
$S_{k}$ and
$X\simeq\alpha_{1}\cdot\dots\cdot\alpha_{k}\simeq\beta_{1}\cdot\dots\cdot\beta_{j}$,
then $k=j$ and there is a unique $s$ for each $t$ such that
$\alpha_{s}=\beta_{t}$.

To prove the first condition, let $X=(x_{1},\dots,x_{p})$ be an element of
$S_{k}^{*}$, where $[X]=p$ for some $p\in\mathbb{N}_{1}$. It is either an atom
or not: if it is an atom, then $X=\alpha_{1}$ for some atom $\alpha_{1}$; if
it is not an atom, then it can be decomposed further into atoms. If so, let
$X=\alpha_{1}\cdot A_{1}$, where $\alpha_{1}$ is an atom and $A_{1}$ is an
element of $S_{k}^{*}$. Since atoms cannot be the empty set, $[A_{1}]<[X]=p$.

If $A_{1}$ is an atom, we may stop there. If $A_{1}$ is not an atom, then it
can be decomposed further into atoms. If so, let $A_{1}=\alpha_{2}\cdot
A_{2}$, where $\alpha_{2}$ is an atom and $A_{2}$ is an element of
$S_{k}^{*}$. Since atoms cannot be the empty set, $[A_{2}]<[A_{1}]<[X]$.

If we assume that there exists an $X$ that is neither an atom nor can be
decomposed into atoms, then this process can occur infinitely many times.
However, by the well-ordering principle, it cannot, since eventually there
will exist an $A_{k}$ for some $k\in\mathbb{N}$ such that $[A_{k}]\leq 0$,
which is not possible. Thus, there are finitely many times one can apply this
process onto some $X\in S_{k}^{*}$. So, there will always exist atoms
$\alpha_{1},\dots,\alpha_{k}\in S_{k}$ such that
$X\simeq\alpha_{1}\cdot\dots\cdot\alpha_{k}$.

To prove the second condition, suppose that
$X\simeq\alpha_{1}\cdot\dots\cdot\alpha_{k}\simeq\beta_{1}\cdot\dots\cdot\beta_{j}$,
where $\alpha_{1},\dots,\alpha_{k}$ and $\beta_{1},\dots,\beta_{j}$ are
primes. Assuming that $k\geq j$, we may consider $\alpha_{1}$ and notice that,
by definition of being a prime, $\alpha_{1}|\beta_{t}$ for some $1\leq t\leq
j$. Since $\alpha_{1}$ and $\beta_{t}$ are both primes, they are associates,
but since all associates are simply themselves, $\alpha_{1}=\beta_{t}$.

We may relabel $\beta_{t}$ as $\beta_{1}$ and $\beta_{1}$ to $\beta_{t}$,
unless $t=1$. So,
$\alpha_{1}\cdot\dots\cdot\alpha_{k}\simeq\beta_{1}\cdot\dots\cdot\beta_{j}\simeq\alpha_{1}\cdot\beta_{2}\cdot\dots\cdot\beta_{j}$
and by the cancellative property, we obtain
$\alpha_{2}\cdot\dots\cdot\alpha_{k}\simeq\beta_{2}\cdot\dots\cdot\beta_{j}$.
We may repeat this process, cancelling out each $\alpha_{s}$ with a
$\beta_{t}$. Supposing that not all atoms of $\alpha$ have been cancelled out
by the end of this process, since $k\geq j$, we obtain
$\alpha_{j+1}\cdot\dots\cdot\alpha_{k}\simeq\emptyset$.

But this implies that the tuples $\alpha_{j+1},\dots,\alpha_{k}=\emptyset$, so
they are not truly atoms. This means that that the only tuples which are
actually primes are $\alpha_{1},\dots,\alpha_{j}$ and
$\beta_{1},\dots,\beta_{j}$, showing that the number of primes of both
factorisations must be equal and that there is a unique $s$ for each $t$ such
that $\alpha_{s}=\beta_{t}$. Using the same logic when assuming $j\geq k$
leads to us proving the second condition fully. ∎

This result implies that satisfying tuples can be decomposed into atomic/prime
satisfying tuples. This is confirmed in the following confirmation:

######  Observation 1.

There exists a prime decomposition for satisfying tuples, unique up to
equivalence.

###### Proof.

Recall that, if $F_{L}(X)=F_{L}(Y)=k$ for some $X,Y\in S_{k}$, then
$F_{L}(X\cdot Y)=F_{L}(Y\cdot X)=k$ and so $X\cdot Y$ and $Y\cdot X$ are also
in $S_{k}$. Usually, the inverse is not true: $X\cdot Y\in S_{k}$ does not
imply $X,Y\in S_{k}$ for any tuple $X,Y$, unless $F_{L}(X\cdot Y)=F_{L}(Y\cdot
X)=k$. To see why, observe that the family of equations produced for any tuple
$D=(d_{1},\dots,d_{n})$ in $S_{k}^{*}$:

$c_{2}=f^{d_{1}}(g(c_{1}))$

$c_{3}=f^{d_{2}}(g(c_{2}))$

$\vdots$

$c_{p}=f^{d_{p-1}}(g(c_{p-1}))$

$c_{p+1}=f^{d_{p}}(g(c_{p}))$

$c_{p+2}=f^{d_{p+1}}(g(c_{p+1}))$

$c_{p+3}=f^{d_{p+2}}(g(c_{p+2}))$

$\vdots$

$c_{n}=f^{d_{n-1}}(g(c_{n-1}))$

$c_{1}=f^{d_{n}}(g(c_{n}))$
where $c_{1}=k$. If there exists a $1<p<n$, such that $c_{p+1}=k$, then the
tuples $(d_{1},\dots,d_{p})$ and $(d_{p+1},\dots,d_{n})$ are also in $S_{k}$.
Let $X=(d_{1},\dots,d_{p})$ and $Y=(d_{p+1},\dots,d_{n})$. Notice that
$D=X\cdot Y=R^{[Y]}(Y\cdot X)$.

Thus, to find atoms in a tuple $D$, we may bitwise rotate that tuple, until
$F_{L}(R^{l}(D))=k$ for $1\leq l<[D]$, where $l$ is the smallest such value,
if that occurs. If it does not occur, then by definition $D$ is an atom. If it
does occur, then we can note that $D$ decomposes into the tuples
$(d_{1},\dots,d_{l})$ and $(d_{l+1},\dots,d_{[D]})$. Since $l$ is the smallest
value such that $F_{L}(R^{l}(D))=k$, then the tuple $(d_{1},\dots,d_{l})$ is
an atom by definition. We may continue this process on the tuple
$(d_{l+1},\dots,d_{[D]})$, decomposing it further into atoms. Thus, we have an
algorithm for decomposing a tuple into a product of atoms over concatenation.
∎

Thus, to investigate properties of satisfying tuples, it is sufficient to
consider only those that are prime. This reduces the number of satisfying
tuples of interest substantially.

##  4 N-Dimensional Objects, representing Cycles

Suppose that we interpreted the $n$-tuple $Y=(y_{1},\dots,y_{n})$ as a
coordinate in $\mathbb{R}^{n}$. By Theorem 2, we know that, if one coordinate
satisfies LCF, then so do all bit wise rotations of it. It may be worth
looking at the object generated by these coordinates. One way to do so is to
connect consecutive points together, creating an $n$-dimensional polygon. Let
$S=(Y,R(Y),\dots,R^{n}(Y))$ be the set of vertices in $\mathbb{R}^{n}$ and
$L=(L_{1},L_{2},\dots,L_{n})$ be the set of edges in $\mathbb{R}^{n}$, such
that $L_{k}$ is the edge from $R^{k-1}(Y)$ to $R^{k}(Y)$, for $1\leq k<n$, and
$L_{n}$ is the edge from $R^{n-1}(Y)$ to $Y$. We denote $|L_{k}|$ to be the
length of edge $L_{k}$. We then observe the trivial observation:

######  Observation 1.

For all $k\in\mathbb{N}_{1}$, where $1\leq k\leq n$, $|L_{k}|$ = $|L_{k+1}|$.

This holds for any n-tuple, regardless of whether it satisfies the LCF or not.
Recall that a cycle is a simple directed circuit in the Loosened Collatz
Graph, and that, in this cycle, $g$ is applied $n$ many times. This is in fact
the length of the associated satisfying tuple, $S$. We can thus denote the
number of times $g$ is applied in a cycle $C$ as $|S|=n$. Also note that, by
Theorem 2, for every vertex that has $g$ applied to it in $C$, it can be
represented as bitwise rotations of a single $n$-tuple. Interpreting these
tuples as coordinates in $\mathbb{R}^{n}$ and the n-dimensional object created
as above, we can provide the following theorem.

######  Theorem 5.

For each and every cycle $C$ in the Collatz Function, there exists a unique
object in $\mathbb{R}^{|S|}$.

###### Proof.

Theorem 2, combined with the interpretation of tuples as coordinates, shows
that every cycle has a unique object. Theorem 3 proves that every cycle can be
represented by a unique tuple, along with its bitwise rotations. Together,
this proves the above theorem. ∎

Thus, we have found a neat one-to-one correspondence between solutions to the
LCF and their geometric counterparts. From this, an equivalent conjecture to
Conjecture 2 states:

######  Conjecture 3.

Excluding the point $(2,2,...)$, all dual objects of satisfying cycles require
their vertices to be 0 on at least one axis.

We also conjecture that this object is invariant under some rotation about the
line $x_{1}=x_{2}=\dots=x_{n}$, where $x_{i}$ corresponds to the $i$th axis.

If either Conjecture 2 or Conjecture 4 are true, then there exist no cycles in
the original Collatz graph aside from the trivial 1,2,4 cycle. Of course,
there is currently no use for this observation, but perhaps those with greater
knowledge and a more varied skill set may use Theorem 5 to investigate the
conjectures.

##  5 Conclusion

We have investigated the Loosened Collatz Graph, derived the LCF and shown how
they relate to tuples satisfying the LCF, which represent circuits. We have
shown the relationship between a tuple and its bitwise rotations and the
satisfaction of a tuple, given its bitwise rotations, and vice versa,
suggesting a possible normal form for such satisfying tuples. We have proven
that the set of tuples which produce the positive integer $k$ in the LCF form
a unique factorisation monoid, which may help aid future researchers into
proving properties of the atoms in the monoid and the structure of tuples in
the set. We have suggested an interpretation of tuples as coordinates, which
may provide geometric insights into the problem, though we currently lack the
knowledge to explore this path. Future researchers may be interested in
proving or disproving the three main conjectures of the paper:

  1. 1.

For all $n\neq 0\text{ (mod 3)}\in\mathbb{N}_{1}$, $n$ is a vertex of a cycle.

  2. 2.

Excluding the tuple $(2,2,...)$, a tuple satisfying the Loosened Collatz
Function must include $0$ in its entry.

## Acknowledgements

We owe thanks to Maiesha Siddika, Owen Mackenzie, Akira Wan, Emils
Bahanovskis, Daniel Espinoza, Tervel Valchanov and others at King’s College
London School of Mathematics for useful discussion at the beginning of
research. We also thank Layo Danbury, Sophia Gregorio, Na Wang, Saanya Verma
and others for their support during our research. We are grateful for the help
Edward Smith’s father provided us with uploading the necessary files to GitHub
and collecting the results for Section 3.2. We are grateful for the services
provided by King’s College London School of Mathematics, as without, we would
not have been able to perform our research.

## References

  * Barina, (2021) Barina, D. (2021).  Convergence verification of the collatz problem.  The Journal of Supercomputing, (77):2681–2688.
  * Böhm and Sontacchi, (1978) Böhm, C. and Sontacchi, G. (1978).  On the existence of cycles of given length in integer sequences like xn+1 = xn/2 if xn even, and xn+1 = 3xn + 1 otherwise.  Proceedings of the Accademia Nazionale dei Lincei. Class of Physical, Mathematical and Natural Sciences, (64):260–264.
  * Chapman, (2014) Chapman, S. T. (2014).  A tale of two monoids: A friendly introduction to nonunique factorizations.  Mathematics Magazine, (87:3):163–173.
  * Petersen, (2014) Petersen, A. (2014).  Unique factorisation in abstract algebra.  Australian Mathematical Sciences Institute.


11institutetext: Dipartimento di Fisica Università di Cagliari and INFN,
Sezione di Cagliari, Italy

# A method to measure the absolute branching fractions of $\Lambda_{c}$ decays

A. Contu  D. Fonnesu  R.G.C. Oldeman  B. Saitta and C. Vacca

(Received: 28 August 2014)

###### Abstract

It is proposed to exploit the decay of the meson $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ and of its charge conjugate $B^{-}$
copiously produced at LHC to obtain a sample of $\Lambda_{c}$ baryons through
the strong decay $\Sigma_{c}\rightarrow\Lambda_{c}\pi$. The sample thus
obtained is not affected by biases typically introduced by selections that
depend on specific decay modes. Therefore it allows a measurement of the
absolute branching fraction for the decay of the $\Lambda_{c}$ baryon into
$pK\pi$ or into other observable final states to be performed in a model
independent manner. The accuracy that can be achieved with this method is
discussed and it is shown that it would be either competitive with or an
improvement over current measurements.

##  1 Introduction

Recently the Belle collaboration Belle  has reported a value of $(6.84\pm
0.24^{+0.21}_{-0.27})\%$ for the absolute branching fraction of the decay
${\mathrm{\Lambda}}_{c}\rightarrow pK\pi$ , obtained from the reconstruction
of the system $D^{*}\bar{p}\pi$ recoiling against the $\Lambda_{c}$ produced
in $e^{+}e^{-}$ annihilation. This measurement is model independent and has a
significantly better precision than earlier results by the CLEO Cleo  and the
ARGUS Argus  collaborations, which were deduced making model dependent
assumptions and are marginally consistent with one another.

In this paper it is suggested to exploit a particular decay of charged $B$
mesons, produced with high yield at LHC, to measure the absolute branching
fraction for the decay $\Lambda_{c}$ into $Kp\pi$ \- or any other observable
decay mode - also in a model independent manner. The proposed method has the
additional advantage of being applicable in a hadron collider environment
since it does not require the reconstruction of the complete event. The paper
is organised as follows. In Section 2 a description is given of the principle
at the basis of the proposed method and in Section 3, by means of a
simulation, the relations imposed by kinematics are exploited. In Section 4
selection efficiencies are evaluated, using the geometrical setup and quoted
performance of the LHCb detector LHCb , to demonstrate the intrisic
feasibility of the proposed measurement. The effects of non-resonant B-decays
into the same final state are also discussed. In Section 5 the accuracy
achievable with current data or with data available in the near future at LHC
is evaluated.

##  2 Principle of $\Lambda_{c}$ reconstruction

Even though the measured branching fraction of the decay
$\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ is only
$2.8\cdot 10^{-4}$ PDG  the abundant production of charged B-mesons at LHC
makes it possible to obtain samples containing O($10^{7}-10^{8}$) decays of
this type. The method proposed in this paper takes advantage of the decay
chain $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\big{(}\overline{\Sigma}_{c}^{--}\rightarrow\overline{\Lambda}_{c}^{-}\pi^{-}\big{)}$
whose diagram is shown in Fig. 1,

Figure 1: Scheme of the decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$.

to measure absolute $\Lambda_{c}$ branching fractions 111Unless otherwise
indicated, charge conjugation is implicitely assumed throughout. . The
principle which the method relies upon is based on the kinematics of the decay
itself and is summarised in the following. Assume that in the decay all
charged particles are observed with the exception of the $\Lambda_{c}$. If the
direction of flight of the $B$-meson is known, it is possible to infer the
existence of the $\Lambda_{c}$ and to determine its momentum 222Up to a
quadratic ambiguity. without observing its decay, thus generating an unbiased
sample of $\Lambda_{c}$’s in which one would search for the mode whose
branching fraction is sought to be measured.

The decay vertex of the $B^{+}$ ($B^{-}$) is identified by the presence of
four charged particles (4-prong decay), namely $p\pi^{+}\pi^{+}\pi^{-}$
($\bar{p}\pi^{-}\pi^{-}\pi^{+}$), having a total charge of $+2$($-2$). The $B$
meson direction of flight is determined from the line joining the production
(primary) and decay vertices.

Furthermore, the pion whose sign of charge is opposite to that of the three
remaining particles certainly originates from the decay of $\Sigma_{c}$
through strong interactions and, in what follows, it will be referred to as
pion from $\Sigma_{c}$, $\pi_{\Sigma}$. In addition the presence of a proton
would serve the purpose of tagging the decays of interest in experiments with
good particle identification.

The B-meson decay vertex will be measured with an accuracy which depends on
the experiment and is separated from the production vertex by a distance which
would depend on the momentum spectrum of the B. The separation between the
production and decay vertex and hence the direction of flight of the B-meson
will be measured with an experiment dependent accuracy as well. These factors
will be taken into account in Section 4.

Let $\hat{e}_{B}$ be a unit vector in the $B$ direction of flight and
$P_{4}=(E_{4},\textbf{p}_{4})$ ($P_{4}^{*}$) be the resultant four-momentum of
all charged particles at the B-decay vertex - not including the $\Lambda_{c}$
\- in the laboratory frame ($B$ rest frame). $P_{3}$ and $P_{3}^{*}$ are the
corresponding quantities of the three like-sign particles at the same vertex.
Let the invariant mass of the two systems be $M_{4}$ and $M_{3}$ respectively
and $\gamma$ be the Lorentz $\gamma$-factor of the decaying $B$-meson.

Assuming that a $\Lambda_{c}$ is the only missing particle in the decay,
through simple algebra, it can be shown that the following two solutions are
obtained for $\gamma$ depending on whether the system of four-particles moves
forward or backward in the $B$ rest frame

$\gamma_{1,2}=\frac{E_{4}\cdot
E_{4}^{*}\mp|\vec{\textbf{p}}_{4}^{L}|\cdot|\vec{\textbf{p}}_{4}^{*L}|}{M_{4}^{2}+|\vec{\textbf{p}}_{4}^{*T}|^{2}}$
(1)

where $\vec{\textbf{p}}_{4}^{T}$ and $\vec{\textbf{p}}_{4}^{L}$ are the
transverse and longitudinal momentum with respect to the B-flight direction
and $E_{4}^{*}$, the energy of the system of four particles in the $B$ rest
frame, is determined by the relation

$E_{4}^{*}=\frac{M_{B}^{2}-M_{\Lambda_{c}}^{2}+M_{4}^{2}}{2M_{B}}$

Hence $E_{B}=\gamma\cdot M_{B}$ and
$\vec{\textbf{P}}_{B}=\sqrt{(E_{B}^{2}-M_{B}^{2})}~{}\hat{e}_{B}$, since the B
flight direction in known. The $\Lambda_{c}$ four-momentum will be determined
by imposing conservation of energy and momentum, $P_{\Lambda}=P_{B}-P_{4}$,
and therefore, if the $\Lambda_{c}$ truly originates from a $\Sigma_{c}$
decay, its momentum would be such that the combination
$(P_{\Lambda}+P_{\pi_{\Sigma}})^{2}$ must be equal to the $\Sigma_{c}$ mass
squared.

This should result in a peaking of the mass distribution around the true value
of the $\Sigma_{c}$ mass when the correct choice for the Lorentz $\gamma$
factor has been made.

In this manner, an unbiased sample of $\Lambda_{c}$ could be selected without
actually observing the decay products of that particle and therefore it would
be sufficient to identify within it the presence of decays into $pK\pi$ to
measure the absolute branching fraction.

##  3 Feasibility of the proposed method

To demonstrate the viability of the proposed method, $pp$ interactions were
generated at centre of mass energy of $14~{}TeV$, using the PYTHIA generator
Pythia . $B^{+}$, produced over the whole solid angle, were forced to decay in
the channel of interest $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ using the software package EVTGEN
EvtGen . Different samples in which the decay was of the non-resonant types
$\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ and
$\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ were also generated
to investigate possible kinematical variables which allow to separate the
different decays.

There are indeed specific experimental advantages in using the suggested decay
chain, some of which can be exploited by detectors with excellent particle
identification, namely:

i) The four-prong decay vertex has charge $+2$. It is therefore relatively
easy to identify and, in real experimental conditions, would help in reducing
background from decays of particles other than the $B^{+}$.

ii) There is a proton at this vertex and therefore it can be efficiently
identified to tag the $B^{+}$ decay. Furthermore, the charge of this proton is
opposite to that of the proton from $\Lambda_{c}$ decays and therefore no bias
is introduced from a specific $\Lambda_{c}$ decay mode.

iii) The pion from the $\Sigma_{c}$ has sign of charge opposite to that of the
other three particles and therefore it can be unambiguously distinguished.

iv) Conditions that events lie within kinematical boundaries can be applied in
the selection to separate decays that occur through the resonance $\Sigma_{c}$
from the non-resonant mode $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$, which has a branching
fraction about 8 times larger, or from $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ . In fact, if the
final state $\bar{\Lambda}_{c}^{-}\pi^{-}p\pi^{+}\pi^{+}$ is reached via the
resonance $\bar{\Sigma}_{c}^{--}$, $M_{4}$ should have values between the
minimum

$(M_{4}^{2})_{min}=\huge{(}E_{3}^{{}^{\prime}}+E_{\pi_{\Sigma}}^{{}^{\prime}})^{2}-(\sqrt{E_{3}^{{}^{\prime}2}-M_{3}^{2}}+\sqrt{E_{\pi_{\Sigma}}^{{}^{\prime}2}-m_{\pi_{\Sigma}}^{2}}\huge{)}^{2}$

and the maximum

$(M_{4}^{2})_{max}=(E_{3}^{{}^{\prime}}+E_{\pi_{\Sigma}}^{{}^{\prime}})^{2}-(\sqrt{E_{3}^{{}^{\prime}2}-M_{3}^{2}}-\sqrt{E_{\pi_{\Sigma}}^{{}^{\prime}2}-m_{\pi_{\Sigma}}^{2}})^{2}$

where $E_{3}^{{}^{\prime}}$ and $E_{\pi_{\Sigma}}^{{}^{\prime}}$ are the
energies, in the $\Sigma$ rest frame, of the system of three-particles and of
the $\pi_{\Sigma}$, respectively.

(a) Non resonant decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$

(b) Non resonant decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$

Figure 2: Distribution of the 4-particle invariant mass squared versus
3-particle invariant mass squared in resonant $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ and non-resonant
$\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ and
$\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ decays. The dashed
line represents the kinematics boundaries for the resonant decay, while within
the contour lines are contained the indicated fractions of non-resonant
decays.

Fig.2 shows the kinematic boundaries defined above in the Dalitz-plane
($M^{2}_{4}-M^{2}_{3}$). Decays in which a $\Sigma_{c}$ is present fall within
these boundaries and are shown by the shaded area, while within the contour
lines are contained the indicated fractions of non-resonant decays
$\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ (a)
and $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ (b). The fraction of
such decays within the kinematic boundaries is $(21.98\pm 0.02)\%$ for
$\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ and
$(50.46\pm 0.03)\%$ for the $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ mode. In Section 4 a
quantitative estimate of the contribution from these decay modes will be
given, once further selections have been applied. At this stage it is
sufficient to observe that requiring a minimum value of $M_{3}^{2}$ and
$M_{4}^{2}$ would be effective in reducing the fraction of non resonant
decays, in particular of $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ whose branching
fraction is not measured at present and only an upper limit exists PDG .

Detector acceptance was simulated in a simple manner, by assuming that
particles with momentum greater than $2~{}GeV/c$ and within the pseudo-
rapidity range $2<\eta<4.5$ would be detectable, as it would be approximately
the case in the LHCb experiment, whose simplified setup will be used to
estimate efficiencies. In the decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\big{(}\overline{\Sigma}_{c}^{--}\rightarrow\overline{\Lambda}_{c}^{-}\pi^{-}\big{)}$
, the spatial distributions of particles are generated according to phase
space, therefore, at generator level, there is no preferential direction for
the $\Lambda_{c}$ in the $B$ rest frame. However, requiring that the decay
products - other than the $\Lambda_{c}$ \- be within a detector acceptance,
introduces asymmetries of order of $20\%$. This is shown in Fig. 3, where the
distributions of the cosine of the angle between the direction of the momentum
of the $\Lambda_{c}$ in the $B$ rest frame and the $B$ direction of flight in
various situations are compared.

There are indeed also losses introduced by this selection and these will be
included in the efficiencies discussed in Section 4. At this stage it is
sufficient to observe that requiring that the three like-sign particles be in
acceptance favours slightly backward-going $\Lambda_{c}$’s, while the further
requirement that the pion from $\Sigma_{c}$ be measurable, i.e. within the
geometrical acceptance as well, would preferentially select forward-going
$\Lambda_{c}$, as shown by the dashed lines in Fig. 3.

Figure 3: Angular distributions of the $\Lambda_{c}$ baryon in the $B$ rest
frame before and after applying selection criteria

This selection of different regions of phase space will have an effect on the
preferred value for the Lorentz $\gamma$ factor. In fact, with the definition
given in Eq. 1, $\gamma_{1}$ is always smaller than $\gamma_{2}$ and therefore
chosing the first over the latter selects $B$’s of lower energy. If for
instance experimental acceptance is such that forward going $\Lambda_{c}$ are
slightly favoured, then $\gamma_{2}$ would be more often the correct solution.
On the other hand, additional selections - that might prove necessary when
dealing with the real experimental conditions - are likely to change the
favoured value. However this would have no effect on the proposed measurement,
as long as no bias is introduced by the decay mode of the $\Lambda_{c}$, which
is the case here since the $\Lambda_{c}$ decay products are not and will not
be considered in the selection.

For each event, the two values of the Lorentz $\gamma$ factor given by Eq. 1
are computed using true quantities at generator level.
$\sqrt{(P_{\Lambda}+P_{\pi_{\Sigma}})^{2}}$ is then computed assuming, as
solution, either $\gamma_{1}$ or $\gamma_{2}$. The mass distribution obtained
in this manner is shown in Fig. 4 as solid histogram. As already mentioned
there is complete symmetry forward-backward at this stage and therefore
chosing either solution, $\gamma_{1}$ or $\gamma_{2}$, leads to the same
result. As expected, a peak around the $\Sigma_{c}$ mass is observed. Its
width is affected by the cases in which the wrong solution for $\gamma$ was
chosen, yet the result was close to its true value and, as a consequence, a
value about $10\%$ larger than expected is observed 333At this level, if the
correct solution were chosen, one would expect to obtain a width of
$2.6~{}MeV/c^{2}$ for the $\Sigma_{c}$, which is the value coded in the
Montecarlo. .

Figure 4: Distributions of the reconstructed $\bar{\Sigma}_{c}^{--}$ mass
before and after applying selection criteria.

Using either value of $\gamma$ and requiring
$2.44~{}GeV/c^{2}<M_{\Sigma}<2.47~{}GeV/c^{2}$, would allow a selection of a
sample of $\Lambda_{c}$ whose size is about 64% of the original sample of
generated events. On the other hand chosing particles that fall within the
geometrical acceptance would produce different mass distributions for the two
choices of $\gamma$, as illustrated in Fig. 4 with slightly different
efficiencies which will be discussed quantitatively in Section 4.

Even though the choice of $\gamma_{1}$ or of $\gamma_{2}$ is irrelevant since
either one would allow the selection of an unbiased sample of $\Lambda_{c}$’s,
efficiencies are different and this would affect the size of the final sample
and ultimately the statistical accuracy. To minimize the statistical error on
the measurement, it would be required in addition that the decay products of
the $\Lambda_{c}$ be detected and hence it is the number of observable decays
that ought to be maximized.

##  4 Effects of experimental resolution and efficiencies

Experimental resolution on the determination of momenta of particles and of
positions of vertices so far have not been taken into account. To estimate its
effect, a simplified geometrical setup of the LHCb experiment was used in the
Montecarlo simulation. Therefore the momentum-dependent smearing quoted in
ref. LHCbP  (which corresponds to $\delta P/P\sim 0.4\%$ for a particle with
$P=10~{}GeV/c$) was applied to particle momenta and gaussian smearings of
$\sigma_{z}=400\mu m$ along the beam direction and of $\sigma_{T}=35\mu m$ in
the transverse beam direction were applied to the position of the $B$ decay
vertex LHCbVX . It was found that, of the two, the error on the vertex
position has the largest effect since it enters in the determination of the
$B$ direction of flight. The effect is shown in Fig. 5 where the
$M_{\Sigma_{c}}$ distribution is displayed for the $\gamma_{2}$ solution and
all four-particles within the geometrical acceptance, assuming the true
$B$-direction or that obtained having applied only the smearing on the
$B$-decay vertex. The result is essentially unchanged when the momentum
smearing is also applied. The effect of the experimental resolution is to
reduce the event sample to about $45\%$ of its original size and it is mostly
due to the fact that the transverse momentum relative to the measured
B-direction exceeds the maximum value allowed by kinematics and hence no
acceptable solutions for $\gamma$ are found.

Figure 5: Distributions of the reconstructed $\bar{\Sigma}_{c}^{--}$ mass
before and after applying resolution smearing.

Tables 1 and 2 summarise the effects of the selections listed below and
applied to the $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ sample as well as to the non-
resonant samples, i.e. to the samples of events in which the final state
$\Lambda_{c}\pi p\pi\pi$, with or without the presence of an additional
neutral pion, is reached directly from B decay and not via the resonance
$\Sigma_{c}$ for the two possible choices of $\gamma$:

  * •

Selection A: $(M_{4}^{2})_{min}<M_{4}^{2}<(M_{4}^{2})_{max}$

  * •

Selection B: Mass of the $\Sigma_{c}$, computed as
$\sqrt{(P_{\Lambda}+P_{\pi_{\Sigma}})^{2}}$, is within $\pm 15~{}MeV/c^{2}$
from its nominal value of $2.455~{}GeV/c^{2}$.

  * •

Selection C: $p\pi\pi$ like-sign within geometrical acceptance.

  * •

Selection D: $\pi_{\Sigma}$ within geometrical acceptance.

  * •

Selection E: Smearing is applied to B-decay vertex.

  * •

Selection F: $M_{3}^{2}>4(GeV/c^{2})^{2}$ and $M_{4}^{2}>6(GeV/c^{2})^{2}$

The criteria were applied in order, i.e. each selection implies that all the
preceeding conditions were satisfied.

Table 1: Accepted solution $\gamma_{1}$ \- Fraction of events kept Condition | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$
---|---|---|---
Selection A | $1$ | $(21.98\pm 0.02)\cdot 10^{-2}$ | $(50.46\pm 0.03)\cdot 10^{-2}$
Selection B | $(62.19\pm 0.03)\cdot 10^{-2}$ | $(1.986\pm 0.005)\cdot 10^{-2}$ | $(2.493\pm 0.006)\cdot 10^{-2}$
Selection C | $(6.501\pm 0.009)\cdot 10^{-2}$ | $(2.00\pm 0.02)\cdot 10^{-3}$ | $(2.39\pm 0.02)\cdot 10^{-3}$
Selection D | $(2.554\pm 0.006)\cdot 10^{-2}$ | $(6.89\pm 0.09)\cdot 10^{-4}$ | $(3.03\pm 0.06)\cdot 10^{-4}$
Selection E | $(1.055\pm 0.004)\cdot 10^{-2}$ | $(3.79\pm 0.07)\cdot 10^{-4}$ | $(2.76\pm 0.06)\cdot 10^{-4}$
Selection F | $(8.52\pm 0.03)\cdot 10^{-3}$ | $(2.96\pm 0.06)\cdot 10^{-4}$ | $(1.07\pm 0.04)\cdot 10^{-4}$
Table 2: Accepted solution $\gamma_{2}$ \- Fraction of events kept Condition | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$
---|---|---|---
Selection A | $1$ | $(21.98\pm 0.02)\cdot 10^{-2}$ | $(50.46\pm 0.03)\cdot 10^{-2}$
Selection B | $(64.02\pm 0.04)\cdot 10^{-2}$ | $(2.014\pm 0.005)\cdot 10^{-2}$ | $(2.581\pm 0.006)\cdot 10^{-2}$
Selection C | $(5.784\pm 0.009)\cdot 10^{-2}$ | $(1.94\pm 0.02)\cdot 10^{-3}$ | $(2.17\pm 0.02)\cdot 10^{-3}$
Selection D | $(3.403\pm 0.007)\cdot 10^{-2}$ | $(1.29\pm 0.01)\cdot 10^{-3}$ | $(1.79\pm 0.02)\cdot 10^{-3}$
Selection E | $(1.536\pm 0.004)\cdot 10^{-2}$ | $(8.3\pm 0.1)\cdot 10^{-4}$ | $(1.43\pm 0.01)\cdot 10^{-3}$
Selection F | $(1.117\pm 0.004)\cdot 10^{-2}$ | $(5.62\pm 0.08)\cdot 10^{-4}$ | $(3.49\pm 0.07)\cdot 10^{-4}$

From the tables it can be concluded that efficiencies at the percent level for
the indicated selections are obtained and that, as expected, $\gamma_{2}$
would be the favoured solution when requiring that $\pi_{\Sigma}$ be within
geometrical acceptance (i.e. having pseudo-rapidity in the range $2<\eta<4.5$,
as previously defined).

As it was also expected, the requirements imposed on kinematics and detector
geometry are less effective for the non-resonant channels, the efficiencies
being smaller by a factor of about 20. This reduction will be partly
compensated for by the larger B-decay branching fraction into these channels.
Quantitatively this is shown in Fig. 6 where the $\Sigma_{c}$ mass
distribution is displayed for the resonant and non-resonant channels properly
weighted with the B-decay branching fractions and the efficiencies quoted in
Table 2 for the selection A. For the decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ , since only an upper
limit exists for the branching fraction, it was assumed to be equal to that of
$\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$. In
the mass range $2.44-2.47~{}GeV/c^{2}$,considered as the signal region, the
total non-resonant fraction, with the above assumption, is $\sim 36\%$ of the
total. In real experimental conditions this signal would be superimposed to a
combinatorial background, therefore it would not be meaningful at this stage
to extract a function describing its shape.

Figure 6: Distributions of the reconstructed $\bar{\Sigma}_{c}^{--}$ mass in
resonant ($\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$)
and non-resonant decays ($\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ and $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ ).

As already pointed out, it is the number of observable decays that should be
maximised. Table 3 shows, for the decay ${\mathrm{\Lambda}}_{c}\rightarrow
pK\pi$, the fraction of events, satisfying the indicated selections, in which
the decay products of the $\Lambda_{c}$ are also within geometrical
acceptance, both for the resonant and non-resonant components. Efficiencies
are of order of $\sim 40\%$ irrespective of the choice of $\gamma$ and depend
weakly on whether the $\Lambda_{c}$ is produced directly from B decay or
through the $\Sigma_{c}$ resonance. This was to be expected, since essentially
only geometrical factors enter in the determination of these fractions.

Table 3: Fraction of events for the indicated selections in which the $\Lambda_{c}$ decay products are also within geometrical acceptance Condition | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$ | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ | $\mbox{B}^{+}\rightarrow p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$
---|---|---|---
Selection D |  |  |
and Solution $\gamma_{1}$ | $39.82\pm 0.12$ | $42.6\pm 0.8$ | $40.9\pm 1.1$
Selection E |  |  |
and Solution $\gamma_{1}$ | $39.17\pm 0.18$ | $40.7\pm 1.0$ | $41.5\pm 1.1$
Selection F |  |  |
and Solution $\gamma_{1}$ | $39.16\pm 0.2$ | $41.4\pm 1.1$ | $42.3\pm 1.9$
Selection D |  |  |
and Solution $\gamma_{2}$ | $40.54\pm 0.10$ | $41.4\pm 0.5$ | $39.5\pm 0.4$
Selection E |  |  |
and Solution $\gamma_{2}$ | $40.03\pm 0.15$ | $40.3\pm 0.6$ | $39.8\pm 0.5$
Selection F |  |  |
and Solution $\gamma_{2}$ | $40.04\pm 0.18$ | $41.9\pm 0.8$ | $41.8\pm 1.0$

##  5 Results and Conclusions

LHCb has measured a cross section of $38.9~{}\mu b$ for the production of
charged B mesons within the experimental acceptance and with transverse
momentum in the range $0-40~{}GeV/c$ LHCb_xsec , in $pp$ collisions at centre
of mass energy of $7~{}TeV$. This would correspond to a cross section of
$~{}193\mu b$ over the whole solid angle, assuming that B-mesons are produced
as in PYTHIA generator. Scaling the production cross-section with $\sqrt{s}$,
this would yield about $3.9\cdot 10^{11}$ charged B-decays per $fb^{-1}$ of
integrated luminosity.

Selecting, for the sake of illustration, the efficiencies corresponding to the
$\gamma_{2}$ solution in Table 2 and assuming that these do not have a strong
dependence on the pp centre-of-mass energy, using an integrated luminosity of
$3~{}fb^{-1}$ (currently available in the LHCb experiment at centre of mass
energies of $7$ and $8~{}TeV$), with the measured branching fraction of
$2.8\cdot 10^{-4}$ for $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\overline{\Sigma}_{c}^{--}$, about $2.5$ million decays of
interest would be reconstructed within the detector: a sizeable, unbiased
sample of $\Lambda_{c}$ decays.

However, to obtain a more realistic estimate of the above number, the effects
of other selections which is necessary to apply and have not been considered
here, should be included. These in general are functions of transverse
momentum and pseudo-rapidity and efficiencies have typically the values quoted
in Ref. LHCb_xsec1 . Assigning the realistic value of $2\%$ to account for all
the effects not included in the simplified simulation, using the efficiencies
of Table 3 for the $\Lambda_{c}$ decay products, about 1000 decays of the type
${\mathrm{\Lambda}}_{c}\rightarrow pK\pi$ from the resonant sample would be
observed in the detector. The statistical error therefore would be comparable
with that of Belle Belle , which is the most precise currently available
measurement.

The presence of large backgrounds could spoil the effectiveness of the
proposed method. These can be estimated only through a detailed and complete
simulation using a specific detector and it is beyond the scope of this
article. However it should be observed that the choice was made - over other
topologies which would have enlarged the sample - of a $B$-decay vertex of
charge $\pm 2$, with a well defined topology, separated from the primary $pp$
interaction vertex, and containing a well identified proton, with the intent
of reducing combinatorial background. Backgrounds from specific decays of
B-mesons have been considered. In particular, as seen in Tables 1 and 2 and in
Fig. 6, non-resonant decays $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$ and $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ , contribute less
than $40\%$ to the total number. The decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}\pi^{0}$ with a missing
neutral pion may be considered representative of the class of decays with
topology identical to that of interest when one or more particles are missing.
The validity of this statement was verified by considering in the simulation
also the decay $\mbox{B}^{\circ}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\pi^{-}\overline{\Lambda}_{c}^{-}$, which would mimic
the signal when missing a $\pi^{-}$. A fraction of about $3.7~{}10^{-4}$ of
events of this type were found for selection F, in the mass region of
interest. As expected, the retained fraction is similar to that accepted for
the decay $\mbox{B}^{+}\rightarrow
p\pi^{+}\pi^{+}\pi^{-}\overline{\Lambda}_{c}^{-}$. The cross section for
$B^{\circ}$ production is similar to that for charged B’s LHCb_xsec . However
for the branching fraction of the decay considered here only an upper limit
exists and therefore it would contribute to background at most $\sim 30\%$ of
the resonant signal.

Systematic effects are experiment dependent as well and therefore can not be
properly estimated in this paper. However it is worth noticing that the method
relies on counting the observed number of ${\mathrm{\Lambda}}_{c}\rightarrow
pK\pi$ decays in a sample whose selection does not rely upon observation of
the $\Lambda_{c}$ decay products. Therefore most of the systematics would
cancel when taking the ratio. The line shape used in the fit to determine the
size of the initial, unbiased sample would be taken from Montecarlo
simulation. The effects of the uncertainty on this shape would not cancel
however and would most likely be the main source of systematic error. Particle
identification (proton in particular) would be used to identify the
$\Lambda_{c}$ decay mode of interest. This would affect only the detection
efficiency listed in Table 3 and therefore it would not cancel in the ratio
and would become important if statistics were limited.

The overall result could be improved by devising dedicated, more efficient
selections at trigger level and when more data and at higher centre of mass
energy become available. Furthermore the decays from $\Lambda_{c}$’s
originating from the non-resonant channels could be added to the sample, since
their detection efficiency is similar to that for $\Lambda_{c}$ coming from
the resonant channel and therefore large corrections would not be required.

## References

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# Reliability Evaluation of Phasor Measurement Unit Considering Failure of
Hardware and Software Using Fuzzy Approach

Evan Carollo and Zikai Xu

###### Abstract

The wide-area measurement system (WAMS) consists of the future power system,
increasing geographical sprawl which is linked by the Phasor measurement
unit(PMU). Thus, the failure of PMU will cause severe results, such as a
blackout of the power system. In this paper, the reliability model of PMU is
considered both hardware and software, where it gives a characteristic of
correlated failure of hardware and software. Markov process is applied to
model PMU, and reliability parameters are given by using symmetrical
triangular membership for Type-1 fuzzy reliability analysis. The paper gives
insightful results revealing the effective approach for analyzing the
reliability of PMU, under a circumstance which lack of sufficient field data.

##  I INTRODUCTION

”The phenomenal development of modern power systems in terms of both
geographical sprawl and technological innovations has emerged as one of the
most complex engineering systems in existence. A modern power system involves
thousands of components for generation, transmission, and distribution before
reaching the consumer. To consistently provide power to residential and
commercial customers around the clock, the generation, transmission, and
distribution systems must have proper coordination. Therefore, it is very
important to periodically monitor the condition or “health” of the system. A
wide-area measurement system (WAMS) is a powerful tool for monitoring the
health of a power system. Phasor measurement units (PMUs) are key components
of a WAMS, providing precise and real-time grid measurements that are time-
stamped according to a common time reference. The synchronized phasor
measurements can be used for improving the reliability of the system.”

”Reliability modeling of PMU has received ample research interest owed to its
critical role in delivering WAMS services. Unfortunately these models are
restricted in terms of possible failure mode of PMU. In fact, none of them
accommodate hardware–software interaction failures in modeling PMU
reliability, which has been an area of research interest, partly because of
its recent identification and partly because of consequences that such
failures have led to some of the contemporary safety critical systems. In
recent years, however, there has been a lot of emphasis on identifying,
modeling, and quantifying the effect of hardware– software interaction
failures on system reliability in myriad cases including integrated circuit
(IC) fabrication technologies, jet propulsion systems and so on. So far, all
the PMU models developed have assumed the central processing unit (CPU) module
as a combined hardware–software unit similar to a digital signal processor or
a microprocessor on which software executes for achieving the intended
functions. This assumption implies that hardware and software are independent
and that the PMU system fails if the hardware or software subsystems fail
alone. Based on this assumption, the software subsystem has been considered to
be in series with the hardware subsystem from a reliability perspective [3].
This assumption imposes a serious limitation for systems, particularly for
embedded systems, where hardware and software work in close proximity with
each other. PMU is an embedded device, and thus its hardware and software
subsystems bear close correlation for proper functioning. In fact, this
interaction between hardware and software subsystems is present throughout the
PMU system, not just in the CPU model. For example, high-precision time clock
signal productions necessitate the inclusion of a high-precision crystal
oscillator within the GPS module of the PMU [2]. The crystal oscillator can
switch between track mode and replacement mode and this switching is a
functional switching as opposed to a physical switching governed by software
logic. It has to be noted that such software controlled functionalities have
not been accounted for in the existing reliability models, since software
subsystem has been accounted only as a part of the CPU module. Similarly,
communication module functions based on the software protocol stacks
implemented and software bugs therein, if any, have to account for is the
overall PMU reliability model. This has encouraged us to analyse the
reliability of PMUs considering the interactions between hardware and
software.”

”The reliability analysis techniques discussed in [2]–[4] do not consider any
data uncertainty in the reliability parameters when evaluating the reliability
of a PMU. Because PMUs have been introduced only recently, statistical data on
the operation of PMUs is very sparse. The uncertainty in sparse statistical
data consequently leads to difficulty in estimating the reliability parameters
of PMU components. Therefore, rather than attempting to estimate a single
value, it is more appropriate to estimate the range of a reliability parameter
to account for uncertainties. Because of the scarcity of statistical data,
practical knowledge and engineering decisions might be used in estimating the
ranges of the reliability parameters. Generally, practical knowledge and
engineering decisions are characterized by fuzzy linguistic descriptions. The
advantage of fuzzy logic is that the representation of knowledge is very
explicit using “IF–THEN” relations.”

The main contribution of the paper is quantitative reliability evaluation of
PMU using type-1 fuzzy sets. Here, a state Markov model has been taken into
consideration which includes the power supply module. Power supply module is
an important component as it supplies power to the PMU. The reliability
parameters are calculated using the fuzzy set symmetrical triangular
membership function. This simplifies the calculations. Moreover, type-1 fuzzy
set handles the data uncertainty. Here, the data uncertainty has been
considered for all the modules for better analysis of PMU reliability.

##  II Phasor Measurement Unit

Figure 1:

The block diagram of Phasor Measurement Unit (PMU) is given in the Fig.1. One
can observe that all the modules of PMU are connected in a series where one
module of failure will cause failure of the system. The current
transformer/potential transformer(CT/PT) model(M1) convert the analogue three-
phase voltages and currents to the safer level. In the next level, these
signal are passed through Anti-aliasing filtering module(M2) to filter out
high-frequency noises, which are subsequently converted to digital signals by
A/D converting Module(M3). The CPU module (M5) obtain the magnitude, phase and
power frequency of the signal processed by M3. The GPS module(M4) attach
highly accurate time synchronisation signal to the measurements. Finally,
Communication module (M6) transmit data to other terminals to further process
PMU data. The power supply module (M0) provides power for all modules in the
system.

Here one can observe that all the components of PMU models consider only
hardware parts; however, the effects of software causing hardware failure or
hardware causing software failure are ignored. In fact, most of research
assume that hardware and software are independent, and the only interaction
between hardware and software are considered in CPU module. Although an
embedded system such as PMU, each module are connection in series and all
controls flow from one module to the next module; the real control behind this
is software. The purpose of this project is to improve the accuracy of
reliability model of PMU considered software and hardware interactions. This
paper provides a unified software and hardware PMU model other than
traditional PMU reliability analysis only focusing on either hardware or
software.

Figure 2:

##  III Unified PMU model

###  III-A Introduction of Model

The scheme of the Unified PMU model is given on Fig.2 where it demonstrate the
Markov model and how hardware and software interact with each other. In the
unified PMU model, there are total 7 states. Each different state can depict
hardware failure, software failure or hardware-software interaction failures
in a PMU. Here, we explain each state’s representations in our unified PMU
model. State 1 stands for UP state of the PMU where PMU is health and working
functionally. From state 1, the system can jump into either software
degradation(SD) state or partial hardware degradation (HD) state. First, we
introduce HD state. There are three possible HD states denoted as HD1, HD2 and
HD3.

  * •

HD1: partial degradation may be detected but may not recover.

  * •

HD2: degradation is detected and recovered by software.

  * •

HD3: degradation is undetected and go faliure

Any HD state can further lead the total failure of the PMU system (TF), where
the last rectangular is shown the TF. At HD3 state, undetected hardware
degradation dose no given any signal to PMU system; and transition from HD3 to
state 4 represents the hardware-software interaction falures. State 5
represents the software degradation(SD). Errors in the SD may accumulate and
finally cause the total failure of the system (TF).

###  III-B Reliability Analysis of The Model

The pre-assumption of the model is a Markov process, where the future state is
independent on past states. Thus, transition paths: 1 to HD1/HD2 to 3; 1 to
HD3 to 4; 1 to SD to 6 are independent to each other. Thus, we can write the
PMU reliability as

$R_{PMU}(t)=R_{HW}(t)R_{SW}R_{HW\&SW}(t).$ (1)

Hardware reliability can be represented by Weibull model

$R_{HW}(t)=e^{-\lambda t^{\beta}},$ (2)

where $\lambda$ is the failure rate of hardware components and $\beta$ is the
shape parameter of the Weibull distribution function.

Software reliability can be definded by non-homogenous process model with mean
time function $m(t)$. If the software startup time T, the software reliability
is

$R_{SW}(t)=e^{(-m(t+T)-m(T))},$ (3)

where t is the testing time and $a$, $b$ are number of faults to be detected
and fault detection rate, respectively.

According [], we can find the probability of hardware-software working at time
$t$ as

$R_{HW\&SW}(t)=Q_{1}(t)+Q_{HD1}(t)+Q_{HD2}(t)+Q_{HD3}(t),$ (4)

where $Q_{1}(t)$ is probability at state 1 at time t. Similarly, $Q_{HD1}(t)$
is the probability at state HD1, $Q_{HD2}(t)$ is the probability at state HD2,
$Q_{HD3}(t)$ is the probability at state HD3. Therefore, we can express the
overall system reliability as

$\displaystyle R_{PMU}(t)$ $\displaystyle=R_{HW}(t)R_{SW}R_{HW\&SW}(t)$ (5)
$\displaystyle=e^{-\lambda t^{\beta}}e^{(-m(t+T)-m(T))}$ (6)
$\displaystyle\times(Q_{1}(t)+Q_{HD1}(t)+Q_{HD2}(t)+Q_{HD3}(t)).$ (7)

Figure 3:

Thus, (7) gives the hardware-software interaction failures. Given PMU failure
data, one can find hardware-software interaction failure; however, PMUs are
launched recently and failure data are limited. Also, we cannot have accurate
reliability index for PMUs. Thus, we use fuzzy approach to obtain reliability
index of PMU, and then we apply Markov simulation to find failure data of PMU.

##  IV  Fuzzy Approach

Reliability parameters are the key factor to analyze PMU; however, failure
data is not sufficient to help us to accurately analyze PMU, causing
uncertainty in reliability parameters of PMU components. Thus, instead of
giving one single value of reliability parameter, a range of reliability
parameter is much more reasonable and accurate results. Here, we use Type-1
fuzzy sets to handle the uncertainty of the data. According to [], we can have
the following steps to determine the Type-1 fuzzy reliability of PMU:

  * (1)

To determine the input data. Here, input data is that the failure and repairer
rates are in a symmetrical triangular membership functions.

  * (2)

$\alpha$-cut sets of input data are established for $\alpha=[0,1]$

  * (3)

The parameters of the equivalent reliability model for any $\alpha$ are
calculated.

  * (4)

$\alpha$-cut are used to calculate fuzzy outputs.

The availability and unavailability of PMU is given in the Fig.3(a) and
Fig.3(b); and Fig.3(c) and Fig.5(d) give a range of repair rate and failure
rate of PMU.

Figure 4:  Figure 5:

Finally, we choose failure rate as 0.6566 F/year and repair rate as 22.2898
Hs/year.

##  V  PMU Failure Data from Simulation

In this project, we apply Markov simulation to estimate failure data of PMU.
First of all, based on failure rate and repair rate given in the previous
section, we use simulation to generate random number for PMU time to failure
and PMU time to repair. Then, based on the time to failure and number of
failure to find hardware and software interaction failure reliability. The
detail steps of simulation as followings and the flow chart of the simulation
is given on Fig.4.

  * (1)

Determine the mission time, we set mission time (TM) 10 years. Total number of
simulation as 10000.

  * (2)

Initialize number of simulation N as 0.

  * (3)

Number of failure and counter set to 0

  * (4)

Check whether counter is less than mission time. If condition is true, another
simulation is applied until condition reach false. Use random number to
generate failure time.

  * (5)

Number of failure and failure time are accumulated. Repair time also
calculate.

  * (6)

Add repair time to counter .

  * (7)

Repeat step 4,5,6 if counter¡ mission time.

  * (8)

Based on the simulation results to obtain reliability indices.

##  VI  Results and Analysis

This section gives results of PMU reliability analysis based on the model
proposed before.

###  VI-A Fitting the PMU failure data to the model

As we discussed in the previous section, we have 6 parts in the system. That
the path from HD3 to 4 is undetected hardware failures leading hardware
related software failures. Assume that all failure rate are considered as 0
except $\lambda_{1}$ and $\lambda_{2}$, we can reduce reliability hardware and
software interaction as

$R_{HW\&SW}(t)=\frac{\lambda_{2}e^{-\lambda_{1}t}-\lambda_{1}e^{-\lambda_{2}t}}{\lambda_{2}-\lambda_{1}}.$
(8)

The estimated value of $(\lambda_{1},\lambda_{2})$ can be calculated by a non-
linear sum of square estimates

$SSE(\lambda_{1},\lambda_{2})=\sum_{i=1}^{8}[X_{i}-\frac{T_{i}}{\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}}],$
(9)

where $X_{i}$ is the aggregate exposure number, and $T_{i}$ is the aggregate
exposure time. As $(\lambda_{1},\lambda_{2}$ is not possible to estimate
uniquely. Thus, we can rewrite$SSE(\lambda_{1},\lambda_{2})$ as
$SSE(G,\lambda_{2})$, and $G=\lambda_{2}/\lambda_{1}$. Therefore the least-
square estimates of $\lambda_{1}$ is

$\lambda_{1}=\frac{(1+1/G)(\sum_{i=1}^{8}X_{i}T_{i})}{\sum_{i=1}^{8}T^{2}},$
(10)

where $\lambda_{2}=G\lambda_{1}$.

###  VI-B Computation Simulation Result

The MATLAB simulation is given in the Fig.5. If we choose $G=2$, we can solve
$\lambda_{1}=8.92\times 10^{-4}$,$\lambda_{2}=3.92\times 10^{-3}$.

##  VII Conclusion

In this paper, we first proposed unified hardware and software PMU. By
applying fuzzy approach, we determine the reliability parameters. We use
MATLAB to run simulation to do the parameter fitting, and finally we solve
failure of hardware and software interaction failure.

Figure 6:

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