##  1 Introduction

In this paper we will develop new estimates associated to the evolution of
interacting moving bodies on the mesoscopic scale. Consider a partial
differential equation of the form:

$\displaystyle\partial_{t}f(x,\xi,t)+\xi.\nabla_{x}f(x,\xi,t)$
$\displaystyle=I(f,x,\xi,t)$ (1.1) $\displaystyle f(x,\xi,0)$
$\displaystyle=f_{0}(x,\xi)$

In the equation defined above, $x,\xi\in{\rm I\\!R}^{n}$ and $t\in[0,\infty)$.
We expect $f(x,\xi,t)$ to be non-negative and interpret it as the density or
amplitude of particles with velocity $\xi$ located at $x$. The interaction
$I(f,x,\xi,t)$ is expected to be mesoscopic in the sense defined below:

######  Definition 1.1.

We say $I$ is a mesoscopic interaction if:

$\displaystyle\int_{{\rm I\\!R}^{n}}I(f,x,\xi,t)\ d\xi$ $\displaystyle=0$
$\displaystyle\int_{{\rm I\\!R}^{n}}I(f,x,\xi,t)|\xi|^{2}\ d\xi$
$\displaystyle=0$ $\displaystyle\int_{{\rm I\\!R}^{n}}I(f,x,\xi,t)\xi\ d\xi$
$\displaystyle=0$

The characteristic of the partial differential equation (1.1) initiating from
arbitrary $x,\xi\in{\rm I\\!R}^{n}$ is the line $(x+t\xi,\xi,t)\subset{\rm
I\\!R}^{n}\times{\rm I\\!R}^{n}\times{\rm I\\!R}$. It is possible to interpret
this equation as an infinite dimensional system of ordinary differential
equations which start from the same initial value $f_{0}(x,\xi)$ and evolve
along these characteristics:

$\displaystyle\frac{d}{dt}f(x+t\xi,\xi,t)$ $\displaystyle=I(f,x+t\xi,\xi,t)$
(1.2) $\displaystyle f(x,\xi,0)$ $\displaystyle=f_{0}(x,\xi)$

We will assume that at time zero the total mass, energy and momentum defined
below are finite:

$\displaystyle({\it mass})$ $\displaystyle{\bf M}$ $\displaystyle=\int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}f(x,\xi,0)\ dxd\xi<\ \infty$ (1.3)
$\displaystyle({\it energy})$ $\displaystyle{\bf E}$ $\displaystyle=\int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}f(x,\xi,0)|\xi|^{2}\ dxd\xi<\ \infty$
$\displaystyle({\it momentum})$ $\displaystyle{\bf V}$
$\displaystyle=\int_{{\rm I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}f(x,\xi,0)\xi\
dxd\xi<\ \infty$

For a subset of the following results we will also assume a localization of
mass at time zero in the sense defined below:

$\displaystyle\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}f(x,\xi,0)|x|^{2}dxd\xi<\infty$ (1.4)

One immediate consequence of the equation above and the definition of a
mesoscopic interaction is that, for the solutions of equation (1.1), the total
amount of mass, momentum and energy remain invariant. For example, the
conservation of mass can be proven by the argument below:

$\frac{d}{dt}\int_{{\rm I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}f(x,\xi,t)\
dxd\xi=\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\frac{d}{dt}f(x+t\xi,\xi,t)\ dxd\xi=\\\ \int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}I(f,x+t\xi,\xi,t)\ dxd\xi=\int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}I(f,x,\xi,t)\ dxd\xi=0$

The Boltzmann equation has a similar structure to the equation (1.1) for a
specific interaction $I$. The notation $Q(f,f)(x,\xi,t)$ represents the
interactions of the particles and is called the Boltzmann collision operator
[2, 3]. In the case which this operator represents collisions of hard elastic
spheres it obtains the form below:

$\displaystyle Q(f,f)(x,\xi,t)=\int_{{\rm
I\\!R}^{n}}\int_{S^{n-1}}(f^{{}^{\prime}}f^{{}^{\prime}}_{*}-ff_{*})|{\bf
n}.(\xi-\xi_{*})|d{\bf n}d\xi_{*}$

In the expression above, ${\bf n}$ is the normal unit vector to the $n-1$
dimensional unit sphere $S^{n-1}$ and, as is customary in the field, we used
the notation below to describe the velocity of particles before and after the
collisions:

$\displaystyle\xi^{{}^{\prime}}=\xi-{\bf n}.(\xi-\xi_{*}){\bf n}$
$\displaystyle\xi^{{}^{\prime}}_{*}=\xi_{*}+{\bf n}.(\xi-\xi_{*}){\bf n}$
$\displaystyle f=f(x,\xi,t),\ f_{*}=f(x,\xi_{*},t)$ $\displaystyle
f^{{}^{\prime}}=f(x,\xi^{{}^{\prime}},t),\
f^{{}^{\prime}}_{*}=f(x,\xi^{{}^{\prime}}_{*},t)$

Assume that $\xi$ and $\xi_{*}$ are velocities of two colliding elastic balls
with unit mass, and let ${\bf n}$ or equivalently $-{\bf n}$, represent the
unit normal vector to the plane which uniquely describes the relative position
of these two spheres upon the collision. Then $\xi^{{}^{\prime}}$ and
$\xi_{*}^{{}^{\prime}}$ are the velocities of the particles after the
collision. For a fixed ${\bf n}$, these velocities are unique solutions to the
conservation laws of momentum and energy written below:

$\displaystyle|\xi|^{2}+|\xi_{*}|^{2}$
$\displaystyle=|\xi^{{}^{\prime}}|^{2}+|\xi^{{}^{\prime}}_{*}|^{2}$
$\displaystyle\xi+\xi_{*}$
$\displaystyle=\xi^{{}^{\prime}}+\xi_{*}^{{}^{\prime}}$

As shown by Boltzmann, the collision operator satisfies the conditions of
Definition 1.1 and therefore is a mesoscopic interaction. We will replicate
his argument here. Consider the well-known change of variables below:

$\displaystyle(\xi,\xi_{*})\Longrightarrow(\xi_{*},\xi),\ \ \
(\xi,\xi_{*})\Longrightarrow(\xi^{{}^{\prime}},\xi_{*}^{{}^{\prime}}),\ \ \
(\xi,\xi_{*})\Longrightarrow(\xi^{{}^{\prime}}_{*},\xi^{{}^{\prime}})$

The transformations above are measure preserving. They represent the intrinsic
symmetries of the conservation laws, in which the role between particles as
well as the velocity of particles before and after a collision are
indistinguishable. We will implement these change of variables for the
Boltzmann collision operator and an arbitrary $\phi(\xi)$, we get:

$\int_{{\rm I\\!R}^{n}}Q(f,f)(x,\xi,t)\phi(\xi)\ d\xi=\frac{1}{4}\int_{{\rm
I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\int_{S^{n-1}}(f^{{}^{\prime}}f^{{}^{\prime}}_{*}-ff_{*})\\\
\times\big{(}\phi(\xi)+\phi(\xi_{*})-\phi(\xi^{{}^{\prime}})-\phi(\xi^{{}^{\prime}}_{*})\big{)}|{\bf
n}.(\xi-\xi_{*})|\ d{\bf n}d\xi_{*}d\xi$

Set $\phi(\xi)$ equal to either 1, $|\xi|^{2}$ or $\xi_{i}$ for $1\leq i\leq
n$. It follows from the conservation laws that
$\phi(\xi)+\phi(\xi_{*})-\phi(\xi^{{}^{\prime}})-\phi(\xi^{{}^{\prime}}_{*})=0$.
We conclude that assumptions of Definition 1.1 are satisfied by $Q$ and hence
it is a mesoscopic interaction.

The existence of global classical solutions to the Boltzmann equation for
small data has been shown before [11]. Additionally, there exist a scattering
theory for small data with convergence in the $L^{\infty}$ setting and
asymptotic completeness results [15]. A concept of renormalized solutions
exists that provides a general existence theory of weak solutions for the
Boltzmann equation [7]. The counterpart of this renormalization in the
microscopic scale is unknown. This missing link makes it infeasible to
completely relate these weak solutions to the underlying physical phenomena
which the Boltzmann equation is intended to describe. The existence of global
in time classical solutions to the Boltzmann equation for the large data or
examples of finite time blow up are still important open problems.

The results of this paper are solely based on the conservation laws and are
independent of the specific structure of the interaction, therefore they are
also true for the Boltzmann equation. The kinetic descriptions of conservation
laws has been studied before [12, 13, 14, 17, 19] using methods like velocity
averaging lemmas and the notion of entropy [8, 9, 16]. In this paper we will
not use these conventional methods. We will generalize some of the recent
results in the kinetic theory from single particle estimates to the
interaction or 2-particle estimates. We will start with a short summery of the
existing results appearing in [15]. Consider the definitions below:

$\displaystyle(angular\ momentum)$ $\displaystyle A(t)$
$\displaystyle=\int_{{\rm I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}f(x,\xi,t)x.\xi\
dxd\xi$ (1.5) $\displaystyle(uncertainty)$ $\displaystyle U(t)$
$\displaystyle=\int_{{\rm I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}f(x,\xi,t)|x||\xi|\
dxd\xi$ $\displaystyle(relative\ angular\ norm)$ $\displaystyle\|f\|_{G}$
$\displaystyle=\sup_{t}\big{(}U(t)-A(t)\big{)}$

For any positive solution of the equation (1.1) subject to a mesoscopic
interaction, angular momentum increases linearly:

$\displaystyle A(t)=A(0)+t{\bf E}$

Furthermore, we have:

$\displaystyle\|f\|_{G}\leq\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}f(x,\xi,0)|x|^{2}dxd\xi+{\bf E}-A(0)$ (1.6)

Which implies, the uncertainty increases on average linearly proportional to
the amount of the total energy and goes to infinity with time:

$\displaystyle\lim_{t\rightarrow\infty}U(t)=\infty$

The uncertainty defined above has a physical interpretation with similarities
to its quantum counterpart [15]. Theses results led to a new physical
intuition based on the notion of a blind cone with respect an observer. These
blind cones have been used in [15] to prove that the total energy within any
bounded set of the spatial variable is integrable over time. Assume
$D\subset{\rm I\\!R}^{n}$ is a bounded set of the spatial variable, we have:

$\displaystyle({\it Kinetic\ Morawetz\ estimate})\ \int_{0}^{\infty}\int_{{\rm
I\\!R}^{n}}\int_{D}f(x,\xi,t)|\xi|^{2}\ dxd\xi dt<\infty$ (1.7)

The previous result has analogies to the Morawetz estimate for the non linear
Schrodinger’s equation [18]. The convergence of the time integral in (1.7)
implies that the total mass of particles with velocity greater than any
positive fixed positive number like $v$ is integrable over time as well.
Including more assumptions will further illuminate the physical intuition
behind the estimate. Assume $f$ is bounded and has a bounded derivative with
respect to the time. The previous argument indicates that inside any bounded
set of the spatial variable like $D\subset{\rm I\\!R}^{n}$, as time goes to
infinity, almost every particle with a magnitude of velocity greater than $v$
will inevitably leave the bounded set. Thus, as time goes to infinity, an idle
observer will almost only identify particles with velocity $\xi=0$ inside any
bounded set. The Galilean invariance of the setting will generalize this
argument to bounded moving regions and moving observers. Each particle has a
tendency to travel with other particles whose velocities are indistinguishable
from each other and as a result avoid interaction with particles that move at
different velocities.

In Section 2, we will generalize (1.7) and create the 2-particle or
interaction Morawetz estimate (Theorem 2.1) for the evolution of mesoscopic
interactions. This estimate has analogies to the interaction Morawetz estimate
for non linear Schrodinger’s equation [4, 5, 18, 6] and has physical
interpretations. This estimate implies that, along the evolution of solutions
to the equation (1.1) and as the time increases, only particles with identical
velocities can remain within an arbitrary fixed distance of one another.

In Section 3, we will generalize the notions appearing in (1.5). We will
demonstrate a new a-priori bound (Definition 3.2) for the 2-particle
interactions and introduce a notion of interaction uncertainty (Definition
3.1). We will show that the interaction uncertainty, similar to the single
particle case, will go to infinity with time (Theorem 3.1). These results will
be used in Section 4.

It is possible to use the kinetic Morawetz estimate (1.7) together with the
bound (1.6) for the relative angular norm to prove concentration of mass
overtime in a specific sense [15]. This concentration implies that the
particles will eventually move away from any observer in a radial manner.
Therefore, the angular part of the gradient will vanish relative to any
observer. Consequently, the total mass concentrates, in an averaged sense over
time, within the collection of blind cones with respect to any fixed observer.

In Section 4, we will create an analogy (Theorem 4.1) to the described
concentration of mass for the case of interactions. We will show that the
interactions, averaged over time, will concentrate within a specific
collection of the blind cones (Definition 4.2) with arbitrary small apex
angle. The relation between Morawetz and interaction Morawetz estimates bears
resemblance to the relationship between concentration of mass and
concentration of interactions. The concentration of interactions within the
collection of blind cones implies that, averaged over time, almost every
collision occurs between particles with arbitrary close velocities or
arbitrary small angle of deflection. These type of interactions are often
called the grazing collisions.

### Acknowledgements

I am grateful for the mentorship of Daniel Tataru during the creation of this
work. I want to express my gratitude for useful discussions with Fraydoun
Rezakhanlou and Mehrdad Shahshahani. This paper would not have been written
without the love and support of Amanda Jeanne Tose (AJT), Kaveh and Zahra.

##  2 Interaction Morawetz estimate

In this section we will develop a kinetic interaction Morawetz estimate based
on the conservation laws of the classical physics.

######  Definition 2.1.

Let $A_{L}(t)$ be the 2-particle localized angular momentum associated to
$f(x,\xi,t)$ at time $t$:

$\displaystyle A_{L}(t)=\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\
dxd\xi dx_{0}d\xi_{0}$

######  Remark.

The quantity defined above is bounded over time.

$\displaystyle A_{L}(t)<({\bf M}+{\bf E})^{2}$ (2.1)

It is possible to interpret $A_{L}(t)$ as a measure of dispersion for the
pairs of particles and has been used in different contexts [10, 1]. The single
particle variant of this quantity appears in [15] and justifies the language
used above.

######  Theorem 2.1.

[Kinetic interaction Morawetz estimate] Assume $f$ is a non negative solution
of the equation (1.1) subject to a mesoscopic interaction. Furthermore, assume
the total mass (M) and energy (E) are as in (1.3). There exist a constant $W$
depending only on the total mass and energy such that for any positive $R$ we
have:

$\frac{1}{R}\int_{0}^{\infty}\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\int_{B(x_{0},R)}\int_{{\rm
I\\!R}^{n}}f(x,\xi,t)f(x_{0},\xi_{0},t)|\xi-\xi_{0}|^{2}\ d\xi
dxd\xi_{0}dx_{0}dt\\\ <W({\bf M},{\bf E})$

###### Proof.

We start by differentiating $A_{L}(t)$ with respect to time:

$\displaystyle\frac{d}{dt}A_{L}=\iiiint\limits_{{\rm
I\\!R}^{4n}}\partial_{t}f(x,\xi,t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\
dxd\xi\ f(x_{0},\xi_{0},t)\ dx_{0}d\xi_{0}$ (2.2)
$\displaystyle+\iiiint\limits_{{\rm
I\\!R}^{4n}}\partial_{t}f(x_{0},\xi_{0},t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\
dx_{0}d\xi_{0}\ f(x,\xi,t)\ dxd\xi$

For the inner double integral appearing in the first term of the right hand
side of the equation above we have:

$\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\partial_{t}f(x,\xi,t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\
dxd\xi\\\ +\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\xi.\nabla_{x}f(x,\xi,t)\
dxd\xi\\\ =\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}I(f,x,\xi,t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\ dxd\xi=0$

Therefore:

$\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\partial_{t}f(x,\xi,t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})dxd\xi\\\
=-\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\xi.\nabla_{x}f(x,\xi,t)\
dxd\xi$

Similarly, for the inner double integral appearing in the second term of the
equation (2.2) we get:

$\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\partial_{t}f(x_{0},\xi_{0},t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})dx_{0}d\xi_{0}\\\
=-\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\xi_{0}.\nabla_{x}f(x_{0},\xi_{0},t)\
dx_{0}d\xi_{0}$

We will continue with integration by parts with respect to $x$ for the two
former equations:

$\partial_{t}\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}f(x,\xi,t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\ dxd\xi\\\
=\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\frac{\xi\cdot(\xi-\xi_{0})|x-x_{0}|^{2}-((x-x_{0})\cdot\xi)((x-x_{0})\cdot(\xi-\xi_{0}))}{|x-x_{0}|^{3}}\\\
\times f(x,\xi,t)\ dxd\xi$

Likewise:

$\partial_{t}\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}f(x_{0},\xi_{0},t)\frac{x-x_{0}}{|x-x_{0}|}\cdot(\xi-\xi_{0})\
dx_{0}d\xi_{0}\\\ =\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\frac{-\xi_{0}\cdot(\xi-\xi_{0})|x-x_{0}|^{2}+((x-x_{0})\cdot\xi_{0})((x-x_{0})\cdot(\xi-\xi_{0}))}{|x-x_{0}|^{3}}\\\
\times f(x_{0},\xi_{0},t)\ dx_{0}d\xi_{0}$

Plug in the the computations above in the equation (2.2). We will obtain the
expression below, where $\theta(x,\xi)$ is the angle between the two vectors:

$\frac{d}{dt}A(t)=\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\frac{1}{|x-x_{0}|}|\xi-\xi_{0}|^{2}\\\
\times\sin^{2}{(\theta(x-x_{0},\xi-\xi_{0})})\ d\xi dxd\xi_{0}dx_{0}$

This shows that $A_{L}(t)$ has a positive derivative and is monotone. Since
this quantity is bounded as (2.1), we get:

$\int_{0}^{\infty}\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\frac{1}{|x-x_{0}|}|\xi-\xi_{0}|^{2}\\\
\times\sin^{2}{(\theta(x-x_{0},\xi-\xi_{0})})\ d\xi dxd\xi_{0}dx_{0}dt<({\bf
M}+{\bf E})^{2}$

We will continue by defining a particular kind of cone. Let
$C_{x_{0},\xi_{0}}(x,c)\subset{{\rm I\\!R}^{n}}$ be the blind cone with apex
angle $c>0$ at point $x\in{{\rm I\\!R}^{n}}$ with respect to a moving observer
at $x_{0}\in{{\rm I\\!R}^{n}}$ with velocity $\xi_{0}\in{{\rm I\\!R}^{n}}$:

$\displaystyle C_{x_{0},\xi_{0}}(x,c)=\\{\xi\in{\rm
I\\!R}^{n}\big{|}\theta(x-x_{0},\xi-\xi_{0})\notin[c,\pi-c]\\}$ (2.3)

We will identify this cone as a subset of the space of velocities at point
$x$. By removing the blind cones $C_{x_{0},\xi_{0}}(x,c)$ from the space of
velocities for every point $x$ and using the previous computations we get that
for any positive $R$:

$\displaystyle\frac{\sin^{2}(c)}{R}\int_{0}^{\infty}\int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}\int_{B(x_{0},R)}\int_{{\rm
I\\!R}^{n}-C_{x_{0},\xi_{0}(x,c)}}f(x,\xi,t)f(x_{0},\xi_{0},t)$ (2.4)
$\displaystyle\times|\xi-\xi_{0}|^{2}\ d\xi dxd\xi_{0}dx_{0}dt<({\bf M}+{\bf
E})^{2}$

Now choose any three distinct observers: $O_{1},O_{2},O_{3}\in\partial B(0,R)$
and for an arbitrary $x\in B(0,R)$ define $P=\partial B(0,R)\cap
C_{O_{1},\xi_{0}}(x,2c)$. If there exists an $O_{i}$ such that $O_{i}\notin P$
then the blind cones $C_{O_{i},\xi_{0}}(x,c)$ and $C_{O_{1},\xi_{0}}(x,c)$
have an empty intersection. Set $P$ belongs to $\partial B(0,R)$ and is made
of two path connected components, consider the longest short path on each
component and set $K$ to be the maximum length of the two. For any fixed $R$
it is possible to choose $c$ small enough such that $K$ becomes as small as
desired. Now set $c$ small enough such that $K$ becomes smaller than the
shortest path on the sphere between any two of the there observers. The pigeon
hole principle implies that, since each path connected component of $P$ can
only contain maximum one of the observers, there exists an $O_{i}$ such that
$O_{i}\notin P$. The previous argument implies that for any $x\in B(0,R)$ the
blind cones with respect to the three observers have an empty intersection:

$\displaystyle C_{O_{1},\xi_{0}}(x,c)\cap C_{O_{2},\xi_{0}}(x,c)\cap
C_{O_{3},\xi_{0}}(x,c)=\emptyset$

Therefore, for any $x_{0}\in{\rm I\\!R}^{n}$ we have:

$\displaystyle C_{x_{0}+O_{1},\xi_{0}}(x,c)\cap
C_{x_{0}+O_{2},\xi_{0}}(x,c)\cap C_{x_{0}+O_{3},\xi_{0}}(x,c)=\emptyset$ (2.5)

Consider the definitions below for $J_{1}$, $J_{2}$ and $J_{3}$. The following
bounds are consequences of (2.4):

$J_{1}=\frac{\sin^{2}(c)}{R}\int_{0}^{\infty}\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\int_{B(x_{0}+O_{1},2R)}\int_{{\rm
I\\!R}^{n}-C_{x_{0}+O_{1},\xi_{0}}(x,c)}f(x,\xi,t)\\\ \times
f(x_{0},\xi_{0},t)|\xi-\xi_{0}|^{2}\ d\xi dxd\xi_{0}dx_{0}dt<({\bf M}+{\bf
E})^{2}$ $J_{2}=\frac{\sin^{2}(c)}{R}\int_{0}^{\infty}\int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}\int_{B(x_{0}+O_{2},2R)}\int_{{\rm
I\\!R}^{n}-C_{x_{0}+O_{2},\xi_{0}}(x,c)}f(x,\xi,t)\\\ \times
f(x_{0},\xi_{0},t)|\xi-\xi_{0}|^{2}\ d\xi dxd\xi_{0}dx_{0}dt<({\bf M}+{\bf
E})^{2}$ $J_{3}=\frac{\sin^{2}(c)}{R}\int_{0}^{\infty}\int_{{\rm
I\\!R}^{n}}\int_{{\rm I\\!R}^{n}}\int_{B(x_{0}+O_{3},2R)}\int_{{\rm
I\\!R}^{n}-C_{x_{0}+O_{3},\xi_{0}}(x,c)}f(x,\xi,t)\\\ \times
f(x_{0},\xi_{0},t)|\xi-\xi_{0}|^{2}\ d\xi dxd\xi_{0}dx_{0}dt<({\bf M}+{\bf
E})^{2}$

From (2.5) we know that it is possible to choose the apex angle $c$ small
enough such that for any point $x\in B(x_{0},R)$ the blind cones with respect
to the three observers $x_{0}+O_{1}$, $x_{0}+O_{2}$ and $x_{0}+O_{3}$
intersect trivially. Also consider that:

$\displaystyle B(x_{0},R)\subset B(x_{0}+O_{1},2R)\cap B(x_{0}+O_{2},2R)\cap
B(x_{0}+O_{3},2R)$

This shows that any subset of $B(x_{0},R)\times{\rm I\\!R}^{n}$ for all
$x_{0}\in{\rm I\\!R}^{n}$ is covered at least once in the domains of
integration for $J_{1},J_{2}$ and $J_{3}$. Therefore the positivity of
integrands completes the proof, there exists some constant $W({\bf M},{\bf
E})$ such that:

$\frac{1}{R}\int_{0}^{\infty}\int_{{\rm I\\!R}^{n}}\int_{{\rm
I\\!R}^{n}}\int_{B(x_{0},R)}\int_{{\rm
I\\!R}^{n}}f(x,\xi,t)f(x_{0},\xi_{0},t)|\xi-\xi_{0}|^{2}\ d\xi
dxd\xi_{0}dx_{0}dt\\\ <J_{1}+J_{2}+J_{3}<W({\bf M},{\bf E})$

∎

######  Remark.

An illustration of the blind cones with respect to an idle observer
($\xi_{0}$=0) at the origin ($x_{0}=0$) can be found in [15].

##  3 New a priori bound and the interaction uncertainty

In this section we will find a new a-priori bound for the evolution of the
mesoscopic interactions. We will introduce the notion of uncertainty
associated to the interactions and demonstrate that as time goes to infinity
the interaction uncertainty goes to infinity as well. These results and the
kinetic interaction Morawetz estimate (Theorem 2.1) will be used in Section 4
to obtain a concentration result for the interactions.

######  Lemma 3.1.

Assume $f$ is a solution of the equation (1.1) subject to a mesoscopic
interaction and that the assumptions (1.3) and (1.4) are true. We have:

$\iiiint\limits_{{\rm I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\ =\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\ (x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi
dx_{0}d\xi_{0}\\\ +t\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\ |\xi-\xi_{0}|^{2}\ dxd\xi
dx_{0}d\xi_{0}$

###### Proof.

Start with the change of variables $x\rightarrow x+t\xi$ and $x_{0}\rightarrow
t\xi_{0}$:

$\iiiint\limits_{{\rm I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\
=\Big{(}\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\ +t\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\ |\xi-\xi_{0}|^{2}\
dxd\xi dx_{0}d\xi_{0}\Big{)}$

Continue with the computation below:

$\frac{d}{dt}\iiiint\limits_{{\rm I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\ =2\iiiint\limits_{{\rm
I\\!R}^{4n}}I(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\ +2t\iiiint\limits_{{\rm
I\\!R}^{4n}}I(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\ |\xi-\xi_{0}|^{2}\
dxd\xi dx_{0}d\xi_{0}\\\ +\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\ |\xi-\xi_{0}|^{2}\
dxd\xi dx_{0}d\xi_{0}$

We will compute each of the three terms appearing in the former equation.
Because $I$ is a mesoscopic interaction (Definition 1.1), for the first term
we have:

$2\iiiint\limits_{{\rm I\\!R}^{4n}}I(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\ =2\iiiint\limits_{{\rm
I\\!R}^{4n}}I(x,\xi,t)f(x_{0},\xi_{0},t)\
(x-x_{0}+t(\xi_{0}-\xi)\cdot(\xi-\xi_{0})))d\xi dxdx_{0}d\xi_{0}=0$

Similarly for the second term we get:

$2\iiiint\limits_{{\rm
I\\!R}^{4n}}I(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)|\xi-\xi_{0}|^{2}\
dxd\xi dx_{0}d\xi_{0}\\\ =2\iiiint\limits_{{\rm
I\\!R}^{4n}}I(x,\xi,t)f(x_{0},\xi_{0},t)\ |\xi-\xi_{0}|^{2}d\xi
dxdx_{0}d\xi_{0}=0$

Finally we will compute the third term:

$\iiiint\limits_{{\rm I\\!R}^{4n}}f(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\
|\xi-\xi_{0}|^{2}\ dxd\xi dx_{0}d\xi_{0}\\\ =\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\ |\xi-\xi_{0}|^{2}\ dxd\xi
dx_{0}d\xi_{0}\\\ +2\iiiint\limits_{{\rm
I\\!R}^{4n}}\int_{0}^{t}I(x+s\xi,\xi,s)f(x_{0},\xi_{0},0)\ |\xi-\xi_{0}|^{2}\
dsdxd\xi dx_{0}d\xi_{0}\\\ +\iiiint\limits_{{\rm
I\\!R}^{4n}}\int_{0}^{t}\int_{0}^{t}I(x+s\xi,\xi,s)I(x_{0}+z\xi_{0},\xi_{0},z)\
|\xi-\xi_{0}|^{2}\ dsdzdxd\xi dx_{0}d\xi_{0}$

As a consequence of Definition 1.1, after a change of variables and changing
the order of integrations, we conclude that the last two terms of the right
hand side of the equation above are zero. Therefore we complete the proof:

$\frac{d}{dt}\iiiint\limits_{{\rm I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\
(x-x_{0})\cdot(\xi-\xi_{0})\ dxd\xi dx_{0}d\xi_{0}\\\ =\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\ |\xi-\xi_{0}|^{2}\ dxd\xi
dx_{0}d\xi_{0}$

∎

######  Definition 3.1.

Assume $f$ is a non negative solution of the equation (1.1). Let $U_{I}(t)$ be
the interaction uncertainty associated to $f$ at time $t$ defined as:

$\displaystyle U_{I}(t)=\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)|x-x_{0}||\xi-\xi_{0}|dxd\xi
dx_{0}d\xi_{0}$

######  Remark.

It is possible to interpret the interaction uncertainty defined above
analogous to its quantum mechanics counterpart, that is by bringing attention
to a fundamental limit on the precision of the physical measurements. Consider
an idle observer observer located at the origin and assume the speed of light
is $C$. For each particle located at $x$ there is a minimum delay of
$T=C^{-1}|x|$ between the actual time of measurement and observation at the
origin. The quantity $T\times
f(0,0,t)f(x,\xi,t)|x|=C^{-1}f(0,0,t)f(x,\xi,t)|x||\xi|=$ represents the
uncertainty of measurement relative to an idle observer at the origin, due to
this interval of delay. The role of the observer and the particle are
symmetric in the context of interactions. Therefore after including all the
particles and observers, we will get the definition above for $C=1$.

######  Definition 3.2.

Assume $f$ is a non negative solution of the equation (1.1). Let $\|\|_{IG}$
be the interaction relative angular norm associated to $f$ defined as:

$\|f\|_{IG}=\sup_{t}\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\Big{(}|x-x_{0}||\xi-\xi_{0}|\\\
-(x-x_{0})\cdot(\xi-\xi_{0})\Big{)}\ d\xi dxd\xi_{0}dx_{0}$

######  Theorem 3.1.

Assume $f$ is a non negative solution of equation (1.1). Additionally, assume
that the assumptions (1.3) and (1.4) are true. We have:

$\|f\|_{IG}<\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\Big{(}|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}\\\
-(x-x_{0}).(\xi-\xi_{0})\Big{)}d\xi dxd\xi_{0}dx_{0}$

Furthermore, the interaction uncertainty goes to infinity with time:

$\displaystyle\lim_{t\rightarrow\infty}U_{I}(t)=\infty$

###### Proof.

Implement the change variables $x\rightarrow x+t\xi$ and $x_{0}+t\xi_{0}$
followed by the triangle inequality, we get:

$\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\Big{(}|x-x_{0}||\xi-\xi_{0}|-(x-x_{0})\cdot(\xi-\xi_{0})\Big{)}\
d\xi dxd\xi_{0}dx_{0}\\\ \leq\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\Big{(}|x-x_{0}||\xi-\xi_{0}|+t|\xi-\xi_{0})|^{2}\\\
-(x-x_{0}+t(\xi-\xi_{0}))\cdot(\xi-\xi_{0})\Big{)}\ d\xi dxd\xi_{0}dx_{0}$

Therefore, using the non negativity of $f$ we have:

$\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\Big{(}|x-x_{0}||\xi-\xi_{0}|-(x-x_{0})\cdot(\xi-\xi_{0})\Big{)}\
d\xi dxd\xi_{0}dx_{0}\\\ <\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x+t\xi,\xi,t)f(x_{0}+t\xi_{0},\xi_{0},t)\Big{(}|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}\\\
-(x-x_{0}).(\xi-\xi_{0})\Big{)}d\xi dxd\xi_{0}dx_{0}$

We will use the structure of the equation in (1.2) and integrate the
interactions along the characteristics. The estimate above leads to the
following inequality:

$\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)\Big{(}|x-x_{0}||\xi-\xi_{0}|-(x-x_{0})\cdot(\xi-\xi_{0})\Big{)}\
d\xi dxd\xi_{0}dx_{0}\\\ <\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\Big{(}|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}\\\
-(x-x_{0}).(\xi-\xi_{0})\Big{)}d\xi dxd\xi_{0}dx_{0}+Z_{1}+Z_{2}$

In the expression above $Z_{1}$ and $Z_{2}$ are defined as:

$Z_{1}=2\iiiint\limits_{{\rm
I\\!R}^{4n}}\int_{0}^{t}f(x,\xi,0)I(x_{0}+s\xi_{0},\xi_{0},s)\Big{(}|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}\\\
-(x-x_{0}).(\xi-\xi_{0})\Big{)}dsd\xi dxd\xi_{0}dx_{0}$
$Z_{2}=\iiiint\limits_{{\rm
I\\!R}^{4n}}\int_{0}^{t}\int_{0}^{t}I(x+s\xi,\xi,s)I(x_{0}+z\xi_{0},\xi_{0},z)\Big{(}|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}\\\
-(x-x_{0}).(\xi-\xi_{0})\Big{)}dsdzd\xi dxd\xi_{0}dx_{0}$

Using Definition 1.1, we conclude that $Z_{1}$ and $Z_{2}$ are zero. Therefore
we complete the proof of the first part of the theorem:

$\|f\|_{IG}<\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,0)f(x_{0},\xi_{0},0)\Big{(}|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}\\\
-(x-x_{0})\cdot(\xi-\xi_{0})\Big{)}\ d\xi dxd\xi_{0}dx_{0}$

We just proved that $\|f\|_{IG}$ is bounded, therefore as a consequence of
Lemma 3.1 and Definition 3.2, we conclude that interaction uncertainty goes to
infinity with time and complete the proof of the theorem:

$\displaystyle\lim_{t\rightarrow\infty}U_{I}(t)=\infty$

∎

##  4 Concentration of interactions

In this section we will use the bound we found for $\|f\|_{IG}$ from Section 3
along with the kinetic interaction Morawetz estimate (Theorem 2.1) to
demonstrate a result about the concentration of 2-particle interactions over
time.

Recall the notion of a blind cone defined in (2.3). We will define a punctured
blind cone by including the velocities that the magnitude of differences
between them and some fixed $\xi_{0}$ are less than some positive constant:

######  Definition 4.1.

Let $C_{x_{0},\xi_{0}}(x,c)$ be as defined in (2.3). Define
$K_{x_{0},\xi_{0}}(x,c)\subset{\rm I\\!R}^{n}$ as the punctured blind cone at
point $x$ with respect to the observer $x_{0}$ that moves with the velocity
$\xi_{0}$ for a fixed positive $v$:

$\displaystyle K_{x_{0},\xi_{0}}(x,c,v)=C_{x_{0},\xi_{0}}(x,c)\cup
B(\xi_{0},v)$

######  Definition 4.2.

Let $\Gamma_{x_{0},\xi_{0}}(c,v)\subset{\rm I\\!R}^{4n}$ be the collection of
punctured blind cones over the spatial variable for some positive $c$ and $v$.

$\displaystyle\Gamma(c,v)=\\{(x_{0},\xi_{0},x,\xi)\in{\rm I\\!R}^{n}\times{\rm
I\\!R}^{n}\times{\rm I\\!R}^{n}\times{\rm I\\!R}^{n}|\ \xi\in
K_{x_{0},\xi_{0}}(x,c,v)\\}$

######  Remark.

An illustration of $\Gamma(c,v)$ for the specific case of an idle observer
($\xi_{0}=0$) at the origin ($x_{0}=0$) can be found in [15].

As a consequence of the conservation of mass, we have that the the total mass
of 2-particle interactions remains constant overtime, therefore the following
average is true:

$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)d\xi dxd\xi_{0}d\xi_{0}dt={\bf M^{2}}$
(4.1)

In the theorem below, we will show that these interactions are actually being
concentrated within the collection of punctured blind cones with arbitrary
small $c$ and $v$. $\Gamma(c,v)\subset{\rm I\\!R}^{4n}$ is much smaller than
${\rm I\\!R}^{4n}$ and as the constants $c$ and $v$ go to zero, it converges
to a measure zero subset.

######  Theorem 4.1.

[Concentration of 2-particle interactions] Assume $f$ is a positive solution
of the equation (1.1) and that the assumptions (1.3) and (1.4) are true. The
2-particle interactions, averaged over time, will almost always occur within
the collection of punctured blind cones $\Gamma(c,v)$ for any positive $c$ and
$v$:

$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\iiiint\limits_{\Gamma(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)d\xi
dxd\xi_{0}d\xi_{0}dt={\bf M^{2}}$

###### Proof.

Consider the following two subsets of ${\rm I\\!R}^{4n}$:

$\displaystyle N_{R}(c,v)=\\{(x_{0},\xi_{0},x,\xi)\in{\rm I\\!R}^{n}\times{\rm
I\\!R}^{n}\times{\rm I\\!R}^{n}\times({\rm
I\\!R}^{n}-K_{x_{0},\xi_{0}}(x,c,v))\ |\ \ |x-x_{0}|>R\\}$ $\displaystyle
M_{R}(c,v)=\\{(x_{0},\xi_{0},x,\xi)\in{\rm I\\!R}^{n}\times{\rm
I\\!R}^{n}\times{\rm I\\!R}^{n}\times({\rm
I\\!R}^{n}-K_{x_{0},\xi_{0}}(x,c,v))\ |\ \ |x-x_{0}|\leq R\\}$

It is possible to divide ${\rm I\\!R}^{4n}$ into 3 mutuality exclusive sets
$N_{R},M_{R}$ and $\Gamma(c,v)$. Therefore the equation (4.1) leads to:

$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)d\xi dxd\xi_{0}d\xi_{0}dt\\\
=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\Big{(}\iiiint\limits_{\Gamma(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}\\\
+\iiiint\limits_{M_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}\\\
+\iiiint\limits_{N_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}\Big{)}dt={\bf M^{2}}$

The kinetic interaction Morawetz estimate (Theorem 2.1) implies the estimate
below for the interactions within $M_{R}(c,v)$:

$\displaystyle\int_{0}^{\infty}\iiiint\limits_{M_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}dt<\infty$

Consequently we get:

$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\iiiint\limits_{{\rm
I\\!R}^{4n}}f(x,\xi,t)f(x_{0},\xi_{0},t)d\xi dxd\xi_{0}d\xi_{0}dt\\\
=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\Big{(}\iiiint\limits_{\Gamma(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}\\\
+\iiiint\limits_{N_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}\Big{)}dt={\bf M^{2}}$

It is possible to use the interaction relative angular norm (Definition 3.2
and Theorem 3.2) to put an upper bound for the interactions within
$N_{R}(c,v)$:

$(Rv-\cos(c))\iiiint\limits_{N_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}\\\
<\iiiint\limits_{N_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)\Big{(}|x-x_{0}||\xi-\xi_{0}|-(x-x_{0})\cdot(\xi-\xi_{0}\Big{)}dxd\xi
dx_{0}d\xi_{0}$

Therefore:

$\displaystyle\iiiint\limits_{N_{R}(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)dxd\xi
dx_{0}d\xi_{0}<\frac{\|f\|_{IG}}{Rv-\cos(c)}$

The inequality above and the previous computations imply:

$\displaystyle{\bf
M^{2}}-\frac{\|f\|_{IG}}{Rv(1-\cos(c))}\leq\liminf_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}(\iiiint\limits_{\Gamma(c,v)}f(x,\xi,t)\
dxd\xi)dt\leq{\bf M}^{2}$

Since the former inequality is valid for any arbitrary $R>0$, we complete the
proof:

$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\iiiint\limits_{\Gamma(c,v)}f(x,\xi,t)f(x_{0},\xi_{0},t)d\xi
dxd\xi_{0}d\xi_{0}dt={\bf M^{2}}$

∎

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#  Lepton flavor violating Higgs decays induced by massive unparticle

E. O. Iltan,
Middle East Technical University, Northern Cyprus Campus,
Guzelyurt, Mersin 10, TURKEY
E-mail address<EMAIL_ADDRESS>
###### Abstract

We predict the branching ratios of the lepton flavor violating Higgs decays
$H^{0}\rightarrow e^{\pm}\mu^{\pm}$, $H^{0}\rightarrow e^{\pm}\tau^{\pm}$ and
$H^{0}\rightarrow\mu^{\pm}\tau^{\pm}$ with the assumption that lepton flavor
violation is due to the unparticle mediation. Here, we consider an effective
interaction which breaks the conformal invariance after the electroweak
symmetry breaking and causes that unparticle becomes massive. The new
interaction results in a modification of the mediating unparticle propagator
and brings additional contribution to the branching ratios of the lepton
flavor violating decays with the new vertex including Higgs field and two
unparticle fields. We observe that the branching ratios of the decays under
consideration lie in the range of $10^{-6}-10^{-4}$.

The standard model (SM) electroweak symmetry breaking mechanism which can
explain the production of the masses of fundamental particles will be tested
at the Large Hadron Collider (LHC) and, hopefully, the Higgs boson $H^{0}$,
which is responsible for this mechanism will be hunt soon. The possible decays
of the Higgs boson to the SM particles are worthwhile to study and, among
them, the lepton flavor violating (LFV) decays reach great interest [1, 2, 4,
3, 5] since the LF violation mechanism is sensitive to the physics beyond the
SM. The addition of the new Higgs doublet to the SM particle spectrum is one
of the possibility to switch on the LFV interactions, arising from the tree
level LFV couplings. In [1, 2, 3], $H^{0}\rightarrow\tau\mu$ decay has been
analyzed and the branching ratio (BR) at the order of magnitude of $0.001-0.1$
has been estimated. In [4], the observable BRs of LF changing $H^{0}$ decays
have been obtained in the SM with right handed neutrinos. Another possibility
to switch on the LF violation is to introduce the intermediate scalar
unparticle (U) with the effective U-lepton-lepton vertex in the loop level. In
[5], the BRs of the LFV Higgs decays $H^{0}\rightarrow e^{\pm}\mu^{\pm}$,
$H^{0}\rightarrow e^{\pm}\tau^{\pm}$ and $H^{0}\rightarrow\mu^{\pm}\tau^{\pm}$
have been estimated, by respecting the unparticle idea. Unparticles,
introduced by Georgi [6, 7], come out as new degrees of freedom due to the SM-
ultraviolet sector interaction; they are massless and have non integral
scaling dimension $d_{u}$, around, $\Lambda_{U}\sim 1\,TeV$.

In the present work we study the LFV SM Higgs decays by considering that the
LF violation exists in the one loop level and it is carried by the effective
U-lepton-lepton vertex. The effective interaction lagrangian, which is
responsible for the LFV decays, is

$\displaystyle{\cal{L}}_{FV}=\frac{1}{\Lambda_{U}^{du-1}}\Big{(}\lambda_{ij}^{S}\,\bar{l}_{i}\,l_{j}+\lambda_{ij}^{P}\,\bar{l}_{i}\,i\gamma_{5}\,l_{j}\Big{)}\,U\,,$
(1)

with the lepton field $l$ and scalar (pseudoscalar) coupling
$\lambda_{ij}^{S}$ ($\lambda_{ij}^{P}$). Here we consider the operators with
the lowest possible dimension since their contributions are dominant in the
low energy effective theory (see [8]). Furthermore, we consider that there
exists an additional interaction which ensures a non-zero mass to unparticle
after the electroweak symmetry breaking [9] as

$\displaystyle{\cal{L}}_{U}=-\frac{\lambda}{\Lambda_{U}^{2\,du-2}}\,U^{2}\,H^{\dagger}\,H\,,$
(2)

and we get

$\displaystyle{\cal{L}}_{U}=-\frac{1}{2}\,\frac{\lambda}{\Lambda_{U}^{2\,du-2}}\,U^{2}\,\Bigg{(}H^{0\,2}+2\,v\,H^{0}+v^{2}\Bigg{)}\,,$
(3)

when the Higgs doublet develops the vacuum expectation value. The interaction
in eq.(3) leads to the lagrangian

$\displaystyle{\cal{L^{\prime}}}_{U}=-\frac{m_{U}^{4-2\,d_{U}}}{v}\,U^{2}\,H^{0}\,,$
(4)

with the unparticle mass

$\displaystyle
m_{U}=\Bigg{(}\frac{\sqrt{\lambda}\,v}{\Lambda_{U}^{du-1}}\Bigg{)}^{\frac{1}{2-d_{U}}}\,,$
(5)

and this term results in an additional diagram driving the LFV decays with the
help of U-lepton-lepton vertices (see Fig.1-(d)). Here, the non-zero
unparticle mass $m_{U}$ is the sign of the broken conformal invariance and one
expects that the unparticle propagator is modified. The propagator is model
dependent (see [10]) and we consider the one in the simple model [11, 12]

$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}U(x)\,U(0)\Big{)}0>=i\frac{A_{d_{u}}}{2\,\pi}\,\int_{0}^{\infty}\,ds\,\frac{s^{d_{u}-2}}{p^{2}-\mu^{2}-s+i\epsilon}\,,$
(6)

with

$\displaystyle
A_{d_{u}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{u}}}\,\frac{\Gamma(d_{u}+\frac{1}{2})}{\Gamma(d_{u}-1)\,\Gamma(2\,d_{u})}\,,$
(7)

and the scale $\mu$ where unparticle sector becomes a particle sector. This
choice has clues about the unparticle nature of the hidden sector, it carries
the information on the effects of the broken scale invariance and ensures a
possibility to estimate the scale invariance breaking effects111Notice that
the modification in the propagator needs a further analysis in order to
understand whether it is based on a consistent quantum field theory and this
is beyond the scope of the present manuscript.. In our calculations we choose
$\mu=m_{U}$ and $d_{u}\sim 1.0$ which is the case that unparticle behaves as
if it is almost gauge singlet scalar222 This is the case that $m_{U}$ lies
near the electroweak scale [9]..

Now, we are ready to present the BR for $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay,

$\displaystyle BR(H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+})=\frac{1}{16\,\pi\,m_{H^{0}}}\,\frac{|M|^{2}}{\Gamma_{H^{0}}}\,,$
(8)

where $M$ is the matrix element of the LFV $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay (see Fig.1) and $\Gamma_{H^{0}}$ is the Higgs
total decay width. The square of the matrix element $|M|^{2}$ reads

$\displaystyle|M|^{2}=2\Big{(}m_{H^{0}}^{2}-(m_{l_{1}^{-}}+m_{l_{2}^{+}})^{2}\Big{)}\,|A|^{2}+2\Big{(}m_{H^{0}}^{2}-(m_{l_{1}^{-}}-m_{l_{2}^{+}})^{2}\Big{)}\,|A^{\prime}|^{2}\,,$
(9)

with the amplitudes

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\,dx\,f_{self}^{S}+\int^{1}_{0}\,dx\,\int^{1-x}_{0}\,dy\,f_{vert}^{S}\,,$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\,dx\,f_{self}^{\prime\,S}+\int^{1}_{0}\,dx\,\int^{1-x}_{0}\,dy\,f_{vert}^{\prime\,S}\,.$
(10)

The functions333$f_{self}^{S}$, $f_{self}^{\prime\,S}$ are the same as the
functions presented in [5] except that the propagators $L_{self}$ and
$L_{self}^{\prime}$ contain the unparticle mass term $m_{U}$. On the other
hand $f_{vert}^{S}$, $f_{vert}^{\prime\,S}$ include additional part
proportional to the parameter $c_{2}$ which comes from the new interaction
(see eq.(4)) leading to the vertex given in Fig.1-d  $f_{self}^{S}$,
$f_{self}^{\prime\,S}$, $f_{vert}^{S}$, $f_{vert}^{\prime\,S}$ are

$\displaystyle f_{self}^{S}$ $\displaystyle=$
$\displaystyle\frac{-i\,c_{1}\,(1-x)^{1-d_{u}}}{16\,\pi^{2}\,\Big{(}m_{l_{2}^{+}}-m_{l_{1}^{-}}\Big{)}\,(1-d_{u})}\,\sum_{i=1}^{3}\,\Big{\\{}(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S}+\lambda_{il_{1}}^{P}\lambda_{il_{2}}^{P})\,m_{l_{1}^{-}}\,m_{l_{2}^{+}}\,(1-x)$
$\displaystyle\times$ $\displaystyle\Big{(}L_{self}^{d_{u}-1}-L_{self}^{\prime
d_{u}-1}\Big{)}-(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{2}^{+}}\,L_{self}^{d_{u}-1}-m_{l_{1}^{-}}\,L_{self}^{\prime
d_{u}-1}\Big{)}\Big{\\}}\,,$ $\displaystyle f_{self}^{\prime\,S}$
$\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,(1-x)^{1-d_{u}}}{16\,\pi^{2}\,\Big{(}m_{l_{2}^{+}}+m_{l_{1}^{-}}\Big{)}\,(1-d_{u})}\,\sum_{i=1}^{3}\,\Big{\\{}(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P})\,m_{l_{1}^{-}}\,m_{l_{2}^{+}}\,(1-x)$
$\displaystyle\times$ $\displaystyle\Big{(}L_{self}^{d_{u}-1}-L_{self}^{\prime
d_{u}-1}\Big{)}-(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S}-\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P})\,m_{i}\,\Big{(}m_{l_{2}^{+}}\,L_{self}^{d_{u}-1}+m_{l_{1}^{-}}\,L_{self}^{\prime
d_{u}-1}\Big{)}\Big{\\}}\,,$ $\displaystyle f_{vert}^{S}$ $\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,m_{i}\,(1-x-y)^{1-d_{u}}}{16\,\pi^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{vert}^{2-d_{u}}}\,\Bigg{\\{}(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S})\,\Big{\\{}(1-x-y)$
$\displaystyle\times$
$\displaystyle\Bigg{(}m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y-m_{l_{2}^{+}}\,m_{l_{1}^{-}}\Bigg{)}+x\,y\,m_{H^{0}}^{2}-\frac{2\,L_{vert}}{1-d_{u}}-m_{i}^{2}\Big{\\}}$
$\displaystyle-$
$\displaystyle(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{1}^{-}}\,(2\,x-1)+m_{l_{2}^{+}}\,(2\,y-1)\Big{)}\Bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{i\,c_{2}\,\Gamma[3-2\,d_{u}]\,(x\,y)^{1-d_{u}}}{16\,\pi^{2}\,\Gamma[2-d_{u}]^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{2vert}^{3-2\,d_{u}}}\,\Bigg{\\{}m_{i}\,(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S})$
$\displaystyle-$
$\displaystyle(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,\Big{(}m_{l_{1}^{-}}\,x+m_{l_{2}^{+}}\,y\Big{)}\Bigg{\\}}\,,$
$\displaystyle f_{vert}^{\prime\,S}$ $\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,m_{i}\,(1-x-y)^{1-d_{u}}}{16\,\pi^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{vert}^{2-d_{u}}}\,\Bigg{\\{}(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,\Big{\\{}(1-x-y)$
(11) $\displaystyle\times$
$\displaystyle\Bigg{(}m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y+m_{l_{2}^{+}}\,m_{l_{1}^{-}}\Bigg{)}+x\,y\,m_{H^{0}}^{2}-\frac{2\,L_{vert}}{1-d_{u}}-m_{i}^{2}\Big{\\}}$
$\displaystyle+$
$\displaystyle(\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{1}^{-}}\,(2\,x-1)+m_{l_{2}^{+}}\,(1-2\,y)\Big{)}\Bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{i\,c_{2}\,\Gamma[3-2\,d_{u}]\,(x\,y)^{1-d_{u}}}{16\,\pi^{2}\,\Gamma[2-d_{u}]^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{2vert}^{3-2\,d_{u}}}\,\Bigg{\\{}m_{i}\,(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})$
$\displaystyle+$
$\displaystyle(\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,\Big{(}m_{l_{1}^{-}}\,x-m_{l_{2}^{+}}\,y\Big{)}\Bigg{\\}}\,,$

where $L_{self}$, $L_{self}^{\prime}$, $L_{vert}$, and $L_{2vert}$ are

$\displaystyle L_{self}$ $\displaystyle=$ $\displaystyle
x\,\Big{(}m_{l_{1}^{-}}^{2}\,(1-x)-m_{i}^{2}\Big{)}+m_{U}^{2}\,(x-1)\,,$
$\displaystyle L_{self}^{\prime}$ $\displaystyle=$ $\displaystyle
x\,\Big{(}m_{l_{2}^{+}}^{2}\,(1-x)-m_{i}^{2}\Big{)}+m_{U}^{2}\,(x-1)\,,$
$\displaystyle L_{vert}$ $\displaystyle=$
$\displaystyle(m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y)\,(1-x-y)-m_{i}^{2}\,(x+y)+m_{H^{0}}^{2}\,x\,y-m_{U}^{2}\,(1-x-y)\,,$
$\displaystyle L_{2vert}$ $\displaystyle=$
$\displaystyle(m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y)\,(1-x-y)-m_{i}^{2}\,(1-x-y)+m_{H^{0}}^{2}\,x\,y-m_{U}^{2}\,(x+y)\,,$
(12)

with

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{g\,A_{d_{u}}\,e^{-i\,\pi\,d_{u}}}{4\,m_{W}\,sin\,(d_{u}\pi)\,\Lambda_{u}^{2\,(d_{u}-1)}}\,,$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{A^{2}_{d_{u}}\,m_{U}^{4-2\,d_{u}}\,e^{-2\,i\,\pi\,d_{u}}}{4\,v\,sin^{2}\,(d_{u}\pi)\,\Lambda_{u}^{2\,(d_{u}-1)}}\,.$
(13)

Here $\lambda_{il_{1(2)}}^{S,P}$ are the scalar and pseudoscalar couplings
related to the $U-i-l_{1}^{-}\,(l_{2}^{+})$ interaction where $i$,
($i=e,\mu,\tau$) is the internal lepton and $l_{1}^{-}\,(l_{2}^{+})$ the
outgoing lepton (anti lepton). Notice that, in the numerical calculations, we
consider the BR due to the production of sum of charged states, namely,

$\displaystyle BR(H^{0}\rightarrow
l_{1}^{\pm}\,l_{2}^{\pm})=\frac{\Gamma(H^{0}\rightarrow(\bar{l}_{1}\,l_{2}+\bar{l}_{2}\,l_{1}))}{\Gamma_{H^{0}}}\,.$
(14)

Discussion

This section is devoted to the analysis of the BRs of the LFV
$H^{0}\rightarrow l_{1}^{-}l_{2}^{+}$ decays in the case that the LF violation
is carried by the U\- lepton-lepton vertex. The LFV decays exist at least in
the loop level with the help of the internal unparticle mediation. The
interaction Lagrangian given in eq.(2) results in a nonzero mass for
unparticle after the electroweak symmetry breaking and the propagator of
unparticle existing in the loop should be modified. In the present work we
take the propagator as (see eq.(6))

$\displaystyle
P(p^{2})=\frac{i\,A_{d_{u}}}{2\,sin\,\pi\,d_{u}}\frac{e^{-i\,d_{u}\,\pi}}{(p^{2}-m_{U}^{2})^{2-d_{u}}}\,,$
(15)

which becomes a massive scalar propagator for $d_{u}=1$.

The LF violation is carried by single unparticle mediation and two unparticles
mediation in the loop (see Fig.1). The possible two unparticles mediation
brings an additional contribution to the LFV decays with the strength which is
a function of unparticle mass $m_{U}$, reaching $246\,GeV$ when $d_{u}\sim
1.0$ for the coupling $\lambda\sim 1.0$. In our numerical calculations we take
the scaling parameter $d_{u}$ not far from $1.0$, namely $1.0\leq d_{u}\leq
1.2$. On the other hand we take the coupling $\lambda$ as $\lambda\leq 1.0$ in
order to guarantee that the calculations are perturbative in case of
$d_{u}\sim 1.0$ and we choose the energy scale $\Lambda_{u}$ as
$\Lambda_{u}\sim 1.0\,(TeV)$. The FV U\- lepton-lepton couplings, the scalar
$\lambda^{S}_{ij}$ and pseudo scalar $\lambda^{P}_{ij}$, are among the free
parameters which we choose $\lambda^{S}_{ij}=\lambda^{P}_{ij}=\lambda_{ij}$.
Furthermore, we first consider that the diagonal $\lambda_{ii}=\lambda_{0}$
and off diagonal $\lambda_{ij}=\kappa\lambda_{0},i\neq j$ couplings are family
blind with $\kappa<1$. Second we assume that the the diagonal couplings
$\lambda_{ii}$ carry the lepton family hierarchy, namely,
$\lambda_{\tau\tau}>\lambda_{\mu\mu}>\lambda_{ee}$, on the other hand, the
off-diagonal couplings, $\lambda_{ij}$ are family blind, universal and
$\lambda_{ij}=\kappa\lambda_{ee}$. In our numerical calculations, we choose
$\kappa=0.5$ and we take the magnitude of the FV coupling(s) at most $1.0$ in
order to ensure that the calculations are the perturbative for $d_{u}=1.0$.

In order to estimate the BR of the LFV decays under consideration one needs
the Higgs mass and its total decay width. The theoretical upper and lower
bounds of Higgs mass read $1.0\,TeV$ and $0.1\,TeV$ [13], respectively. This
is due to the fact that one does not meet the unitarity problem and the
instability of the Higgs potential both. Furthermore, the electroweak
measurements predict the range of the Higgs mass as
$m_{H^{0}}=129^{+74}_{-49}$ [14] which is not in contradiction with the
theoretical results. The total Higgs decay width is another parameter which
should be restricted and it is estimated by using the possible decays for the
chosen Higgs mass444For the light (heavy) Higgs boson, $m_{H^{0}}\leq
130\,GeV$ ($m_{H^{0}}\sim 180\,GeV$), the leading decay mode is $b\bar{b}$
pair [15, 16, 17] ($H^{0}\rightarrow WW\rightarrow
l^{+}l^{\prime-}\nu_{l}\nu_{l^{\prime}}$ [18, 19, 20]).. Notice that
throughout our calculations we choose $m_{H^{0}}=120\,(GeV)$ and we use the
input values given in Table (1).

Parameter | Value
---|---
$m_{e}$ |  $0.0005$ (GeV)
$m_{\mu}$ |  $0.106$ (GeV)
$m_{\tau}$ |  $1.780$ (GeV)
$\Gamma(H^{0})|_{m_{H^{0}}=120\,GeV}$ |  $0.0029$ (GeV)
$G_{F}$ | $1.1663710^{-5}(GeV^{-2})$
Table 1: The values of the input parameters used in the numerical
calculations.

In Fig.2, we present the BR$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ with respect
to the scale parameter $d_{u}$ for the flavor blind (FB) couplings
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$. Here, the solid (long
dashed-short dashed-dotted) line represents the BR for
$\lambda=0.0\,(0.2-0.5-1.0)$. The possible interaction of unparticle with the
Higgs scalar leads to a nonzero mass for unparticle after the spontaneous
symmetry breaking and the mass term leads to a suppression in the BR. The
additional term coming from the $U-U-H^{0}$ vertex does not result is an
enhancement in the BR. The BR reaches to the values of the order of $10^{-4}$
for $\lambda=0$ and $d_{u}\sim 1.0$. For $\lambda\sim 1.0$ and near $d_{u}\sim
1.0$ 555This is the case that unparticle mass is near the vacuum expectation
value, namely $m_{U}\sim 246\,GeV$. the BR is of the order of $10^{-6}$. Fig.3
represents the BR$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ with respect to
$\lambda$ for the scale parameter $d_{u}=1$. Here, the solid (long dashed-
short dashed) line represents the BR for
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$
($\lambda_{ee}=0.1,\,\lambda_{\mu\mu}=0.5,\,\lambda_{\tau\tau}=1.0$-$\lambda_{ee}=0.01,\,\lambda_{\mu\mu}=0.1,\,\lambda_{\tau\tau}=1.0$).
This figure shows the strong sensitivity of the BR to the $U-U-H^{0}$
interaction strength $\lambda$, especially for $\lambda<0.3$.

Fig.4 (5) shows the
BR$(H^{0}\rightarrow\tau^{\pm}\,e^{\pm}\,(\tau^{\pm}\,\mu^{\pm}))$ with
respect to the scale parameter $d_{u}$, for the FB couplings
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$. Here, the solid-long
dashed-short dashed-dotted lines represent the BR for
$\lambda=0.0-0.2-0.5-1.0$. In the case of $d_{u}\sim 1.0$, the BR is almost
$5.0\times 10^{-6}$ ($6.0\times 10^{-6}$) for $\lambda\sim 1.0$ and enhances
up to $4.0\times 10^{-4}$ for $\lambda=0$ and $d_{u}\sim 1.0$. Similar to the
previous decay the mass term leads to a suppression in the BR and the
additional term coming from the $U-U-H^{0}$ vertex is not enough to enhance
the BR over the numerical values which is obtained for the massless unparticle
case.

In Fig.6 (7) we present the
BR$(H^{0}\rightarrow\tau^{\pm}\,e^{\pm}\,(\tau^{\pm}\,\mu^{\pm}))$ with
respect to $\lambda$ for the scale parameter $d_{u}=1$. Here, the solid (long
dashed-short dashed) line represents the BR for
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$
($\lambda_{ee}=0.1,\,\lambda_{\mu\mu}=0.5,\,\lambda_{\tau\tau}=1.0$-$\lambda_{ee}=0.01,\,\lambda_{\mu\mu}=0.1,\,\lambda_{\tau\tau}=1.0$).
It is observed that the BR is suppressed more than one order in the range
$0.0<\lambda<1.0$ and this suppression is strong for $\lambda<0.3$.

Conclusion

As a summary, the mass of unparticle which arises with unparticle Higgs scalar
interaction results in that the BRs of the LFV $H^{0}\rightarrow
l_{1}^{\pm}\,l_{2}^{\pm}$ decays are suppressed. The BRs are of the order of
$10^{-6}$ for $\lambda\sim 1.0$ and $d_{u}\sim 1.0$. If the unparticle-Higgs
scalar interaction is switched off unparticle remains massless and the BRs of
the decays studied reach to the values of the order of $10^{-4}$ for FB
U-lepton-lepton couplings. With the possible production of the Higgs boson
$H^{0}$ at the LHC the theoretical results of the BRs of the LFV Higgs decays
will be tested and the new physics which drives the flavor violation,
including the unparticle sector will be searched.

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Figure 1: One loop diagrams contribute to $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay with scalar unparticle mediator. Solid line
represents the lepton field: $i$ represents the internal lepton, $l_{1}^{-}$
($l_{2}^{+}$) outgoing lepton (anti lepton), dashed line the Higgs field,
double dashed line unparticle field.

Figure 2:  $d_{u}$ dependence of the BR $(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$
for $\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$. Here, the solid
(long dashed-short dashed-dotted) line represents the BR for
$\lambda=0.0\,(0.2-0.5-1.0)$. Figure 3:  $\lambda$ dependence of the BR
$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ for $d_{u}=1$. Here, the solid (long
dashed-short dashed) line represents the BR for
$\lambda_{ee}=\lambda_{\mu\mu}=\lambda_{\tau\tau}=1.0$
($\lambda_{ee}=0.1,\,\lambda_{\mu\mu}=0.5,\,\lambda_{\tau\tau}=1.0$-$\lambda_{ee}=0.01,\,\lambda_{\mu\mu}=0.1,\,\lambda_{\tau\tau}=1.0$).
Figure 4: The same as Fig.2 but for $H^{0}\rightarrow\tau^{\pm}\,e^{\pm}$
decay. Figure 5: The same as Fig.2 but for
$H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm}$ decay. Figure 6: The same as Fig.3 but
for $H^{0}\rightarrow\tau^{\pm}\,e^{\pm}$ decay. Figure 7: The same as Fig.3
but for $H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm}$ decay.