# On a Side Condition for
Wronskian-Involving Differential Equations

Nicoleta Bîlă 111Department of Mathematics and Computer Science, Fayetteville
State University, 1200 Murchison Road, Fayetteville, NC 28301, E-mail:
<EMAIL_ADDRESS>
###### Abstract

The purpose of this paper is to make a few connections among specific concepts
occurring in differential geometry and the theory of differential equations
with the aim of identifying an intriguing class of undetermined nonlinear
ordinary differential equations whose solutions satisfy a specific side
condition consisting in a homogeneous third-order linear ordinary differential
equation. A method for solving this class of Wronskian-involving differential
equations based on the proposed side condition is presented. The Tzitzeica
curve equation arising in the theory of space curves is considered as an
example, and new closed and integral-form solutions for this equation are
obtained.

MSC 2000: 34A05, 34A30, 34A34, 53A04, 53A15.

Keywords: Nonlinear Ordinary Differential Equations; Linear Ordinary
Differential Equations; Wronskian; Tzitzeica Curves.

##  1 Introduction

Since there is no general theory for integrating nonlinear ordinary
differential equations (ODEs), it will always be a challenge to explore
methods involving intriguing side conditions that would allow one “to undo”
the nonlinearity encapsulated among the unknown functions and their
derivatives and obtain particular solutions to the studied equation. The
classical Lie method [7] is one of the most well-known techniques that may be
applied in this situation and, for instance, would allow one to reduce the
order of an ODE by one whenever the equation is invariant with respect to a
particular one-parameter Lie group of transformations. However, this reduction
may become more complicated in the case of underdetermined nonlinear ODEs of
higher order that have fewer equations than unknowns and contain arbitrary
constants or functions. In this paper, the focus is on a specific class of
nonlinear differential equations (4) that expresses a relation between the
Wronskian (1) of the unknown functions and the Wronskian (2) of their first
derivatives. This class of autonomous ODEs is analyzed in the particular case
when its solutions satisfy the auxiliary equation (9). For instance, the
Tzitzeica curve equation (23) belongs to this class. It is important to point
out that the side condition (9) is not found by using the classical Lie method
but rather is related to how the nonlinearity among the unknown functions and
their derivatives has occurred while deriving the equation.

Introduced in 1812 by Józef Maria Hoëné-Wroński and later on mentioned by
Thomas Muir in [6], the Wronskian determinant (or, simply, the Wronskian) of a
set of $n$ smooth functions is the determinant of the $n\times n$ matrix whose
entries are the functions and their derivatives up to the $(n-1)$st order.
Namely, the functions are listed in the first row, their first derivatives are
placed on the second row, and so on, and their $(n-1)$st derivatives are
written on the last row ([4], p. 221). In particular, for $n=3$, the Wronskian
$W(x,y,z)(t)$ of the smooth functions $x(t)$, $y(t)$, and $z(t)$ is given by
(1). If the functions represent the set of fundamental solutions of a $n$th-
order homogeneous linear ODE, then their Wronskian does not vanish and
satisfies the Abel-Liouville-Ostrogradski formula (see, e.g., [4], p. 239).
Interestingly, this formula allows us to determine the Wronskian associated
with the solutions without effectively integrating the linear differential
equation.

On the other hand, the Wronskian also occurs in the theory of space curves.
For a smooth regular space curve (20), the Wronskian $W(x,y,z)(t)$ of the
functions defining parametrically the curve may arise, for instance, in the
calculation of the distance $d(t)$ from the origin of the system of
coordinates to the osculating plane at an arbitrary point of the curve while
the Wronskian $W(x^{\prime},y^{\prime},z^{\prime})(t)$ of their first
derivatives (2) is used in the computation of the curve’s torsion $\tau(t)$ as
in (21). Therefore, for example, any condition imposed upon on $d(t)$ and
$\tau(t)$ yields a Wronskian-involving underdetermined nonlinear differential
equation for the curve’s defining functions.

In this paper, the class of nonlinear ODEs (4) involving the Wronskians (1)
and (2) of the unknown functions $x(t)$, $y(t)$, and $z(t)$ and, respectively,
of their first derivatives $x^{\prime}(t)$, $y^{\prime}(t)$, and
$z^{\prime}(t)$ is introduced and analyzed in the context of the family of
solutions satisfying the auxiliary linear differential equation (9) such that
the condition (3) holds. The motivation of considering differential equations
of this kind lies in the study of the condition (22) that was introduced for
specific curves by the Romanian mathematician Gheorghe Tzitzeica in 1911
during his work on affine invariants [11]. Nowadays, a space curve satisfying
the relation (22) that may be written in an equivalent form as (23) is called
a Tzitzeica curve. Along with Tzitzeica curves, Tzitzeica surfaces are affine
invariants. A Tzitzeica surface is a surface for which the ratio of its
Gaussian curvature and the fourth power of the distance from the origin to the
tangent plane at any arbitrary point of the surface is constant [10]. It may
be shown that the asymptotic curves on a Tzitzeica surface with negative
Gaussian curvature are Tzitzeica curves (see, e.g., [2]). Although during the
past years the Tzitzeica curves have been of interest to many geometers, their
related nonlinear ODE has not been studied in detail so far, and, hence, there
are known only a few examples of Tzitzeica curves defined explicitly in
algebraic, transcendental, or integral forms (see [2], [3], [5], and [13]). In
[2], Agnew et al. used the Mathematica software to find and illustrate the
asymptotic curves on the Tzitzeica surface of revolution $z(x^{2}+y^{2})=1$
and expressed them in cylindrical coordinates in terms of logarithmic and
exponential functions. In [5], Crâşmăreanu determined elliptic and hyperbolic
cylindrical Tzitzeica curves written in integral form. In the paper by
Williams [13], the nonlinear Tzitzeica curve equation has been derived
explicitly, and new closed-form solutions have been presented. Bîlă and Eni
[3] showed that the nonlinear ODE (23) admits particular solutions obtained by
augmenting it with a side condition consisting in a third-order homogeneous
linear ODE with constant coefficients. In this paper, it is shown that the
Tzitzeica curve equation also admits solutions that satisfy the side condition
(9). Although this is a slight generalization of the auxiliary condition
introduced in [3], by Proposition 3.1, the new attached equation yields a
larger family of solutions to the Tzitzeica curve equation. The new
interesting solutions (16) and (26) are expressed in closed-form or in terms
of Airy’s Ai and Bi functions. In Fig. 1, the software Geogebra has been used
to visualize the Tzitzeica curve (26) on the Tzitzeica surface of equation
$yz=1-4x^{2}$, surface that was found in [12] by applying the classical Lie
method to the Tzitzeica surface partial differential equation. Additionally,
for various functions $\gamma(t)$ in (9), new Tzitzeica curves expressed in
terms of other special functions [1] such as Bessel or generalized
hypergeometric functions may be investigated. Theorem 2.1 gives a
generalization of these results for the class of nonlinear ODEs (4).

The structure of the paper is the following. In Section 2, the side condition
(9) is explored in the case of the class of underdetermined nonlinear
Wronskian-involving ODEs (4) for which (3) holds. A new method for solving
these types of nonlinear differential equations with the help of the auxiliary
condition (9) is introduced. The novel approach is exemplified in Section 3
for the Tzitzeica curve equation (23) for which intriguing solutions are
obtained. In the last section, conclusions of this work are presented.

##  2 A class of Wronskian-Involving Equations

###  2.1 On a related side condition

If $x(t)$, $y(t)$, and $z(t)$ are smooth functions defined on an open interval
$I\subset\mathbb{R}$, then their Wronskian is defined as

$W(x,y,z)(t)=\left|\begin{array}[]{ccc}x(t)&y(t)&z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|.$
(1)

Similarly, the Wronskian of the functions’ first derivatives is given by

$W(x^{\prime},y^{\prime},z^{\prime})(t)=\left|\begin{array}[]{ccc}x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\\\
x^{\prime\prime\prime}(t)&y^{\prime\prime\prime}(t)&z^{\prime\prime\prime}(t)\end{array}\right|.$
(2)

In what follows, assume that

$W(x,y,z)(t)\neq 0\quad\text{and}\quad
W(x^{\prime},y^{\prime},z^{\prime})(t)\neq 0,$ (3)

for all $t\in I$. This supposition implies that the functions $x(t)$, $y(t)$,
and $z(t)$ and, respectively, their first derivatives $x^{\prime}(t)$,
$y^{\prime}(t)$, and $z^{\prime}(t)$ are linearly independent (see [4], p.
221). Consider the differential equation

${\cal{F}}\left(W(x,y,z)(t),W(x^{\prime},y^{\prime},z^{\prime})(t)\right)=0$
(4)

involving the Wronskians (1) and (2), where ${\cal{F}}$ is a smooth real-
valued function in two variables. The equation above is a third-order
underdetermined differential equation in the unknown functions $x(t)$, $y(t)$,
and $z(t)$. Next, the problem of finding solutions to (4) is explored in the
context of a specific side condition that would allow the Wronskian (2) to be
expressed in terms of the Wronskian (1). Suppose that the functions $x(t)$,
$y(t)$, and $z(t)$ satisfy the following homogeneous third-order linear ODE

$u^{\prime\prime\prime}+\beta(t)u^{\prime\prime}+\gamma(t)u^{\prime}+\delta
u=0,$ (5)

where $\beta(t)$ and $\gamma(t)$ are smooth functions defined on the interval
$I$, and $\delta$ is a nonzero constant. For simplicity, in this paper,
$\delta$ is considered a nonzero constant. However, the method explained below
may be modified to accommodate $\delta$ variable too. Since the functions
$x(t)$, $y(t)$, and $z(t)$ are linearly independent, they form a fundamental
set of solutions to (5). Therefore, the general solution to this equation may
be written as $u(t)=C_{1}x(t)+C_{2}y(t)+C_{3}z(t)$, where $C_{i}$ are
arbitrary real constants. If the leading derivatives
$x^{\prime\prime\prime}=-\beta(t)x^{\prime\prime}-\gamma(t)x^{\prime}-\delta
x$,
$y^{\prime\prime\prime}=-\beta(t)y^{\prime\prime}-\gamma(t)y^{\prime}-\delta
y$, and
$z^{\prime\prime\prime}=-\beta(t)z^{\prime\prime}-\gamma(t)z^{\prime}-\delta
z$ are substituted into (2), then after applying a few properties of
determinants, this Wronskian may be expressed as

$W(x^{\prime},y^{\prime},z^{\prime})(t)=-\delta W(x,y,z)(t),\quad t\in I.$ (6)

Replacing the above relation into the differential equation (4) yields

${\cal{G}}\left(W(x,y,z)(t);\delta\right)=0,$ (7)

where ${\cal{G}}$ denotes the resulting function that depends on $W(x,y,z)(t)$
(the notation used here shows that $\delta$ occurs as well in the reduced
equation due to (6)). On the other hand, according to Abel-Liouville-
Ostrogradski formula (see, e.g., [4], p. 239), the Wronskian of the set of
fundamental solutions of (5) satisfies the first-order linear ODE

$W^{\prime}(x,y,z)(t)=-\beta(t)W(x,y,z)(t).$ (8)

The differentiation of (7) with respect to $t$ leads to
$W^{\prime}(x,y,z)(t)=0$ for any $t\in I$. By (3), the Wronskian $W(x,y,z)(t)$
does not vanish, and, thus, the equation (8) implies $\beta(t)=0$ for all
$t\in I$. In conclusion, the equation (5) reduces to

$u^{\prime\prime\prime}+\gamma(t)u^{\prime}+\delta u=0,$ (9)

where $\gamma(t)$ is a smooth function and $\delta$ is a nonzero constant. In
this way, the equation (4) has been reduced to the homogeneous third-order
linear ODE (9) along with the equation (7) provided that (3) holds. Observe
that now the Wronskian $W(x,y,z)(t)=C_{0}$ is constant, and the equation (7)
turns into the compatibility condition

${\cal{G}}\left(C_{0};\delta\right)=0.$ (10)

In conclusion, the following result has been proven.

######  Theorem 2.1.

Any three linearly independent solutions $x(t)$, $y(t)$, and $z(t)$ of the
homogeneous third-order linear ODE (9) conditioned to (3) satisfy the
nonlinear differential equation (4) provided that (10) holds.

###  2.2 Special side conditions

In what follows, the ODE (9) is integrated in a few particular cases that
yield to explicit or integral form solutions. Recall that in each of these
cases the Wronskian of the solutions is constant. As it has been shown in the
previous section, the equation (4) has been reduced to the side condition (9)
and compatibility condition (10).

######  Remark 2.1.

If $\gamma(t)=0$, then the equation (9) becomes $u^{\prime\prime\prime}+\delta
u=0$, and its set of fundamental solutions is

$\displaystyle x(t)$ $\displaystyle=\exp\left(-\delta^{1/3}t\right),$
$\displaystyle y(t)$
$\displaystyle=\exp\left(\frac{1}{2}\delta^{1/3}t\right)\cos\left(\frac{\sqrt{3}}{2}\delta^{1/3}t\right),$
$\displaystyle z(t)$
$\displaystyle=\exp\left(\frac{1}{2}\delta^{1/3}t\right)\sin\left(\frac{\sqrt{3}}{2}\delta^{1/3}t\right),$
(11)

with $t\in\mathbb{R}$. In this case, $W(x,y,z)(t)=-3\delta\sqrt{3}/2$.

######  Remark 2.2.

For $\gamma(t)=\tilde{\gamma}$, where $\tilde{\gamma}$ is a nonzero real
constant, the equation (9) is reduced to the homogeneous linear ODE with
constant coefficients

$u^{\prime\prime\prime}+\tilde{\gamma}u^{\prime}+\delta u=0$ (12)

whose solutions have been discussed in detail in the context of the Tzitzeica
curve equation in [3]. The characteristic equation related to (12) is the
depressed cubic equation $v^{3}+\tilde{\gamma}v+\delta=0$. The character of
the solutions depends on the sign of the associated determinant

$D=-4\gamma^{3}-27\delta^{2}.$

a) If $D>0$, the depressed cubic equation has three real distinct solutions
$v_{i}\neq 0$, $i=1,2,3$ whose sum is zero. The conditions $v_{3}\neq v_{i}$,
with $i=1,2$, imply $v_{2}\neq-2v_{1}$ and $v_{2}\neq-v_{1}/2$. In this case,
the fundamental set of solutions of (12) is

$x(t)=\exp\left(v_{1}t\right),\quad y(t)=\exp\left(v_{2}t\right),\quad
z(t)=\exp\left(-\left(v_{1}+v_{2}\right)t\right),$ (13)

with $t\in\mathbb{R}$, for which
$W(x,y,z)(t)=(v_{2}-v_{1})(2v_{1}+v_{2})(v_{1}+2v_{2})$.

b) For $D=0$, the characteristic equation has three real solutions out of
which two are equal, i.e., $v_{1}=v_{2}\neq 0$ and $v_{3}=-2v_{1}$. Therefore,

$x(t)=\exp(v_{1}t),\quad y(t)=t\exp(v_{1}t),\quad z(t)=\exp(-2v_{1}t),$ (14)

with $t\in\mathbb{R}$, form a fundamental solution set to (12). In this case,
their Wronskian is given by $W(x,y,z)(t)=9v_{1}^{2}$.

c) If $D<0$, the depressed cubic equation has two nonreal complex conjugate
solutions, $v_{1,2}=m\pm in$ and one real nonzero solution $v_{3}=-2m$, where
$m,n\neq 0$ are real numbers. The integration of (12) yields

$x(t)=\exp(mt)\cos(nt),\;\;y(t)=\exp(mt)\sin(nt),\;\;z(t)=\exp(-2mt),$ (15)

where $t\in\mathbb{R}$, and for which $W(x,y,z)(t)=n(9m^{2}+n^{2})$.

######  Remark 2.3.

If $\gamma(t)=\delta t$, then (9) may be integrated once and becomes
$u^{\prime\prime}+\delta tu=C$, where $C\in\mathbb{R}$ (here $C$ is nonzero
whenever three linearly independent solutions to (9) are needed). The
solutions of the latter ODE are given in integral form in terms of the Airy’s
Ai and Bi functions of the first and second kind, respectively ([8], p. 214),
that is,

$\displaystyle x(t)$ $\displaystyle=\text{Ai}\left(-\delta^{1/3}t\right),$
$\displaystyle y(t)$ $\displaystyle=\text{Bi}\left(-\delta^{1/3}t\right),$
$\displaystyle z(t)$ $\displaystyle=\pi\delta^{-1/3}\left(x(t)\int
y(s)ds-y(t)\int x(s)ds\right),$ (16)

with $t\in\mathbb{R}$, and their Wronskian is $W(x,y,z)(t)=-\delta^{1/3}/\pi$.

###  2.3 On an integration method involving a side condition

Assume that $x_{0}(t)$ is a known solution to ODE (9). In this situation, a
method for integrating the ODE (9) may be introduced as follows.

Step 1. Substitute $u=x_{0}(t)$ into the ODE (9) and solve the equation for
$\gamma(t)$. Denote $\gamma_{0}(t;\delta)$ the resulting solution that depends
on $\delta$ too.

Step 2. Replace $\gamma_{0}(t;\delta)$ in (9) and obtain
$u^{\prime\prime\prime}+\gamma_{0}(t;\delta)u^{\prime}+\delta u=0$. By using
the change of function

$u(t)=x_{0}(t)\int w(s)ds,$ (17)

the previous equation is reduced to

$x_{0}(t)w^{\prime\prime}+3x^{\prime}_{0}(t)w^{\prime}+\left[3x^{\prime\prime}_{0}(t)+\gamma_{0}(\delta;t)x_{0}(t)\right]w=0$
(18)

which is a homogeneous linear second-order ODE for the function $w(t)$.

Step 3. Integrate (18) and determine its general solution
$w(t)=C_{1}w_{1}(t)+C_{2}w_{2}(t)$, where $w_{1}(t)$ and $w_{2}(t)$ represent
its fundamental solution set.

Step 4. Replace $w(t)$ into (17) to have

$u(t)=x_{0}(t)\int w(s)ds=C_{1}x_{0}(t)\int w_{1}(s)ds+C_{2}x_{0}(t)\int
w_{2}(s)ds+C_{3}x_{0}(t),$

where $t\in I$. In conclusion,

$x(t)=x_{0}(t),\quad y(t)=x_{0}(t)\int w_{1}(s)ds,\quad z(t)=x_{0}(t)\int
w_{2}(s)ds,$ (19)

with $t\in I$, form the fundamental set of solutions of the equation of (9).

##  3 Solutions to Tzitzeica Curve Equation

###  3.1 Tzitzeica Curve Equation

In this section, the compatibility of the side condition (9) for the
underdetermined nonlinear ODE (4) is presented in detail in the case of the
Tzitzeica curve equation (23). Consider

$\mathbf{r}(t)=\left(x(t),y(t),z(t)\right),\quad t\in I$ (20)

a smooth, regular space curve, where $I\subset\mathbf{R}$ is an interval.
Assume that the curve’s curvature

$k(t)=\frac{||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||}{||\mathbf{r^{\prime}}(t)||^{3}},\quad
t\in I$

and torsion

$\tau(t)=\frac{\langle\mathbf{r^{\prime}}(t),\mathbf{r^{\prime\prime}}(t),\mathbf{r^{\prime\prime\prime}}(t)\rangle}{||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||^{2}},\quad
t\in I$ (21)

do not vanish on $I$. Here
$||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||$ is the
magnitude of the cross product of the tangent vector $\mathbf{r^{\prime}}(t)$
and the acceleration vector $\mathbf{r^{\prime\prime}}(t)$,
$\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)$,
$||\mathbf{r^{\prime}}(t)||$ is the magnitude of $\mathbf{r^{\prime}}(t)$, and

$\langle\mathbf{r^{\prime}}(t),\mathbf{r^{\prime\prime}}(t),\mathbf{r^{\prime\prime\prime}}(t)\rangle=\left|\begin{array}[]{ccc}x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\\\
x^{\prime\prime\prime}(t)&y^{\prime\prime\prime}(t)&z^{\prime\prime\prime}(t)\end{array}\right|$

is the mixed product of vectors $\mathbf{r^{\prime}}(t)$,
$\mathbf{r^{\prime\prime}}(t)$, and $\mathbf{r^{\prime\prime\prime}}(t)$ (see,
for instance, [9], p. 48). The curvature shows the amount by which a curve
deviates from being a line, and its torsion shows how sharply the curve is
twisting out of its osculating plane. If the torsion of the curve is nonzero
then $W(x^{\prime},y^{\prime},z^{\prime})(t)\neq 0$ and, hence, the functions
$x^{\prime}(t)$, $y^{\prime}(t)$, and $z^{\prime}(t)$ are linearly
independent. On the other hand, the osculating plane at an arbitrary point of
the curve (20) is the plane generated by the vectors $\mathbf{r^{\prime}}(t)$
and $\mathbf{r^{\prime\prime}}(t)$. Therefore, its equation in the determinant
form is given by

$\left|\begin{array}[]{ccc}x-x(t)&y-y(t)&z-z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|=0.$

If $d(t)$ denotes the distance from the origin to the osculating plane of the
curve, it can be shown that

$d(t)=\frac{1}{||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||}\left|\begin{array}[]{ccc}x(t)&y(t)&z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|.$

The distance $d(t)$ does not vanish iff $W(x,y,z)(t)\neq 0$. In what follows,
in addition to the condition that the curvature and the torsion of the curve
(20) are nonzero, the distance $d(t)$ is assumed nonzero on $I$ as well.
Therefore, the curve’s defining functions $x(t)$, $y(t)$, and $z(t)$ are
assumed to satisfy the condition (3).

A Tzitzeica curve is defined as a space curve with the property that the ratio
of its torsion $\tau(t)$ and the square of the distance $d(t)$ from the origin
to the osculating plane at an arbitrary point of the curve is constant, i.e,

$\frac{\tau(t)}{d^{2}(t)}=\alpha,$ (22)

for all $t\in I$, where $\alpha\neq 0$ is a nonzero real number called here
the curve’s constant. Substituting $\tau(t)$ and $d(t)$ into (22) yields

$\left|\begin{array}[]{ccc}x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\\\
x^{\prime\prime\prime}(t)&y^{\prime\prime\prime}(t)&z^{\prime\prime\prime}(t)\end{array}\right|=\alpha\left|\begin{array}[]{ccc}x(t)&y(t)&z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|^{2}.$

The above equation can be rewritten in the terms of the Wronskians (1) and (2)
as follows

$W(x^{\prime},y^{\prime},z^{\prime})(t)=\alpha\left[W(x,y,z)(t)\right]^{2}.$
(23)

Based on Theorem 2.1, the Wronskian-involving equation (23) admits particular
solutions satisfying the side condition (9) provided that (3) holds.

Assume that the functions $x(t)$, $y(t)$, and $z(t)$ are solutions to the ODE
(5). In this case, the Wronskian of their first derivatives (2) satisfies the
relation (6) which may be used to rewrite the equation (23) in the form

$-\delta W(x,y,z)(t)=\alpha\left[W(x,y,z)(t)\right]^{2},$ (24)

or, equivalently, as follows

$W(x,y,z)(t)\left[\alpha W(x,y,z)(t)-\delta)\right]=0.$

Thus, in this case, the above relation represents the compatibility condition
(10) . Taking into account that $W(x,y,z)(t)$ does not vanish on $I$, it
follows

$\alpha=-\frac{\delta}{W(x,y,z)(t)}.$ (25)

Since the Tzitzeica curve equation (24) is a particular case of the Wronskian-
involving equation (4), according to Theorem 2.1, the following result has
been proven.

######  Proposition 3.1.

Any three linearly independent solutions $x$, $y$, and $z$ of the homogeneous
third-order linear ODE (9) for which the condition (3) holds satisfy the
Tzitzeica curve equation (24), where the curve’s constant is given by (25).

###  3.2 Examples of Tzitzeica Curves

A few examples of Tzitzeica curves that derive from the side condition (9) are
presented below. However, more examples may be given by exploring other
options for the function $\gamma(t)$ in this auxiliary condition.

Example 1. According to Remark 2.1, the functions (11) represent a fundamental
set of solutions for the side condition (9) in the case $\gamma(t)=0$. Since
the Tzitzeica curve’s equation (24) reduces to (25), then (11) represents a
Tzitzeica curve with $\alpha=-2\sqrt{3}/9$ that lies on the surface
$S:x(y^{2}+z^{2})=1$. After applying the affine transformation
$T:\tilde{x}=z,\tilde{y}=x,\tilde{z}=y$ to the surface $S$, this becomes
$\tilde{S}:z(x^{2}+y^{2})=1$. It may be shown that $\tilde{S}$ is a Tzitzeica
surface with negative Gaussian curvature, and, therefore, its asymptotic
curves are Tzitzeica curves. Agnew et al [2] have found the asymptotic curves
(expressed in cylindrical coordinates) for $\tilde{S}$ by using the software
Mathematica. It may be shown that the transformed Tzitzeica curve (11) via the
transformation $T$ is an asymptotic curve on the surface $\tilde{S}$.

Example 2. Consider the side condition (12) in Remark 2.2. Then there are
three classes of transcendental Tzitzeica curves that are classified in terms
of the sign of the determinant $D$ of the depressed equation. The solutions
(13), (14), and (15) represent Tzitzeica curves for which their associated
curve’s constants are given, respectively, by (25). These curves have been
found and discussed in detailed in [3].

Example 3. The solution (16) represents a new Tzitzeica curve for which, by
(25), the curve’s constant is $\alpha=\pi\delta^{2/3}$. Interestingly, this
curve is expressed in terms of Airy Ai and Bi functions.

Example 4. This example refers to the method introduced in Subsection 2.3.
Assume that $x_{0}(t)=t^{-3/2}$ is a known solution to (9). At Step 1, after
substituting $x_{0}(t)$ back in the equation and solving for $\gamma(t)$, it
follows

$\gamma(t)=\gamma_{0}(t;\delta)=\frac{8\delta t^{3}-105}{12t^{2}}.$

In particular, for $\delta=-27/8$, the ODE (9) becomes

$u^{\prime\prime\prime}-\frac{9t^{3}+35}{4t^{2}}u^{\prime}-\frac{27}{8}u=0.$

At Step 2, the change of variable (17) reduces the above equation to

$w^{\prime\prime}-\frac{9}{2t}w^{\prime}+\left(-\frac{9t}{4}+\frac{5}{2t^{2}}\right)w=0.$

The set of fundamental solutions is found at Step 3 to be

$w_{1}(t)=\frac{3}{2}\left(t^{2}-\sqrt{t}\right)\exp\left({t^{3/2}}\right),\quad
w_{2}(t)=-\frac{3}{2}\left(t^{2}+\sqrt{t}\right)\exp\left(-{t^{3/2}}\right),$

for $t>1$. Therefore, at Step 4, the following Tzitzeica curve is found

$\displaystyle x(t)$ $\displaystyle=t^{-3/2},$ $\displaystyle y(t)$
$\displaystyle=\left(1-2t^{-3/2}\right)\exp\left({t^{3/2}}\right),$
$\displaystyle z(t)$
$\displaystyle=\left(1+2t^{-3/2}\right)\exp\left({-t^{3/2}}\right),$ (26)

where $t>1$. The Wronskian (1) becomes $W(x,y,z)(t)=27/4$, and, from (25), the
curve’s constant takes the value $\alpha=1/2$. The new Tzitzeica curve (26)
lies on the Tzitzeica surface $M:\;yz=1-4x^{2}$ with negative Gaussian
curvature, surface that was found by using the symmetry analysis theory in
[12] (see Fig. 1). The curve (26) is not an asymptotic curve on $M$ as this is
an hyperboloid with one sheet whose asymptotic curves are lines.

Figure 1: The Tzitzeica curve defined by (26) for $t\in[1.001,8]$ represented
on the Tzitzeica surface of equation $yz=1-4x^{2}$.

##  4 Discussion and Conclusions

The motivation of this work is based on the study of the Tzitzeica curve
equation (24) which is an underdetermined nonlinear autonomous ordinary
differential equation arising from the condition stating that the ratio of the
curve’s torsion $\tau(t)$ and the square of the distance $d(t)$ from the
origin to its osculating plane at an arbitrary point of the curve is constant.
This paper is a continuation of author’s work on closed-form solutions for the
Tzitzeica curves that she has initiated with her students (see [3] and [13]).
It is shown that the auxiliary equation (9) provides a large family of
solutions for the class of Wronskian-involving equations (4) which includes
the Tzitzeica curve nonlinear ODE (24). The side condition (9) has been
derived by using a linear combination of derivatives of the unknown functions
that would allow specific properties of determinants to be applied. Another
key observation is the relation (6) that shows that the Wronskian (2) may be
expressed in terms of the Wronskian (1). In Section 2, a method based on the
side condition (9) is introduced and allows one to find a Tzitzeica curve if a
solution $x_{0}(t)$ to (9) is known. As a consequence, new solutions for (4)
and, in particular, for (24) are obtained. It is intriguing how the auxiliary
condition (9) may provide a large variety of solutions to the nonlinear ODEs
(4) and, hence, to (24). In the future, it will be interesting to make the
connection of these results with the classical Lie symmetries associated with
the Tzitzeica curve equation and with other particular Wronskian-type
equations.

Acknowledgments. Part of these results have been presented to the online XVth
International Conference on Differential Geometry and Dynamical Systems, held
on August 26–29, 2021, Bucharest, Romania, and, hence, the author would like
to thank the organizers, in particular, Prof. Dr. C. Udrişte and Prof. Dr. V.
Bălan. The author would also like to thank Prof. Dr. M. Crâşmăreanu for
reading the manuscript and making interesting comments that have improved the
presentation of the paper.

## References

  * [1] Abramowitz, M. and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover (1972)
  * [2] A. F. Agnew, A. Bobe, W. G. Boskoff and B. D. Suceava, Tzitzeica curves and surfaces, The Mathematica Journal, 12, 1-18 (2010)
  * [3] N. Bîlă and M. Eni, Particular solutions to the Tzitzeica curve equation, Differential Geometry - Dynamical Systems, 24, 38-47 (2022)
  * [4] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 8th edition, John Wiley & Sons, Inc. (1986)
  * [5] M. Crâşmăreanu, Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan J. Geom. Appl.,7, No. 1, 37-42 (2002)
  * [6] T. Muir, A treatise on the theorie of determinants, London. Macmillan (1882)
  * [7] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York (1986)
  * [8] A. D. Polyanin, V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations Second Edition, Chapman & Hall/CRC Press, Boca Raton (2003)
  * [9] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag London Limited (2012)
  * [10] G. Tzitzeica, Sur une nouvelle classes de surfaces, Comptes Rendus, Acad. Sci. Paris, Paris, 144, 1257-1259 (1907)
  * [11] G. Tzitzeica, Sur certaines courbes gauches, Ann. de l’Ec. Normale Sup., 28, 9-32 (1911)
  * [12] C. Udrişte, N. Bîlă, Symmetry group of Tzitzeica surfaces PDE, Balkan J. Geom. Appl., 4(2), 123-140 (1999)
  * [13] L. R. Williams, On The Tzitzeica Curve Equation, Explorations: The Undergraduate Research and Creative Activities Journal for the State of North Carolina, VIII, 105-115 (2013)


# Study of semi-boosted top quark reconstruction performance on the line shape
of a $t\bar{t}$ resonance

J. Pácalt  J. Kvita

###### Abstract

We study the top quark pair events production in $pp$ collisions in the
$\ell$+jets channel at the energy of $\sqrt{s}=14$ TeV for Standard Model as
well as new physics processes. We explore the usage of semi-boosted topologies
where the top quark decays into a high-transverse momentum (boosted) hadronic
$W$-jet and an isolated $b$-jet and study their performance in the $t\bar{t}$
events kinematic reconstruction. An important event fraction is recovered and
the correlation of selected kinematic variables between the detector and
particle level is studied. Quality of the reconstructed mass line shape of a
hypothetical scalar resonance decaying into $t\bar{t}$ is evaluated and
compared for regimes of a different degree of the transverse boost. Unfolding
performance is checked in terms of comparing the excess of events in spectra
before and after the unfolding, concluding with the proof of a signal
significance loss after the unfolding procedure for both energy and angle
related observables, with possible applications in current LHC experiments.

##  1 Introduction

This work studies the top quarks pair kinematic reconstruction in a collider
detector close to that of the ATLAS detector [1] at the Large Hadron Collider
(LHC) [2] at CERN using a parameterized detector simulation provided by
Delphes [3]. The LHC nominally collides protons at four interaction points
where the detectors are located. The two main multi-purpose detector
facilities are the ATLAS and CMS [4] detectors which are versatile particle
detectors with ability to discern all kinds of particles with the exception of
neutrinos. Both are of similar phase-space coverage, resolution and detection
and identification capabilities based on different experimental technologies.

Quarks and gluons originating in collisions are not detected directly because
they become confined in hadrons or, in case of the top quark, decay before
their arrival to the detector. The process of hadronization forms showers of
particles collimated in the direction of the original particle, resulting in a
hadronic jet reaching the detector. The degree of collimation is proportional
to the momentum of the parent particle and this results in particular
perpetual positions of hadronic showers in the detector. If the primarily
particle has a large momentum with respect to the beam (transverse momentum,
${p_{\rm T}}$), the corresponding particle shower is more collimated leading
to a reconstructed hadronic final state with an imprint of the parent particle
four-vector, including its mass. Also, stable particles of different origin
can overlap in a jet.

The energy of particles used in colliders increases with the advance of the
experimental technology. This leads to enrichment of events with particles of
higher transverse momenta. This paper studies the process of top and anti-top
quark pair($t\bar{t}$) production $pp\rightarrow t\bar{t}$ at the LHC at CERN
at the center-of-mass energy $\sqrt{s}=14$ TeV. This paper also considers a
process with a hypothetical massive heavy scalar particle $y_{0}$ as a
mediator for the process $pp\rightarrow y_{0}\rightarrow t\bar{t}$ through a
triangle loop for the enhancement of events in the phase space of large
transverse momenta.

##  2 The ${t\bar{t}}$ final states topologies

The $t\bar{t}$ events are categorized into three channels, according to the
decaying products of the top quarks. The top quark decay is described by the
following process: $t\rightarrow W^{+}q$ $(q=b,s,d)$. The rest of allowed
decay processes are weak neutral currents which are heavily suppressed and
their contribution is negligible. Furthermore, the decay of the top quark is
mainly to the bottom quark thanks to the large value of the CKM mixing matrix
element between the bottom and top quarks. The $W$ boson has two main decay
modes; hadronic (68%) and leptonic (32%) [5]. There are two $W$ boson decays
in each $t\bar{t}$ event and the $t\bar{t}$ decay channels can thus be
categorized based on the combination of $W$ decay modes to all-hadronic, semi-
leptonic and dilepton channels. This analysis focuses on the semi-leptonic
channel.

The degree of collimation of the produced particle showers and their angular
separation in the detector defines the topology of an event. In the resolved
topology, $t\bar{t}$ decay products are reconstructed as individual jets and a
lepton, see Fig. 1a). Events in this topology are usually produced at lower
invariant masses of the $t\bar{t}$ pair. In the semi-boosted topology, decay
products on the side where the $W$ boson is decaying hadronically are
collimated enough to form one jet in the detector with exception of the jet
from the $b$ quark, see Fig. 1b). In the semi-boosted mixed topology, which is
a special case of semi-boosted topology, the angularly isolated jet is one of
the $W$ boson hadronic decay products, see Fig. 1c). In the boosted topology,
all products from the hadronically decaying top quark are collimated and form
one large jet in the detector, see Fig. 1d). The fractions of the topologies
are correlated to the energy spectrum of the $t\bar{t}$ pair forming a gradual
transition from the resolved to boosted topologies. The number of events of
the resolved topology drops significantly with increasing energy of the
process. In an intermediate energy regime, the number of events in the boosted
topology is not yet large enough to fill the gad after the resolved topology.

Finding ways to improve the events reconstruction efficiency by adding the
semi-boosted topologies, which reside in the aforementioned transition energy
region, is one of the aims of this paper. This can help gain statistics in
$t\bar{t}$ analyses. All these four topologies mentioned are explored in this
paper. In addition, we study the resolution of the mass peak of an
hypothetical scalar resonance decaying to ${t\bar{t}}$ and we check the
performance of the unfolding procedure in terms of its ability to retain an
excess of a possible new physics signal.

|
---|---
a) | b)
|
c) | d)
Figure 1: A schematic of $pp\rightarrow y_{0}\rightarrow t\bar{t}$ decays in
the resolved a), the semi-boosted mixed b), the semi-boosted c) and the
boosted d) topologies in the $\ell$+jets channel. Red (green) cones represent
small-$R$ (large-$R$) jets.

The semiboosted topologies have been used in analyses at the LHC as _e.g._ by
CMS [6, 7, 8] or ATLAS [9, 10, 11, 12], although mostly at lower energies or
in cases where the boosted $W$-tagged hadronic jet plays a key rôle in the
analysis. We argue that their potential is still worth exploring in the
${t\bar{t}}$ final states also at the highest energies of the LHC to come. We
employ the semiboosted topologies in the ${t\bar{t}}\rightarrow\ell$+jets
final states and emphasize their ability to recover a non-negligible event
fraction as well as explore their usage in unfolding differential
distributions, i.e. in precision measurements as well in searches for new
physics.

##  3 Samples

Events were generated for the processes $pp\rightarrow{t\bar{t}}{}$ (SM) and
$pp\rightarrow y_{0}^{\prime}\rightarrow{t\bar{t}}{}$ with the addition of a
$y_{0}$ scalar particle to the Standard Model [13, 14, 15, 16, 17, 18, 19, 20]
using the MadGraph5 version 2.6.4 simulation toolkit [21]. The spin-0 model
[19] contains an s-channel color-singlet scalar mediator with purely flavour-
diagonal couplings proportional to the masses of the SM particles, therefore
leaving top quark as the only relevant SM fermion coupling to the new
hypothetical scalar. The parton shower and the hadronization processes were
simulated using Pythia8 [22]. Masses of the hypothetical $y_{0}$ particle,
which serves effectively as a source of semi-boosted and boosted top quarks,
were selected as 500, 600, 700, 800, 900 and 1000 GeV to sample through the
region where the number of events in the resolved topology starts to decline
rapidly (500 GeV) to where the number of the boosted topology events is
becoming dominant (1000 GeV). The decay width of $y_{0}$ of 1%, 10% and 30% of
its mass for each sample were studied, results and values in the tables and
plots are shown for the decay width of 10%.

The SM $t\bar{t}$-sample without the hypothetical $y_{0}$ particle ensures the
correspondence with data measured in LHC experiments and is used as a
background to $y_{0}$ and for corrections for the unfolding procedure. The top
quark mass in simulation was set to 173 GeV (MadGraph5 default).

To ensure the strength of the evidence from the contribution of different
samples, all samples were weighted to the same luminosity ($\sim
12~{}\mathrm{fb}^{-1}$) for stacking and unfolding purposes, and which
corresponds to the luminosity of the $t\bar{t}$ sample.

The cross-sections of the samples used are summarized in Table 3. The numbers
of events in the table are presented for one statistically independent sample
and the generated samples also include charge conjugated processes in the
decay, _i.e._ the top and anti-top quark decays were swapped. All samples were
generated at the next-to-leading order (NLO) accuracy in perturbative quantum
chromodynamics (QCD), allowing also hard process with the additional
high-${p_{\rm T}}$ jet production.

The samples for the description of the $W+$jets and $WW+$jets backgrounds were
prepared within the same framework as the signal samples. The addition of the
associated production of one $W$ boson and two $b$ quarks
($Wbb\rightarrow\ell\nu bb$ \+ jets) background and two $W$ bosons and two $b$
quarks ($WWbb\rightarrow\ell\nu bb+$jets) background brings the analysis close
to those over data while the $y_{0}$ sample represents a signal of new
physics.

The ATLAS-like detector was simulated using the Delphes version 3.4.1 package
[3] with a modified ATLAS card111The modification is the addition of
information about $B$-hadrons and in the reconstruction of both small as well
as large-$R$ jets.. This simulation is able to perform particle propagation
through the magnetic field as well as hadronic and electromagnetic calorimeter
simulation including the response of the detector, muon identification system
and missing energy. The Delphes package has its own reconstruction procedure
leading to detector-level jets with a simulated realistic energy response
based on the performance of the ATLAS detector at LHC. Jets with two distance
parameters 0.4 and 1.0 were reconstructed using the anti-$k_{t}$ algorithm
[23] to form small-$R$ jets (small jets) and large-$R$ jets (large jets),
using the FastJet algorithm [24]. The Delphes package has its built-in jet
energy scale correction for jets, which is in our case used for small jets
only as pre-correction, with a private jet energy scale correction applied on
top of it for small-$R$ and as a fully private correction for the large-$R$
jets. More details can be found in Appendix A and in the selection section.
For the large-$R$ jets see Section 4.3 while for small-$R$ jets see Section
4.4.

The cross-section, processes details and the generated number of events for
samples generated by the MadGraph5 package; c.c. stands for the charge
conjugation and $\ell$ for an electron or muon. A cut of
$p_{\mathrm{T,SJ}}>20$ GeV is applied at the generator level. In the left
column, values on the $y_{0}$ lines indicate its generated mass.

Sample | Cross-section [pb] | Generated process | Events
---|---|---|---
$Wbb$+jets | 153.40000 |  $pp\rightarrow W^{+}+j,W^{+}\rightarrow\ell^{+}+\nu_{\ell}$+c.c. | 655,855
$WWbb$+jets | 180.30000 |  $pp\rightarrow W^{+}W^{-}b\bar{b},W^{+}\rightarrow\ell^{+}+\nu_{\ell},W^{-}\rightarrow jj$+c.c. | 1,000,000
$y_{0}$ 1000 GeV | 0.031 |  $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 900 GeV | 0.053 |  $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 800 GeV | 0.091 |  $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 700 GeV | 0.160 |  $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 600 GeV | 0.270 |  $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 500 GeV | 0.410 |  $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$t\bar{t}+$jets | 178.60000 |  $pp\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 2,934,961

##  4 Object and event selection

Events considered in the analysis are reconstructed at two levels; once with
the Delphes ATLAS-like detector simulation, forming detector level spectra,
and at the particle level. The event selection and the requirements differ
slightly for the reconstruction level and for the boosted, semi-boosted, semi-
boosted mixed and resolved topologies and are described below. Unless stated
otherwise, the same object and event selection applies to the particle-level
objects and selections.

###  4.1 Missing transverse energy requirement

The missing transverse energy ($E_{\mathrm{T,miss}}$) is a measure of energy
imbalance in the plane transverse to the beam and is equal to the value of
negative vector sum in the transverse plane of energies of all objects leaving
a calorimeter deposit. By definition this should equal to zero thanks to the
law of energy conservation, but the energy taken away by the undetected
neutrinos is not counted for in the detector and their contribution to
$\ell/\mu$ channels via leptonic $\tau$ decays is small. The magnitude of the
missing transverse energy is required to be $E_{\mathrm{T,miss}}>25$ GeV for
all topologies as well as for both the detector and the particle levels. This
ensures that only events in which neutrinos carry away a considerable amount
of energy are chosen for the analysis. This is a standard requirement for the
missing energy in most of top quark analyses in channels involving a charged
lepton.

###  4.2 Lepton selection

A requirement on the lepton (muon or electron) transverse momentum ensures the
selected lepton comes from the hard process and a cut of
$p_{\mathrm{T},\ell}>25$ GeV is used, a typical value in real experiment also
due to trigger requirements. Tau leptons are not considered in this analysis
as they decay before they enter the detector. In case more leptons fulfilling
the ${p_{\rm T}}$ requirement, only the electron or muon with the highest
transverse momentum is taken into account. This requirement is the same for
all topologies.

Charged leptons may radiate low energy photons which are highly collimated. E.
g. for electrons the separation of such photons and the lepton is below the
resolution of the detector and thus the photon energy is included by
construction at the detector level. The lepton dressing procedure is performed
at the particle level reconstruction to correct for this phenomenon, in which
the photon four-vectors, fulfilling the condition of the angular separation
threshold $\Delta
R_{\mathrm{\gamma,\ell}}=\sqrt{\Delta\eta_{\mathrm{\gamma,\ell}}^{2}+\Delta\phi_{\mathrm{\gamma,\ell}}^{2}}<0.1$,
are added to the lepton four-vector.

###  4.3 Large jet selection

Jets are the experimental signatures of hadronic final states of quarks and
gluons, which form particle showers entering the detector. In a typical
collider detector, jet constituents are clustered energy deposits in
calorimeters, or stable particles at the particle level. The jet four-vector
is the result of the reconstruction with the anti-$k_{t}$ algorithm.

We call jets reconstructed with a distance parameter $\Delta R=1$ as large
jets or large-$R$ jets. A private jet energy scale correction is derived on
the $t\bar{t}$ sample and applied to the detector level large jet before the
selection in order to correct jet energies to the particle level. The
magnitude of the jet energy scale correction is about 5% depending on jet
pseudorapidity $\eta$222The pseudorapidity is defined as a function of the
polar angle $\theta$ as $\eta\equiv-\ln\tan\frac{\theta}{2}$. and
$p_{\mathrm{T}}$. In the event selection, the transverse momentum of large
jets is required to be $p_{\mathrm{T,LJ}}>100$ GeV. This condition helps to
reduce the number of events with jets not coming from top quark or $W$ decays.
Furthermore, all large jets are considered in the pseudorapidity range
$|\eta|<2.5$. This constraint ensures in practice better jet identification as
the forward (large $|\eta|$) region is not well instrumented for tracking and
has a worse energy resolution. The isolation criterion of jets to be isolated
from the lepton ensures that the selected lepton is not contained within the
large jet by following the requirement of $\Delta
R_{\mathrm{LJ,\ell}}=\sqrt{\Delta\eta_{\mathrm{LJ,\ell}}^{2}+\Delta\phi_{\mathrm{LJ,\ell}}^{2}}>1$.
All these requirements are applied to all three topologies333There is no large
jet in the resolved topology. and both the detector and the particle levels.
Each large jet is then probed for the top quark and $W$ boson tagging (see
Appendix B for tagging and mistag efficiencies), first for the hypothesis as
coming from the top quark decay, then, in the semi-boosted topology, as coming
from the $W$ boson decay and in case none of the tagging was successful, the
event is then considered as a candidate for the semi-boosted mixed or the
resolved topology.

Tagging for the boosted topology is based on the constraint on the mass of the
large jet $110$ GeV$<M_{\mathrm{LJ}}<240$ GeV and a constraint combining the
large jet mass and a jet substructure variable $\tau_{3,2}$ [25], roughly a
consistency measure of finding three sub-jets inside the studied large jet
rather than two sub-jets, as $M_{\mathrm{LJ}}/\tau_{3,2}>256$ GeV. The value
for the second constraint was added to avoid background events, e.g. a large
jet from the $W$ boson. The selection is depicted in Fig. 2 (right) by the
area inside the dotted lines. The jet substructure variable
$\tau_{\mathrm{N}}$ is defined as

$\tau_{\mathrm{N}}=\frac{1}{d_{\mathrm{0}}}\sum_{k}p_{\mathrm{T},k}\min{R_{1,k},R_{2,k},...,R_{N,k}},$
(1)

where $d_{0}$ is a normalization parameter computed as

$d_{\mathrm{0}}=\Delta R\,\sum_{k}p_{\mathrm{T},k}$ (2)

with $\Delta R$ being the jet distance parameter used (here 1.0). The
subjettinesses are then combined into a ratio, such as
$\tau_{2,1}=\frac{\tau_{2}}{\tau_{1}}$, which has the ability to distinguish
the compatibility of the jet substructure with the one subjet hypothesis
(ratio closer to unity) in comparison to the scenario with two subjets (ratio
closer to zero). The substructure variable $\tau_{3,2}$ is defined in a
similar manner and effectively compares the jet substructure consistency with
two or three subjets.

|
---|---
Figure 2: Distribution of the jet substructure variable $\tau_{2,1}$ (left)
and $\tau_{3,2}$ (right) in the dependence on the large jet mass
($M_{\mathrm{LJ}}$) for the sample with $M_{\mathrm{y_{0}}}=700$ GeV at the
detector level. The large jets in the red dotted area are selected for the
reconstruction of the boosted $W$ boson and the top quark. The mass window is
set around the expected masses of the $W$ boson and top quark, respectively.

Tagging of large jets for the semi-boosted topology is based on the large jet
mass window $60$ GeV $<M_{\mathrm{LJ}}<120$ GeV and the jet substructure
variable $\tau_{2,1}$, reflecting the consistency of the jet to contain two
sub-jets inside the studied large jets rather than a single sub-jet, as
$\tau_{2,1}<0.6$. The selection is shown in Fig. 2 (left).

The large jet in the semi-boosted topology can be tagged as coming from the
$W$ boson, but there is no expected peak in the large jet mass spectrum in the
semi-boosted mixed topology, and thus it cannot be tagged based on its mass.

The case of the semi-boosted mixed topology is also considered, where the
isolated small jet does not originate from the $b$-quark but from a light
quark from the $W$ boson decay. To ensure the consistency with a $b$-quark
being a part of the large jet, there is a requirement on the large jet to
overlap with a small jet originating from a $b$-quark by requiring the
condition $\Delta
R_{\mathrm{LJ,SJ}}=\sqrt{\Delta\eta_{\mathrm{LJ,SJ}}^{2}+\Delta\phi_{\mathrm{LJ,SJ}}^{2}}<0.5$
at the detector level, see Section 4.4 for further information about small
jets. A similar condition is set at the particle level for the large jet to
contain a $B$-hadron within $\Delta R_{\mathrm{LJ,B-had}}<0.5$.

###  4.4 Small jets selection

The general requirements for small jets (small-$R$ jets), which are jets
reconstructed with a distance parameter $\Delta R=0.4$, are on the transverse
momentum $p_{T}>25$ GeV, pseudorapidity $|\eta|<2.5$ and the isolation from
the selected lepton $\Delta R_{\mathrm{SJ,\ell}}>0.5$. The jet energy scale
correction is applied on the detector level small jet objects before the
selection, which was derived on the $t\bar{t}$ sample on top of the Delphes
default jet energy scale. The magnitude of this residual jet energy scale
correction is about 2%. The identification of the small jets as coming from
the $b$-quark, $b$-tagging, is done by the Delphes simulation at the detector
level using the efficiency parameterization taken from [26], leading to the
$b$-tagging efficiency of 67.4% (72.8%) for jets of ${p_{\rm T}}=50$ (100)
GeV. At the particle level, $b$-tagging is done by the requirement of
containing a $B$-hadron within the jet as $\Delta R_{\mathrm{SJ,B-had}}<0.2$
for $B$ hadrons of ${p_{\rm T}}>5$ GeV as recommended by the LHC Top WG [27].

####  4.4.1 Small jet for the reconstruction of the leptonically decaying top
quark

The small jet for the reconstruction of the leptonically decaying top quark
has to fulfill the angular condition $\Delta R_{\mathrm{SJ,\ell}}<2$, which
ensures that it lies in the vicinity of the selected lepton, and must be
$b$-tagged. The condition of a large jet isolation from the lepton $\Delta
R_{\mathrm{SJ,LJ}}>1.5$ applies to all topologies with the exception of the
resolved topology, where there is no large jet. Such a selected small jet is
then removed from the jet collection and from further consideration. This is
the only selected small jet in case of the boosted topology.

####  4.4.2 Small jet for the hadronically decaying top quark, semi-boosted
topology

For the reconstruction of the hadronically decaying top quark in the semi-
boosted topology a small $b$-tagged jet is required in the vicinity to the
selected large jet $1<\Delta R_{\mathrm{SJ,LJ}}<1.5$. Thus a partial overlap
between the selected large jet and the considered small jet is allowed, _i.e._
the selected $b$-tagged small jet should be partially contained in the
selected large jet.

####  4.4.3 Small jet for the hadronically decaying top quark, semi-boosted
mixed topology

The conditions for the semi-boosted mixed topology are similar to the
conditions for the semi-boosted topology. The vicinity condition to the large
jet remains unchanged but the small jet is required not to be $b$-tagged while
a $b$-tag is required for the selected large jet.

####  4.4.4 Small jet for the hadronically decaying top quark, resolved
topology

The resolved topology selection is tried as the last option before the event
is discarded. The reconstruction of the hadronically decaying top quark in the
resolved topology requires three small jets, one of them $b$-tagged. The
algorithm first takes two small non-$b$-tagged jets with the highest
transverse momentum and tests their invariant mass $M_{\mathrm{SJ,SJ}}<120$
GeV to avoid dijets not corresponding to the mass of the $W$ boson. Then it
adds the four-vector of the remaining $b$-tagged jet444One $b$-tagged jet is
used in the reconstruction of the leptonically decaying top quark.. If all
such three jets are found, the event is accepted.

##  5 Reconstruction

The events passing the selection described in the previous chapter are
entering the top anti-top quark pair four-vector reconstruction described in
this section and illustrated on the $y_{0}\rightarrow t\bar{t}$ sample with
$M_{\mathrm{y_{0}}}=700$ GeV, although all samples were processed the same
way.

###  5.1 Leptonically decaying top quark

The reconstruction of the leptonically decaying top quark is the same for all
four studied topologies, starting with setting the transverse momentum of the
neutrino ($p_{\mathrm{T,\nu}}$) with the missing transverse energy
$E_{\mathrm{T,miss}}$. The missing energy together with the four-vector of the
selected lepton is used to calculate the longitudinal component of the
neutrino momentum ($p_{z,\nu}$) from the $W$ boson mass constraint
$M_{W}=M_{\ell\nu}$, which leads to a quadratic equation with two solutions in
general. The solution which leads to the more central neutrino in the rapidity
is accepted in the reconstruction. If the solution leads to a complex number
result, the imaginary part is discarded. This procedure is often used in top
quark analyses, e.g. in the ATLAS experiment [1]. The $W$ boson is
reconstructed as the sum of four-vectors of the lepton and the reconstructed
neutrino. Finally, the top quark four-vector is formed from the reconstructed
$W$ boson four-vector and the selected $b$-tagged small jet as described in
Section 4.4.1. The mass of the reconstructed leptonically decaying top quark
in the studied topologies is shown in Fig. 3 at both the detector and
particles levels.

|
---|---
Figure 3: Comparison between topologies for the shapes of the reconstructed
leptonically decaying top quark mass ($M_{\mathrm{t,lep}}$) for the sample
with $M_{\mathrm{y_{0}}}=700$ GeV at the particle (left) and the detector
(right) levels. The vertical dashed lines indicate the position of the maximum
value in the spectrum for each of the topologies, which happens to be the same
bin for all topologies at the particle level.

###  5.2 Hadronically decaying top quark, boosted topology

The recognition of the boosted event is done by selecting a large jet
fulfilling conditions specified in Section 4.3. Since there is no
reconstructed $W$ boson candidate in this case, the top-tagged large jet is
considered to contain most of the products coming from the top quark decay. To
verify this, the mass of the large jet corresponding to the reconstructed
hadronically decaying top quark is shown in Fig. 4 at both the detector and
the particle levels, showing a peak around the top quark mass.

###  5.3 Hadronically decaying top quark, semi-boosted topology

The reconstruction of the hadronically decaying top quark in the semi-boosted
topology uses the selected $W$-tagged large jet and one small $b$-tagged jet,
fulfilling the conditions mentioned in Section 4.4.2. The large jet is
considered as the hadronically decaying $W$ boson, with its mass shown in Fig.
5. The reconstructed top quark is formed by adding the selected $b$-tagged
small jet four-vector and its mass is shown in Fig. 4, again exhibiting the
expected peak which is sharper at the particle level due to finite detector
resolution.

###  5.4 Hadronically decaying top quark, semi-boosted mixed topology

The reconstruction of the hadronically decaying top quark in the semi-boosted
mixed topology is performed by summing one large jet and one non-$b$-tagged
small jet four-vectors, see Section 4.4.3 for details. The invariant mass of
such a large jet does not produce a peak but in the combination with the
selected small jet the resulting invariant mass peak should correspond to the
one of the top quark as shown in Fig. 4.

###  5.5 Hadronically decaying top quark, resolved topology

The resolved topology has the largest combinatorial ambiguity as it involves
largest multiplicity of objects for the reconstruction, starting with the $W$
boson reconstruction from two highest transverse momentum small jets which are
not tagged as $b$-jets. The reconstructed $W$ boson candidate mass, shown in
Fig. 5, is required to be 60–120 GeV, otherwise the event is discarded. The
third selected small jet for the reconstruction in the resolved topology is
required to be $b$-tagged and is added to the reconstructed $W$ boson forming
finally the four-vector of the hadronically decaying top quark with its mass
shown in Fig. 4.

|
---|---
Figure 4: Comparison between topologies for the hadronically decaying top quark mass ($M_{\mathrm{t,had}}$) for the sample with $M_{\mathrm{y_{0}}}=700$ GeV at the particle (left) and the detector (right) level. The vertical dashed lines indicate the position of the maximum value in the spectrum for each of the topologies. |
---|---
Figure 5: Comparison between topologies for the shapes of the reconstructed
hadronically decaying $W$ boson mass ($M_{\mathrm{W,had}}$) for the sample
with $M_{\mathrm{y_{0}}}=700$ GeV at the particle level (left) and the
detector level (right). The vertical dashed lines indicate the position of the
maximum value in the spectrum for each of the topologies.

###  5.6 Top anti-top quark pair system

The four-vector of the top anti-top quark (${t\bar{t}}{}$) pair system is
reconstructed as the combination of the leptonically and hadronically decaying
top quarks. Its mass and the contributing fractions of events from the
particular topologies are shown in Fig. 6. The fractions depend on the mass of
the hypothetical $y_{0}$ particle as shown in Fig. 7. This plot illustrates
the existence of the transition region between the resolved and the boosted
topology which was mentioned in Section 1 and which benefits from the
implementation of the semi-boosted and semi-boosted mixed topologies via their
non-negligible event fractions.

|
---|---
Figure 6: Contributions of different topologies to the reconstruction of the
${t\bar{t}}$ invariant mass ($M_{\mathrm{t\bar{t}}}$) for the sample with
$M_{\mathrm{y_{0}}}=700$ GeV in descendant order: boosted (blue), semi-boosted
(red), semi-boosted mixed (purple), and resolved (green) at the particle
(left) and the detector (right) level. The corresponding percentage is
presented in the legend for each topology.
---
Figure 7: The fraction of events contributing to the $t\bar{t}$ reconstruction
from each topology over samples with various masses of the hypothetical
$y_{0}$ particle ($M_{y_{0}}$) at the detector (solid lines, full markers) and
particle (dotted lines, open markers) levels.

###  5.7 Migration of events between topologies

The kinematic reconstruction at the detector and particle levels is
accomplished in parallel under the same selection requirements but the
resulting event topologies are not necessarily the same at the two
reconstruction levels. This is described by a migration matrix between the two
levels in terms of the topologies as illustrated in Fig. 8 (left). Similar
migrations further apply to values and bins of any studied observable. The
example for the reconstructed ${t\bar{t}}{}$ pair transverse momentum
($p_{\mathrm{T,t\bar{t}}}$) is shown in Fig. 8 (right), this plot also shows
migration of events between different bins. A matching condition, requiring
the same topology at the detector and particle levels, is applied for the
purpose of unfolding and only those events are taken into account to study
migration between bins in selected spectra in the subsequent unfolding
procedure.

|
---|---
Figure 8: The migration of events between the detector and the particle levels
between the topologies (left) and an example of the migration of events for
the transverse momentum of the $t\bar{t}$ system $p_{\mathrm{T,t\bar{t}}}$
between the resolved (R), semi-boosted mixed (SBM), semi-boosted (SB) and
boosted (B) topologies (right) for the sample with $M_{\mathrm{y_{0}}}=700$
GeV. The bin range for each of the sub-spectrum is 0–1000 GeV.

##  6 Results

The results are summarized in this section consisting of the resolution of
different samples, the performance of the unfolding process for selected
variables in dedicated stacked samples; and the significance studies of a
Beyond the Standard Model (BSM) signal over the Standard Model background
before and after unfolding.

###  6.1 Resolution of the ${t\bar{t}}{}$ resonance mass peak

As expected for the $y_{\mathrm{0}}\rightarrow t\bar{t}$ BSM process, the
reconstructed $t\bar{t}$ mass ($M_{\mathrm{t\bar{t}}}$) spectrum peaks around
the value of the mass of the studied hypothetical particle $y_{0}$ by
construction. The width of the distribution is a measure of the resolution of
the $t\bar{t}$ resonance mass in given topology. A Gaussian curve was used to
determine its width at both the detector and the particle levels and is shown
in Fig. 9 as absolute (top) as well as relative (bottom), i.e. when divided by
the $y_{0}$ mass in the corresponding sample.

|
---|---
Figure 9: The comparison of the $t\bar{t}$ system mass resolution for the
samples with different masses of the hypothetical particle $y_{0}$ in the
particular topologies the absolute (left) and relative (right) resolution with
respect to $M_{\mathrm{y_{0}}}$. The horizontal shift in the position of
markers around each mass point in the right plot is on the purpose to avoid
the loss of information due to their overlap.

The relative resolution is comparable between all the topologies with the
exception for the semi-boosted mixed topology at the detector level where the
resolution is slightly worse, being approximately 15% while the resolution of
the other topologies is 9–13%.

###  6.2 The unfolding procedure and significance tests

The unfolding procedure corrects for finite detector resolution effects in the
measured spectra. The Fully Bayesian Unfolding (FBU) [28] was chosen for the
purposes of this procedure as implemented with the PyMC3 package [29]. In
short, FBU uses the Bayesian theorem to estimate the truth (here particle)
level spectrum from the measured detector-level spectrum using the migration
matrix. As such, the matrix must be evaluated using simulated events and is
normalized so that it describes the migration of events in a selected spectrum
from particle-level to detector level bins. FBU returns a binned posterior
corresponding to the estimated probability distribution of the variable in
each particle-level bin. Its maximum can be chosen as the truth-level estimate
for the observable in given bin. Among the main advantages of this method are
that the migration matrix does not need to be inverted (which is a numerically
unstable task) nor the problems is regularized and therefore becoming modified
as e.g. in the singular value decomposition method [30]. Further advantages
are the absence of iterations (and a need to terminate them at some point) as
in the iterative Bayesian unfolding [31]; and the full control over the result
as the full probability density is revealed in each bin.

In general, the unfolding process can be described by the following formula

$\hat{T}_{\mathrm{i}}=\frac{1}{f_{\mathrm{i,eff}}}M_{\mathrm{ij}}^{-1}f_{\mathrm{j,acc}}(D_{\mathrm{j}}-B_{\mathrm{j}})\,,$
(3)

where $\hat{T}_{i}$ is the estimate of the particle level spectrum in bin $i$,
$f_{\mathrm{i,eff}}$ is the efficiency correction, $M_{\mathrm{ij}}^{-1}$
stands for the main unfolding procedure using the migration matrix555Fully
Bayesian Unfolding does not use the inverted migration matrix for the
estimation of the particle level spectra, the notation in the formula stands
for a shorthand of the unfolding procedure in general. $M_{\mathrm{ij}}$,
$f_{\mathrm{j,acc}}$ is the acceptance correction in detector level bin $j$,
$D_{\mathrm{j}}$ is the measured detector level spectrum and $B_{\mathrm{j}}$
is the estimated background. The efficiency and acceptance correction factors
are defined as

$f_{\mathrm{i,eff}}=\frac{P_{\mathrm{t\bar{t},i}}^{\mathrm{match}}}{P_{\mathrm{t\bar{t},i}}}\quad\mathrm{and}\quad
f_{\mathrm{j,acc}}=\frac{D_{\mathrm{t\bar{t},j}}^{\mathrm{match}}}{D_{\mathrm{t\bar{t},j}}}\,,$
(4)

where, in the context of this study, $P_{\mathrm{t\bar{t},i}}$ is the
particle-level spectrum in bin $i$ using the $t\bar{t}$ sample as the model
process, while $P_{\mathrm{t\bar{t},i}}^{\mathrm{match}}$ is the particle
level spectrum in bin $i$ for events matched to the detector level events,
_i.e._ only events reconstructed at both particle and detector levels in the
same topology contribute to this spectrum. Similarly,
$D_{\mathrm{t\bar{t},j}}$ is the detector-level spectrum in bin $j$ and
$D_{\mathrm{t\bar{t},j}}^{\mathrm{match}}$ is the detector level spectrum in
bin $j$ for events matched to the particle level. Both correction factors are
in the range between 0 and 1 by definition and were prepared using a
statistically independent $t\bar{t}$ sample w.r.t. the ${t\bar{t}}{}$ sample
used to compose the spectra to be unfolded.

The test spectra entering the unfolding procedure have the addition of the
$y_{0}$ signal sample with an amplified cross section by an ad hoc number of
$10^{3}$ ($5\cdot 10^{3}$ in case of resolved topology) to study the impact of
the unfolding on the strength of a well-present signal of similar significance
in all topologies.

###  6.3 Unfolding selected spectra

Unfolding of three selected spectra is described in this section on the
example in the semi-boosted topology, namely the transverse momentum of the
hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$), the
invariant mass of the reconstructed top anti-top quark pair
($M_{\mathrm{t\bar{t}}}$) and the production angle of the hadronically
decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$).

The simulated samples were divided into two statistically independent halves.
The sum of first such halves is taken as the pseudo-data serving as input into
the unfolding procedure while their statistically independent counterparts are
stacked to form the expected total prediction.

The comparison between the pseudo-data (black markers) and stacked histograms
of the prediction (filled) for the transverse momentum of the hadronically
decaying top quark in the semi-boosted topology is shown in Fig. 10 (left).
The histogram and the stacked spectra agree well, their difference is within
the statistical uncertainties. The binning was selected to respect the
resolution, falling statistics, and so that the unfolding proceeds fast enough
but still delivers useful information about spectra shape. The $t\bar{t}$
system invariant mass spectrum was chosen due to the possibility to see the
hints of events from the sample with the hypothetical particle $y_{0}$. The
spectrum entering the unfolding procedure and its statistically independent
counterpart is shown in Fig. 10 (right).

The production angle of the hadronically decaying top quark was chosen to
study the performance of this variable against the different topologies as an
example of an angular variable. The general assumption is that boosted
topology has jets preferably in smaller $|\eta|$ range due to the usually
larger transverse momentum of the particles. This trend of the central $\eta$
preference drops in semi-boosted topologies and diminishes in the resolved
topology. The spectrum used in unfolding and its statistically independent
counterpart is also shown in Fig. 10 (bottom).

The corrections used in the unfolding procedure were evaluated with the
statistically independent sample from the one forming the pseudo-data using
the $t\bar{t}$ sample. The corresponding acceptance and efficiency corrections
are shown in Fig. 11 and the migration matrices are presented in Fig. 12.

The unfolding results for the three spectra are shown in Fig. 13. A $\chi^{2}$
test was computed between the unfolded and the $t\bar{t}$ particle level
spectra with the $y_{0}$ signal included, resulting in
$\chi^{2}_{\mathrm{t\bar{t}+y_{0}}}$. As the input pseudo-data do contain the
$y_{0}$ signal, this comparison is a closure test of the unfolding procedure
ability to recover the particle level spectrum which consists of
${t\bar{t}}{}$ as well as the $y_{0}$ signal sample. Another $\chi^{2}$ test
was performed between the unfolded spectrum and the $t\bar{t}$-only particle
level spectrum, resulting in $\chi^{2}_{\mathrm{t\bar{t}}}$. This tests
evaluates the incompatibility of the unfolded pseudo-data with the
${t\bar{t}}$-only hypothesis. The middle panels of Fig. 13 show the ratio of
the unfolded spectra over the particle level spectra from the $t\bar{t}$
sample (black full markers)where the disagreement with the $t\bar{t}$-only
particle level spectrum is caused by the presence of the $y_{0}$ sample in the
pseudo-data.

|
---|---
Figure 10: Comparison of the detector level spectra from two statistically independent parts (full markers and filled stack) for the $t\bar{t}$ sample with the addition of $Wbb$ and $WWbb$ backgrounds and an admixture of events from the sample with $M_{\mathrm{y_{0}}}=700$ GeV for the transverse momentum of the hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, left), for the top anti-top quark pair invariant mass ($M_{\mathrm{t\bar{t}}}$, right) and for the crossing angle of the hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom), all spectra are reconstructed in the semi-boosted topology. The hatched bands in the top panel represent the statistical uncertainty in each sample. |
---|---
Figure 11: The acceptance and the efficiency for the two statistically independent $t\bar{t}$ samples for the reconstructed hadronically decaying top quark transverse momentum ($p_{\mathrm{T}}^{\mathrm{t,had}}$, left), the invariant mass of the reconstructed $t\bar{t}$ system ($M_{\mathrm{t\bar{t}}}$, right) and for the production angle of the hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom). All spectra are reconstructed in the semi-boosted topology. Indices 1 and 2 denote the two statistically independent samples. |
---|---
Figure 12: The migration matrices for the reconstructed transverse momentum of the hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, left) for the reconstructed invariant mass of $t\bar{t}$ system ($M_{\mathrm{t\bar{t}}}$, right) and for the crossing angle of the hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom), all in the semi-boosted topology. All matrices were derived from the $t\bar{t}$ sample and used in the unfolding procedure. The correlation factor $\rho$ is calculated between the detector and particle levels. |
---|---
Figure 13: The comparison between the unfolded spectrum (black full markers),
the detector level spectrum (blue) with the acceptance and efficiency
correction applied to be comparable to the particle level one; and the
$t\bar{t}$-only particle level spectrum (red) for the transverse momentum of
the hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, top
left), for the invariant mass of the reconstructed $t\bar{t}$ spectrum
($M_{\mathrm{t\bar{t}}}$, top right) and for the production angle of the
hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom),
all in the semi-boosted topology. The $\chi^{2}_{\mathrm{t\bar{t}}}$ test is
performed between the unfolded and $t\bar{t}$ particle level spectra while
$\chi^{2}_{\mathrm{t\bar{t}+y_{0}}}$ between the unfolded and the $t\bar{t}$
particle level spectra with the $y_{0}$ signal included (closure test). The
middle panels show the ratio of the unfolded spectrum over the particle level
spectrum of the $t\bar{t}$ sample (full markers). Here the disagreement with
the $t\bar{t}$-only particle level spectrum is caused by the addition of the
$y_{0}$ signal before unfolding, the yellow band shows the statistical
uncertainty in the particle level spectrum from the $t\bar{t}$ sample. The
bottom panels show the detector (open markers, dashed line) and unfolded (full
markers, solid line) $y_{0}$ signal significances in each bin. The detector-
level $y_{0}$ signal significance is calculated using the original detector-
level spectrum, _i.e._ without applying the acceptance and efficiency
correction.

###  6.4 Significance at the detector and unfolded levels

The strength of the $y_{0}$ signal is quantified by the significance which
considers the total statistical uncertainties in samples used in given bin.
The significance $S$ in bin $i$ before unfolding is defined as

$S_{\mathrm{i,det}}\equiv(P_{\mathrm{i}}^{t\bar{t}+y_{0}+B}-T_{\mathrm{i}}^{t\bar{t}}-B_{\mathrm{i},1}-\ldots-
B_{\mathrm{i},k})/\sqrt{\sigma_{P_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}+\sigma_{B_{\mathrm{i},1}}^{2}+\ldots}\,,$
(5)

where $P_{\mathrm{i}}$ is the number of the the pseudo data events consisting
from the signal and the background added to the expected $t\bar{t}$ sample in
bin $i$; $T_{\mathrm{i}}$ is the detector level spectrum from the
statistically independent $t\bar{t}$ sample, $B_{\mathrm{i},k}$ is the
background contribution to the studied spectra from the $k$-th background
sample; and $\sigma_{P_{\mathrm{i}}}$, $\sigma_{T_{\mathrm{i}}}$ and
$\sigma_{B_{\mathrm{i},k}}$ are the statistical uncertainties in the pseudo
data, $t\bar{t}$ and the $k$-th background samples, respectively (all at the
detector level). The composition of the sample is denoted in the upper index,
_e.g._ $t\bar{t}+y_{0}+B$ stands for the sum of the background, $t\bar{t}$ and
the $y_{0}$ samples. Systematic uncertainties are not part of this study as
their effect would be largely coherent across topologies, thus not changing
the conclusions nor hierarchy of the observed patterns.

A similar significance is defined after the unfolding procedure, before which
the background was subtracted, as

$S_{\mathrm{i,unf}}\equiv(U_{\mathrm{i}}^{t\bar{t}+y_{0}}-T_{\mathrm{i}}^{t\bar{t}})/\sqrt{\sigma_{U_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}}\,,$
(6)

where $U_{\mathrm{i}}$ is the number of the unfolded pseudo-data events in bin
$i$, $T_{\mathrm{i}}$ is the particle level spectrum from the statistically
independent $t\bar{t}$ sample; and $\sigma_{U_{\mathrm{i}}}$ and
$\sigma_{T_{\mathrm{i}}}$ are the statistical uncertainties in the unfolded
spectrum bin $i$ and in the statistically independent $t\bar{t}$ sample at the
particle level, respectively. The binned detector and unfolded significances
are shown under the ratio plots of the unfolded spectra in Fig. 13. The
integral significance, strength of the signal over the whole spectrum, is
defined similarly for both the detector and the unfolded level. The detector
level integral significance is defined as

$S_{\mathrm{I,det}}\equiv\sum_{i=0}^{m}(P_{\mathrm{i}}^{t\bar{t}+y_{0}+B}-T_{\mathrm{i}}^{t\bar{t}}-B_{\mathrm{i},1}-\ldots-
B_{\mathrm{i},k})/\sqrt{\sum_{i=0}^{m}(\sigma_{P_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}})\,,$
(7)

where $m$ is number of bins in given spectrum. The detector level integral
significance is the same for all variables.

The integral significance at the unfolded level is defined as

$S_{\mathrm{I,unf}}\equiv\sum_{i=0}^{m}(U_{\mathrm{i}}^{t\bar{t}+y_{0}}-T_{\mathrm{i}}^{t\bar{t}})/\sqrt{\sum_{i=0}^{m}(\sigma_{U_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}})\,.$
(8)

The unfolded integral significance varies slightly over spectra as in the
unfolding procedure the integral of the spectrum may not be preserved. The
values of both the detector and the unfolded integral significance are
presented in the legend in Fig. 14.

The significances were calculated for the three selected spectra in all
topologies and at both detector and particle levels. The comparison between
significances before and after the unfolding procedure over the studied
topologies are shown in Fig. 14 for the spectra of
$p_{\mathrm{T}}^{\mathrm{t,had}}$, $M_{\mathrm{t\bar{t}}}$ and
$\cos\theta^{*}_{\mathrm{t,had}}$.

The significance uncertainties were estimated by using 100 pseudo experiments
for each spectrum with a smeared content in each detector-level bin. The
smearing was performed by drawing a random number from the Gaussian
distribution within the $\sigma$ parameter equal to the statistical
uncertainty in the total detector-level spectrum in the given bin and with the
mean parameter set to zero. Each such spectrum was unfolded using the same
procedure and corrections and the binned significances were evaluated. The
resulting standard deviation of significances in each bin is considered as the
statistical uncertainty in the unfolded significance. The statistical
uncertainty of the significances is already presented as error bars in Fig.
14.

|
---|---
Figure 14: The detector (open markers, dashed line) and unfolded (full
markers, solid line) significances for the transverse momentum of the
hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, top left),
invariant mass of the reconstructed $t\bar{t}$ pair ($M_{\mathrm{t\bar{t}}}$,
top right) and the crossing angle of the hadronically decaying top quark
($\cos\theta^{*}_{\mathrm{t,had}}$, bottom) plotted for all the topologies in
each bin. The orange band defines the area where the absolute value of the
significance is below three, corresponding to the 3-$\sigma$ interval. The
lower pads present the ratios of the unfolded over the detector level
significances, without uncertainties which are highly correlated.

The signal significances in the $p_{\mathrm{T}}^{\mathrm{t,had}}$ spectrum
peaks at different values of $p_{\mathrm{T}}^{\mathrm{t,had}}$ depending on
the topology as the event selection in each topology biases the spectrum and
effectively selects different ranges in $M_{\mathrm{t\bar{t}}}$, too. On the
other hand, significance in the $M_{\mathrm{t\bar{t}}}$ spectrum peaks around
the value of the generated $y_{0}$ mass of 700 GeV as expected, with a slight
tail to lower values in the resolved topology which is the least suitable one
to reconstruct a resonance of such a large mass. In contrast, the
$\cos\theta^{*}_{\mathrm{t,had}}$ is very flat also for the signal sample and
there is no clear isolated excess of signal events in this spectrum, with the
exception of the boosted topology which selects, by construction,
high-${p_{\rm T}}$ large jets and thus also top quarks, consequently more
localized in the central rapidity region, producing a pear around zero in
$\cos\theta^{*}_{\mathrm{t,had}}$.

The three selected spectra are thus good candidate observables to illustrate
different behavior and spread of significances over bins, also presenting a
selection of observables of a dimension of energy as well as dimensionless
(angular). The binned significances are in general lower after the unfolding,
for which an explicit proof is delivered by this study. The cause of this is
as follows.

While a sharper spectrum may be recovered by unfolding, the procedure in
general correlates information among bins by maximizing a likelihood function
in case of FBU, or minimizing (possibly modified and regularized) $\chi^{2}$
or iterating and sequentially improving the result for the case of other
methods. An increase in the correlation across bins of the unfolded spectrum
is a known and important fact and a correlation matrix should preferably be
published along with unfolded spectra from real experiments, as done _e.g._ in
[32, 33] where a correlation matrix between the observables was also
evaluated. We observe that in case of the FBU method the posteriors variance
usually increases, leading to larger absolute as well as relative
uncertainties of the unfolded spectrum w.r.t. the particle level one. This
effectively decreases the significance of the observed signal excess. An
increase of the statistical uncertainties with the number of iterations in
case of the Iterative Bayesian Unfolding [31] was also reported by other
analyses [34]. We note that in our case the BSM signal significances decrease
by 20–40%. Other more explicit regularization methods like the SVD [30]
provide a spectrum with a smaller statistical uncertainty by definition
(effectively ditching small regularized response matrix eigenvalues which
would lead to large variations), but are prone to unfolding biases towards the
underlying simulation spectrum. Also the FBU extension with a regularization
term leads to more narrow posteriors (smaller statistical uncertainty) [35].
This places the standard FBU (without an explicit regularization term) among
high-level unfolding methods with realistic statistical uncertainties.

##  7 Conclusions

The results of the semi-boosted and semi-boosted mixed reconstruction
algorithm show potential to enhance the number of events in the $t\bar{t}$
analyses in the semi-leptonic decay channel. The estimates show the enrichment
in events between 20% and 50% in the $t\bar{t}$ pair mass region ranging from
500 GeV to 1000 GeV. The resolution in the semi-boosted topology and the
resolved or boosted topology is comparable, only the semi-boosted mixed has a
worse resolution roughly by factor of 1.5. The performance of the unfolding
procedure including a simple background model shows results corresponding well
to the particle level and the significance of the enhanced signal of the
hypothetical $y_{0}$ particle is still visible after the unfolding. Values of
the detector and the unfolded integral significance are comparable, yet there
is 20–30% decrease in significance between the detector and the unfolding
levels caused by the unfolding in the reconstructed $t\bar{t}$ mass spectrum
and 5–20% in the reconstructed transverse momentum of the hadronically
decaying top quark spectrum and the production angle of the top quarks, i.e.
both for energy-dependent as well as angular variables.

The concrete proof of diminishing a BSM signal significance by the unfolding
procedure (using FBU) is, to our knowledge, shown explicitly for the first
time in this study. Our findings thus support why the model-dependent searches
for new physics at the LHC are mostly done at the detector level, using
concrete detector-level (fully-simulated) BSM signals. We attribute the
decrease of the significance to the increase of the statistical uncertainty in
the unfolded spectrum, _i.e._ to the posterior widening in the case of the FBU
method. Due to this, although the $M_{{t\bar{t}}}$ spectrum becomes sharper
after unfolding, revealing a narrower peak of the $y_{0}$ resonance, the
binned significance does not increase due to larger statistical uncertainties
of the unfolded spectrum caused by unfolding-induced correlation across bins.
On the other hand, the significance stays substantial even after unfolding,
opening doors to a comparison of any theory prediction at the particle-level
to the unfolded data.

The described algorithm proves that semi-boosted and semi-boosted mixed
topologies are sensitive to the possible presence of BSM signals. The
selection criteria chosen close to those in real analyses make the studied
algorithms applicable also in current LHC experiments.

##  8 Acknowledgments

The authors gratefully acknowledge the support from the Czech Science
Foundation project GAČR 19-21484S and project IGA_PrF_2021_004 of the Faculty
of Science of the Palacky University Olomouc, Czech Republic. This work was
performed as a part of fulfillment of doctoral studies of J. Pacalt at the
Applied physics programme at the Faculty of Science of the Palacky University.

## References

  * [1] G. Aad et al.  The ATLAS Experiment at the CERN Large Hadron Collider.  JINST, 3:S08003, 2008.
  * [2] Lyndon R Evans and Philip Bryant.  LHC Machine.  JINST, 3:S08001. 164 p, 2008.  This report is an abridged version of the LHC Design Report (CERN-2004-003).
  * [3] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi.  DELPHES 3, A modular framework for fast simulation of a generic collider experiment.  JHEP, 02:057, 2014.
  * [4] S. Chatrchyan et al.  The CMS Experiment at the CERN LHC.  JINST, 3:S08004, 2008.
  * [5] Particle Data Group.  Review of Particle Physics.  Progress of Theoretical and Experimental Physics, 2020(8), 08 2020\.  083C01.
  * [6] Vardan Khachatryan et al.  Identification techniques for highly boosted W bosons that decay into hadrons.  JHEP, 12:017, 2014.
  * [7] Serguei Chatrchyan et al.  Search for Anomalous $t\bar{t}$ Production in the Highly-Boosted All-Hadronic Final State.  JHEP, 09:029, 2012.  [Erratum: JHEP 03, 132 (2014)].
  * [8] Serguei Chatrchyan et al.  Search for Resonant $t\bar{t}$ Production in Lepton+Jets Events in $pp$ Collisions at $\sqrt{s}=7$ TeV.  JHEP, 12:015, 2012.
  * [9] Georges Aad et al.  Identification of boosted, hadronically decaying W bosons and comparisons with ATLAS data taken at $\sqrt{s}=8$ TeV.  Eur. Phys. J. C, 76(3):154, 2016.
  * [10] Morad Aaboud et al.  Measurement of jet-substructure observables in top quark, $W$ boson and light jet production in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector.  JHEP, 08:033, 2019.
  * [11] Georges Aad et al.  A new method to distinguish hadronically decaying boosted $Z$ bosons from $W$ bosons using the ATLAS detector.  Eur. Phys. J. C, 76(5):238, 2016.
  * [12] Morad Aaboud et al.  Performance of top-quark and $W$-boson tagging with ATLAS in Run 2 of the LHC.  Eur. Phys. J. C, 79(5):375, 2019.
  * [13] Y. Afik, F. Maltoni, K. Mawatari, P. Pani, G. Polesello, Y. Rozen, and M. Zaro.  DM+$b\bar{b}$ simulations with DMSimp: an update.  In Dark Matter at the LHC 2018: Experimental and theoretical workshop, 11 2018.
  * [14] Chiara Arina, Mihailo Backović, Jan Heisig, and Michele Lucente.  Solar $\gamma$ rays as a complementary probe of dark matter.  Phys. Rev. D, 96(6):063010, 2017.
  * [15] Andreas Albert et al.  Recommendations of the LHC Dark Matter Working Group: Comparing LHC searches for dark matter mediators in visible and invisible decay channels and calculations of the thermal relic density.  Phys. Dark Univ., 26:100377, 2019.
  * [16] Sabine Kraml, Ursula Laa, Kentarou Mawatari, and Kimiko Yamashita.  Simplified dark matter models with a spin-2 mediator at the LHC.  Eur. Phys. J. C, 77(5):326, 2017.
  * [17] Goutam Das, Celine Degrande, Valentin Hirschi, Fabio Maltoni, and Hua-Sheng Shao.  NLO predictions for the production of a spin-two particle at the LHC.  Phys. Lett. B, 770:507–513, 2017.
  * [18] Matthias Neubert, Jian Wang, and Cen Zhang.  Higher-Order QCD Predictions for Dark Matter Production in Mono-$Z$ Searches at the LHC.  JHEP, 02:082, 2016.
  * [19] Mihailo Backović, Michael Krämer, Fabio Maltoni, Antony Martini, Kentarou Mawatari, and Mathieu Pellen.  Higher-order QCD predictions for dark matter production at the LHC in simplified models with s-channel mediators.  Eur. Phys. J. C, 75(10):482, 2015.
  * [20] Olivier Mattelaer and Eleni Vryonidou.  Dark matter production through loop-induced processes at the LHC: the s-channel mediator case.  Eur. Phys. J. C, 75(9):436, 2015.
  * [21] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro.  The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations.  JHEP, 07:079, 2014.
  * [22] Torbjörn Sjöstrand, Stefan Ask, Jesper R. Christiansen, Richard Corke, Nishita Desai, Philip Ilten, Stephen Mrenna, Stefan Prestel, Christine O. Rasmussen, and Peter Z. Skands.  An introduction to pythia 8.2.  Computer Physics Communications, 191:159–177, Jun 2015.
  * [23] Matteo Cacciari, Gavin P Salam, and Gregory Soyez.  The anti-ktjet clustering algorithm.  Journal of High Energy Physics, 2008(04):063–063, Apr 2008.
  * [24] Matteo Cacciari, Gavin P. Salam, and Gregory Soyez.  FastJet User Manual.  Eur. Phys. J., C72:1896, 2012.
  * [25] Benjamin Nachman, Pascal Nef, Ariel Schwartzman, Maximilian Swiatlowski, and Chaowaroj Wanotayaroj.  Jets from Jets: Re-clustering as a tool for large radius jet reconstruction and grooming at the LHC.  JHEP, 02:075, 2015.
  * [26] Expected performance of the ATLAS $b$-tagging algorithms in Run-2.  Technical report, CERN, Geneva, Jul 2015.  All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2015-022.
  * [27] LHC Top WG.  Particle level objects and pseudo-top-quark definitions.  https://twiki.cern.ch/twiki/bin/view/LHCPhysics/ParticleLevelTopDefinitions, 2014\.
  * [28] Georgios Choudalakis.  Fully bayesian unfolding.  https://arxiv.org/abs/1201.4612v4, 2012.
  * [29] John Salvatier, Thomas V Wiecki, and Christopher Fonnesbeck.  Probabilistic programming in python using pymc3.  PeerJ Computer Science, 2:e55, 2016.
  * [30] Andreas Höcker and Vakhtang Kartvelishvili.  Svd approach to data unfolding.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 372(3):469–481, Apr 1996.
  * [31] G. D’Agostini.  Improved iterative bayesian unfolding.  https://arxiv.org/abs/1010.0632, 2010.
  * [32] Georges Aad et al.  Measurements of normalized differential cross sections for $t\bar{t}$ production in pp collisions at $\sqrt{s}=7$ TeV using the ATLAS detector.  Phys. Rev. D, 90(7):072004, 2014.
  * [33] M. Aaboud et al.  Measurements of top-quark pair differential cross-sections in the lepton+jets channel in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector.  JHEP, 11:191, 2017.
  * [34] Georges Aad et al.  Differential top-antitop cross-section measurements as a function of observables constructed from final-state particles using pp collisions at $\sqrt{s}=7$ TeV in the ATLAS detector.  JHEP, 06:100, 2015.
  * [35] P. Baron and J. Kvita.  Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method.  Symmetry, 12:2100, 2020.

##  Appendix A Jet energy scale derivation and closure tests

The jet energy scale (JES) procedure corrects for a finite energy response of
the detector to hadronic final states (jets) as function of their angle in the
detector and energy. The goal is to correct detector-level jet energies to the
particle level. The method to calculate the jet response thus uses information
from Monte Carlo simulation and forms a ratio between value of the jet energy
measured at the simulated detector $E_{\mathrm{det}}$ to the energy of a
angularly matched the particle level jet ($E_{\mathrm{ptcl}}$). In detail,
first a jet at the detector level is chosen, then the particle level jet
candidate with the smallest distance parameter defined as

$\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}<R_{\mathrm{cut}}\,.$ (9)

is chosen as the matching jet, with the $R_{\mathrm{cut}}$ parameter set to
0.2 for small ($R=0.4$) jets and 0.3 for large ($R=1$) jets.

The correction is binned in the detector-level jet pseudorapidity and
transverse momentum

$\mathrm{JES}(\eta,p_{\mathrm{{T}}})=\left\langle\frac{E_{\mathrm{ptcl}}^{\mathrm{jet}}}{E_{\mathrm{det}}^{\mathrm{jet}}}\right\rangle,$
(10)

and so the inverse value of the JES correction is the response of the
detector-level jet. Histograms of the jet response corresponding to different
energy intervals are filled, each fitted by a Gaussian function. The mean of
the fit is plotted against the reconstructed energy and fitted by a polynomial
logarithmic function which is used to interpolate the JES correction to any
energy. The visualization of the derived JES correction in dependence on the
jet ${p_{\rm T}}$ and $\eta$ is in Fig. 15 for small jets (left) and large
jets (right).

|
---|---
Figure 15: The visualization of derived JES correction functions for large
jets (left) and for small jets (right).

A closure test was performed, in which the JES factors are applied to jets on
a statistically independent sample to the one used to derive the JES
corrections, but otherwise generated under the same settings. The correction
is re-derived on this already pre-corrected sample, thus the corrected
detector level jet energy over the matched particle level jet energy is
expected to be around unity by construction.

The closure test results as function of the transverse momentum and
pseudorapidity $\eta$ of jets are shown for both small jets and large jets in
Fig. 16 where the mean deviation from unity is well within 5%.

|
---|---
|
Figure 16: Jet energy correction closure tests for the large jets (right) and
small jets (left) as function of the transverse momentum (top) and $\eta$
(bottom) of jets.

##  Appendix B Top quark and $W$ boson tagging efficiencies

The tagging of the large-$R$ jets originating from the top quark or the $W$
boson is a common practice in high energy physics and was used also in this
paper. In order to evaluate the tagging efficiencies, a comparison to the
generator level information is needed to define a truth jet label as top, $W$
or light otherwise. The angularly closest large-$R$ jet to the direction of
the original top quark was used as a probe for the truth top tagging
efficiency $\varepsilon_{\mathrm{top}}$ which is defined as follows

$\varepsilon_{\mathrm{top}}=\frac{N_{\mathrm{jet,top}}^{\mathrm{match\&tag}}}{N_{\mathrm{jet,top}}^{\mathrm{match}}}\,,$
(11)

where $N_{\mathrm{jet,top}}^{\mathrm{match\&tag}}$ is the number of jets
matched to the original top quark and top-tagged by the tagging technique; and
$N_{\mathrm{jet,top}}^{\mathrm{match}}$ is number of all jets tagged as
possibly originating from top quark, as marked by the algorithm. The truth $W$
boson tagging efficiency is defined in a similar manner. The mistag (fake)
efficiency was also evaluated, which describes the false positivity of the
tagger on jets not originating from the top quark or the $W$ boson. It is
defined by the following formula in case of the top tagging

$\varepsilon_{\mathrm{mis,top}}=\frac{N_{\mathrm{jet,top}}^{\mathrm{\neg
match\&tag}}}{N_{\mathrm{jet,top}}^{\mathrm{\neg match}}}\,,$ (12)

where $\varepsilon_{\mathrm{mis,top}}$ is the mistag efficiency of the top
quark tagger, $N_{\mathrm{jet,top}}^{\mathrm{\neg match\&tag}}$ is the number
of large jets which are not matched to the generator level top quark but are
top-tagged by the tagger; and $N_{\mathrm{jet,top}}^{\mathrm{\neg match}}$ is
the number of all large jets not matched to the generator level top quark. The
$W$ mistag efficiency is calculated in a similar way for the both detector and
particle levels.

Both tag and mistag efficiencies are shown in Fig 17 for the $W$ (left) and
top (right) tagger, studied at both detector and particle levels in dependence
on the transverse momentum of the large jet. The sample for the mistag
efficiency was the production of $2j2b$ events and contained 32.5M events,
with events generated in exclusive ${p_{\rm T}}$ ranges of the leading and
sub-leading jets in order to populate the phase space of higher transverse
momenta.

|
---|---
Figure 17: Tagging (blue) and mistag (red) efficiencies for the $W$ boson
(left) and for the top quark (right) in dependence on the transverse momentum
of the large jet studied on the $t\bar{t}$ sample at both particle and
detector levels.