# On non-Newtonian Helices in Multiplicative Euclidean Space
$\mathbb{E}_{*}^{3}$

Aykut HAS$,^{1*}$ Beyhan YILMAZ,2
1,2Department of Mathematics, Faculty of Science, Kahramanmaras Sutcu Imam
University, Kahramanmaras, Turkey 111 Corresponding Author: Aykut HAS e-mail:
<EMAIL_ADDRESS>
Contributing Authors:Beyhan YILMAZ<EMAIL_ADDRESS>Muhittin
Evren AYDIN<EMAIL_ADDRESS>0000-0000-0000-0000
0000-0000-0000-0000

###### Abstract

In this article, spherical indicatrices of a curve and helices are re-examined
using both the algebraic structure and the geometric structure of non-
Newtonian (multiplicative) Euclidean space. Indicatrices of a multiplicative
curve on the multiplicative sphere in multiplicative space are obtained. In
addition, multiplicative general helix, multiplicative slant helix and
multiplicative clad and multiplicative g-clad helix characterizations are
provided. Finally, examples and drawings are given.

###### keywords:

Non-Newtonian calculus , spherical indicatrices , helices , multiplicative
differential geometry.

###### MSC:

53A04 , 11U10 , 08A05.

## 1 Introduction

Classical analysis, which is a widely used mathematical theory today, was
defined by Gottfried Leibniz and Isaac Newton in the second half of the 17th
century based on the concepts of derivatives and integrals. Constructed upon
algebra, trigonometry, and analytic geometry, the classical analysis consists
of concepts such as limits, derivatives, integrals, and series. These concepts
are regarded as simple versions of addition and subtraction, leading to the
designation of this analysis as summational analysis. Classical analysis finds
applications in various fields, including natural sciences, computer science,
statistics, engineering, economics, business, and medicine, where mathematical
modeling is required, and optimal solution methods are sought. However, there
are situations in some mathematical models where classical analysis falls
short. Therefore, alternative analyses have been defined based on different
arithmetic operations while building upon classical analysis. For instance, in
1887, Volterra V. developed an approach known as Volterra-type analysis or
multiplicative analysis since it is founded on the multiplication operation
[1]. In multiplicative analysis, the roles of addition and subtraction
operations in classical analysis are assumed by the multiplication and
division operations, respectively. Following the definition of Volterra
analysis, Grossman M. and Katz R. conducted some new studies between 1972 and
1983. This led to the development of the non-Newtonian analysis, which also
involves fundamental definitions and concepts [2, 3]. These analyses have been
referred to as geometric analysis, bigeometric analysis, and anageometric
analysis. Multiplicative analysis has emerged as an alternative approach to
classical analysis and has become a significant area of research and
development in the field of mathematics. These new analyses may allow for a
more effective resolution of various problems by examining different
mathematical structures. Furthermore, these studies contribute to the
expansion of the boundaries of mathematical analysis and find applications in
various disciplines.

Arithmetic is an integer field which is a subset of the real numbers. An
arithmetic system is the structure obtained by algebraic operations defined in
this field. In fact, this field can be considered as a different
interpretation of the real number field such that a countable number of
infinitely ordered objects can be formed and these structures are equivalent
or isomorphic to each other. The generator function, which is used to create
arithmetic systems, is a one-to-one and bijective transformation whose domain
is real numbers and whose value set is a subset of positive real numbers. The
unit function $I$ and the function $e^{x}$ are examples of generator
functions. Just as each generator produces a single arithmetic, each
arithmetic can be produced with the help of a single generator. Multiplicative
analysis has its own multiplicative space. In this special space, the
classical number system has turned into a multiplicative number system
consisting of positive real numbers, denoted by $\mathbb{R}_{*}$. Likewise,
the basic mathematical operations in classical analysis have also turned into
their purely multiplicative versions. This is clearly shown in the table
below.

$a+_{*}b$ | $e^{\log a+\log b}$ | $ab$
---|---|---
$a-_{*}b$ | $e^{\log a-\log b}$ | $\frac{a}{b}$
$a\cdot_{*}b$ | $e^{\log a\log b}$ | $a^{\log b}$
$a/_{*}b$ | $e^{\log a/\log b}$ | $a^{\frac{1}{\log b}},~{}b\neq 1$
(1)

Table 1. Basic multiplicative operations.

Multiplicative analysis, contrast to not a completely new topic, has recently
started to be explored and discovered more in today’s context. The main reason
behind this lies in the successful modeling of problems that cannot be
addressed using classical analysis, achieved through the application of
multiplicative analysis. This characteristic has led many mathematicians to
prefer multiplicative analysis for solving challenging problems that are
otherwise difficult to model within their respective fields. Stanley D. took
the lead in this regard and re-announced geometric analysis as multiplicative
analysis [4]. On this subject, fractal growths of fatigue defects in materials
are studied by Rybaczuk M. and Stoppel P. [5] and the physical and fractional
dimension concepts are studied by Rybaczuk M. and Zielinski W. [6]. In
addition, there are many studies on multiplicative analysis in the field of
pure mathematics. For example, the non-Newtonian efforts in complex analysis
are [7, 8], in numerical analysis [9, 10, 11], in differential equations [12,
13, 14]. Also, Bashirov et al. reconsider multiplicative analysis with some
basic definitions, theorems, propositions, properties and examples [15]. The
multiplicative Dirac system and multiplicative time scale are studied by Y.
Emrah et al. [16, 17, 18].

Georgiev S. brought a completely different perspective to multiplicative
analysis with the books titled "Multiplicative Differential Calculus",
"Multiplicative Differential Geometry" and "Multiplicative Analytic Geometry"
published in 2022 [19, 20, 21]. Unlike previous studies, Georgiev S. used
operations as purely multiplicative operations and almost reconstructed the
multiplicative space. These books have been recorded as the initial studies in
particular for multiplicative geometry. Georgiev S.’s book [20] serves as a
guide for researchers in this field by encompassing numerous fundamental
definitions and theorems pertaining to curves, surfaces, and manifolds. The
book elucidates how to associate basic geometric objects such as curves,
surfaces, and manifolds with multiplicative analysis, shedding light on their
properties in multiplicative spaces. Additionally, it emphasizes the
connections between multiplicative geometry and other mathematical domains,
making it a valuable resource for researchers working in various branches of
mathematics. Afterward, Nurkan S.K. et al. tried to construct geometry with
geometric calculus. In addition, Gram-Schmidt vectors are obtained [22]. On
the other hand, Aydın M.E. et al. studied rectifying curves in multiplicative
Euclidean space. The multiplicative rectifying curves are fully classified and
visualized through multiplicative spherical curves [23].

A helix curve is the curve that a point follows as it rotates around a fixed
axis in a three-dimensional space. The helix curve is formed as a result of
this rotational movement, and the rotation time around the axis determines the
stability of the curve. While the helix curve is important in terms of
geometry, it is also increasing in different branches of science. For example,
helix is a term used for the connections of DNA. The double helix structure of
DNA is called an image helix [24]. In computer graphics and 3D applications,
helix curves are used in sections of complex surfaces and their results [25].
The helix is used in blades and aerospace engineering for the design and
performance analysis of propellers and rotor blades [26]. In addition, helices
have been traditionally studied by many researchers with their different
properties [27, 28, 29, 30, 31].

In this study, spherical indicatrices and helix curves, which are important
for differential geometry, are examined in multiplicative space. Spherical
indicatrices, general helix, slant helix, clad helix and g-clad helix are
rearranged with reference to multiplicative operations. Moreover, in the
multiplicative Euclidean space, basic concepts such as orthogonal vectors,
orthogonal system, curves, Frenet frame, etc. are mentioned. In addition, it
is aimed to make these basic concepts more memorable by visualizing them.

## 2 Multiplicative Calculus and Multiplicative Space

In this section, basic definitions and concepts regarding multiplicative
analysis are presented within the framework of the information provided by
Georgiev S. [19, 20, 21]. As mentioned in the Introduction, multiplicative
analysis has its own distinct space. Let

$\mathbb{R}_{*}=\\{exp(a):a\in\mathbb{R}\\}=\mathbb{R}^{+}.$

Given by equation (1), a multiplicative structure is formed by the field
$(\mathbb{R}_{*},+_{*},\cdot_{*})$. Each element of the space $\mathbb{R}_{*}$
is referred to as a multiplicative number and is denoted by
$a_{*}\in\mathbb{R}_{*}$, where $a_{*}=exp(a)$. In addition, the unit elements
of multiplicative addition and multiplicative multiplication operations are
$0_{*}=1$ and $1_{*}=e$, respectively. Also, we say that a number
$a\in\mathbb{R}_{*}$ is the multiplicative positive number if $a>1$ or
$a>0_{*}$. A number $a\in\mathbb{R}_{*}$ is said to be multiplicative negative
if it is not equal to $0_{*}$ and $a<0_{*}$ or $0<a<1$. Inverse elements of
multiplicative addition and multiplicative multiplication are, respectively,

$-_{*}a=1/a,\quad a^{-1_{*}}=e^{\frac{1}{\log a}}.$

The multiplicative absolute value function in multiplicative space is given as
follows in line with the information above

$\left|a\right|_{*}=\left\\{\begin{array}[]{ll}a,&a>0_{*}\\\ 0_{*},&a=0_{*}\\\
-_{*}a,&a<0_{*},\end{array}\right.$

where it can be used as $1/a$ instead of $-_{*}a$ and $a<0_{*}$ in stead of
$a\in(0,1)$. Also, some important features of the power function on
multiplicative space are as follows for all $a\in\mathbb{R}_{*}$ and
$k\in\mathbb{N}$

$a^{k_{*}}=e^{(\log a)^{k}},\quad
a^{\frac{1}{2}_{*}}=\sqrt[*]{a}=e^{\sqrt{\log a}}.$

Each ordered coordinate vector $(u_{1},u_{2},...,u_{n})$ of the space
$\mathbb{R}_{*}^{n}$ determines a point $P$. This point $P$ is called the
$n$-dimensional multiplicative vector. The multiplicative addition and
multiplicative scalar multiplication are in the multiplicative vector space
$\mathbb{R}_{*}^{n}$ for each $\mathbf{u},\mathbf{v}\in\mathbb{R}_{*}^{n}$ and
$k\in\mathbb{R}_{*}$, as follows

$\displaystyle\mathbf{u}+_{*}\mathbf{v}$ $\displaystyle=$
$\displaystyle(u_{1}+_{*}v_{1},...,u_{n}+_{*}v_{n})=(u_{1}v_{1},..,u_{n}v_{n}),$
$\displaystyle k\cdot_{*}\mathbf{u}$ $\displaystyle=$
$\displaystyle(k\cdot_{*}u_{1},...,k\cdot_{*}u_{n})=(u_{1}^{\log
k},...,u_{n}^{\log k})=e^{\log k\log\mathbf{u}}.$

where $\log\mathbf{u}=(\log u_{1},...,\log u_{n}).$

Let $\mathbf{u}$ and $\mathbf{v}$ be any two multiplicative vectors in the
multiplicative vector space $\mathbb{R}_{*}^{n}$. The multiplicative inner
product of the $\mathbf{u}$ and $\mathbf{v}$ vectors is

$\langle\mathbf{u},\mathbf{v}\rangle_{*}=e^{\langle\log\mathbf{u},\log\mathbf{v}\rangle}.$

where $\langle,\rangle$ is the Euclidean inner product. Moreover, if the
multiplicative vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal to each
other in the multiplicative sense a relation can be given as

$\langle\mathbf{u},\mathbf{v}\rangle_{*}=0_{*}.$

In Fig. 1, we present the graph of the multiplicative orthogonal vectors.

Figure 1: Multiplicative orthogonal vectors
$\mathbf{u}=(e^{\frac{1}{2}},e^{-\frac{3}{4}},e^{\frac{3}{2}})$ and
$\mathbf{v}=(e^{\frac{3}{4}},e,e^{\frac{1}{4}})$.

For a vector $\mathbf{u}\in\mathbb{R}_{*}^{n}$, the multiplicative norm of
$\mathbf{u}$ is defined as follows,

$\|\mathbf{u}\|_{*}=e^{\sqrt{\langle\log\mathbf{u},\log\mathbf{u}\rangle}}.$

The multiplicative cross product of $\mathbf{u}$ and $\mathbf{v}$ in
$\mathbb{R}_{*}^{3}$ is defined by

$\mathbf{u}\times_{*}\mathbf{v}=(e^{\log u_{2}\log v_{3}-\log u_{3}\log
v_{2}},e^{\log u_{3}\log v_{1}-\log u_{1}\log v_{3}},e^{\log u_{1}\log
v_{2}-\log u_{2}\log v_{1}}).$

It is direct to prove that the multiplicative cross product holds the standard
algebraic and geometric properties. For example,
$\mathbf{u}\times_{*}\mathbf{v}$ is multiplicative orthogonal to $\mathbf{u}$
and $\mathbf{v}$. In addition, $\mathbf{u}\times_{*}\mathbf{v}=\mathbf{0}_{*}$
if and only if $\mathbf{u}$ and $\mathbf{v}$ are multiplicative collinear. In
Fig. 2, we present the graph of the multiplicative orthogonal system.

Figure 2: Multiplicative orthogonal system
$\mathbf{u}=(e^{\frac{1}{2}},e^{-\frac{3}{4}},e^{\frac{3}{2}})$,
$\mathbf{v}=(e^{\frac{3}{4}},e,e^{\frac{1}{4}})$ and
$\mathbf{u}\times_{*}\mathbf{v}=(e^{-\frac{27}{16}},e,e^{\frac{17}{16}})$.

Let $\mathbf{u}$ and $\mathbf{v}$ represent two unit multiplicative directions
in $\mathbb{R}_{*}^{3}$. Denote by $\theta$ the multiplicative angle between
the multiplicative unit vectors $\mathbf{u}$ and $\mathbf{v}$. Hence

$\theta=\arccos_{*}(e^{\langle\log\mathbf{u},\log\mathbf{v}\rangle}).$

Trigonometric functions in terms of multiplicative space, are

$\sin_{*}\theta=e^{\sin\log\theta},\quad\cos_{*}\theta=e^{\cos\log\theta}.$

Similarly, trigonometric functions $\tan_{*}\theta$ and $\cot_{*}\theta$ are
also available at hand. In addition, multiplicative trigonometric functions
provide multiplicative trigonometric relations in parallel with classical
trigonometric relations. For example, there is the idendity
$\sin_{*}^{2*}\theta+_{*}\cos_{*}^{2*}\theta=1_{*}$. For other relations, see
[19].

Let $x\in I\subset\mathbb{R}_{*}$ and $f(x)\subset\mathbb{R}_{*}$ be a
multiplicative function. The first multiplicative derivative of $f$ at $x\in
I$ is defined as follows, which will be denoted by $f^{*}(x)$,

$f^{*}(x)=\lim_{h\rightarrow 0_{*}}(f(x+_{*}h)-_{*}f(x))/_{*}h.$

In terms of the usual arithmetical operations,

$\displaystyle f^{*}(x)$ $\displaystyle=$ $\displaystyle\lim_{h\rightarrow
1}\left(\frac{f(xh)}{f(x)}\right)^{1/\log h}$ $\displaystyle=$
$\displaystyle\lim_{h\rightarrow 1}e^{\frac{\log\frac{f(xh)}{f(x)}}{\log h}}.$

If the L’ Hospital’s rule applies here, we get

$f^{*}(x)=e^{\frac{xf^{\prime}(x)}{f(x)}}.$

In addition, if the function $f$ is differentiable in the multiplicative sense
and continuous, it is called * (multiplicative) differentiable function.

It can be said that the multiplicative derivative provides some properties
such as linearity, Leibniz rule and chain rules as in the classical derivative
as follows

  1. 1.

$(a\cdot_{*}f)^{*}(x)=a\cdot_{*}f^{*}(x),~{}\text{for
all}~{}a\in\mathbb{R_{*}}$,

  2. 2.

$(f(x)\mp_{*}g(x))^{*}=f^{*}(x)\mp_{*}g^{*}(x)$,

  3. 3.

$(f(x)\cdot_{*}g(x))^{*}=f^{*}(x)\cdot_{*}g(x)+_{*}g^{*}(x)\cdot_{*}f(x)$,

  4. 4.

$(f(x)/_{*}g(x))^{*}=(f^{*}(x)\cdot_{*}g(x)-_{*}g^{*}(x)\cdot_{*}f(x))/_{*}g^{2_{*}}(x)$,

  5. 5.

$(f\circ g)^{*}(x)=f^{*}(g(x))\cdot_{*}g^{*}(x)$,

  6. 6.

$f^{*^{(k)}}(x)=(f^{*^{(k-1)}}(x))^{*},k\in\mathbb{N},$

where $f,g$ are $*$ (multiplicative)-differentiable for each
$x\in\mathbb{R}_{*}$.

The definition of the multiplicative integral is given as the inverse operator
of the multiplicative derivative. The multiplicative indefinite integral of
the function $f(x)$ is defined by

$\int_{*}f(x)\cdot_{*}d_{*}x=e^{\int\frac{1}{x}\log f(x)dx},\quad
x\in\mathbb{R}_{*}.$

The geometric location of points with equal multiplicative distances from a
point in multiplicative space is called a multiplicative sphere. The equation
of the sphere with centered at $C(a,b,c)$ and radius $r$ is

$\|P-_{*}C\|_{*}=r,$

where $P=(x,y,z)$ is the representation point of the multiplicative sphere, so

$e^{(\log x-\log a)^{2}+(\log y-\log b)^{2}+(\log z-\log c)^{2}}=e^{(\log
r)^{2}}.$

In Fig. 3 we show the multiplicative sphere with centered at multiplicative
origin $O(0_{*},0_{*},0_{*})$ and radius $1_{*}$.

Figure 3: A multiplicative sphere with centered at multiplicative origin
$O(0_{*},0_{*},0_{*})$ and radius $1_{*}$.

## 3 Differential Geometry of Curves in Multiplicative Space

A multiplicative parametrization of class $C_{*}^{k}$ ($k\geq 1_{*}$) for a
curve $\mathbf{x}$ in $\mathbb{R}_{*}^{3}$ (i.e., the component-functions of
$\mathbf{x}$ are $k$-times continuously multiplicative differentiable), is a
multiplicative vector valued function
$\mathbf{x}:I\subset\mathbb{R}_{*}\to\mathbb{E}_{*}^{3}$, where $s$ is mapped
to $\mathbf{x}(s)=(x_{1}(s),x_{2}(s),x_{3}(s))$. In particular, a parametric
multiplicative curve $\mathbf{x}$ is regular if and only if
$\|\mathbf{x}^{*}(s)\|_{*}\neq 0_{*}$ for any $s\in I$. Looking at it
dynamically, the multiplicative vector $\mathbf{x}^{*}(s)$ represents the
multiplicative velocity of the multiplicative curve at time $s$. For a
multiplicative curve $\mathbf{x}$ to have multiplicative naturally parameters,
the necessary and sufficient condition is that the curve is from the class
$C_{*}^{k}$ and $\|\mathbf{x}^{*}(s)\|_{*}=1_{*}$ for each $s\in I$.

Given $s_{0}\in I$, the multiplicative arc length of a multiplicative regular
parametrized curve $\mathbf{x}(s)$ from the point $s_{0}$, is by definition

$h(s)=\int_{*s_{0}}^{s}\|\mathbf{x}^{*}(t)\|_{*}\cdot_{*}d_{*}t.$ (2)

As an example, the multiplicative circle curve in multiplicative plane with
center $(0_{*},0_{*},0_{*})$ and radius $r=e^{-2}$ is given by the equation
$\mathbf{x}(s)=e^{-2}\cdot_{*}(e^{\frac{1}{2}}\cos_{*}2s,e^{\frac{1}{2}}\cdot_{*}\sin_{*}2s,e^{\sqrt{3}})$
in $\mathbb{R}_{*}^{3}$. It can be plotted as in Fig 4.

Figure 4: A multiplicative circle in the plane $z=e^{\frac{\sqrt{3}}{2}}$ with
centered at $(0_{*},0_{*},0_{*})$, radius $r=1/e^{2}$ and $0_{*}<s<e^{2\pi}$.

The multiplicative Frenet trihedron of a naturally parameterized
multiplicative curve $\mathbf{x}(s)$ are

$\mathbf{t}(s)=\mathbf{x}^{*}(s),\quad\quad\mathbf{n}(s)=\mathbf{x}^{**}(s)/_{*}\|\mathbf{x}^{**}(s)\|_{*},\quad\quad\mathbf{b}(s)=\mathbf{t}(s)\times_{*}\mathbf{n}(s).$

The vector field $\mathbf{t}(s)$ (resp. $\mathbf{n}(s)$ and $\mathbf{b}(s)$)
along $\mathbf{x}(s)$ is said to be multiplicative tangent (resp.
multiplicative principal normal and multiplicative binormal). It is direct to
prove that $\\{\mathbf{t}(s),\mathbf{n}(s),\mathbf{b}(s)\\}$ is mutually
multiplicative orthogonal and
$\mathbf{n}(s)\times_{*}\mathbf{b}(s)=\mathbf{t}(s)$ and
$\mathbf{b}(s)\times_{*}\mathbf{t}(s)=\mathbf{n}(s)$. We also point out that
the arc length parameter and multiplicative Frenet frame are independent from
the choice of multiplicative parametrization [20].

To give an example, the multiplicative Frenet vectors of the multiplicative
curve

$\mathbf{x}(s)=\left((e^{3}/_{*}e^{5})\cdot_{*}\cos_{*}s,(e^{3}/_{*}e^{5})\cdot_{*}\sin_{*}s,e^{4}/_{*}e^{5}\cdot_{*}e^{s}\right)$

are

$\displaystyle\mathbf{t}(s)$ $\displaystyle=$
$\displaystyle(-_{*}(e^{3}/_{*}e^{5})\cdot_{*}\sin_{*}s,(e^{3}/_{*}e^{5})\cdot_{*}\cos_{*}s,e^{4}/_{*}e^{5})$
$\displaystyle\mathbf{n}(s)$ $\displaystyle=$
$\displaystyle(-_{*}\cos_{*}s,-_{*}\sin_{*}s,0_{*})$
$\displaystyle\mathbf{b}(s)$ $\displaystyle=$
$\displaystyle((e^{4}/_{*}e^{5})\cdot_{*}\sin_{*}s,-_{*}(e^{4}/_{*}e^{5})\cdot_{*}\cos_{*}s,e^{3}/_{*}e^{5})$

In Fig. 5, we present the graph of the multiplicative Frenet frame on the
multiplicative curve $\mathbf{x}(s)$.

Figure 5: Multiplicative curve and its multiplicative Frenet frame.

The multiplicative Frenet formulae of $\mathbf{x}$ are given by

$\displaystyle\mathbf{t}^{*}$ $\displaystyle=$
$\displaystyle\kappa\cdot_{*}\mathbf{n},$ $\displaystyle\mathbf{n}^{*}$
$\displaystyle=$
$\displaystyle-_{*}\kappa\cdot_{*}\mathbf{t}+_{*}\tau\cdot_{*}\mathbf{b},$
$\displaystyle\mathbf{b}^{*}$ $\displaystyle=$
$\displaystyle-_{*}\tau\cdot_{*}\mathbf{n},$

where $\kappa=\kappa(s)$ and $\tau=\tau(s)$ are the curvature and the torsion
functions of $\mathbf{x}$, calculated by

$\displaystyle\kappa(s)$ $\displaystyle=$
$\displaystyle\|\mathbf{x}^{**}(s)\|_{*}=e^{\langle\log\mathbf{x}^{**},\log\mathbf{x}^{**}\rangle^{\frac{1}{2}}},$
(3) $\displaystyle\tau(s)$ $\displaystyle=$
$\displaystyle\langle\mathbf{n}^{*}(s),\mathbf{b}(s)\rangle_{*}=e^{\langle\log\mathbf{n}^{*}(s),\log\mathbf{b}(s)\rangle}.$
(4)

## 4 Main Results

### 4.1 Multiplicative Spherical Indicatries

Consider a multiplicative curve $\mathbf{x}(s)\in\mathbb{R}_{*}^{3}$. The
multiplicative Frenet vectors of $\mathbf{x}$ also evolve along the curve as a
multiplicative vector field. The thing to note here is that since the
multiplicative Frenet vectors of the multiplicative curve $\mathbf{x}$ are
multiplicative unit vectors, they form a curve on the multiplicative sphere.
In this section, such curves will be examined.

The multiplicative curve $\mathbf{x}(s)$ is associated with multiplicative
Frenet vectors $\\{\mathbf{t},\mathbf{n},\mathbf{b}\\}$. Now, let us consider
the unit multiplicative tangent vectors along $\mathbf{x}(s)$. These vectors
collectively form another curve, denoted by $\mathbf{x}_{t}=\mathbf{t}$. This
new curve resides on the surface of a multiplicative sphere with a radius of
$1_{*}$ and centered at the multiplicative origin $O=(0_{*},0_{*},0_{*})$. The
multiplicative curve $\mathbf{x}_{t}$ is often referred to as the
multiplicative spherical indicatrix associated with the unit multiplicative
tangent vector $\mathbf{t}$. We will call this curve multiplicative tangent
indicatrix of the original multiplicative curve $\mathbf{x}$, in line with the
more conventional notation. With similar thought, we will call multiplicative
curves $\mathbf{x}_{n}=\mathbf{n}$ and $\mathbf{x}_{b}=\mathbf{b}$ as
multiplicative normal indicatrix and multiplicative binormal indicatrix of
$\mathbf{x}$, respectively.

######  Proposition 4.1.

Let $\mathbf{x}_{t}$ be the multiplicative tangent indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$. The multiplicative
naturally parameter $s_{t}$ of $\mathbf{x}_{t}$ is given by

$s_{t}=e^{\int^{s}\frac{1}{u}\kappa(u)du},$

where $s$ is multiplicative naturally parameter of $\mathbf{x}$ and
$\kappa(s)$ is multiplicative curvature of $\mathbf{x}$

###### Proof.

Let $\mathbf{x}(s)$ be the multiplicative naturally parameterized curve. Also,
let $\mathbf{x}_{t}(s)=\mathbf{t}(s)$ be the multiplicative tangent indicatrix
of $\mathbf{x}(s)$. Considering Eq. (2), we get the multiplicative naturally
parameter of $\mathbf{x}_{t}$ as follows

$s_{t}=\int_{*}^{s}\|\mathbf{t}^{*}(u)\|_{*}\cdot_{*}d_{*}u.$

Then from the definition of multiplicative curvature, we get

$s_{t}=\int_{*}^{s}\kappa(u)\cdot_{*}d_{*}u$

or equivalently

$s_{t}=e^{\int^{s}\frac{1}{u}\kappa(u)du}.$

∎

######  Theorem 4.2.

Let $\mathbf{x}_{t}$ be the multiplicative tangent indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$ with $\kappa\neq
0_{*}$ on $I$. The multiplicative Frenet vectors $\\{T_{t},N_{t},B_{t}\\}$ of
$\mathbf{x}_{t}$ satisfy

$\displaystyle T_{t}$ $\displaystyle=$ $\displaystyle\mathbf{n},$
$\displaystyle N_{t}$ $\displaystyle=$
$\displaystyle(-_{*}\mathbf{t}+_{*}f\cdot_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}},$ $\displaystyle B_{t}$ $\displaystyle=$
$\displaystyle(f\cdot_{*}\mathbf{t}+_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}},$

where $f=f(s)$ and $f=\tau/_{*}\kappa$.

###### Proof.

Since $\mathbf{x}_{t}$ is the multiplicative tangent indicatrix of
$\mathbf{x}$, we have

$\mathbf{x}_{t}=\mathbf{t}.$

Taking the multiplicative derivative of both sides of the above equation with
respect to $s$,

$(d_{*}\mathbf{x}_{t}/_{*}d_{*}s_{t})\cdot_{*}(d_{*}s_{t}/_{*}d_{*}s)=\kappa\cdot_{*}\mathbf{n}.$

Using Proposition 4.1 and putting $T_{t}=d_{*}\mathbf{x}_{t}/_{*}d_{*}s_{t}$,
we get

$T_{t}=\mathbf{n}.$ (5)

If we take the multiplicative derivative of Eq. (5) with respect to $s$ and
apply multiplicative Frenet formulas, we have

$\kappa\cdot_{*}(d_{*}T_{t}/_{*}d_{*}s_{t})=-_{*}\kappa\cdot_{*}\mathbf{t}+_{*}\tau\cdot_{*}\mathbf{b}$

and

$d_{*}T_{t}/_{*}d_{*}s_{t}=-_{*}\mathbf{t}+_{*}(\tau/_{*}\kappa)\cdot_{*}\mathbf{b}.$

Considering the multiplicative norm, the following equation is obtained:

$\displaystyle\|d_{*}T_{t}/_{*}d_{*}s_{t}\|_{*}$ $\displaystyle=$
$\displaystyle
e^{(\langle-\log\mathbf{t},-\log\mathbf{t}\rangle+(\frac{\log\tau}{\log\kappa})^{2}\langle\log\mathbf{b},\log\mathbf{b}\rangle)^{\frac{1}{2}}},$
(6) $\displaystyle=$ $\displaystyle
e^{\sqrt{(1+(\frac{\log\tau}{\log\kappa})^{2})}}.$

In that case, we can see that

$N_{t}=(d_{*}T_{t}/_{*}d_{*}s_{t})/_{*}\|d_{*}T_{t}/_{*}d_{*}s_{t}\|_{*}=(-_{*}\mathbf{t}+_{*}(\tau/_{*}\kappa)\cdot_{*}\mathbf{b})/_{*}e^{(1+(\frac{\log\tau}{\log\kappa})^{2})^{\frac{1}{2}}}.$

Setting $\tau/_{*}\kappa=f$,

$N_{t}=(-_{*}\mathbf{t}+_{*}f\cdot_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}.$ (7)

On the other hand, if we take into account Eqs. (5) and (7) along with the
multiplicative Frenet formulas, we obtain the final Frenet vector as

$B_{t}=[\mathbf{n}\times_{*}(-_{*}\mathbf{t}+_{*}f\cdot_{*}\mathbf{b})]/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}.$

When we organize the multiplicative operations, we obtain

$B_{t}=(f\cdot_{*}\mathbf{t}+_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}.$ (8)

∎

######  Proposition 4.3.

Let $\mathbf{x}_{t}$ be the multiplicative tangent indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$ with $\kappa\neq
0_{*}$ on $I$. The multiplicative curvatures of $\mathbf{x}_{t}$ are

$\kappa_{t}=e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}\quad\text{and}\quad\tau_{t}=\sigma\cdot_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}},$

where $\sigma=f^{*}/_{*}(e^{\log\kappa(1+(\log f(s))^{2})^{\frac{3}{2}}}).$

###### Proof.

The first equality follows by Eq. (6),

$\kappa_{t}=e^{\langle\log\mathbf{x}_{t}^{**},\log\mathbf{x}_{t}^{**}\rangle^{\frac{1}{2}}}=e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}.$

Next considering calculate the multiplicative derivative of $N_{t}$ with
respect to $s_{t}$, considering Eq. (7). Also, is chosen $e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}=\lambda$ in Eq. (7), so

$(d_{*}N_{t}/_{*}d_{*}s_{t})\cdot_{*}(d_{*}s_{t}/_{*}d_{*}s)=[(-_{*}\kappa\cdot_{*}\mathbf{n}+_{*}f^{*}\cdot_{*}\mathbf{b}-_{*}f\cdot_{*}\tau\cdot_{*}\mathbf{n})\cdot_{*}\lambda-_{*}\lambda\cdot_{*}(-_{*}\mathbf{t}+_{*}f\cdot_{\mathbf{b}})]/_{*}\lambda^{2*}$

after this we can write

$N^{*}_{t}\cdot_{*}\kappa=[\lambda^{*}\cdot_{*}\mathbf{t}-_{*}\lambda\cdot_{*}(\kappa+_{*}f\cdot_{*}\tau)\cdot_{*}\mathbf{n}+_{*}(\lambda\cdot_{*}f^{*}-_{*}\lambda^{*}\cdot_{*}f)\cdot_{*}\mathbf{b}]/_{*}\lambda^{2*}$

and so

$N^{*}_{t}=[\lambda^{*}\cdot_{*}\mathbf{t}-_{*}\lambda\cdot_{*}(\kappa+_{*}f\cdot_{*}\tau)\cdot_{*}\mathbf{n}+_{*}(\lambda\cdot_{*}f^{*}-_{*}\lambda^{*}\cdot_{*}f)\cdot_{*}\mathbf{b}]/_{*}\lambda^{2*}\cdot_{*}\kappa.$
(9)

Then from Eqs. (8) and (9), we obtain

$\displaystyle\tau_{t}=\langle\log N^{*}_{t},\log B_{t}\rangle_{*}$
$\displaystyle=$
$\displaystyle(f\cdot_{*}\lambda^{*})/_{*}\lambda^{3*}\cdot_{*}\kappa+_{*}(\lambda\cdot_{*}f^{*}-_{*}\lambda^{*}\cdot_{*}f)/_{*}\lambda^{3*}\cdot_{*}\kappa$
$\displaystyle=$
$\displaystyle(f^{*}\cdot_{*}\lambda)/_{*}\lambda^{3*}\cdot_{*}\kappa.$

Here again let’s consider the choice $e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}=\lambda$, so we get

$[f^{*}/_{*}(e^{(1+(\log
f(s))^{2})^{\frac{3}{2}}}\cdot_{*}\kappa)]\cdot_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}.$

Finally, if a choice is made in the form
$\sigma=f^{*}/_{*}(e^{\log\kappa(1+(\log
f(s))^{2})^{\frac{3}{2}}}\cdot_{*}\kappa)$, the above-mentioned equation
becomes

$\tau_{t}=\sigma\cdot_{*}e^{(1+(\log f(s))^{2})^{\frac{1}{2}}}.$

∎

Using similar arguments, we may have the following results.

######  Proposition 4.4.

Let $\mathbf{x}_{n}$ be the multiplicative normal indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$. The multiplicative
arc parameter $s_{n}$ of the multiplicative curve $\mathbf{x}_{n}$ provides

$s_{n}=\int_{*}\kappa(s)\cdot_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}\cdot_{*}d_{*}s$ (10)

where $f=f(s)$ and $f=\tau/_{*}\kappa$.

######  Theorem 4.5.

Let $\mathbf{x}_{n}$ be the multiplicative normal indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$. The multiplicative
Frenet vectors $\\{T_{n},N_{n},B_{n}\\}$ of $\mathbf{x}_{n}$ as follows

$\displaystyle T_{n}$ $\displaystyle=$
$\displaystyle(-_{*}\mathbf{t}+_{*}f\cdot_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}},$ $\displaystyle N_{n}$ $\displaystyle=$
$\displaystyle(\sigma/_{*}e^{(1+(\log\sigma(s))^{2})^{\frac{1}{2}}})\cdot_{*}[((f\cdot_{*}\mathbf{t}+_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}})-_{*}\mathbf{n}/_{*}\sigma],$ $\displaystyle B_{n}$
$\displaystyle=$
$\displaystyle(e/_{*}e^{(1+(\log\sigma(s))^{2})^{\frac{1}{2}}})\cdot_{*}[((f\cdot_{*}\mathbf{t}+_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}-_{*}\mathbf{n}\cdot_{*}\sigma],$

where $\sigma=f^{*}/_{*}(e^{\log\kappa(1+(\log f(s))^{2})^{\frac{3}{2}}}).$

######  Proposition 4.6.

Let $\mathbf{x}_{n}$ be the multiplicative normal indicatrix of the
multiplicative curve $\mathbf{x}$. The multiplicative curvatures of the normal
indicatrix $\mathbf{x}_{n}$ are described as follows

$\kappa_{n}=e^{(1+(\log\sigma(s))^{2})}\quad\text{and}\quad\tau_{n}=\Gamma\cdot_{*}e^{(1+(\log
f(s))^{2})}$ (11)

where $\Gamma=\sigma^{*}/_{*}(e^{\log\kappa(1+(\log
f(s))^{2})(1+(\log\sigma(s))^{2})^{\frac{3}{2}}}.$

######  Proposition 4.7.

Let $\mathbf{x}_{b}$ be the multiplicative binormal indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$. The multiplicative
arc parameter $s_{b}$ of the multiplicative curve $\mathbf{x}_{b}$ provides

$s_{b}=\int_{*}\tau(s)d_{*}s.$

######  Theorem 4.8.

Let $\mathbf{x}_{b}$ be the multiplicative binormal indicatrix of a
multiplicative naturally parameterized curve $\mathbf{x}$. The multiplicative
Frenet vectors $\\{T_{b},N_{b},B_{b}\\}$ of $\mathbf{x}_{b}$ satisfy

$\displaystyle T_{b}$ $\displaystyle=$ $\displaystyle-_{*}\mathbf{n},$
$\displaystyle N_{b}$ $\displaystyle=$
$\displaystyle(\mathbf{t}-_{*}f\cdot_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}},$ $\displaystyle B_{b}$ $\displaystyle=$
$\displaystyle(f\cdot_{*}\mathbf{t}+_{*}\mathbf{b})/_{*}e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}},$

where $f=f(s)$ and $f=\tau/_{*}\kappa$.

######  Proposition 4.9.

Let the multiplicative curve, denoted as $\mathbf{x}_{b}$, be the binormal
indicatrix of the multiplicative curve $\mathbf{x}$. Then, the multiplicative
curvatures of the $\mathbf{x}_{b}$ are described as follows

$\kappa_{b}=e^{(1+(\log
f(s))^{2})^{\frac{1}{2}}}/_{*}f\quad\text{and}\quad\tau_{b}=(-_{*}\sigma\cdot_{*}e^{(1+(\log\sigma(s))^{2})^{\frac{1}{2}}})/_{*}f$

where $\sigma=f^{*}/_{*}(\kappa\cdot_{*}(e^{1+(\log
f(s))^{2}})^{\frac{3}{2}*}).$

######  Example 4.10.

Let $\mathbf{x}$ $:I\subset\mathbb{R_{*}}\rightarrow\mathbb{E}_{*}^{3}$ be
multiplicative naturally parametrized curve in $\mathbb{R}_{*}^{3}$
parameterized by

$\mathbf{x}(s)=\left(e^{s},e^{\frac{e^{2}}{2}},e^{\frac{e^{3}}{6}}\right).$

In Fig. 6, we present the graph of the multiplicative spherical indicatrices
of $\mathbf{x}$.

Figure 6: Multiplicative spherical indicatrices.

### 4.2 Multiplicative Helices

######  Definition 4.1.

Consider the multiplicative curve
$\mathbf{x}:I\subset\mathbb{R}_{*}\to\mathbb{E}_{*}^{3}$ with $\kappa\neq
0_{*}$. If the multiplicative tangent vector field of the curve $\mathbf{x}$
makes a constant multiplicative angle with a constant multiplicative vector,
then the curve $\mathbf{x}$ is referred to as a multiplicative general helix
[20].

######  Theorem 4.11.

Let $\mathbf{x}:I\subset\mathbb{R}_{*}\to\mathbb{E}_{*}^{3}$ be a
multiplicative curve with $\kappa\neq 0_{*}$. The multiplicative space curve
$\mathbf{x}$ is a multiplicative general helix if and only if the
multiplicative ratio of multiplicative torsion and multiplicative curvature is
constant. In other words, it is

$\kappa/_{*}\tau=c,\quad c\in\mathbb{R}_{*}.$

###### Proof.

The proof of the theorem is explained by Georgiev S.(see [20]). ∎

######  Example 4.12.

Let $\mathbf{x}$ $:I\subset\mathbb{R_{*}}\rightarrow\mathbb{E}_{*}^{3}$ be
multiplicative naturally parametrized general helix curve in
$\mathbb{R}_{*}^{3}$ parameterized by

$\mathbf{x}(s)=\left(e^{3}/_{*}e^{5}\cdot_{*}\cos_{*}s,e^{3}/_{*}e^{5}\cdot_{*}\sin_{*}s,e^{4}/_{*}e^{5}\cdot_{*}e^{s}\right).$

With the help of multiplicative curvature formulas from Eq. (4), we give

$\kappa(s)=e^{3}/_{*}e^{5}\quad\text{and}\quad\tau=e^{4}/_{*}e^{5}.$

Since

$\tau/_{*}\kappa=e^{(\log e^{4}/\log e^{5})/(\log e^{3}/\log
e^{5})}=e^{4}/_{*}e^{3}$

is a multiplicative constant, $\mathbf{x}$ is a multiplicative helix. In Fig.
7, we present the graph of the multiplicative general helix.

Figure 7: Multiplicative general helix.

######  Definition 4.2.

Let $\mathbf{x}:I\subset\mathbb{R}_{*}\to\mathbb{E}_{*}^{3}$ be the
multiplicative curve with $\kappa\neq 0_{*}$. If the multiplicative normal
vector field of the curve $\mathbf{x}$ makes a constant multiplicative angle
with a constant multiplicative vector, then the curve $\mathbf{x}$ is referred
to as a multiplicative slant helix.

######  Theorem 4.13.

Let $\mathbf{x}:I\subset\mathbb{R}_{*}\to\mathbb{E}_{*}^{3}$ be multiplicative
curve with $\kappa\neq 0_{*}$. The multiplicative curve $\mathbf{x}$ is a
multiplicative slant helix if and only if the following equality is a
multiplicative constant function.

$\sigma(s)=[\kappa^{2*}(s)/_{*}(\kappa^{2*}(s)+_{*}\tau^{2*}(s))^{\frac{3}{2}*}]\cdot_{*}(\tau(s)/_{*}\kappa(s))^{*}.$
(12)

###### Proof.

Suppose that multiplicative naturally parametrized curve
$s\mapsto\mathbf{x}(s)$ is a multiplicative slant helix. Since the
multiplicative normal vector field of the multiplicative curve $\mathbf{x}$
makes a constant multiplicative angle with $\mathbf{v}$, which is a constant
multiplicative vector, we have

$\langle\mathbf{n},\mathbf{v}\rangle_{*}=\cos_{*}\theta,$ (13)

where $\theta$ constant multiplicative angle. Taking a multiplicative
derivative of Eq. (13), we get

$\langle\mathbf{n}^{*},\mathbf{v}\rangle_{*}=0_{*},$ (14)

and

$-_{*}\kappa\cdot_{*}\langle\mathbf{t},\mathbf{v}\rangle_{*}+_{*}\tau\cdot_{*}\langle\mathbf{b},\mathbf{v}\rangle_{*}=0_{*}.$

As can be seen from the elements of multiplicative Frenet frame and Eq. (13),
there is a constant angle between $\mathbf{n}$ and fixed direction
$\mathbf{v}$ and there is also a constant angle between $\mathbf{b}$ and fixed
direction $\mathbf{v}$. Then the following equations are provided,

$\displaystyle\langle\mathbf{t},\mathbf{v}\rangle_{*}$ $\displaystyle=$
$\displaystyle(c\cdot_{*}\tau)/_{*}\kappa,$ (15)
$\displaystyle\langle\mathbf{b},\mathbf{v}\rangle_{*}$ $\displaystyle=$
$\displaystyle c,\textbf{ }c\in\mathbb{R}_{*}.$ (16)

In terms of the multiplicative Frenet frame, we can write the decomposition
for $\mathbf{v}$ as

$\displaystyle\mathbf{v}$ $\displaystyle=$ $\displaystyle
e^{\langle\log\mathbf{t},\log\mathbf{v}\rangle\log\mathbf{t}+\langle\log\mathbf{n},\log\mathbf{v}\rangle\log\mathbf{n}+\langle\log\mathbf{b},\log\mathbf{v}\rangle\log\mathbf{b}}$
$\displaystyle=$
$\displaystyle\langle\mathbf{t},\mathbf{v}\rangle_{*}\cdot_{*}\mathbf{t}+_{*}\langle\mathbf{n},\mathbf{v}\rangle_{*}\cdot_{*}\mathbf{n}+_{*}\langle\mathbf{b},\mathbf{v}\rangle_{*}\cdot_{*}\mathbf{b}.$

The constant direction $\mathbf{v}$ from Eqs. (13), (15) and (16) is obtained
as follows

$\mathbf{v}=(c\cdot_{*}\tau)/_{*}\kappa\cdot_{*}\mathbf{t}+_{*}\cos_{*}\theta\cdot_{*}\mathbf{n}+_{*}c\cdot_{*}\mathbf{b}.$
(17)

Since $\mathbf{v}$ is the multiplicative unit vector, taking the
multiplicative norm of both sides of the above equation, we get

$\displaystyle e^{\langle\log\mathbf{v},\log\mathbf{v}\rangle^{\frac{1}{2}}}$
$\displaystyle=$
$\displaystyle((c\cdot_{*}\tau)/_{*}\kappa)^{2*}\cdot_{*}e^{\langle\log\mathbf{t},\log\mathbf{t}\rangle^{\frac{1}{2}}}+_{*}\cos_{*}^{2*}\theta\cdot_{*}e^{\langle\log\mathbf{n},\log\mathbf{n}\rangle^{\frac{1}{2}}}$
$\displaystyle+_{*}$ $\displaystyle
c^{2*}\cdot_{*}e^{\langle\log\mathbf{b},\log\mathbf{b}\rangle^{\frac{1}{2}}}$

or

$c^{2*}\cdot_{*}(\tau^{2*}/_{*}\kappa^{2*}+_{*}e)=\sin_{*}^{2*}\theta.$

If the necessary algebraic operations are performed here, we obtain

$c=(\kappa/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{1}{2}*}\cdot_{*}\sin_{*}\theta.$

Therefore, we can easily write $\mathbf{v}$ as

$\mathbf{v}=\tau/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{1}{2}*}\cdot_{*}sin_{*}\theta\cdot_{*}\mathbf{t}+_{*}\cos_{*}\theta\cdot_{*}\mathbf{n}+_{*}\kappa/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{1}{2}*}\cdot_{*}sin_{*}\theta\cdot_{*}\mathbf{b}.$
(18)

Take the multiplicative derivative of Eq. (14), we get

$\langle\mathbf{n}^{**},\mathbf{v}\rangle_{*}=0_{*}.$ (19)

From multiplicative Frenet frame and Eqs. (18) and (19), we have

$\displaystyle\langle-_{*}\kappa^{*}\cdot_{*}\mathbf{t}-_{*}(\kappa^{2*}+_{*}\tau^{2*})\cdot_{*}\mathbf{n}+_{*}\tau^{*}\mathbf{b},\tau/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{1}{2}*}\cdot_{*}sin_{*}\theta\cdot_{*}\mathbf{t}$
$\displaystyle+_{*}\cos_{*}\theta\cdot_{*}\mathbf{n}+_{*}\kappa/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{1}{2}*}\cdot_{*}sin_{*}\theta\cdot_{*}\mathbf{b}\rangle_{*}=0_{*}.$

Here the following equation exists

$(\kappa\cdot_{*}\tau^{*}-\tau\cdot_{*}\kappa^{*})/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{3}{2}*}\cdot_{*}\tan_{*}\theta+e=0_{*},$

and finally, we get

$\tan_{*}\theta=(\kappa\cdot_{*}\tau^{*}-\tau\cdot_{*}\kappa^{*})/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{3}{2}*}.$

Since the multiplicative angle $\theta$ is constant, after the necessary
adjustments, we obtain that

$\kappa^{2*}/_{*}(\kappa^{2*}+_{*}\tau^{2*})^{\frac{3}{2}*}\cdot_{*}(\tau/_{*}\kappa)^{*}=c,\quad
c\in\mathbb{R}_{*}.$

∎

######  Example 4.14.

Let $\mathbf{x}$ $:I\subset\mathbb{R_{*}}\rightarrow\mathbb{E}_{*}^{3}$ be
multiplicative naturally parametrized slant helix curve in
$\mathbb{R}_{*}^{3}$ as

$\mathbf{x}(s)=\left(x_{1}(s),x_{2}(s),x_{3}(s)\right)$

where

$\displaystyle x_{1}(s)$ $\displaystyle=$ $\displaystyle
e^{9}/_{*}e^{400}\cdot_{*}e^{\sin\log
25s}+_{*}e^{25}/_{*}e^{144}\cdot_{*}e^{\sin\log 9s}$ $\displaystyle x_{2}(s)$
$\displaystyle=$ $\displaystyle-_{*}e^{9}/_{*}e^{400}\cdot_{*}e^{\cos\log
25s}+_{*}e^{25}/_{*}e^{144}\cdot_{*}e^{\cos\log 9s}$ $\displaystyle x_{3}(s)$
$\displaystyle=$ $\displaystyle e^{15}/_{*}e^{136}\cdot_{*}e^{\sin\log 17s}$

In Fig. 8, we present the graph of the multiplicative slant helix.

Figure 8: Multiplicative slant helix.

######  Definition 4.3.

Let a regular multiplicative curve $\mathbf{x}$ be given in the multiplicative
space with $\kappa\neq 0_{*}$. The multiplicative curve $\mathbf{x}$ is called
the multiplicative clad helix if the multiplicative spherical image of the
multiplicative principal normal vector
$\mathbf{n}:I\rightarrow\mathbb{S}^{2}_{*}$ ($\mathbb{S}^{2}_{*}$ denotes the
multiplicative sphere) of the curve $\mathbf{x}$ is part of the multiplicative
cylindrical helix in $\mathbb{S}^{2}_{*}$.

Therefore we remark that a multiplicative slant helix is a multiplicative clad
helix. We have the following characterization of clad helices.

######  Theorem 4.15.

Let $\mathbf{x}$ be a multiplicative naturally parametrized curve with
$\kappa\neq 0_{*}$. Then $\mathbf{x}$ is a multiplicative clad helix if and
only if

$\displaystyle\Gamma=\sigma^{*}/_{*}[\kappa\cdot_{*}(e+_{*}f^{2*})\cdot_{*}(e+_{*}\sigma^{2*})^{\frac{3}{2}*}]$

is a constant function. Here, $f=\tau/_{*}\kappa$ and
$\sigma=f^{*}/_{*}(\kappa\cdot_{*}(e+_{*}f^{2*})^{\frac{3}{2}*}).$

###### Proof.

With the multiplicative normal indicatrix of the multiplicative curve
$\mathbf{x}$ being $\mathbf{x}_{n}$, we know from Eq. (11) that the
multiplicative curvatures of $\mathbf{x}_{n}$ are as follows

$\kappa_{n}=(e+_{*}\sigma^{2*})^{\frac{1}{2}*},$
$\tau_{n}=\Gamma\cdot_{*}(e+_{*}\sigma^{2*})^{\frac{1}{2}*}.$

It follows that $\Gamma=\tau_{n}/_{*}\kappa_{n}$. For a part of
$\mathbf{x}_{n}$ to be a multiplicative cylindrical helix,
$\tau_{n}/_{*}\kappa_{n}$ must be a multiplicative constant. This means that
$\Gamma$ is a multiplicative constant. ∎

######  Definition 4.4.

Let a regular multiplicative curve $\mathbf{x}$ be given in the multiplicative
space with $\kappa\neq 0_{*}$. The multiplicative curve $\mathbf{x}$ is called
the multiplicative g-clad helix if the multiplicative spherical image of the
multiplicative principal normal vector
$\mathbf{n}:I\rightarrow\mathbb{S}^{2}_{*}$ of the curve $\mathbf{x}$ is part
of the multiplicative slant helix in $\mathbb{S}^{2}_{*}$.

We have the following characterization of g-clad helices.

######  Theorem 4.16.

Let $\mathbf{x}$ be a multiplicative naturally parametrized curve with
$\kappa\neq 0_{*}$. Then $\mathbf{x}$ is a multiplicative g-clad helix if and
only if

$\psi(s)=\Gamma^{*}(s)/_{*}\left((\kappa^{2*}(s)+_{*}\tau^{2*}(s))^{\frac{1}{2}*}\cdot_{*}(e+_{*}\sigma^{2*}(s))^{\frac{1}{2}*}\cdot_{*}(e+_{*}\Gamma^{2*}(s))^{\frac{3}{2}*}\right)$

is a constant function.

###### Proof.

With the multiplicative normal indicatrix of the multiplicative curve
$\mathbf{x}$ being $\mathbf{x}_{n}$, from Eq. (11) the multiplicative
curvatures of $\mathbf{x}_{n}$ are as follows

$\displaystyle\kappa_{n}$ $\displaystyle=$
$\displaystyle(e+_{*}\sigma^{2*})^{\frac{1}{2}*},$ $\displaystyle\tau_{n}$
$\displaystyle=$
$\displaystyle\Gamma\cdot_{*}(e+_{*}\sigma^{2*})^{\frac{1}{2}*}.$

If the necessary algebraic operations are performed here, we get

$\kappa_{n}^{2*}+_{*}\tau_{n}^{2*}=(e+_{*}\sigma^{2*})\cdot_{*}(e+_{*}\Gamma^{2*}).$

From Eq. (12), we know that

$\left(\kappa_{n}^{2*}/_{*}(\kappa_{n}^{2*}+_{*}\tau_{n}^{2*})^{\frac{3}{2}*}\right)\cdot_{*}(\tau_{n}/_{*}\kappa_{n})^{*}=c,c\in\mathbb{R}_{*}.$

So, we can easily see that

$\Gamma^{*}/_{*}\left((\kappa^{2*}+_{*}\tau^{2*})^{\frac{1}{2}*}\cdot_{*}(e+_{*}\sigma^{2*})^{\frac{1}{2}*}\cdot_{*}(e+_{*}\Gamma^{2*})^{\frac{3}{2}*}\right).$
(20)

If the normal indicatrix of the multiplicative curve $\mathbf{x}$ is a slant
helix, Eq. (20) is a constant function. This completes the proof. ∎

######  Conclusion 1.

Considering Theorems (4.15) and (4.16) that a multiplicative slant helix is a
multiplicative helix with the condition $\sigma\equiv 0_{*}$, a multiplicative
clad helix is a multiplicative slant helix with the condition $\Gamma\equiv
0_{*}$ and a multiplicative g-clad helix is a multiplicative clad helix with
the condition $\psi\equiv 0_{*}$. Hence, we have the following relation

$\left\\{\begin{array}[]{ll}&\textrm{the family of}\\\
&\textrm{multiplicative}\\\
&\textrm{helices}\end{array}\right\\}\subset\left\\{\begin{array}[]{ll}&\textrm{the
family of}\\\ &\textrm{multiplicative}\\\ &\textrm{slant
helices}\end{array}\right\\}\subset\left\\{\begin{array}[]{ll}&\textrm{the
family of}\\\ &\textrm{multiplicative}\\\ &\textrm{clad helices}\\\
\end{array}\right\\}\subset\left\\{\begin{array}[]{ll}&\textrm{the family
of}\\\ &\textrm{multiplicative}\\\ &\textrm{g-clad helices}\\\
\end{array}\right\\}.$

######  Example 4.17.

Let $\mathbf{x}$ $:I\subset\mathbb{R_{*}}\rightarrow\mathbb{E}_{*}^{3}$ be
multiplicative naturally parametrized clad helix curve in $\mathbb{R}_{*}^{3}$
parameterized by

$\mathbf{x}(s)=\left(x_{1}(s),x_{2}(s),x_{3}(s)\right)$

where

$\displaystyle x_{1}(s)$ $\displaystyle=$ $\displaystyle
e^{18}\cdot_{*}\cos_{*}3s\cdot_{*}\cos_{*}(e^{6}\cdot_{*}\cos_{*}3s),$
$\displaystyle x_{2}(s)$ $\displaystyle=$ $\displaystyle
e^{-18}\cdot_{*}\cos_{*}3s\cdot_{*}\sin_{*}(e^{6}\cdot_{*}\cos_{*}3s),$
$\displaystyle x_{3}(s)$ $\displaystyle=$ $\displaystyle\sin_{*}2s.$

In Fig. 9, we present the graph of the multiplicative clad helix.

Figure 9: Multiplicative clad helix.

## References

  * [1] Volterra V., Hostinsky B., Operations Infinitesimales Lineares, Herman, Paris, 1938.
  * [2] Grossman M., Katz R., Non-Newtonian Calculus, 1st ed., Lee Press, Pigeon Cove Massachussets, 1972.
  * [3] Grossman M., Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foundation, Massachusetts, 1983.
  * [4] Stanley D., A multiplicative calculus, Primus, 9(4) (1999), 310-326.
  * [5] Rybaczuk M., Stoppel P., The fractal growth of fatigue defects in materials, International Journal of Fracture, 103 (2000), 71-94.
  * [6] Rybaczuk M., Zielinski W., The concept of physical and fractal dimension I. The projective dimensions, Chaos, Solitons and Fractals, 12(13) (2001), 2517-2535.
  * [7] Uzer A., Multiplicative type complex calculus as an alternative to the classical calculus, Computers and Mathematics with Applications, 60 (2010) 2725–2737.
  * [8] Bashirov A., Riza M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1) (2011), 75-85.
  * [9] Yazici M., Selvitopi H., Numerical methods for the multiplicative partial differential equations, Open Math., 15 (2017), 1344–1350.
  * [10] K. Boruah, B. Hazarika, Some Basic Properties of Bigeometric Calculus and its Applications in Numerical Analysis, Afrika Matematica, 32 (2021), 211-227.
  * [11] Aniszewska, D., Multiplicative Runge-Kutta Methods. Nonlinear Dynamics, 50 (2007), 262-272.
  * [12] Bashirov A., Mısırlı E., Tandogdu Y., Ozyapıcı A., On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ., 26(4) (2011), 425-438.
  * [13] Yalçın N., Celik E., Solution of multiplicative homogeneous linear differential equations with constant exponentials, New Trends in Mathematical Sciences, 6(2) (2018), 58-67.
  * [14] Waseem M., Noor M.A., Shah F.A., Noor K.I., An efficient technique to solve nonlinear equations using multiplicative calculus, Turkish Journal of Mathematics, 42 (2018), 679-691.
  * [15] Bashirov A.E., Kurpınar E.M., Özyapıcı A., Multiplicative calculus and its applications. J. Math. Anal. Appl., 337 (2008), 36–48.
  * [16] Gulsen T., Yilmaz E., Goktas S., Multiplicative Dirac system. Kuwait J.Sci., 49(3) (2022), 1-11.
  * [17] Goktas S., Kemaloglu H., Yilmaz E., Multiplicative conformable fractional Dirac system. Turk. J. Math., 46 (2022), 973–990.
  * [18] Goktas S., Yilmaz E., Yar A.C., Multiplicative derivative and its basic properties on time scales. Math. Meth. Appl. Sci., 45 (2022), 2097-2109.
  * [19] Georgiev S.G., Zennir K., Multiplicative Differential Calculus (1st ed.), Chapman and Hall/CRC., New York, 2022.
  * [20] Georgiev S.G., Multiplicative Differential Geometry (1st ed.), Chapman and Hall/CRC., New York, 2022.
  * [21] Georgiev S.G., Zennir K., Boukarou A., Multiplicative Analytic Geometry (1st ed.), Chapman and Hall/CRC., New York, 2022.
  * [22] Nurkan S.K., Gurgil I., Karacan M.K., Vector properties of geometric calculus. Math. Meth. Appl. Sci., (2023), 1–20.
  * [23] Aydin M.E., Has A., Yilmaz B., A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curves. ArXiv, (2023), https://doi.org/10.48550/arXiv.2307.16782
  * [24] Watson J.D., Baker T., Stephen P.B. et al., Molecular Biology of the Gene. Pearson Publishing, London, 2013.
  * [25] Hughes J., Dam A.V., McGuire M., Sklar D. et al., Computer Graphics: Principles and Practice, Addison-Wesley Professional Publishing, New York, 2013.
  * [26] Sadraey M.H., Aircraft Design: A Systems Engineering Approach. Wiley, New York, 2013
  * [27] Izumuya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28 (2004), 153-163.
  * [28] Takahashi, T., Takeuchi, N., Clad helices and developable surface, Tokyo Gakugei University Bulletin, Natural Science, 66 (2014), 1-9.
  * [29] Yılmaz, B., Has, A., New Approach to Slant Helix. International Electronic Journal of Geometry, 12 (2019), 111-115.
  * [30] Mak, M., Framed clad helices in Euclidean 3-space, Filomat, 37(28) (2023), 9627–9640.
  * [31] Kaya, S., Ateş, O., Gök, İ., Yaylı, Y., Timelike clad helices and developable surfaces in Minkowski 3-space, Rend. Circ. Mat. Palermo, II. Ser 68 (2019), 259–273.


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-228 LHCb-PAPER-2017-036 September 12, 2017

${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ resonance parameters with the decays
$\chi_{c1,c2}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$

LHCb collaboration†††Authors are listed at the end of this Letter.

The decays ${\chi_{{c}1}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ and
${\chi_{{c}2}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ are observed and used to study the resonance parameters
of the ${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ mesons. The masses of these states
are measured to be

$\displaystyle m({\chi_{{c}1}})=3510.71\pm 0.04\mathrm{\,(stat)}\pm
0.09\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}\,,$ $\displaystyle
m({\chi_{{c}2}})=3556.10\pm 0.06\mathrm{\,(stat)}\pm
0.11\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}\,,$

where the knowledge of the momentum scale for charged particles dominates the
systematic uncertainty. The momentum-scale uncertainties largely cancel in the
mass difference

$m({\chi_{{c}2}})-m({\chi_{{c}1}})=45.39\pm 0.07\mathrm{\,(stat)}\pm
0.03\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}\,.\\\ $

The natural width of the ${\chi_{{c}2}}$ meson is measured to be

$\Gamma({\chi_{{c}2}})=2.10\pm 0.20\mathrm{\,(stat)}\pm
0.02\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}\,.\\\ $

These results are in good agreement with and have comparable precision to the
current world averages.

Published in Phys. Rev. Lett. 119, 221901 (2017).

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

Studies of the properties and production of quarkonia at hadron colliders
provide an important testing ground for Quantum Chromodynamics [1].
Measurements of the spectra test potential models [2] whilst the production
rate can be calculated pertubatively in nonrelativistic effective field
theories such as NRQCD [3]. Most studies of ${\chi_{{c}1}}$ and
${\chi_{{c}2}}$ mesons at hadron colliders have exploited the radiative decays
$\chi_{c1,c2}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\gamma$
with the subsequent decay ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\mu^{+}\mu^{-}$ [4, 5, 6, 7, 8]. The branching fractions for
these processes are large, allowing a signal to be observed despite high
background.

Recently, the BESIII collaboration [9] reported the first observation of the
electromagnetic Dalitz decays [10] of $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$
mesons into the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}e^{+}e^{-}$
final state. This Letter reports the first observation of the
${\chi_{{c}1}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ and
${\chi_{{c}2}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ decay modes, using
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$
decays. These decays are used to measure the ${\chi_{{c}1}}$ and
${\chi_{{c}2}}$ masses together with the ${\chi_{{c}2}}$ natural width. The
event topology with four muons in the final state provides a clean signature
that is ideal for studies in hadron collisions.

This analysis uses the LHCb data set collected in $pp$ collisions up to the
end of 2016. The data collected at centre-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$ corresponds to integrated liminosities of 1
and 2$\mbox{\,fb}^{-1}$ and is collectively referred to as Run 1, while data
collected at centre-of-mass energy of 13$\mathrm{\,Te\kern-1.00006ptV}$
corresponds to 1.9$\mbox{\,fb}^{-1}$ and is referred to as Run 2.

The LHCb detector is a single-arm spectrometer covering the pseudorapidity
range $2<\eta<5$, described in detail in Refs. [11, 12]. The detector includes
a high-precision tracking system consisting of a silicon-strip vertex detector
[13], a large-area silicon-strip detector located upstream of a dipole magnet
with a bending power of about 4 Tm, and three stations of silicon-strip
detectors and straw drift tubes [14] placed downstream of the magnet. The
tracking system measures the momentum of charged particles with a relative
uncertainty that varies from 0.5% at low momentum to 1.0% at
$200\mathrm{\,Ge\kern-1.00006ptV}$ (natural units with $c=\hbar=1$ are used
throughout this Letter). The momentum scale is calibrated using samples of
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$ and
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
decays collected concurrently with the data sample used for this analysis [15,
16, 17]. The use of the large ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
data sample allows to correct for variations of the momentum scale at the
level of $10^{?4}$ or less that occur over time whilst the use of the
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
decay allows the momentum scale to be determined as a function of the $K^{+}$
kinematics. The procedure is validated using samples of
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\pi^{+}\pi^{-}$,
$\psi(2S)\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$, $\psi(2S)\rightarrow\mu^{+}\mu^{-}$, other fully
reconstructed ${b}$-hadron and $\mathchar 28935\relax(nS),n=1,2,3$ decays.
Based upon these studies the accuracy of the procedure is evaluated to be
$3\times 10^{-4}$. Muons are identified by a system composed of alternating
layers of iron and multiwire proportional chambers [18]. The online event
selection is performed by a trigger [19], which consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by a
software stage, which applies a full event reconstruction. The events used in
this analysis are selected by a hardware trigger that requires one or two
muons with transverse momentum, $p_{\mathrm{T}}$, larger than
$1.5\mathrm{\,Ge\kern-1.00006ptV}$. At the software trigger stage, a pair of
oppositely charged muons with an invariant mass consistent with the known
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [20] is required. In Run
1 the full event information for selected events was stored. To keep the rate
within the available bandwidth it was necessary to require
$\mbox{$p_{\mathrm{T}}$}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})>3\mathrm{\,Ge\kern-1.00006ptV}$. For Run 2, a new data taking scheme
was introduced [21] allowing real-time alignment to be performed in the
trigger [22] that, together with an increase in the online computing
resources, made possible the full track reconstruction in the online system
[23, 24]. Consequently, lower-level information could be discarded, reducing
the event size and allowing all events selected at the hardware stage that
contain a ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate to be
stored without any $p_{\mathrm{T}}$ requirement.

Offline, ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidates are
combined with a pair of oppositely charged muons to form
$\chi_{c1,c2}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ candidates. Several criteria are applied to reduce the
background and maximize the sensitivity for the mass measurement. Selected
muon candidates are required to be within the range $2<\eta<4.9$.
Misreconstructed tracks are suppressed by the use of a neural network trained
to discriminate between these and real particles. Muon candidates are selected
with a neural network trained using simulated samples to discriminate muons
from hadrons and electrons. Finally, to improve the mass resolution, a
kinematic fit is performed [25]. In this fit the mass of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate is constrained to
the known mass of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson
[20] and the position of the $\chi_{c1,c2}$ candidate decay vertex is
constrained to be the same as that of the primary vertex. The $\chi^{2}$ per
degree of freedom of this fit is required to be less than four, which
substantially reduces the background while retaining almost all the signal
events.

In the simulation, $pp$ collisions are generated using Pythia [26] with a
specific LHCb configuration [27]. For this study, signal decays are generated
using EvtGen [28] with decay amplitudes that depend on the invariant dimuon
mass, $m(\mu^{+}\mu^{-})$, using the model described in Ref. [29]. This model
assumes that the decay proceeds via the emission of a virtual photon from a
pointlike meson and is known to provide a good description of the
corresponding dielectron mode [9]. Final-state radiation is accounted for
using Photos [30]. The interaction of the generated particles with the
detector, and its response, are implemented using the Geant4 toolkit [31,
*Agostinelli:2002hh] as described in Ref. [33].

The signal yields and parameters of the $\chi_{c1,c2}$ resonances are
determined with an extended unbinned maximum likelihood fit performed to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mu^{+}\mu^{-}$ invariant mass
distribution. In this fit, the ${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ signals are
modelled by relativistic Breit-Wigner functions with Blatt-Weisskopf form
factors [34] with a meson radius parameter of
$3\mathrm{\,Ge\kern-1.00006ptV}^{-1}$. Jackson form factors [35] are
considered as an alternative to estimate the uncertainty associated with this
choice. The orbital angular momentum between the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson and the
$\mu^{+}\mu^{-}$ pair is assumed to be 0 (1) for the
${\chi_{{c}1}}\,({\chi_{{c}2}})$ cases.

The relativistic Breit-Wigner functions are convolved with the detector
resolution. Three resolution models are found to describe the simulated data
well: a double-Gaussian function, a double-sided Crystal Ball function [36,
37] and a symmetric variant of the Apollonios function [38]. The double-
Gaussian function is used by the default model and the other functions are
considered to estimate the systematic uncertainty. The parameters of the
resolution model are determined by a simultaneous fit to the ${\chi_{{c}1}}$
and ${\chi_{{c}2}}$ simulated samples. All the parameters apart from the core
resolution parameter, $\sigma$, are common between the two decay modes. For
all the models in the simulation it is found that
$\alpha\equiv\sigma^{{\chi_{{c}2}}}/\sigma^{{\chi_{{c}1}}}=1.13\pm 0.01$. This
is close to the value expected, $\alpha=1.11$, from the assumption that the
resolution scales with the square root of the energy release.

Combinatorial background is modelled by a second-order polynomial function.
The total fit function consists of the sum of the background and the
${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ signals. The free parameters are the
yields of the two signal components, the yield of the background component,
the two background shape parameters, the ${\chi_{{c}1}}$ and ${\chi_{{c}2}}$
masses, $\sigma^{{\chi_{{c}1}}}$ and the natural width of the ${\chi_{{c}2}}$
resonance, $\Gamma({\chi_{{c}2}})$. The other resolution parameters are fixed
to the simulation values. Since the natural width of the ${\chi_{{c}1}}$ state
$\Gamma({\chi_{{c}1}})=0.84\pm 0.04\mathrm{\,Me\kern-1.00006ptV}$ [20] is less
than the detector resolution ($\sigma^{{\chi_{{c}1}}}=1.41\pm
0.01\mathrm{\,Me\kern-1.00006ptV}$), this study has limited sensitivity to its
value. By applying Gaussian constraints on the natural width of the
$\chi_{{c}1}$ state (to the value from Ref. [20]) and $\alpha$ (to the value
found in the simulation) the ${\chi_{{c}2}}$ width is determined in a data-
driven way using the observed resolution for the ${\chi_{{c}1}}$ state.

The fit of this model to the full data sample is shown in Fig. 1 and the
resulting parameters of interest are summarized in Table 1. The fitted value
of $\sigma^{{\chi_{{c}1}}}$ is $1.51\pm 0.04\mathrm{\,Me\kern-1.00006ptV}$,
which agrees at the level of $5\%$ with the value found in the simulation.
Figure 2 shows the $m({\mu^{+}\mu^{-}})$ mass distribution for selected
candidates where the background has been subtracted using the sPlot technique
[39]. The data agree well with the model described in Ref. [29].

Figure 1:  Mass distribution for selected ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{\mu^{+}\mu^{-}}$ candidates. The fit is shown in thick orange, the ${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ signal components are shown by the thin red solid curve and the background component by the dashed blue curve.  Table 1:  Signal yields and resonance parameters from the nominal fit. No correction for final-state radiation is applied to the mass measurements at this stage.  Fit parameter | Fitted value
---|---
$N({\chi_{{c}1}})$ |  | $\phantom{0.00}4755\pm 81\phantom{0.0}$
$N({\chi_{{c}2}})$ |  | $\phantom{0.00}3969\pm 96\phantom{0.0}$
$m({\chi_{{c}1}})$ | $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $3510.66\pm 0.04$
$m({\chi_{{c}2}})$ | $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $3556.07\pm 0.06$
$\Gamma({\chi_{{c}2}})$ | $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $\phantom{000}2.10\pm 0.20$
Figure 2:  Background-subtracted $m({\mu^{+}\mu^{-}})$ distribution for
$\chi_{{c}1}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
$\mu^{+}\mu^{-}$ (solid red circles) and $\chi_{{c}2}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\mu^{+}\mu^{-}$ (open blue
squares) decays. The distributions are normalized to unit area. The curves
show the expected distribution from the simulation, which uses the model
described in Ref. [29].

The dominant source of systematic uncertainty on the mass measurements comes
from the knowledge of the momentum scale. This is evaluated by adjusting the
momentum scale by the $3\times 10^{-4}$ uncertainty on the calibration
procedure and rerunning the mass fit. Uncertainties of
$88\mathrm{\,ke\kern-1.00006ptV}$ and $102\mathrm{\,ke\kern-1.00006ptV}$ are
assigned to the ${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ mass measurements,
respectively. A further uncertainty arises from the knowledge of the
correction for energy loss in the spectrometer, which is known with 10 %
accuracy [12]. Based on the studies in Ref. [17] a
$20\mathrm{\,ke\kern-1.00006ptV}$ uncertainty is assigned.

The distortion of the lineshape due to final-state radiation introduces a bias
on the mass. This bias is evaluated using the simulation to be $47\pm
7\mathrm{\,ke\kern-1.00006ptV}\,(29\pm 10\mathrm{\,ke\kern-1.00006ptV})$ for
the ${\chi_{{c}1}}\,({\chi_{{c}2}})$ where the uncertainty is statistical. The
central values of the mass measurements are corrected accordingly and the
uncertainties are propagated.

Other uncertainties arise from the fit modelling and are studied using a
simplified simulation. Several variations of the relativistic Breit-Wigner
distribution are considered. Using Jackson form factors, modifying the meson
radius parameter and varying the orbital angular momentum, the observed
${\chi_{{c}1}}\,({\chi_{{c}2}})$ mass changes by at most
$15\,(24)\mathrm{\,ke\kern-1.00006ptV}$, which is assigned as a systematic
uncertainty. Similarly, fitting with a double-sided Crystal Ball or Apollonios
model, variations of $7\mathrm{\,ke\kern-1.00006ptV}$ and
$2\mathrm{\,ke\kern-1.00006ptV}$ are seen for the ${\chi_{{c}1}}$ and
${\chi_{{c}2}}$ masses and assigned as systematic uncertainties. Finally,
varying the order of the polynomial background function results in a further
uncertainty of $2\mathrm{\,ke\kern-1.00006ptV}$. The uncertainties due to the
momentum scale and energy loss correction largely cancel in the mass
difference. The assigned systematic uncertainties on the mass measurements are
summarized in Table 2.

Table 2: Systematic uncertainties on the mass and mass difference measurements. Source of uncertainty | $m({\chi_{{c}1}})\,\left[\\!\mathrm{\,ke\kern-1.00006ptV}\right]$ | $m({\chi_{{c}2}})\,\left[\\!\mathrm{\,ke\kern-1.00006ptV}\right]$ | $m({\chi_{{c}2}})-m({\chi_{{c}1}})\,\left[\\!\mathrm{\,ke\kern-1.00006ptV}\right]$
---|---|---|---
Momentum scale | $88$ | $102$ | $18$
Energy loss correction | $20$ | $\phantom{0}20$ | —
Final-state radiation | $\phantom{0}7$ | $\phantom{0}10$ | $12$
Resonance shape | $15$ | $\phantom{0}24$ | $25$
Background model | $<2\phantom{0}$ | $<2$ | $<2\phantom{0}$
Resolution model | $\phantom{0}7$ | $\phantom{00}2$ | $\phantom{0}6$
Sum in quadrature | 92 | 107 | 34

The main uncertainty on the determination of the natural width of the
${\chi_{{c}2}}$ is due to the knowledge of the detector resolution. This is
accounted for in the statistical uncertainty since the resolution scale is
determined using the ${\chi_{{c}1}}$ signal in data. Similarly, the
uncertainty on the knowledge of the ${\chi_{{c}1}}$ width is propagated via
the Gaussian constraint in the mass fit. By running fits with and without the
constraint the latter is evaluated to be $40\mathrm{\,ke\kern-1.00006ptV}$.
Further uncertainties of $10\mathrm{\,ke\kern-1.00006ptV}$ and
$20\mathrm{\,ke\kern-1.00006ptV}$ arise from the assumed Breit-Wigner
parameters and resolution model, respectively. Other systematic uncertainties,
e.g. due to the background model, are negligible. The stability of the results
is studied by dividing the data into different running periods and also into
kinematic bins and repeating the fit. None of these tests shows evidence of a
systematic bias.

In summary, the decays
${\chi_{{c}1}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ and
${\chi_{{c}2}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mu^{+}\mu^{-}$ are observed and the mass of the ${\chi_{{c}1}}$ meson
together with the mass and natural width of the ${\chi_{{c}2}}$ are measured.
The results for the mass measurements are

$\displaystyle m({\chi_{{c}1}})$ $\displaystyle=3510.71\pm 0.04\pm
0.09\mathrm{\,Me\kern-1.00006ptV},$ $\displaystyle m({\chi_{{c}2}})$
$\displaystyle=3556.10\pm 0.06\pm 0.11\mathrm{\,Me\kern-1.00006ptV},$
$\displaystyle m({\chi_{{c}2}})-m({\chi_{{c}1}})$
$\displaystyle=\phantom{00}45.39\pm 0.07\pm
0.03\mathrm{\,Me\kern-1.00006ptV},$

where the first uncertainty is statistical and the second is systematic. The
dominant systematic uncertainty is due to the knowledge of the momentum scale
and largely cancels in the mass difference. It can be seen in Table 3 that the
measurements are in good agreement with and have comparable precision to the
best previous ones, made using $p\overline{p}$ annihilation at threshold by
the E760 [40] and E835 experiments [41] at Fermilab. They are considerably
more precise than the best measurement based on the final-state reconstruction
[42]. It should be noted that the world average for the ${\chi_{{c}1}}$ mass
has a scale factor of 1.5 to account for the poor agreement between the
results [20]. The result for the ${\chi_{{c}2}}$ natural width is

$\Gamma({\chi_{{c}2}})=2.10\pm 0.20\mathrm{\,(stat)}\pm
0.02\mathrm{\,(syst)}\mathrm{\,Me\kern-1.00006ptV}\,.\\\ $

It has similar precision to and is in good agreement with previous
measurements [20].

Table 3: LHCb measurements compared to both previous measurements from Ref. [40] and the current world averages from Ref. [20]. The quoted uncertainties includes statistical and systematic uncertainties. Quantity | LHCb | Best previous |
---|---|---|---
$\,\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | measurement | measurement | World average
$m({\chi_{{c}1}})$ | $3510.71\pm 0.10$ | $3510.72\pm 0.05$ | $3510.66\pm 0.07$
$m({\chi_{{c}2}})$ | $3556.10\pm 0.13$ | $3556.16\pm 0.12$ | $3556.20\pm 0.09$
$\Gamma({\chi_{{c}2}})$ | $\phantom{000}2.10\pm 0.20$ | $\phantom{000}1.92\pm 0.19$ | $\phantom{000}1.93\pm 0.11$

The observations presented here open up a new avenue for hadron spectroscopy
at the LHC. These decay modes can be used to measure the production of
${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ states with a similar precision to the
converted photon study presented in Ref. [6]. Importantly, it will be possible
to extend measurements down to very low
$\mbox{$p_{\mathrm{T}}$}(\chi_{c1,c2})$ probing further QCD predictions [43,
44, 45]. In addition, measurements of the transition form factors [46] will
provide inputs on the interaction between charmonium states and the
electromagnetic field. With larger data samples, studies of the Dalitz decays
of other heavy-flavour states will become possible. For example, measurement
of the transition form factor of the $X(3872)$ via its Dalitz decay may help
elucidate the nature of this enigmatic state [9].

## Acknowledgements

We thank Jielei Zhang for useful discussions concerning Dalitz decays. We
express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom),
RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF
(Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities
behind the multiple open-source software packages on which we depend.
Individual groups or members have received support from AvH Foundation
(Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union),
ANR, Labex P2IO, ENIGMASS and OCEVU, and Région Auvergne-Rhône-Alpes (France),
RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith
Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust
(United Kingdom)

## References

  * [1] N. Brambilla et al., _Heavy quarkonium: Progress, puzzles, and opportunities_ , Eur. Phys. J. C71 (2011) 1534, arXiv:1010.5827
  * [2] E. Eichten et al., _Charmonium: The model_ , Phys. Rev. D17 (1978) 3090, Erratum ibid. B21 (1980) 313
  * [3] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, _Effective field theories for heavy quarkonium_ , Rev. Mod. Phys. 77 (2005) 1423, arXiv:hep-ph/0410047
  * [4] CDF collaboration, A. Abulencia et al., _Measurement of $\sigma_{\chi_{c2}}{\cal B}(\chi_{c2}\rightarrow J/\psi\gamma)/\sigma_{\chi_{c1}}{\cal B}(\chi_{c1}\rightarrow J/\psi\gamma)$ in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV_, Phys. Rev. Lett. 98 (2007) 232001, arXiv:hep-ex/0703028
  * [5] LHCb collaboration, R. Aaij et al., _Measurement of the cross-section ratio $\sigma({\chi_{{c}2}})/\sigma({\chi_{{c}1}})$ for prompt $\chi_{c}$ production at $\sqrt{s}=7$$~{}T\kern-0.50003pteV$_, Phys. Lett. B714 (2012) 215, arXiv:1202.1080
  * [6] LHCb collaboration, R. Aaij et al., _Measurement of the relative rate of prompt ${\chi_{{c}0}}$, ${\chi_{{c}1}}$ and ${\chi_{{c}2}}$ production at $\sqrt{s}=7$$~{}T\kern-0.50003pteV$_, JHEP 10 (2013) 115, arXiv:1307.4285
  * [7] CMS collaboration, S. Chatrchyan et al., _Measurement of the relative prompt production rate of $\chi_{c1}$ and $\chi_{c2}$ in $pp$ collisions at $\sqrt{s}=7$ TeV_, Eur. Phys. J. C72 (2012) 2251, arXiv:1210.0875
  * [8] ATLAS collaboration, G. Aad et al., _Measurement of $\chi_{c1}$ and $\chi_{c2}$ production with $\sqrt{s}$ = 7 TeV $pp$ collisions at ATLAS_, JHEP 07 (2014) 154, arXiv:1404.7035
  * [9] BESIII collaboration, M. Ablikim et al., _Observation of $\psi(3686)\rightarrow e^{+}e^{-}\chi_{cJ}$ and $\chi_{cJ}\rightarrow e^{+}e^{-}J/\psi$_, Phys. Rev. Lett. 118 (2017) 221802, arXiv:1701.05404
  * [10] R. H. Dalitz, _On an alternative decay process for the neutral $\pi$ meson, Letters to the Editor_, Proc. Phys. Soc. A64 (1951) 667
  * [11] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [12] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [13] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) P09007, arXiv:1405.7808
  * [14] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [15] LHCb collaboration, R. Aaij et al., _Measurement of ${b}$-hadron masses_, Phys. Lett. B708 (2012) 241, arXiv:1112.4896
  * [16] LHCb collaboration, R. Aaij et al., _Measurements of the ${{\mathchar 28931\relax}^{0}_{b}}$, ${{\mathchar 28932\relax}^{-}_{b}}$, and ${{\mathchar 28938\relax}^{-}_{b}}$ baryon masses_, Phys. Rev. Lett. 110 (2013) 182001, arXiv:1302.1072
  * [17] LHCb collaboration, R. Aaij et al., _Precision measurement of ${D}$ meson mass differences_, JHEP 06 (2013) 065, arXiv:1304.6865
  * [18] A. A. Alves Jr. et al., _Performance of the LHCb muon system_ , JINST 8 (2013) P02022, arXiv:1211.1346
  * [19] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [20] Particle Data Group, C. Patrignani et al., _Review of particle physics_ , Chin. Phys. C40 (2016) 100001, and 2017 update
  * [21] B. Sciascia, _LHCb Run 2 trigger performance_ , PoS BEAUTY 2016 (2016) 029
  * [22] G. Dujany and B. Storaci, _Real-time alignment and calibration of the LHCb Detector in Run II_ , J. Phys. Conf. Ser. 664 (2015) 082010
  * [23] R. Aaij et al., _Tesla: An application for real-time data analysis in High Energy Physics_ , Comput. Phys. Commun. 208 (2016) 35, arXiv:1604.05596
  * [24] A. Dziurda, _Full offline reconstruction in real time with the LHCb detector_ , EPJ Web Conf. 127 (2016) 00007
  * [25] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [26] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [27] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [28] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [29] A. Faessler, C. Fuchs, and M. I. Krivoruchenko, _Dilepton spectra from decays of light unflavored mesons_ , Phys. Rev. C61 (2000) 035206, arXiv:nucl-th/9904024
  * [30] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [31] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [32] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [33] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [34] J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics, Springer, New York, 1952.  doi: 10.1007/978-1-4612-9959-2
  * [35] J. D. Jackson, _Remarks on the phenomenological analysis of resonances_ , Nuovo Cim. 34 (1964) 1644
  * [36] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [37] LHCb collaboration, R. Aaij et al., _Observation of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$-pair production in ${p}{p}$ collisions at $\sqrt{s}=7$$~{}T\kern-0.50003pteV$_, Phys. Lett. B707 (2012) 52, arXiv:1109.0963
  * [38] D. Martínez Santos and F. Dupertuis, _Mass distributions marginalized over per-event errors_ , Nucl. Instrum. Meth. A764 (2014) 150, arXiv:1312.5000
  * [39] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [40] E760 collaboration, T. A. Armstrong et al., _Study of the $\chi_{c1}$ and $\chi_{c2}$ charmonium states formed in ${\overline{p}}{p}$ annihilations_, Nucl. Phys. B373 (1992) 35
  * [41] E865 collaboration, M. Andreotti et al., _Measurement of the resonance parameters of the $\chi_{1}(1^{3}P_{1})$ and $\chi_{2}(1^{3}P_{2})$ states of charmonium formed in $p\bar{p}$ annihilations_, Nucl. Phys. B717 (2005) 34, arXiv:hep-ex/0503022
  * [42] BES collaboration, M. Ablikim et al., _Precise measurement of spin-averaged $\chi_{cJ}(1P)$ mass using photon conversions in $\psi(2S)\rightarrow\gamma\chi_{cJ}$_, Phys. Rev. D71 (2005) 092002, arXiv:hep-ex/0502031
  * [43] Y.-Q. Ma, K. Wang, and K.-T. Chao, _QCD radiative corrections to $\chi_{cJ}$ production at hadron colliders_, Phys. Rev. D83 (2011) 111503, arXiv:1002.3987
  * [44] A. K. Likhoded, A. V. Luchinsky, and S. V. Poslavsky, _Production of heavy quarkonia in hadronic experiments_ , Phys. Atom. Nucl. 78 (2015) 1056, [Yad. Fiz. 78 (2015) 1119]
  * [45] D. Boer and C. Pisano, _Polarized gluon studies with charmonium and bottomonium at LHCb and AFTER_ , Phys. Rev. D86 (2012) 094007, arXiv:1208.3642
  * [46] L. G. Landsberg, _Electromagnetic decays of light mesons_ , Phys. Rept. 128 (1985) 301

LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B.
Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, M. Atzeni42, G. Auriemma26, M. Baalouch5, I.
Babuschkin56, S. Bachmann12, J.J. Back50, A. Badalov38,m, C. Baesso62, S.
Baker55, V. Balagura7,b, W. Baldini17, A. Baranov35, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V.
Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1,
A. Beiter61, L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K. Belous37,
I. Belyaev32,40, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S. Beranek9, A.
Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt42, F. Betti15, M.-O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42,
S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bizzeti18,u, M. Bjørn57, T.
Blake50, F. Blanc41, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31, I. Bordyuzhin32, S. Borghi56, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
T. Britton61, J. Brodzicka27, D. Brundu16, E. Buchanan48, C. Burr56, A.
Bursche16,f, J. Buytaert40, W. Byczynski40, S. Cadeddu16, H. Cai64, R.
Calabrese17,g, R. Calladine47, M. Calvi21,i, M. Calvo Gomez38,m, A.
Camboni38,m, P. Campana19, D.H. Campora Perez40, L. Capriotti56, A.
Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i,
L. Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina21, M. Cattaneo40, G.
Cavallero20,40,h, R. Cenci24,t, D. Chamont7, M.G. Chapman48, M. Charles8, Ph.
Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S. Chen16, S.F.
Cheung57, S.-G. Chitic40, V. Chobanova39,40, M. Chrzaszcz42,27, A. Chubykin31,
P. Ciambrone19, X. Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52, M.
Clemencic40, H.V. Cliff49, J. Closier40, J. Cogan6, E. Cogneras5, V.
Cogoni16,f, L. Cojocariu30, P. Collins40, T. Colombo40, A. Comerma-Montells12,
A. Contu40, A. Cook48, G. Coombs40, S. Coquereau38, G. Corti40, M. Corvo17,g,
C.M. Costa Sobral50, B. Couturier40, G.A. Cowan52, D.C. Craik58, A.
Crocombe50, M. Cruz Torres1, R. Currie52, C. D’Ambrosio40, F. Da Cunha
Marinho2, E. Dall’Occo43, J. Dalseno48, A. Davis3, O. De Aguiar Francisco40,
S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d,
P. De Simone19, C.T. Dean53, D. Decamp4, L. Del Buono8, H.-P. Dembinski11, M.
Demmer10, A. Dendek28, D. Derkach35, O. Deschamps5, F. Dettori54, B. Dey65, A.
Di Canto40, P. Di Nezza19, H. Dijkstra40, F. Dordei40, M. Dorigo40, A. Dosil
Suárez39, L. Douglas53, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany8,
P. Durante40, R. Dzhelyadin37, M. Dziewiecki12, A. Dziurda40, A. Dzyuba31, S.
Easo51, M. Ebert52, U. Egede55, V. Egorychev32, S. Eidelman36,w, S.
Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53, S. Ely61, S.
Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S. Farry54, D.
Fazzini21,i, L. Federici25, D. Ferguson52, G. Fernandez38, P. Fernandez
Declara40, A. Fernandez Prieto39, F. Ferrari15, F. Ferreira Rodrigues2, M.
Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M. Fiorini17,g, M. Firlej28, C.
Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, K. Fohl40, M. Fontana16,40, F.
Fontanelli20,h, D.C. Forshaw61, R. Forty40, V. Franco Lima54, M. Frank40, C.
Frei40, J. Fu22,q, W. Funk40, E. Furfaro25,j, C. Färber40, E. Gabriel52, A.
Gallas Torreira39, D. Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2,
P. Gandini22, Y. Gao3, L.M. Garcia Martin70, J. García Pardiñas39, J. Garra
Tico49, L. Garrido38, P.J. Garsed49, D. Gascon38, C. Gaspar40, L. Gavardi10,
G. Gazzoni5, D. Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, Ph.
Ghez4, S. Gianì41, V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V.
Gligorov8, D. Golubkov32, A. Golutvin55, A. Gomes1,a, I.V. Gorelov33, C.
Gotti21,i, E. Govorkova43, J.P. Grabowski12, R. Graciani Diaz38, L.A. Granado
Cardoso40, E. Graugés38, E. Graverini42, G. Graziani18, A. Grecu30, R. Greim9,
P. Griffith16, L. Grillo21, L. Gruber40, B.R. Gruberg Cazon57, O. Grünberg67,
E. Gushchin34, Yu. Guz37, T. Gys40, C. Göbel62, T. Hadavizadeh57, C.
Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C. Haines49, B. Hamilton60, X.
Han12, T.H. Hancock57, S. Hansmann-Menzemer12, N. Harnew57, S.T. Harnew48, C.
Hasse40, M. Hatch40, J. He63, M. Hecker55, K. Heinicke10, A. Heister9, K.
Hennessy54, P. Henrard5, L. Henry70, E. van Herwijnen40, M. Heß67, A.
Hicheur2, D. Hill57, C. Hombach56, P.H. Hopchev41, W. Hu65, Z.C. Huard59, W.
Hulsbergen43, T. Humair55, M. Hushchyn35, D. Hutchcroft54, P. Ibis10, M.
Idzik28, P. Ilten58, R. Jacobsson40, J. Jalocha57, E. Jans43, A. Jawahery60,
F. Jiang3, M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N.
Jurik57, S. Kandybei45, M. Karacson40, J.M. Kariuki48, S. Karodia53, N.
Kazeev35, M. Kecke12, F. Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel44, E.
Khairullin35, B. Khanji12, C. Khurewathanakul41, T. Kirn9, S. Klaver56, K.
Klimaszewski29, T. Klimkovich11, S. Koliiev46, M. Kolpin12, R. Kopecna12, P.
Koppenburg43, A. Kosmyntseva32, S. Kotriakhova31, M. Kozeiha5, L. Kravchuk34,
M. Kreps50, F. Kress55, P. Krokovny36,w, F. Kruse10, W. Krzemien29, W.
Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K. Kuonen41, T.
Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56, A. Lai16, G. Lanfranchi19,
C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A. Leflat33,40, J.
Lefrançois7, R. Lefèvre5, F. Lemaitre40, E. Lemos Cid39, O. Leroy6, T.
Lesiak27, B. Leverington12, P.-R. Li63, T. Li3, Y. Li7, Z. Li61, T.
Likhomanenko68, R. Lindner40, F. Lionetto42, V. Lisovskyi7, X. Liu3, D. Loh50,
A. Loi16, I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, A. Luchinsky37, M.
Lucio Martinez39, H. Luo52, A. Lupato23, E. Luppi17,g, O. Lupton40, A.
Lusiani24, X. Lyu63, F. Machefert7, F. Maciuc30, V. Macko41, P. Mackowiak10,
S. Maddrell-Mander48, O. Maev31,40, K. Maguire56, D. Maisuzenko31, M.W.
Majewski28, S. Malde57, B. Malecki27, A. Malinin68, T. Maltsev36,w, G.
Manca16,f, G. Mancinelli6, D. Marangotto22,q, J. Maratas5,v, J.F. Marchand4,
U. Marconi15, C. Marin Benito38, M. Marinangeli41, P. Marino41, J. Marks12, G.
Martellotti26, M. Martin6, M. Martinelli41, D. Martinez Santos39, F. Martinez
Vidal70, L.M. Massacrier7, A. Massafferri1, R. Matev40, A. Mathad50, Z.
Mathe40, C. Matteuzzi21, A. Mauri42, E. Maurice7,b, B. Maurin41, A. Mazurov47,
M. McCann55,40, A. McNab56, R. McNulty13, J.V. Mead54, B. Meadows59, C.
Meaux6, F. Meier10, N. Meinert67, D. Melnychuk29, M. Merk43, A. Merli22,40,q,
E. Michielin23, D.A. Milanes66, E. Millard50, M.-N. Minard4, L. Minzoni17,
D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez1, T. Mombächer10, I.A.
Monroy66, S. Monteil5, M. Morandin23, M.J. Morello24,t, O. Morgunova68, J.
Moron28, A.B. Morris52, R. Mountain61, F. Muheim52, M. Mulder43, D. Müller56,
J. Müller10, K. Müller42, V. Müller10, P. Naik48, T. Nakada41, R.
Nandakumar51, A. Nandi57, I. Nasteva2, M. Needham52, N. Neri22,40, S.
Neubert12, N. Neufeld40, M. Neuner12, T.D. Nguyen41, C. Nguyen-Mau41,n, S.
Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A. Nogay68, D.P. O’Hanlon50,
A. Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19, R. Oldeman16,f, C.J.G.
Onderwater71, A. Ossowska27, J.M. Otalora Goicochea2, P. Owen42, A.
Oyanguren70, P.R. Pais41, A. Palano14, M. Palutan19,40, A. Papanestis51, M.
Pappagallo14,d, L.L. Pappalardo17,g, W. Parker60, C. Parkes56, G.
Passaleva18,40, A. Pastore14,d, M. Patel55, C. Patrignani15,e, A. Pearce40, A.
Pellegrino43, G. Penso26, M. Pepe Altarelli40, S. Perazzini40, P. Perret5, L.
Pescatore41, K. Petridis48, A. Petrolini20,h, A. Petrov68, M. Petruzzo22,q, E.
Picatoste Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26, F. Pisani40, A.
Pistone20,h, A. Piucci12, V. Placinta30, S. Playfer52, M. Plo Casasus39, F.
Polci8, M. Poli Lener19, A. Poluektov50, I. Polyakov61, E. Polycarpo2, G.J.
Pomery48, S. Ponce40, A. Popov37, D. Popov11,40, S. Poslavskii37, C.
Potterat2, E. Price48, J. Prisciandaro39, C. Prouve48, V. Pugatch46, A. Puig
Navarro42, H. Pullen57, G. Punzi24,p, W. Qian50, R. Quagliani7,48, B.
Quintana5, B. Rachwal28, J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S.
Rangel2, I. Raniuk45,†, F. Ratnikov35, G. Raven44, M. Ravonel Salzgeber40, M.
Reboud4, F. Redi55, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz70, V.
Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, V. Rives
Molina38, P. Robbe7, A. Robert8, A.B. Rodrigues1, E. Rodrigues59, J.A.
Rodriguez Lopez66, A. Rogozhnikov35, S. Roiser40, A. Rollings57, V.
Romanovskiy37, A. Romero Vidal39, J.W. Ronayne13, M. Rotondo19, M.S.
Rudolph61, T. Ruf40, P. Ruiz Valls70, J. Ruiz Vidal70, J.J. Saborido Silva39,
E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino Guimaraes1, C.
Sanchez Mayordomo70, B. Sanmartin Sedes39, R. Santacesaria26, C. Santamarina
Rios39, M. Santimaria19, E. Santovetti25,j, G. Sarpis56, A. Sarti19,k, C.
Satriano26,s, A. Satta25, D.M. Saunders48, D. Savrina32,33, S. Schael9, M.
Schellenberg10, M. Schiller53, H. Schindler40, M. Schmelling11, T.
Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40, H.F. Schreiner59, M.
Schubiger41, M.-H. Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A.
Semennikov32, E.S. Sepulveda8, A. Sergi47, N. Serra42, J. Serrano6, L.
Sestini23, P. Seyfert40, M. Shapkin37, I. Shapoval45, Y. Shcheglov31, T.
Shears54, L. Shekhtman36,w, V. Shevchenko68, B.G. Siddi17, R. Silva
Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49,
N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T. Smith52, J. Smith49, M.
Smith55, l. Soares Lavra1, M.D. Sokoloff59, F.J.P. Soler53, B. Souza De
Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S.
Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12, O.
Stenyakin37, M. Stepanova31, H. Stevens10, S. Stone61, B. Storaci42, S.
Stracka24,p, M.E. Stramaglia41, M. Straticiuc30, U. Straumann42, J. Sun3, L.
Sun64, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, T. Szumlak28, M.
Szymanski63, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G. Tellarini17,g, F.
Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand4, M.
Tobin41, S. Tolk49, L. Tomassetti17,g, D. Tonelli24, F. Toriello61, R.
Tourinho Jadallah Aoude1, E. Tournefier4, M. Traill53, M.T. Tran41, M.
Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N.
Tuning43,40, A. Ukleja29, A. Usachov7, A. Ustyuzhanin35, U. Uwer12, C.
Vacca16,f, A. Vagner69, V. Vagnoni15,40, A. Valassi40, S. Valat40, G.
Valenti15, R. Vazquez Gomez40, P. Vazquez Regueiro39, S. Vecchi17, M. van
Veghel43, J.J. Velthuis48, M. Veltri18,r, G. Veneziano57, A. Venkateswaran61,
T.A. Verlage9, M. Vernet5, M. Vesterinen57, J.V. Viana Barbosa40, B. Viaud7,
D. Vieira63, M. Vieites Diaz39, H. Viemann67, X. Vilasis-Cardona38,m, M.
Vitti49, V. Volkov33, A. Vollhardt42, B. Voneki40, A. Vorobyev31, V.
Vorobyev36,w, C. Voß9, J.A. de Vries43, C. Vázquez Sierra39, R. Waldi67, C.
Wallace50, R. Wallace13, J. Walsh24, J. Wang61, D.R. Ward49, H.M. Wark54, N.K.
Watson47, D. Websdale55, A. Weiden42, C. Weisser58, M. Whitehead40, J.
Wicht50, G. Wilkinson57, M. Wilkinson61, M. Williams56, M.P. Williams47, M.
Williams58, T. Williams47, F.F. Wilson51,40, J. Wimberley60, M. Winn7, J.
Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K.
Wraight53, K. Wyllie40, Y. Xie65, M. Xu65, Z. Xu4, Z. Yang3, Z. Yang60, Y.
Yao61, H. Yin65, J. Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V.
Zhukov33, J.B. Zonneveld52, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Universita & INFN, Milano-Bicocca, Milano, Italy
22Sezione di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
71Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased