\section{Introduction}

	Let $(U_t)$ be a one-dimensional time-homogeneous diffusion process satisfying the stochastic differential equation
		\[	dU_t = \nu(U_t) dt + \sigma(U_t) dW_t, \qquad U_0 = u_0,	\]
	where $(W_t)$ denotes a Brownian motion.	The aim of this note is to bound, from above and below, the transition probability density function for $(U_t)$, $p_U(t, u_0, w) := \frac{d}{dw} P_{U}(t,u_0,w)$, where
$P_{U}(t,u_0,w) := \Pr(U_t \leq w | U_0 = u_0)$. While the focus is on the one-dimensional case, the results are easily extended to some special cases in $\mathbb{R}^n$, $n \geq 2$ (see remark at the end of this section). Some simple bounds for the distribution function are also considered.

	Except for a few special cases, the transition functions are unknown for general diffusion processes, so finding approximations to them is an important alternative approach. We use Girsanov's theorem and then a transformation of the Radon-Nikodym density of the type suggested in \cite{Baldi_etal_0802} to relate probabilities for a general diffusion $(U_t)$ to those of a `reference diffusion'. Using a reference diffusion with known transition functions, we are able to derive various bounds for the transition functions under mild conditions on the original process. The results have a simple form and are readily evaluated.

	As an aside, the generator of the diffusion $(U_t)$ is given by
		\[	Af(x) = \nu(x) \partf{f}{x} + \frac{1}{2} \sigma^2(x) \partfd{f}{x}{2},	\]
	and the transition probability density function is the minimal fundamental solution to the parabolic equation
		\[	\left(A - \partf{}{t}\right)u(t,x) = 0.	\]
	Thus the results presented here also bound solutions to certain types of parabolic partial differential equations.

	Several papers on this topic are available in the literature, especially for bounding the transition probability density. Most recently, \cite{Qian_etal_0504} proposed upper and lower bounds for diffusions whose drift satisfied a linear growth constraint. This appears to be the first such paper to relax the assumption of a bounded drift term. The results in \cite{Qian_etal_0504} will be compared with those obtained in the current paper, although the former can not be used for processes not satisfying the linear growth constraint. To the best of our knowledge, the bounds presented in the current paper are the only ones to relax this constraint, and also appear to generally offer a tightening of the bounds previously available. For further background on diffusions with bounded drift, see e.g.\ \cite{Qian_etal_1103} and references therein.

	In addition, the same ideas allow us to obtain bounds for other functions related to the diffusions. This is not the focus of this note and is not discussed in great detail here, but as an example at the end of Section~\ref{sec:main_result} we consider the density of the process and its first crossing time. This has application in many areas, such as the pricing of financial barrier options. Bounds for other probabilities may be derived in the same manner.

	Consider a one-dimensional time-homogeneous non-explosive diffusion $(U_t)$ governed by the stochastic differential equation (SDE)
		\begin{align}
		\label{eq:original_diff}
			dU_t = \nu (U_t) dt + \sigma(U_t) dW_t,
		\end{align}
	where $(W_t)$ is a Brownian motion and $\sigma(y)$ is differentiable and non-zero inside the diffusion interval (that is, the the smallest interval $I \subseteq \mathbb{R}$ such that $U_t \in I$ a.s.). As is well-known, one can transform the process to one with unit diffusion coefficient by letting
		\begin{align}
		\label{eq:fn_transform}
			F(y) := \int_{y_0}^y \frac{1}{\sigma(u)} du
		\end{align}
	for some $y_0$ from the diffusion interval of $(U_t)$ and then considering $X_t := F(U_t)$ (see e.g.\ \cite{Rogers_xx85}, p.161). By It\^o's formula, $(X_t)$ will have unit diffusion coefficient and a drift coefficient $\mu(y)$ given by the composition
		\[	\mu(y) := \left( \frac{\nu}{\sigma} - \frac{1}{2} \sigma'\right) \circ F^{-1}(y).	\]
	From here on we work with the transformed diffusion process $(X_t)$ governed by the SDE
		\begin{align*}
			dX_t = \mu(X_t) dt + dW_t, \qquad X_0 = F(U_0) =: x.
		\end{align*}
	Conditions mentioned throughout refer to the transformed process $(X_t)$ and its drift coefficient $\mu$.

	We will consider the following two cases only (the results extend to diffusions with other diffusion intervals with one finite endpoint by employing appropriate transforms):
		\begin{enumerate}
			\item[] [A] The diffusion interval of $(X_t)$ is the whole real line $\mathbb{R}$.
			\item[] [B] The diffusion interval of $(X_t)$ is $(0, \infty)$.
		\end{enumerate}
	The results extend to diffusions with other diffusion intervals with one finite endpoint by employing appropriate transforms.

	For the diffusion $(X_t)$ we will need a reference diffusion $(Y_t)$ with certain characteristics. The reference diffusion must have the same diffusion interval as $(X_t)$ and a unit diffusion coefficient, so that Girsanov's theorem may be applied to $(X_t)$. To be of any practical use, the reference process must also have known transition functions. In case [A], we use the Brownian motion as the reference process, while in case [B] we use the Bessel process of an arbitrary dimension $d \geq 3$.

	Recall the definition of the Bessel process $(R_t)$ of dimension $d = 3, 4, \ldots,$ starting at a point $x>0$. This process gives the Euclidean norm of the $d$-dimensional Brownian motion originating at $(x,0, \ldots, 0)$, that is,
		\[  R_t = \sqrt{\bigl(x +W_t^{(1)}\bigr)^2 + \cdots + \bigl(W_t^{(d)}\bigr)^2},	\]
	where the $\bigl(W_t^{(i)}\bigr)$ are independent standard Brownian motions, $i = 1, \ldots, d$. As is well known (see e.g.\ \cite{Revuz_etal_xx99}, p.445), $(R_t)$ satisfies the SDE
		\begin{align*}

			dR_t = \frac{d-1}{2} \frac{1}{R_t}dt + dW_t.
		\end{align*}
	Note that for non-integer values of $d$ the Bessel process of `dimension' $d$ is defined using the above SDE. The process has the transition density function
		\begin{align*}

			p_R(t,y,z) = z \left(\frac{z}{y}\right)^{\eta} t^{-1} e^{-(y^2 + z^2)/2t} {\cal I}_{\eta} \left(\frac{yz}{t} \right),
		\end{align*}
	where $\eta = d/2 -1$ and ${\cal I}_{\eta}(z)$ is the modified Bessel function of the first kind. For further information, see Chapter XI in \cite{Revuz_etal_xx99}.

	We denote by $\Pr_x$ and $\mathbb{E}_x$ probabilities and expectations conditional on the process in question ($(X_t)$ or some other process, which will be obvious from the context) starting at $x$. We work with the natural filtration $\cal{F}_s := \sigma\left(X_u: u \leq s\right)$.

	Finally, note that the present work can be easily extended to a class of $n$-dimensional diffusions for $n \geq 2$. If $(X_t)$ is an $n$-dimensional diffusion satisfying the SDE
		\[	dX_t = \mu(X_t) dt + dW_t,	\]
	$(W_t)$ being an $n$-dimensional Brownian motion, then the majority of results can be extended assuming $\mu(\cdot)$ is curl-free. The extension is straight-forward, and in this note we shall only concern ourselves with the one-dimensional case.

\section{Main Results}
\label{sec:main_result}

	This section states and proves a result relating probabilities for the diffusion $(X_t)$ to expectations under an appropriate reference measure. In the case [A], the result may be known, and we state it here for completeness. The extension to case [B] is straight-forward. We then apply this proposition to obtain bounds for transition densities and distributions.

	\textbf{Relation to the Reference Process}

	We define the functions $G(y)$ and $N(t)$ as follows, according to the diffusion interval of $(X_t)$:
 		\begin{enumerate}
			\item[][A] If the diffusion interval of $(X_t)$ is $\mathbb{R}$, then we define, for some fixed $y_0 \in \mathbb{R}$,
          \begin{equation}
          	\begin{array}{rl}
            	G(y) \!\!\!&:= \displaystyle\int_{y_0}^{y} \mu(z) dz,\\
            	\vphantom{.}\\
							N(t) \!\!\!&:= \displaystyle\int_0^t \left(\mu'(X_u) + \mu^2(X_u)\right) du.
            \end{array}
          \label{eq:bm_G}
        	\end{equation}
			\item[][B] If the diffusion interval of $(X_t)$ is $(0, \infty)$, then we define, for some fixed $d \geq 3$ (the dimension of the reference Bessel process) and $y_0 > 0$,
					\begin{align*}
							G(y) &:= \int_{y_0}^{y} \left(\mu(z) - \frac{d-1}{2z} \right) dz,\\
							N(t) &:= \int_0^t \left( \mu'(X_u) - \frac{(d-1)(d-3)}{4X_u^2} + \mu^2(X_u)  \right) du.
					\end{align*}
			\end{enumerate}
		\begin{rem}
			For diffusions on $(0, \infty)$, the choice of $d$ is arbitrary subject to $d \geq 3$. Therefore this choice can be used to optimise any bounds presented in the next subsection.
		\end{rem}

\begin{prop}
\label{th:gen_int}
	Assume the the drift coefficient $\mu$ of $(X_t)$ is absolutely continuous. Then, for any $A \in \mathcal{F}_t$,
		\[	\Pr_x(A) = \hat{\Exp}_x\left[ e^{G(X_t) - G(x)} e^{-(1/2) N(t)}\mathbbm{1}}      %{1\hspace{-2.5mm}{1}_A \right],	\]
	where $\hat{\Exp}_x$ denotes expectation with respect to the law of the reference process.
\end{prop}
\begin{rem}
	In terms of the original process $(U_t)$ defined in \eqref{eq:original_diff}, the condition of absolute continuity of $\mu(y)$ requires $\nu(z)$ and $\sigma'(z)$ to be absolutely continuous.
\end{rem}

\begin{proof}

	The proof is a straight-forward application of Girsanov's theorem and its idea is similar to the one used in \cite{Baldi_etal_0802}. We present the proof for case [A], the proof for case [B] is completed similarly (see \cite{Downes_etal_xx08} for the general approach).

	Define $\mathbb{Q}_x$ to be the reference measure such that under $\mathbb{Q}_x$, $X_0 = x$ and
		\[	dX_s = d\widetilde{W}_s,	\]
	for a $\mathbb{Q}_x$ Brownian motion $(\widetilde{W}_s)$. Set
		\begin{align*}
			\zeta_s &:= \frac{d\Pr_x}{d\mathbb{Q}_x} = \exp \left\{ \int_0^s \mu (X_u) d\widetilde{W}_u - \frac{1}{2} \int_0^s \mu^2(X_u) du \right\},
		\end{align*}
	so by Girsanov's theorem under $\Pr_x$ we regain the original process $(X_s)$ satisfying
		\[	dX_s = \mu(X_s) ds + dW_s,	\]
	for a $\Pr_x$ Brownian motion $(W_s)$. The regularity conditions allowing this application of Girsanov's theorem are satisfied (see e.g.\ Theorem~7.19 in \cite{Liptser_etal_xx01}), since under both $\Pr_x$ and $\mathbb{Q}_x$ the process $(X_s)$ is non-explosive and $\mu(y)$ is locally bounded so we have, for any $t>0$,
	  \[	\Pr_x \left( \int_0^t \mu^2(X_s) ds < \infty \right) = \mathbb{Q}_x \left( \int_0^t \mu^2(X_s)  ds < \infty \right) = 1.	\]
	We then have, under $\mathbb{Q}_x$, using It\^o's formula and \eqref{eq:bm_G},
		\begin{align}
		\label{eq:dG}
			d G (X_s) &= \mu(X_s) dX_s + \frac{1}{2} \mu'(X_s) (dX_s)^2\notag\\
				&= \mu(X_s) d\widetilde{W}_s + \frac{1}{2} \mu'(X_s) ds.
		\end{align}
	Note that in order to apply It\^o's formula, we only require $\mu$ to be absolutely continuous with Radon-Nikodym derivative $\mu'$ (see e.g.\ Theorem~19.5 in \cite{Kallenberg_xx97}). This also implies the above is defined uniquely only up to a set of Lebesgue measure zero, and we are free to assign an arbitrary value to $\mu'$ at points of discontinuity.

	Rearranging \eqref{eq:dG} gives
		\[	\int_0^s \mu (X_u) d\widetilde{W}_u = G(X_s) - G(X_0) - \frac{1}{2} \int_0^s \mu'(X_u) du.	\]
	Hence
		\[	\zeta_s = \exp \left\{ G(X_s) - G(X_0) - \frac{1}{2} \int_0^s \left( \mu'(X_u) + \mu^2 (X_u) \right) du \right\},	\]
	which together with
		\begin{align*}
			\Pr_x(A) = \mathbb{E}_x[\mathbbm{1}}      %{1\hspace{-2.5mm}{1}_A] = \int \mathbbm{1}}      %{1\hspace{-2.5mm}{1}_A d\Pr_x = \int \mathbbm{1}}      %{1\hspace{-2.5mm}{1}_A \zeta_t d\mathbb{Q}_x = \hat{\Exp}_x [ \zeta_t \mathbbm{1}}      %{1\hspace{-2.5mm}{1}_A ],
		\end{align*}
	completes the proof of the proposition.

\end{proof}

	\textbf{Bounds for Transition Densities and Distributions}

	Define $L$ and $M$ as follows, according to the diffusion interval of $(X_t)$:
 		\begin{enumerate}
			\item[][A] If the diffusion interval of $(X_t)$ is $\mathbb{R}$, then
					\begin{align*}
						L &:= \displaystyle \mbox{ess sup}\left(\mu'(y) + \mu^2(y)\right),\\
						M &:= \displaystyle \mbox{ess inf}\left(\mu'(y) + \mu^2(y)\right),
					\end{align*}
				where the essential supremum/infimum is taken over $\mathbb{R}$.
			\item[][B] If the diffusion interval of $(X_t)$ is $(0, \infty)$, then, for some fixed $d \geq 3$ (the dimension of the reference Bessel process), we put
					\begin{align*}
						L &:= \displaystyle \mbox{ess sup} \left( \mu'(y) - \frac{(d-1)(d-3)}{4y^2} + \mu^2(y)  \right),\\
						M &:= \displaystyle \mbox{ess inf} \left( \mu'(y) - \frac{(d-1)(d-3)}{4y^2} + \mu^2(y)  \right),
					\end{align*}
				where the essential supremum/infimum is taken over $(0, \infty)$.
		\end{enumerate}

	Note that in what follows, in the case [B], the dimension of the reference Bessel process may be chosen so  as to optimise the particular bound. Recall also that $(Y_t)$ denotes the reference process (the Weiner process in case [A], the $d$-dimensional Bessel process in case [B]).

	\begin{cor}
	\label{cor:trans_dens}
			The transition density of the diffusion $(X_t)$ is bounded according to
				\begin{align}
				\label{eq:trans_dens}
					e^{-tL/2} \leq \frac{p_X(t,x,w)}{e^{G(w) - G(x)} p_Y(t,x,w)} \leq e^{-tM/2}.
				\end{align}
	\end{cor}
	\begin{rem}
		The bound is sharp: for a constant drift coefficient $\mu$, equalities hold in \eqref{eq:trans_dens}.
	\end{rem}
	\begin{proof}
		Recall (see the proof of Proposition~\ref{th:gen_int}) we only required $\mu$ to be absolutely continuous, and its value on a set of Lebesgue measure zero is irrelevant. Hence $L$ (respectively $M$) gives an upper (lower) bound for the integrand in $N(t)$ for all paths. Applying Proposition~\ref{th:gen_int} with $A = \{X_t \in [w, w+h)\}$, $h>0$, gives
			\begin{align*}
				\inf_{w \leq y \leq w+h} e^{G(y) - G(x)} e^{-tL/2} \Pr_x(Y_t \in [w, &w+h)) \leq \Pr_x(X_t \in [w, w+h))\\
				&\leq \sup_{w \leq y \leq w+h} e^{G(y) - G(x)} e^{-tM/2} \Pr_x(Y_t \in [w, w+h)).
			\end{align*}
		Taking the limits as $h \rightarrow 0$ gives the required result.
	\end{proof}
	In the case of bounded $L$ and $M$ this immediately gives an asymptotic expression for the density $p_X(t,x,w)$ as $t \rightarrow 0$.
	\begin{cor}
		If $-\infty < L, M < \infty$, then, as $t \rightarrow 0$,
			\[	p_X(t,x,w) \sim e^{G(w) - G(x)} p_Y(t,x,w),	\]
		uniformly in $x$, $w$.
	\end{cor}

	While the tightest bounds for the transition distribution are obtained by integrating the bounds for the density given above, this does not, in general, yield a simple closed form expression. We mention other, less tight bounds that are simple and are obtained by a further application of Proposition~\ref{th:gen_int}.

	\begin{cor}
	\label{cor:trans_dist}
		The transition distribution function of the diffusion $(X_t)$ admits the following bound: for any $w \in \mathbb{R}$,
			\begin{align*}
				 \inf_{ \ell \leq y \leq w} e^{G(y) - G(x)} e^{-tL/2} P_Y(t,x,w) \leq P_X(t,x,w)
				 \leq \sup_{\ell \leq y \leq w} e^{G(y) - G(x)} e^{-tM/2} P_Y(t,x,w),
			\end{align*}
		where $\ell$ is the lower bound of the diffusion interval.
	\end{cor}

	The assertion of the corollary immediately follows from that of Proposition~\ref{th:gen_int} with $A = \{X_t \leq w \}$.

	By considering other events (e.g.\ $A = \{ X_t > w \}$), other similar bounds can be derived.

	\textbf{Further Probabilities}

	While the focus of this note is on bounds for the transition functions, Proposition \ref{th:gen_int} can be used to obtain other useful results. For example, consider
		\[	\eta_X(t, x, y, w)	:= \frac{d}{dw} \Pr_x \left( \sup_{0 \leq s \leq t} X_s \geq y, X_t \leq w \right).	\]
	Such a function has applications in many areas, for example the pricing of barrier options in financial markets. Using ideas similar to the proof of Corollary~\ref{cor:trans_dens} immediately gives
		\begin{cor}
		\label{cor:other_probs}
			For the diffusion $(X_t)$,
			\[	e^{ -tL/2} \leq \frac{\eta_X(t,x,y,w)}{e^{G(w) - G(x)} \eta_Y(t,x,y,w)} \leq e^{ -tM/2}.	\]
		\end{cor}
	Note that for such probabilities the bounds may be improved, if desired, by replacing $L$ and $M$ with appropriate constants on a case-by-case basis. For example, if we are considering the probability our diffusion stays between two constant boundaries at the levels $c_1 < c_2$, then the supremum (for $L$) and infimum (for $M$) need only be taken over the range $c_1 \leq y \leq c_2$.

	Other probabilities may be considered in a similar way.

\section{Numerical Results}
\label{sec:num_results}

	Here we illustrate the precision of the results from the previous section for transition densities. Bounds from Corollary~\ref{cor:trans_dens} are compared with known transition density functions and previously available bounds for the Ornstein-Uhlenbeck process in the case [A]. For the case [B], we only compare the bounds obtained in the current paper with exact results, since there appears to be no other known bounds in the literature. We also construct a `truncated Ornstein-Uhlenbeck' process in order to compare our results with other bounds available in the literature. For the Ornstein-Uhlenbeck process we also consider an example to illustrate Corollary~\ref{cor:other_probs}.

	\textbf{The Ornstein-Uhlenbeck Process}

	We consider an Ornstein-Uhlenbeck process $(S_t)$, which satisfies the SDE
		\[	dS_t = -S_t dt + dW_t.	\]
	This process has the transition density
		\[	p_S(t,x,w) = \frac{ e^{t}}{\sqrt{\pi (e^{2 t}-1)}} \exp\left(\frac{\left(w e^{t} - x\right)^2}{1-e^{2 t}} \right),	\]
	see e.g.\ (1.0.6) in \cite{Borodin_etal_xx02}, p.522, and we begin by comparing this with the bound obtained in Corollary~\ref{cor:trans_dens}. Since $\mu(z) = -z$, we have
		\[	M = -1, \qquad G(w) - G(x) = \frac{1}{2}(x^2 - w^2),	\]
	giving the estimate
		\[	p_S(t,x,w) \leq e^{\frac{1}{2}(x^2 - w^2 + t)} p_W(t,x,w).	\]
	Clearly in this case the bound is tighter for smaller values of $|x|$ and $t$. Figure~\ref{fig:OU_dens_cent} displays a plot of the right-hand side of this bound together with the exact density for $x=0$ and $t=1,2$.

\begin{figure}[!htb]
	\begin{centering}
			\includegraphics[width = 0.90 \textwidth, height = 2.5 in]{ou_dens}
		\caption{Transition density for an Ornstein-Uhlenbeck process, alongside its upper bound, with $x=0$. The left-hand side displays the functions for $t=1$, the right for $t=2$.}
		\label{fig:OU_dens_cent}
	\end{centering}
\end{figure}

	To compare our results with other known bounds for transition functions, we look at the bound given by (3.3) in \cite{Qian_etal_0504} (which, to the best of the author's knowledge, is the only bound available for such a process). Figure~\ref{fig:OU_dens_comp} compares this bound with that obtained in Corollary~\ref{cor:trans_dens} and the exact transition density. The values $x=0$ and $t=1$ are used (for the bound in \cite{Qian_etal_0504}, $q=1.2$ seemed to give the best result, see \cite{Qian_etal_0504} for further information on notation). Note that \cite{Qian_etal_0504} gives a sharper bound for $w$ close to zero, but quickly grows to very large values as $|w|$ increases, and in general the bounds presented in this note offer a large improvement. This is typical for all values of $t$, with the effect becoming more pronounced as $t$ decreases. A meaningful lower bound for this process is unavailable by the methods of the present paper, since $L = -\infty$.

\begin{figure}[!htb]
	\begin{centering}
			\includegraphics[width = 0.45 \textwidth, height = 2.5 in]{ou_comp}
		\caption{Transition density for an Ornstein-Uhlenbeck process (solid line) compared to bounds given in \cite{Qian_etal_0504} (dashed line) and Corollary~\ref{cor:trans_dens} (dotted line), for $x=0$ and $t=1$.}
		\label{fig:OU_dens_comp}
	\end{centering}
\end{figure}

	For this example, we briefly look at the bound obtained in Corollary~\ref{cor:other_probs}. We have, see e.g.\ (1.1.8) in \cite{Borodin_etal_xx02}, p. 522,
		\[	\eta_S(t,x,0,z) = \frac{1}{\sqrt{\pi (1 - e^{-2 t})}} \exp \left( -\frac{(|z| - x e^{-t})^2}{1 - e^{-2 t}} \right).	\]
	Figure~\ref{fig:ou_eta} compares this as a function of $t \in [0,1]$ with the bound obtained in Corollary~\ref{cor:other_probs},
		\begin{align*}
			\eta_S(t,x,0,z) &\leq \exp \left\{\frac{1}{2} (x^2 - z^2 + t) \right\} \eta_W(t,x,0,z)\\
					&= \exp \left\{ \frac{1}{2} (x^2 - z^2 + t) \right\} \frac{1}{\sqrt{2 \pi t}} \exp \left\{ - \frac{1}{2t} (|z| - x)^2 \right\},
		\end{align*}
	where $\eta_W(t,x,0,z)$ is given by (1.1.8) on p. 154 of \cite{Borodin_etal_xx02}.

\begin{figure}[!htb]
	\begin{centering}
			\includegraphics[width = 0.45 \textwidth, height = 2.5 in]{sup_ou}
		\caption{Bound for $\eta_S(t,x,0,z)$ compared with its true value, for $x = z = -0.5$.}
		\label{fig:ou_eta}
	\end{centering}
\end{figure}

	\textbf{The Truncated Ornstein-Uhlenbeck Process}

	Other density bounds available in the literature hold only for processes which have bounded drift. For completeness we compare one such bound with the results of this paper. We use the bound in \cite{Qian_etal_1103}, which is the most recent for bounded drift and seems to give the best results over a large domain. To use these results, however, we need a process with bounded drift. As such, we have chosen the `truncated Ornstein-Uhlenbeck' process, which we define as a process $(\overline{S}_t)$ satisfying the SDE
		\[	d\overline{S}_t = \mu(\overline{S}_t) dt + dW_t,	\]
	where, for a fixed $c > 0$,
		\[	\mu(z) =
					\begin{cases}
						c, & \qquad z < -c,\\
						-z, & \qquad |z| \leq c,\\
						-c, & \qquad z > c.
					\end{cases}
		\]
	For this process we again have $M=-1$ and, assuming $|x| \leq c$,
		\[	G(w) - G(x) =
					\begin{cases}
						\frac{1}{2}(c^2 + x^2)  + cw, & \qquad w < -c,\\
						\frac{1}{2}(x^2 - w^2), & \qquad |w| \leq c,\\
						\frac{1}{2}(c^2 + x^2)  - cw, & \qquad w > c.
					\end{cases}
		\]
	Figure~\ref{fig:trunc_ou} displays the bounds from Corollary~\ref{cor:trans_dens} together with those in \cite{Qian_etal_1103} for different values of $c$ with $x=0$ and $t=1$. Smaller values of $c$ move the bounds closer together, however for the given choice of $x$ and $t$ they do not touch until we use the (rather severe) truncation $c \approx 0.45$. In general the method outlined in this note provides a dramatic improvement. We have also plotted an estimate for the transition density using simulation. The simulation was performed using the predictor-corrector method (see e.g.\ \cite{Kloeden_etal_xx94} p.198), with $10^5$ simulations and $100$ time-steps.

\begin{figure}[!htb]
	\begin{centering}
			\includegraphics[width = 0.90 \textwidth, height = 2.5 in]{outrunc_dens}
		\caption{Simulated density and bounds for the transition density of the truncated Ornstein-Uhlenbeck process $(\overline{S}_t)$. The solid lines give the simulated densities, the dotted lines the bound given in Corollary~\ref{cor:trans_dens} and the dashed lines the bounds from \cite{Qian_etal_1103}. Both graphs display the functions for $x=0$ and $t=1$, while the left graph displays them for $c=1$, the right for $c=2$.}
		\label{fig:trunc_ou}
	\end{centering}
\end{figure}

	\textbf{A Diffusion on $(0, \infty)$}

	Finally we consider a process from the case [B]. The author believes this is the first paper to present a bound on transition densities without the linear growth constraint. The process $(V_t)$ satisfying the SDE
		\begin{align}
		\label{eq:feller_sde}
			dV_t = (p V_t + q)dt + \sqrt{2 r V_t} dW_t
		\end{align}
	with $p$, $q \in \mathbb{R}$ and $r>0$, has a known transition density (see (26) in \cite{Giorno_etal_0686}). After applying the transform $Z_t = F(V_t)$, with $F(y) = \sqrt{\frac{2}{r}y}$ by \eqref{eq:fn_transform}, we obtain the process
		\[	dZ_t = \mu(Z_t) dt + dW_t,	\]
	where
		\[	\mu(y) = \frac{p}{2} y + \frac{1}{y}\left( \frac{q}{r} - \frac{1}{2} \right).	\]
	For $q > r$ this dominates the drift of a Bessel process of order $2q/r > 2$ so is clearly a diffusion on $(0, \infty)$.

	We take the values $q=2.5$, $r=1$ and $p=1$. Using these values, we have
		\[	M = \inf_{0 \leq y \leq \infty} \left[ \frac{y^2}{4} + 2.5 + \frac{1}{y^2} \left(2 - \frac{(d-1)(d-3)}{4}\right) \right],	\]
	and
		\begin{align*}

			G(y) - G(x) = \frac{1}{4}(y^2 - x^2) + c \log \left( \frac{y}{x} \right),
		\end{align*}
	where $d$ is the order of the reference Bessel process and $c = 2 - (d-1)/2$.

	It remains to choose the order of the reference Bessel process. It is not clear how to define the `best' order of the reference process for a range of $z$ values, as for fixed $t$ and $x$ the upper bound for $p_Z(t,x,w)$ is minimised for different values of $d$ depending on the value of $z$. In Figure~\ref{fig:rplus_dens} we have taken $t=x=0.5$ and used $d=4.7$, however depending on the relevant criterion improvements can be made. Again, a meaningful lower bound for this process is unavailable by the methods of this paper, since $L= -\infty$.

	\begin{figure}[!htb]
	\begin{centering}
			\includegraphics[width = 0.45 \textwidth, height = 2.5 in]{plus_dens}
		\caption{Transition density for the diffusion \eqref{eq:feller_sde}, alongside its upper bound, with $x=0.5$ and $t=0.5$.}
		\label{fig:rplus_dens}
	\end{centering}
	\end{figure}

	\newpage

    {\bf Acknowledgements:} This research was supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems. The author is grateful for many useful discussions with K. Borovkov which lead to improvements in the paper.


\section{Introduction}

An intriguing open question in condensed matter physics, one with potential practical significance, is whether there exists a lower bound for the density of a crystal. In other words, how large can the average distance between nearest neighbor atoms be, before crystalline long-range order is suppressed by thermal and quantum fluctuations? It remains unclear whether a lower limit exists, or
how to approach it experimentally. We begin our discussion by rehashing a few basic facts about the crystalline phase of matter.
\\ \indent
Crystallization occurs at low temperature ($T$) in almost all known substances, as the state of lowest energy (ground state) is approached. Classically, the ground state is one
in which the potential energy of interaction among the constituent particles is minimized,
a condition  that corresponds to an orderly arrangement of particles in regular, periodic lattices.
In most cases, quantum mechanics affects this fundamental conclusion only quantitatively, as zero-point motion of particles results in lower equilibrium densities and melting temperatures, with respect to what one would predict classically; typically, these corrections are relatively small. Only in helium is the classical picture upended by quantum mechanics, as the fluid resists crystallization all the way down to  temperature $T=0$,K, under the pressure of its own vapor.
\\ \indent
In many
condensed matter systems, the interaction among atoms
is dominated by pairwise contributions, whose ubiquitous features are a strong repulsion at interparticle distances less than a characteristic length $\sigma$, as well as an attractive part, which can be described by an effective energy well depth $\epsilon$. The dimensionless parameter $\Lambda=\hbar^2/[m\epsilon\sigma^2]$, where $m$ is the particle mass, quantifies the relative importance of quantum-mechanical effects in the ground state of the system. \\ \indent
It has been established  for a specific model pair potential incorporating the above basic features, namely the
Lennard-Jones potential,
and for Bose statistics,
that the equilibrium phase is a crystal if $\Lambda < \Lambda_c \approx 0.15$; it is a (super)fluid if $0.15\lesssim \Lambda\lesssim 0.46$, while, for $\Lambda > 0.46$, the system only exists in the gas phase \cite{pnas}. The value of $\Lambda_c$ is estimated \cite{nosanow} to be slightly higher ($\approx$ 20 \%) for systems obeying Fermi statistics. For substances whose elementary constituents are relatively simple atoms or molecules, this result provides a useful, general criterion to assess their propensity to crystallize, as measured by their proximity in parameter space to a quantum phase transition to a fluid phase. It can be used to infer, at least semi-quantitatively, macroscopic properties of the crystal, such as its density and melting temperature, both of which generally decrease \cite{pnas} as $\Lambda\to\Lambda_c$.
\\ \indent
The two stable isotopes of helium have the highest known values of $\Lambda$, namely 0.24 (0.18) for $^3$He ($^4$He); for all other
naturally occurring substances, $\Lambda$ is considerably lower. The next highest value is that of molecular hydrogen (H$_2$), namely $\Lambda = 0.08$, quickly decreasing for heavier elements and compounds. At low temperature, H$_2$ forms one of the least dense crystals known, of density $\rho=0.0261$ \AA$^{-3}$ (mass density 0.086 gr/cc). The low mass of a H$_2$ molecule (half of that of a $^4$He atom) and its bosonic character (its total spin is $S=0$ in its ground state)
led to the speculation that liquid H$_2$ may turn superfluid at low temperature \cite{ginzburg}, just like $^4$He. In practice, no superfluid phase is observed (not even a metastable one), as molecular interactions, and specifically \cite{boninsegni18} the relatively large value of $\sigma$ ($\sim 3$ \AA) cause H$_2$ to crystallize at a temperature $T=13.8$\,K.
\\ \indent
Present consensus is
that H$_2$ is a non-superfluid, insulating crystal at low temperature, including in reduced dimensions \cite{2d,1d}; only small clusters of parahydrogen ($\sim 30$ molecules or less) are predicted \cite{sindzingre,mezz1,mezz2,2020} to turn superfluid at $T\sim 1$ K,
for which some experimental evidence has been reported \cite{grebenev}.
If, hypothetically, the mass of the molecules could be progressively reduced, while leaving the interaction unchanged, thus increasing $\Lambda$ from its value for H$_2$ all the way to $\Lambda_c$, several intriguing scenarios might arise, including a low temperature superfluid liquid phase, freezing into a low-density crystal at $T=0$, and even a supersolid phase, namely one enjoying at the same time crystalline order and superfluidity \cite{supersolid}. Obviously, the value of $\Lambda$ can also be modified by changing one or both of the interaction parameters ($\epsilon$ and $\sigma$) independently of the mass; in this work, however, we focus for simplicity on the effect of mass on the physics of the system.
\\ \indent
One potential way to tune the mass is via substitution of protons or
electrons with muons. For example, an
assembly of molecules of muonium hydride (HMu)
differ from H$_2$ by the replacement of one of the two protons with an antimuon ($\mu^+$), whose mass is approximately 11\% of that of a proton.
Quantum chemistry calculations
have shown that it is very similar in size to H$_2$, and has the same quantum numbers in the ground state
\footnote{Specifically, the average distance between the proton and the muon in a HMu molecule is estimated at 0.8 \AA, whereas that between the two protons in a H$_2$ molecule is 0.74 \AA. See Refs, \cite{suff,zhou}}.
It is therefore not inconceivable that the interaction between two HMu might be quantitatively close to that between two H$_2$
(we further discuss this aspect below). In this case the value of the parameter $\Lambda$
is $\sim 0.14$, i.e., very close to $\Lambda_c$ for a Bose system. This leads to the speculation that this substance may crystallize into a highly quantal solid, displaying a strikingly unique behavior, compared to ordinary crystals.
In order to investigate such a scenario, and more specifically to gain insight into the effect of mass reduction, we studied  theoretically the low temperature phase diagram of a hypothetical condensed phase of HMu,
by means of first principle computer simulations of a microscopic model derived from that of H$_2$.
\\ \indent
The main result is that the equilibrium phase of the system at low temperature is a crystal of very
low density, some $\sim 5\%$ lower than the $T=0$ equilibrium density of {\em liquid} $^4$He. Despite the low density, however, such a crystal melts at a fairly high temperature, close to 9 K, i.e., only a few K lower than that of H$_2$. No superfluid phases are observed, either fluid or crystalline, as exchanges of indistinguishable particles, known to underlie superfluidity \cite{feynman}, are strongly suppressed in this system, much as they are in H$_2$, by the relatively large size of the hard core repulsion at short distances (i.e., $\sigma$). As a result, the behavior of this hypothetical system can be largely understood along classical lines.
This underscores once again the crucial role of exchanges of identical particles in destabilizing the classical picture, which is not qualitatively altered by zero-point motion of particles alone \cite{role}; it also reinforces the conclusion \cite{boninsegni18} reached elsewhere that it is the size of the hard core diameter of the intermolecular interaction that prevents superfluidity in H$_2$, {\em not} the depth $\epsilon$ of its attractive part.
\\ \indent
To complement our investigation, we also studied nanoscale HMu
clusters of varying sizes, comprising up to few hundred molecules. We find the behavior of these clusters to be much closer to that of $^4$He (rather than H$_2$) clusters. For example, at $T=1$ K the structure of HMu clusters is liquid-like, and their superfluid response  approaches 100\%, even for the largest cluster considered (200 HMu molecules). Thus, while mass reduction does not bring about substantial physical differences between the behavior of bulk HMu and that of H$_2$, it significantly differentiate the physics of nanoscale size clusters.
\section{Model}\label{mod}
We model the HMu molecules as point-like, identical particles of mass $m$ and spin $S=0$, thus obeying Bose statistics.
For the bulk studies, the system is enclosed in a cubic cell of volume $V=L^3$ with periodic boundary conditions in the three directions, giving a
density of $\rho=N/V$.
For the cluster studies, the $N$ particles  are
placed in a supercell of large enough size to remove boundary effects.
The quantum-mechanical many-body Hamiltonian reads as follows:
\begin{eqnarray}\label{u}
\hat H = - \lambda \sum_{i}\nabla^2_{i}+\sum_{i<j}v(r_{ij})
\end{eqnarray}
where the first (second) sum runs over all particles (pairs of particles), $\lambda\equiv\hbar^2/2m=21.63$ K\AA$^{2}$ (reflecting the replacement of a proton with a $\mu^+$ in a H$_2$ molecule), $r_{ij}\equiv |{\bf r}_i-{\bf r}_j|$ and $v(r)$ denotes the pairwise interaction between two HMu molecules, which is assumed spherically symmetric.
\begin{figure}[h]
\centering
\includegraphics[width=2.0in]{com.pdf}
\caption{Geometry utilized in the calculation of the effective interaction between two HMu molecules. Shown is the line connecting the centers of mass of the two molecules, as well the three angles describing their relative orientation, i.e., the two polar angles $\theta_1, \theta_2$, as well as the azimuthal angle $\phi$.}
\label{geom}
\end{figure}
\\ \indent
In order to decide on an adequate model potential to adopt in our calculation, we use as our starting point the H$_2$  intermolecular potential, for which a considerable amount of theoretical work has been carried out \cite{SG,DJ,Szal}. We consider here for definiteness the {\em ab initio} pair potential proposed in  Ref. \onlinecite{Szal}.
The
most important effect of the replacement of one of the protons of the H$_2$ molecule with a $\mu^+$ is the shift of the center of mass of the molecule away from the midpoint, and toward the proton. We assume that this effect provides the leading order correction with respect to the H$_2$ intermolecular potential, and we set out to obtain a corrected version of the interaction for the new geometry, as illustrated in Fig.~\ref{geom}. We use the program provided in Ref. \cite{Szal} to generate a potential as a function of the distance between the midpoints and the angular configurations,
and then transform that to a potential as a function of the distance between the centers of mass and the angular configurations. Finally, we average over the angular configurations to obtain a one-dimensional isotropic potential. We take the distance between the proton and the $\mu^+$ to be that computed in Ref. \cite{zhou}, which differs only slightly from the distance between the two protons in the H$_2$ molecule.
\\ \indent
In Fig. \ref{fig:pot}, we compare the potential energy of interaction between two HMu molecules resulting from this procedure with that of obtained for two H$_2$ molecules, i.e., with the center of mass at the midpoint. The comparison suggests that the displacement of the center of mass of the molecule results in a slight stiffening of the potential at short distance. Indeed, in the range of average interparticle separations explored in this work (namely the 3.5-3.7 \AA \ range), the differences between the interactions are minimal. Also shown in Fig. \ref{fig:pot} is the Silvera-Goldman potential \cite{SG}, which is arguably the most widely adopted in theoretical studies of the condensed phase of H$_2$, and has proven to afford a quantitatively accurate \cite{op} description of structure and energetics of the crystal. As one can see, it has a significantly smaller diameter and is considerably ``softer'' than the potential of Ref. \onlinecite{Szal}; the reason is that it incorporates, in an effective way, non-additive contributions (chiefly triplets), whose overall effect is to soften the repulsive core of the pair-wise part computed {\em ab initio}.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{pot}
\caption{The intermolecular interaction energy $E$ (in K) as a function of distance (\AA) between the centers of mass of the molecules, for
the  H$_2$ (red, dashed) and  HMu
(black, solid line) cases, obtained with the aid of the programs provided in Ref. \cite{Szal}. Also shown (blue, solid line) is the Silvera-Goldman potential.}
\label{fig:pot}
\end{figure}
\\ \indent
It therefore seems reasonable to utilize the Silvera-Goldman potential \cite{SG} to carry out our study, as the use of a quantitatively more accurate potential is not likely to affect the conclusions of our study in a significant way. Furthermore, because  the majority of the theoretical studies of condensed H$_2$ have been carried out using the Silvera-Goldman potential, its use in this study allows us to assess the effect of mass alone.
\\ \indent

\section*{Methodology}
The low temperature phase diagram of the thermodynamic system described by Eq.  (\ref{u}) as a function of  density and temperature has been studied in this work by means of  first principles numerical simulations, based on the continuous-space Worm Algorithm \cite{worm1,worm2}.  Since this technique is by now fairly well-established, and extensively described in the literature, we shall not review it here; we used a variant of the algorithm in which the number of particles $N$ is fixed \cite{mezz1,mezz2}.
Details of the simulation are  standard; we made use of the fourth-order approximation for the short imaginary time ($\tau$)  propagator (see, for instance, Ref. \onlinecite{jltp2}), and all of the results presented here are extrapolated to the $\tau\to 0$ limit. We generally found numerical estimates for structural and energetic properties of interest here, obtained with a value of the time step $\tau\sim 3.0\times 10^{-3}$ K$^{-1}$ to be indistinguishable from the extrapolated ones, within the statistical uncertainties of the calculation.
We carried out simulations of systems typically comprising a number $N$ of particles. In the cluster calculations, $N$ can be chosen arbitrarily.
For the bulk, the precise value of $N$
depends on the type of crystalline structure assumed; typically, $N$ was set to 128 for simulations of body-centered cubic (bcc) and face-centered cubic (fcc) structures, 216 for hexagonal close-packed (hcp). However, we also performed a few targeted simulations with twice as many particles, in order to gauge the quantitative importance of finite size effects.
\\ \indent
Physical quantities of interest for the bulk calculations include the energy per particle and pressure as a function of density and temperature, i.e., the thermodynamic equation of state in the low temperature limit. We estimated the contribution to the energy and the pressure arising from pairs of particles at distances greater than the largest distance allowed by the size of the simulation cell (i.e., $L/2$), by approximating the pair correlation function $g(r)$ with 1, for $r > L/2$. We have also computed the pair correlation function and the related static structure factor, in order to assess the presence of crystalline order, which can also be  detected through visual inspection of the imaginary-time paths.
\\ \indent
 We probed for possible superfluid order through the direct calculation of the superfluid fraction using
 the well-established winding number estimator \cite{pollock}. In order to assess the propensity of the system to develop a superfluid response, and its proximity to a superfluid transition, we also rely on a more indirect criterion, namely we monitor the frequency of cycles of permutations of identical particles involving a significant fraction of the particles in the system.
While there is no quantitative connection between permutation cycles and the superfluid fraction \cite{mezzacapo08}, a global superfluid phase requires exchanges of macroscopic numbers of particles (see, for instance, Ref. \onlinecite{feynman}).

\section {Results}
\subsection{Bulk}
We simulated crystalline phases of HMu assuming different structures, namely bcc, fcc and hcp. At low temperature, all of these crystals remain stable in the simulation. The energy difference between different crystal structures is typically small, of the order of the statistical errors of our calculation, i.e., few tenths of a K. This is similar to what is observed in H$_2$ \cite{op}). Consequently, we did not attempt to establish what the actual equilibrium structure is, as this is not central to our investigation.
\\
\indent
As a general remark, we note that the estimates of the physical quantities in which we are interested remain unchanged below a temperature $T=2$ K. Thus, results for temperatures lower than 2 K can be considered ground state estimates.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{energy}
\caption{Energy per particle $e$ in (K) as a function of density $\rho$ in \AA$^{-3}$, computed at temperature $T=1$ K. Solid line is  a  quadratic fit to the data. These energies are computed assuming a bcc solid structure.}
\label{fig:energy}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{sq.pdf}
\caption{Static structure factor computed at temperature $T=1$ K, for a hcp crystal of HMu of density $\rho=0.0209$ \AA$^{-3}$. The simulated system comprises $N=216$ molecules.}
\label{fig:sq}
\end{figure}

We begin by discussing the equation of state of the system in the $T\to 0$ limit.
Fig. \ref{fig:energy} shows the energy per HMu molecule computed as a function of density for a temperature $T=1$ K. The solid line is a quadratic fit to the data, whose fitting parameters yield the equilibrium density $\rho_0=0.02090(5)$ \AA$^{-3}$, corresponding to an average intermolecular distance 3.63 \AA, as well as the ground state energy $e_0=-45.75(2)$ K. The ground state energy is almost exactly \cite{op} one half of that of H$_2$, with a kinetic energy of 70.4(1) K, virtually identical to that of H$_2$ at equilibrium, i.e., at a 30\% higher density.
\\ \indent
The equilibrium density $\rho_0$ is lower than that of superfluid $^4$He, which is 0.02183 \AA$^{-3}$.
The system is in the crystalline phase, however,
as we can ascertain through the calculation of the static structure factor $S(q)$, shown in Fig. \ref{fig:sq}. The result shown in the figure pertains to a case in which particles are initially arranged into a hcp crystal. The sharp peaks occurring at values of $q$ corresponding to wave vectors in the reciprocal lattice signals long-range crystalline order.

\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{gr.pdf}
\caption{Pair correlation function $g(r)$ for HMu at density $\rho_0=0.0209$ \AA$^{-3}$ and $T=1$ K (squares). Also shown for comparison are the corresponding correlation functions for para-H$_2$ at $\rho=0.0261$ \AA$^{-3}$ (circles), and solid $^4$He at $\rho=0.0287$ \AA$^{-3}$ (triangles), at the same temperature. In all three cases the crystalline structure is hcp. Statistical errors are smaller than symbol sizes.}
\label{fig:gr}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{press.pdf}
\caption{Pressure of the HMu bcc crystal as a function of density at
$T=1$ K (circles). The line is a linear fit to the data. Statistical errors are smaller than symbol sizes.}
\label{fig:press}
\end{figure}

It is interesting to compare the structure of a HMu crystal to that of two other reference quantum solids, namely H$_2$ (more precisely, para-H$_2$) at its equilibrium density, namely $\rho=0.0261$ \AA$^{-3}$, and solid $^4$He near melting, at density $\rho=0.0287$ \AA$^{-3}$. The pair correlation functions shown in Fig. \ref {fig:gr} were computed for hcp crystals at temperature $T=1$ K (the results for H$_2$ and $^4$He were obtained in Ref. \onlinecite{marisa}). While the fact that the peaks appear at different distances reflects the difference in density, $^4$He being the most
and HMu the least dense crystals, the most noticeable feature is that the height of the peaks is much more pronounced in H$_2$ than in HMu and $^4$He, whose peaks height and widths are
similar.
\\ \indent
Fig. \ref{fig:press} shows the pressure of a HMu crystal computed at $T=1$ K, as a function of density. In the relatively narrow density range considered, a linear fit is satisfactory, and allows us to compute the speed of sound $v = (m\rho\kappa)^{-1/2}$, where $\kappa = \rho^{-1}(\partial\rho/\partial P)$ is the compressibility. We
obtained the speed of sound $1300\pm100$ m/s at the equilibrium density.
\\ \indent
Next, we discuss the possible superfluid properties of the crystal, as well as of the fluid into which the crystal melts upon raising the temperature. There is no evidence that the crystalline phase of HMu  may display a finite superfluid response in the $T\to 0$ limit. Indeed, the observation is that the behavior of this crystal is virtually identical to that of solid H$_2$, as far as superfluidity is concerned. The main observation is that exchanges of identical particles, which underlie superfluidity, are strongly suppressed in HMu, much like they are in H$_2$, mainly due to the relatively large diameter of the repulsive core of the pairwise interaction. Indeed, exchanges are essentially non-existent in solid HMu at the lowest temperature considered here ($T=1$ K), much like in solid H$_2$; the reduction of particle mass by a factor two does not quantitatively alter the physics of the system in this regard.
\\ \indent
As temperature is raised, we estimate the melting temperature to be around $T\sim 9$ K. We arrive to that conclusion by computing the pressure as a function of temperature for different densities (at and below $\rho_0$), and by verifying that the system retains solid order even at negative pressure, all the way up to $T=8$ K. No evidence is seen of any metastable, under-pressurized fluid phase; on lowering the density, the system eventually breaks down in solid clusters, just like in H$_2$ \cite{2d}. Above 8 K, the pressure of the solid phase jumps and stable fluid phases appear at lower densities, signaling the occurrence of melting. These fluid phases do not display any superfluid properties; exchanges of two or three particles occur with a frequency of approximately 0.1\%. It is known that {quantum mechanical effects} in H$_2$ contribute about one half of the Lindemann ratio at melting \cite{marisa}; we did not perform the same calculation for this system, but in this system quantum mechanics should be more important, on account of the lighter mass. However, qualitatively melting appears to occur very similarly as in H$_2$.
Also, our results suggest that thermal expansion, which is negligible \cite{cabrillo} in solid H$_2$, is  likely very small in this crystal.
\subsection{Clusters}
It is also interesting to study the physics of nanoscale size clusters of HMu, and compare their behavior to that of parahydrogen clusters, for which, as mentioned above, superfluid behavior is predicted at temperature of the order of 1 K, if their size is approximately 30 molecules or less. Crystalline behavior emerges rather rapidly for parahydrogen clusters of greater size \cite{cqp}, with ``supersolid'' behavior occurring for specific clusters \cite{supercl}. This calculation is carried out with the same methodology adopted for the bulk phase of the system, the only difference being that the simulation cell is now taken large enough that a single cluster forms. In this respect, the behavior of HMu clusters is closer to that of parahydrogen than $^4$He clusters,  in that no external potential is required to keep them together (as in the case of $^4$He), at least at sufficiently low temperature ($\lesssim 4$ K).
\\ \indent
However, the reduced molecular mass makes the physics of HMu clusters both quantitatively and qualitatively different from that of parahydrogen clusters. The first observation is that the superfluid response is greatly enhanced; specifically, it is found that clusters with as many as $N=200$ molecules are close to 100\% superfluid at $T=1$ K, and remain at least 50\% superfluid up to $T\lesssim4$ K, at which point they begin to evaporate, i.e., they do not appear to stay together as normal clusters.
This is in stark contrast with parahydrogen clusters, where exchanges are rare in clusters of more than 30 molecules at this temperature, and they would involve at most $\sim 10$ particles. Consistently, the superfluid response is insignificant in parahydrogen clusters, and they stay together almost exclusively due to the potential energy of interaction.
Our results highlight the importance of  the energetic contribution of quantum-mechanical exchanges in the stabilization of HMu clusters at low temperature. Indeed, at $T=1$ K we observe cycles of exchanges involving {\em all} of the particles in HMu clusters comprising as many as 200 molecules.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{profiles.pdf}
\caption{Radial density profiles (computed with respect to the center of mass) for HMu clusters comprising $N=30$, 100 and 200 molecules, at temperature $T=1$ K. At this temperature, all of these clusters are essentially 100\% superfluid. Also shown for comparison is the same quantity for a cluster of $N=30$ parahydrogen molecules, at the same temperature; this cluster has no significant superfluid response.}
\label{profiles}
\end{figure}
\\ \indent
Fig. \ref{profiles} shows  density profiles computed with respect to the
center of mass
of the system for three different clusters of HMu, comprising $N=30$, 100 and 200 molecules. As mentioned above, all these clusters are essentially fully superfluid at this temperature. These  density profiles are qualitatively similar, nearly featureless and reminiscent of those computed for $^4$He droplets \cite{sindzingre89}. The comparison with the density profile computed for a cluster of 30 parahydrogen molecules, also shown in Fig. \ref{profiles}, illustrates how the latter is much more compact and displays pronounced oscillations, which are indicative of a well-defined, solid-like shell structure. \\ \indent
As the number of molecules increases, clusters of HMu ought to evolve into solid-like objects, their structure approaching that of the bulk. The calculation of the radial superfluid density \cite{kwonls,mezzacapo08} suggests that crystallization begins to occur at the center of the cluster; for example, at $T=1$ K the largest HMu cluster studied here ($N=200$) displays a suppressed superfluid response inside a central core of approximately 5 \AA\ radius, in which crystalline order slowly begins to emerge, while the rest of the cluster is essentially entirely superfluid. We expect the non-superfluid core to grow with the size of the cluster. In other words, large clusters consist of a rigid, insulating core and a superfluid surface layer, with a rather clear demarcation between the two. This is qualitatively different from the behavior observed in parahydrogen clusters, in which crystallization occurs for much smaller sizes, and ``supersolid'' clusters simultaneously displaying  solid-like and superfluid properties can be identified \cite{supercl}.
\section{Conclusions}
We have investigated the low temperature phase diagram of a bulk assembly of muonium hydride molecules, by means of first principle quantum simulations. Our model assumes a pairwise, central interaction among HMu molecules which is identical to that of H$_2$ molecules.
By a mapping of the H$_2$ inter-atomic potentials derived from ab initio
calculations, we showed that this model provides a reasonable description
of the HMu-HMu interation.
It is certainly possible to carry out a more accurate determination of the pair potential, but
the main effect of the lower $\mu^+$ mass should be that of increasing the diameter of the repulsive core of the interaction, in turn suppressing exchanges even more. As  illustrated in Ref. \onlinecite{role}, exchanges of identical particles play a crucial role in destabilizing the  classical picture in a many-body system; when exchanges are suppressed, quantum zero point motion can only alter such a picture quantitatively, not qualitatively. Alternatively, this can be understood by the fact that $\Lambda$ is decreased
as the hard core radius is increased, making the system more classical. Obviously, regarding the interaction as spherically symmetric is also an approximation, but one that affords quantitatively accurate results for parahydrogen \cite{op}, for which the potential energy of interaction among molecules plays a quantitatively more important role in shaping the phase diagram of the condensed system.
\\ \indent
Perhaps the most significant observation of this study is the different physics of bulk and nanoscale size clusters of HMu. Bulk HMu is very similar to H$_2$; despite its very low density (lower than that of superfluid $^4$He), the equilibrium crystalline phase is stable below a temperature of about 9 K, much closer to the melting temperature of H$_2$ (13.8 K) than the mass difference may have led one to expect.
No evidence is observed of any superfluid phase, neither liquid nor crystalline, underscoring once again how, in order for a supersolid phase to be possible in continuous space, { it requires some physical mechanism to cause a ``softening" of the} repulsive part of the pair potential at short distances \cite{su}, even if only along one direction, like in the case of a dipolar interaction \cite{patterned}. On the other hand, clusters of HMu including up to a few hundred molecules display superfluid behavior similar to that of $^4$He clusters. This suggests that, as the value of the quantumness parameter $\Lambda$ approaches $\Lambda_c$ from below, one may observe nanoscale superfluidity in clusters of rather large size.
\\ \indent
We conclude by discussing the possible experimental realization of the system described in this work. The replacement of elementary constituents of matter, typically electrons, with other subatomic particles of the same charge, such as muons \cite{Egan}, has been discussed for a long time, and some experimental success has been reported. Even a bold scenario consisting of  replacing {\em all} electrons \cite{wheeler} in atoms with muons (the so-called ``muonic matter'') has been considered; recently a long-lived ``pionic helium'' has been created \cite{pionic}.  Thus,
it also seems plausible to replace  a proton in a H$_2$ molecule with an antimuon; indeed, muonium chemistry has been an active area of research for several decades \cite{muonium}.
In order for ``muonium condensed matter" to be feasible,
a main challenge to overcome is the very short lifetime of the $\mu^+$, of the order of a $\mu$s.

\section*{Acknowledgements}
This work was supported by the Natural Sciences and Engineering Research Council of Canada,
a Simons Investigator grant (DTS) and the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, DTS).
Computing support of Compute Canada
and of the Flatiron Institute
are
gratefully acknowledged.
The Flatiron Institute is a division of the Simons Foundation.


\section{Introduction}
A current exciting  challenge in particle physics is the explanation of  the smallness of the neutrino masses  through new physics at TeV scale. In this regard, the inverse seesaw mechanism(ISS) \cite{ISS} became the paradigm of successful TeV scale seesaw mechanism. Its minimal implementation requires the introduction to the electroweak  standard model (SM) of two sets of three  neutral fermion singlets , $N=(N_1\,,\,N_2\,,\,N_3) $ and $S=(S_1\,,\,S_2\,,\,S_3)$,  composing the following mass terms in the flavor basis,
\begin{equation}
{\cal L}_{mass} \supset \bar \nu M_D  N + \bar N M_N S + \frac{1}{2} \bar S^C \mu_N S + h.c,
\label{masstermsoriginal}
\end{equation}
where $\nu=(\nu_1\,,\,\nu_2\,,\,\nu_3)$ is the set of standard neutrinos. In the basis $(\nu\,,\,N^C\,,\,S)$ the neutrino mass  may be put in the following $9 \times 9$  matrix form,
\begin{equation}
M_\nu=
\begin{pmatrix}
0 & M_D & 0 \\
M^T_D & 0 & M_N\\
0 & M^T_N & \mu_N
\end{pmatrix}.
\label{ISSmatrix}
\end{equation}
In the regime  $\mu_N << M_D < M_N$, the mechanism provides $m_\nu = M_D^T M_N^{-1}\mu_N (M_N^T)^{-1} M_D$ for the mass matrix of the standard neutrinos. Taking $M_D$ at electroweak scale, $M_N$ at TeV and $\mu_N$ at keV scale, the mechanism provides standard neutrinos at eV scale. The new set of fermion singlets $(N\,,\,S)$ develop mass at $M_N$ scale and may be probed at the LHC.

The challenge concerning the ISS mechanism is to find scenarios that  realize it.  This means to propose  models that generate the mass terms in Eq. (\ref{masstermsoriginal}).  In this regard, as the ISS mechanism works in the TeV scale, it seems to be natural to look for realization of the ISS mechanism  in the framework of theories that we expect will manifest at TeV scale\cite{ISSnonsusy1, ISSnonsusy2}, which is the case of supersymmetry ( SUSY). Thus it seems to be interesting to look for  scenarios that realize   the ISS mechanism in the context of SUSY\cite{ISSSUSY1,ISSSUSY2,ISSSUSYR}.

We know already that a natural way of obtaining small neutrino mass in the context of the MSSM is to consider  that R-parity, $R \equiv (-1)^{2S+3(B-L)}$, is violated  through bilinear terms  like $\mu_i \hat L_i \hat H_u$ in the superpotential\cite{standardRPV}.  Thus we wonder  if  R-parity violation (RPV) is an interesting framework for the  realization of the SUSYISS mechanism.  For this, we implement the SUSYISS mechanism in a framework  where R-parity and lepton number are violated  explicitly but baryon number is conserved in a way that we call the minimal realization of the SUSYISS mechanism once  the necessary  set of superfields required to realize it  is the  original one, $\hat N^C_i$ and $ \hat S_i$, only.

Moreover, it has been extensively discussed  that the minimal supersymmetric standard model (MSSM) faces difficulties in accommodating a Higgs of mass of 125 GeV, as discovered by ATLAS and CMS\cite{atlasCMS} while keeping the principle of naturalness\cite{higgsmassanalysis}. This is so because, at tree level, the MSSM predicts a Higgs with a mass whose value cannot exceed  $91$ GeV. Thus robust loop corrections are necessary in order to lift this value to $125$ GeV. Consequently stops with mass far above $1$TeV are required. To accept this is to put the naturalness principle aside. We show that the SUSYISS mechanism developed here  accommodates a $125$ GeV Higgs mass  without resort to robust loop corrections.

\section{The mechanism}

The supersymmetric version of the ISS (SUSYISS) mechanism\cite{ISSSUSY1} requires the assumption of two sets of three singlet  superfields $\hat N^C_i\,,\, \hat S_i$ ($i=1,2,3$) composing, with the  MSSM superfields, $\hat L^T_ i =(\hat \nu_i\,,\,\hat e_i)^T\,,\, \hat H_d^T=(\hat H^-_d \,,\,\hat H^0_d )^T\,,\, \hat H_u^T=(\hat H^+_u \,,\,\hat H^0_u)^T $,  the following  extra terms in the superpotential, $W \supset \hat L \hat H_u \hat N^C + \hat S M_N \hat N^C + \frac{1}{2} \hat S \mu_N \hat S$. A successful  extension of the MSSM  that realizes the  SUSYISS mechanism  must generate these terms. This would be an interesting result in particle physics since we would be providing an origin for the energy scales $M_N$ and $\mu_N$\cite{ISSSUSY2}.

The mechanism we propose here is minimal in the sense that it  requires the  addition to the MSSM of the two canonical  singlet superfields $\hat N^C_i $ and $ \hat S_i$, only. Moreover, we  impose that the superpotential be invariant under the following set of discrete symmetries, $Z_3 \otimes Z_2$, according to the following transformation: under $Z_3$ the transformations are,
\begin{equation}
(\hat S_i \,,\, \hat N^C_i\,,\, \hat e^C_i)\,\, \rightarrow \,\,w(\hat S_i \,,\, N^C_i\,,\, \hat e^C_i), \,\,\,\,\hat L_i \,\,\rightarrow \,\, w^2 \hat L_i,
\label{z3}
\end{equation}
with $w=\exp^{i2\pi/3}$. Under   $Z_2$ we have,   $\hat S_i \rightarrow - \hat S_i$, with all the remaining superfields transforming trivially by $Z_3\otimes Z_2$.

Thus the superpotential of the  SUSYISS mechanism we propose here involves the following  terms,
\small
\begin{eqnarray}
\hat{W}&=&\mu \hat{H}^{a}_{u}\hat{H}_{da} +
Y^{ij}_{u}\epsilon_{ab}\hat{Q}^{a}_{i}\hat{H}^{b}_{u}\hat{u}^{c}_{j} +
Y^{ij}_{d}\hat{Q}^{a}_{i}\hat{H}^{b}_{d}\hat{d}^{c}_{j} +
Y^{ij}_{e}\hat{L}^{a}_{i}\hat{H}^{b}_{d}\hat{e}^{c}_{j} \nonumber \\
&+&
Y^{ij}_{\nu}\epsilon_{ab}\hat{L}^{a}_i\hat{H}^{b}_{u}\hat{N}^{c}_{i} + \frac{1}{2}\lambda^{ijk}_{s}\hat{N}^{c}_{i}\hat{S}_{j}\hat{S}_{k} + \frac{1}{3}\lambda^{ijk}_{v}\hat{N}^{c}_{i}\hat{N}^{c}_{j}\hat{N}^{c}_{k},
\label{superpotential}
\end{eqnarray}
where $a \,,\,b$ are $SU(2)$ indices and $i$ and $j$ are generation indices.  $\hat{Q}_{i}$,  $\hat{u}^{c}_i$, $\hat{d}^{c}_{i}$ and $\hat{e}^{c}_{i}$ are the standard superfields of the MSSM. Perceive that the $Z_3 \otimes Z_2$ symmetry permits that lepton number as well as R-parity be explicitly violated in this model by  terms in the superpotential that involve the singlet superfields  $\hat N^C_i $ and $ \hat S_i$, only .

Now we make an important assumption. We assume  that the  scalars that compose the   superfields  $\hat N^C_i $ and $\hat S_i$ develop nonzero vacuum expectation value (VEV),  $\langle \tilde S \rangle = v_{S_i}$ and $ \langle \tilde N^C_i \rangle =v_{N_i}$, respectively. This assumption provides the source of the canonical mass terms $M_N$ and $\mu_N$ of the SUSYISS mechanism. Note that,   from the last  two  terms in the superpotential above, we have that   the VEV of the scalar $ \tilde S$ becomes the source of the mass scale $M_N$ while the VEV of the scalar $ \tilde N^C$ becomes the source of the mass scale $\mu_N$ . In other words, the superpotential above together with the assumption that the scalars $\hat N^C_i $ and $\hat S_i $ develop non zero  VEVs has the required ingredients  to realize the SUSYISS mechanism.

Another important point of the model is to discuss  the possible values  $v_{S_i}$ and $v_{N_i}$ may take. For this we have to obtain the potential of the model.  The soft breaking sector will play an important role in the form of the potential.

The most general soft breaking sector of our interest involves the following terms,
\begin{eqnarray}
-\cal{ L}_{\mbox{soft}} &=& M^{2}_{Q_{ij}}\tilde{Q^a_i}^{*}\tilde{Q^a_j}
+ M^{2}_{u^{c}_{ij}}\tilde{u^c_i}^{*}\tilde{u^c_j} +
M^{2}_{d^{c}_{ij}}\tilde{d^c_i}^{*}\tilde{d^c_j} \nonumber \\
&+&
M^{2}_{L_{ij}}\tilde{L^a_i}^{*}\tilde{L^a_j} +
M^{2}_{e^{c}_{ij}}\tilde{e^c_i}^{*}\tilde{e^c_j} +
M^{2}_{h_{u}}H^{a*}_u H^{a}_u \nonumber \\
&+& M^{2}_{h_{d}}H^{a*}_d H^{a}_d + M^{2}_{\tilde N_i}
\tilde{N_i}^{*C}
\tilde{N}^{C}_i + M^{2}_{\tilde S_i}
\tilde{S^*_i} \tilde{S_i}
\nonumber \\
&-&
[\left(A_{u}Y_{u}\right)_{ij}\epsilon_{ab}\tilde{Q}^{a}_{i}H^{b}_{u}\tilde{u}^{c}_{j}
+ \left(A_{d}Y_{d}\right)_{ij}\tilde{Q}^{a}_{i}H^{a}_{d}\tilde{d}^{c}_{j}
\nonumber \\
&+&
\left(A_{e}Y_{e}\right)_{ij}\tilde{L}^{a}_{i}H^{a}_{d}\tilde{e}^{c}_{j} +
h.c.] - [B\mu H^{a}_u H^{a}_d + h.c.] \nonumber \\
&+& \frac{1}{2}
\left(M_{3}\tilde{\lambda}_{3}\tilde{\lambda}_{3}
+ M_{2}\tilde{\lambda}_{2}\tilde{\lambda}_{2} +
M_{1}\tilde{\lambda}_{1}\tilde{\lambda}_{1} + h.c.\right) \nonumber \\
&+& (A_{y}Y_{\nu})^{ij}
\epsilon_{ab}\tilde{L}^{a}_i  H^{b}_{u}\tilde{N}^{*C}_j \nonumber \\
&+& [\frac{1}{2}
(A_{s}\lambda_{s})^{ijk}\tilde{N}^{*C}_i\tilde{S_j}\tilde{S_k} +
\frac{1}{3}(A_{v}\lambda_{v})^{ijk}\tilde{N}^{* C}_i\tilde{N}^{*C}_j
\tilde{N}^{*C}_k  \nonumber \\ &+& h.c.].
\label{softterms}
\end{eqnarray}
Note that the last two trilinear terms violate explicitly lepton number and the energy scale parameters $A_s$ and $A_v$ characterize such violation.

A common assumption in developing ISS mechanisms it  to assume that the new neutral singlet fermions are degenerated in masses and self-couplings. However, for our case here, it seems to be more convenient,  instead of considering the degenerated case, to consider the case of only one generation of superfields. The extension for the case of three generations is straightforward and the results are practically the same.

The potential of the model is  composed by the terms $V=V_{soft} + V_D + V_F$. The soft
term, $V_F$,  is given  above in Eq. (\ref{softterms}). The relevant contributions to $V_D$  are,
\begin{equation}
V_{D}= \frac{1}{8}(g^{2}+g^{\prime 2})(\tilde{\nu}\tilde{\nu}^* + H^0_d H^{0*}_d -
H^0_u H^{0*}_u)^2.
\label{Dterm}
\end{equation}

In what concerns the  F-term, the relevant contributions are given by the following terms,

\begin{eqnarray}
V_{F} &=& \left|\frac{\partial \hat{W}}{\partial \hat{H^{0}_{u}}
}\right|^{2}_{H_{u}} + \left|\frac{\partial \hat{W}}{\partial
\hat{H^{0}_{d}} }\right|^{2}_{H_{d}} + \left|\frac{\partial \hat{W}}{\partial
\hat{\nu}} \right|^{2}_{\tilde{\nu}} + \left|\frac{\partial \hat{W}}{\partial
\hat{N^{C}} }\right|^{2}_{\tilde{N}} + \left|\frac{\partial
\hat{W}}{\partial \hat{S}_{L}}\right|^{2}_{\tilde{S}} \nonumber \\
&=& \mu^{2}\left|H^{0}_{u}\right|^{2} + \mu^{2}\left|H^{0}_{d}\right|^{2} +
Y^{2}_{\nu}\left|\tilde{N}^{C}\right|^{2}\left|\tilde{\nu}\right|^{2} + Y_{v}\mu
H^{0*}_{d}\tilde{N}^{C*}\tilde{\nu}
\nonumber \\
&+& Y^{2}_{\nu}\left|H^{0}_{u}\right|^{2}\left|\tilde{\nu}\right|^{2}
+ \frac{1}{4}\lambda^{2}_{s}\left|\tilde{S}\right|^{4} + 4
\lambda^{2}_{v}\left|\tilde{N}^{C}\right|^{4} +
\lambda^{2}_{s}\left|\tilde{N}^{C}\right|^{2}\left|\tilde{S}\right|^{2}\nonumber \\
&+&
\frac{Y_{\nu}\lambda_{s}H^{0}_{u}\tilde{\nu}\tilde{S}^{* 2}}{2} +
2Y_{\nu}\lambda_{v}\left|\tilde{N}^{C}\right|^{2}H^{0}_{u}\tilde{\nu} +
Y^{2}_{\nu}\left|\tilde{N}^{C}\right|^{2}\left|H^{0}_{u}\right|^{2} \nonumber \\
&+&
\lambda_{s}\lambda_{v}\left|\tilde{N}^{C}\right|^{2}\tilde{S}^{2} + h.c.
\label{Fterm}
\end{eqnarray}

With the potential of the model in hand,  we are ready  to obtain the  set of constraint equations for the neutral scalars $H^0_u\,,\, H^0_d\,,\,\tilde \nu\,,\, \tilde S\,,\,\tilde N^C$,
\begin{eqnarray}
&&  v_u\left( M^{2}_{h_u} + \mu^{2} +
\frac{1}{4}(g^{2}+g^{\prime 2})(v^{2}_{u}-v^{2}_{d}-v^2_\nu)  +Y^2_\nu v^2_N +Y^2_\nu v^2_\nu \right) +\nonumber \\
&& -B\mu v_d+ \frac{1}{2}Y_{\nu}\lambda_{s}v_\nu v^{2}_{S}+Y_\nu A_y v_\nu v_N  + 2Y_\nu\lambda_v v_\nu v^2_N =0,\nonumber
\\
&& v_d\left(M^{2}_{h_d} + \mu^{2} -
\frac{1}{4}(g^{2}+g^{\prime 2})(v^{2}_{u}-v^{2}_{d}-v^2_\nu) \right) -B\mu v_u+
Y_\nu \mu v_\nu v_N=0,\nonumber
\\
&&  v_\nu \left(M^{2}_{\tilde \nu} +
\frac{1}{4}(g^{2}+g^{\prime 2})(v^2_\nu + v^{2}_{d}-v^{2}_{u})+Y^{2}_{\nu}v^{2}_{u} +
Y^{2}_{\nu}v^{2}_{N}\right) + \nonumber \\
&& + \frac{1}{2}\lambda_{s}Y_{\nu}v_{u}v^{2}_{S} + Y_\nu A_y v_u v_N + 2Y_\nu
\lambda_v v_u v^2_N + Y_\nu \mu v_d v_N =0, \nonumber \\
&& M^{2}_{\tilde S} + \lambda_{s}Y_{\nu}v_{u}v_\nu + \frac{1}{2}
\lambda^{2}_{s} v^{2}_{S} +\lambda_s A_s v_N + \lambda^2_s v^2_N  +
2\lambda_s\lambda_v v^2_N=0,\nonumber
\\
&&v_{N}\left( M^{2}_{\tilde N} + Y^{2}_{\nu}v^{2}_{u} + \lambda^{2}_{s}v^{2}_{S} +
3\lambda_v A_v v_N + 8\lambda^2_v v^2_N + 4\lambda_{v}Y_{\nu} v_{u} v_\nu
+ 2\lambda_{v}\lambda_{s}
v^{2}_{S} + Y^{2}_{\nu}v^2_\nu \right) +\nonumber \\
&&+ Y_\nu v_\nu (A_{y}v_{u}   + \mu v_{d})+\frac{1}{2} A_{s} \lambda_{s} v^{2}_{S}=0.
\label{constraints}
\end{eqnarray}

Let us first focus on the third relation in the equation above. Observe that the dominant term inside the parenthesis is $M^{2}_{\tilde \nu} $. Outside the parenthesis, on considering for while that $v_N < v_S$, the dominant term is $\frac{1}{2}\lambda_{s}Y_{\nu}v_{u}v^{2}_{S}  $. In view of this, from the third relation above  we have that,
\begin{equation}
v_\nu \approx -\frac{1}{2}
\frac{\lambda_{s}Y_{\nu}v_{u}v^{2}_{S}}{M^2_{\tilde \nu}}.
\label{vnu}
\end{equation}
 For  $M_{\tilde \nu} > v_S$, we have $v_\nu < v_{u , d, S} $, as expected.

Let us now focus on  the fifth relation of the Eq. (\ref{constraints}). The dominant term inside the parenthesis is $M^{2}_{\tilde N}$, while  outside the parenthesis the dominant term  is $\frac{1}{2} A_{s} \lambda_{s} v^{2}_{S}$. Thus the fifth relation  provides,
\begin{equation}
v_N \approx -\frac{1}{2}\frac{ A_{s} \lambda_{s} v^{2}_{S}}{M^2_{\tilde N}} .
\label{smallseesaw}
\end{equation}
 This expression for $v_N$ is similar to the $v_\nu $ case and suggests that $v_N$ is also small.

 Let us now focus on the forth relation. Taking $v_\nu\,,\,v_N \ll v_S$, we have that the dominant terms in that relation are,
 \begin{equation}
 M^{2}_{\tilde S} + \frac{1}{2}
\lambda^{2}_{s} v^{2}_{S} =0.
\label{vS}
\end{equation}
Perceive that  $M_S$ dictates the value of $v_S$. As the neutral singlet scalar $\tilde S$ belongs to an extension of the MSSM, then it is reasonable to expect that  its soft mass term $M_S$ lies at TeV scale. Consequently  $v_S$  must assume values around TeV. In regard to the first and second relations they control the standard VEVs $v_u$ and $v_d$.

Let us return to the expression to $v_N$ in Eq. (\ref{smallseesaw}). As the neutral singlet scalar $\tilde N$ also belongs to an extension of the MSSM, then it is reasonable to expect that  its soft mass term $M_{\tilde N}$  lies at TeV scale, too. In this case perceive that the value of $v_N$ get dictated by the soft trilinear term $A_s$. Thus a small $v_N$ means a tiny $A_s$. As $A_s$ is a trilinear soft breaking term, then it must be generated by some spontaneous SUSY breaking scheme. The problem is that we do not know how SUSY is spontaneously  broken. Thus there is no way to infer exactly the value  of $A_s$.   Moreover, note that $A_s$ is a soft trilinear term involving only neutral  scalar singlets by MSSM which turns its estimation even more complex. We argue here that it is somehow natural to expect that such terms be  small.

For this we have to think in terms of spontaneous SUSY breaking schemes.  For example, in the framework of gauge mediated supersymmetry breaking (GMSB) all soft trilinear terms are naturally suppressed once they arise from  loops. In our case  the new singlets are sterile by the standard gauge group.  The  minimal scenario where such soft trilinear terms could arise would be one that involve the GMSM of the B-L gauge extension of the MSSM. To build such extension and evaluate $A_s$ in such a scenario is out of the scope of this paper. However, whatever be the case,  in the framework of GMSB scheme   $A_s$ must be naturally small and consequently  $v_N$, too.  In this point we call the attention to the fact that the idea behind the ISS mechanism is that lepton number is explicitly violated at low energy scale.  This suggests  that the GMSB seems to be the adequate spontaneous SUSY breaking scheme to be adopted in realizing  SUSYISS mechanism.

Let us discuss the case of  gravity mediated supersymmetry breaking.  As in the ISS mechanism  lepton number is assumed to be explicitly violated at low energy scale, it is expected that    $v_N\,,\,v_S\,,\, v_\nu \, ,\, A_s \,,\,A_v$ are all null at GUT scale.  Considering this, the authors of  Ref. \cite{SUSYISSvalle} evaluated  the running of soft trilinear terms involving scalar singlets  from GUT to down scales in a different realization of the SUSYISS model. As a result they obtained that these terms  develop small values at electroweak scale.   Our case is somehow similar to the case of Ref.  \cite{SUSYISSvalle} and it seems reasonable to expect that, in the general case of three generations, on doing such  evaluation of the running of the soft trilinear terms, our mechanism recover the  result of Ref.  \cite{SUSYISSvalle}.  As we are just presenting the idea by means  of only one generation, such evaluation  of the running of $A_s$ is out of the scope of this work.

Thus it seems to be reasonable to expect that, whatever be the spontaneous SUSY breaking scheme adopted, the soft trilinear terms that violate explicitly lepton number involving  neutral singlet fields as  $\tilde S$ and $\tilde N$ have the tendency to develop  small values.  In what follow we assume  that $A_s$ and $A_v$  lies at keV scale.

There is still an issue to consider in respect to the scalar potential. As
can be easily verified, the value of the potential at origin of the fields is zero.
In order to guarantee that electroweak symmetry will be broken,  we need the potential in the  minimum  to be negative. Taking the constraints in Eq. (\ref{constraints}) to eliminate the soft masses in the
scalar potential, we have,
\begin{eqnarray}
\langle V \rangle_{mim} = &-&\frac{1}{8}\left(g^{2}+g'^{2}\right)\left(v^{2}_{\nu} + v^{2}_{d} -
v^{2}_{u}\right)^2 - Y^{2}_{\nu}\left( v^{2}_{\nu}v^{2}_{N} +
v^{2}_{\nu}v^{2}_{u} + v^{2}_{u}v^{2}_{N}\right) -
\lambda^{2}_{s}v^{2}_{S}v^{2}_{N} - \frac{1}{4}\lambda^{2}_{s}v^{4}_{S} - A_{y}Y_{\nu}v_{\nu}v_{N}v_{u} \nonumber
\\ &-&
\frac{1}{2}A_{s}\lambda_{s}v_{N}v^{2}_{S} - A_{v}\lambda_{v}v^{3}_{N}
-Y_{\nu}\lambda_{s}v_{\nu}v_{u}v^{2}_{S} -
4Y_{\nu}\lambda_{v}v_{\nu}v_{u}v^{2}_{N} -
2\lambda_{s}\lambda_{v}v^{2}_{N}v^{2}_{S} - 4\lambda^{2}_{v}v^{4}_{N} \nonumber
\\ &-& Y_{\nu}\mu v_{\nu}v_{N}v_{d}.
\end{eqnarray}
For the magnitudes of VEVs discussed above, the dominant term is $-
\frac{1}{4}\lambda^{2}_{s}v^{4}_{S}$, which is negative. For the case of one generation considered here this is a strong evidence of the   stability of the potential.

After all these considerations, we are ready to go to the central part of this work that is  to develop the neutrino sector. For this we have, first, to obtain the mass matrix that involves the neutrinos. Due to the RPV the gauginos and Higgsinos  mix with the neutrinos $\nu$, $N$ and $S$. Considering the basis $(\lambda_{0},\lambda_{3},\psi_{h^{0}_{u}}, \psi_{h^{0}_{d}},\nu,N^{c},S)$, we obtain the following mass matrix for these neutral fermions,
\begin{eqnarray}
 \left(\begin{array}{ccccccc}  M_{1} &
  0 & \frac{g' v_{u}}{\sqrt{2}} & -\frac{g' v_{d}}{\sqrt{2}} & -\frac{g'
  v_\nu}{\sqrt{2}} & 0 & 0 \\
 0 & M_{2} & -\frac{g v_{u}}{\sqrt{2}} & \frac{g v_{d}}{\sqrt{2}} & \frac{g v_\nu}{\sqrt{2}} & 0 & 0\\
 \frac{g' v_{u}}{\sqrt{2}}  & -\frac{g v_{u}}{\sqrt{2}}  &  0 & \mu & Y_{\nu} v_{N} & Y_{\nu} v_\nu & 0 \\
 -\frac{g' v_{d}}{\sqrt{2}}  & \frac{g v_{d}}{\sqrt{2}} & \mu & 0 & 0 & 0 & 0 \\
 -\frac{g'  v_\nu}{\sqrt{2}} & \frac{g v_\nu}{\sqrt{2}} & Y_\nu v_N & 0 & 0 & Y_{\nu} v_{u} & 0 \\
 0 & 0 & Y_\nu v_\nu & 0 & Y_{\nu} v_{u} & 6 \lambda_{v} v_{N} & \lambda_{s} v_{S} \\
 0 & 0 & 0 & 0 & 0 & \lambda_{s} v_{S}  & \lambda_{s} v_{N}
 \end{array}\right),
 \label{generalneutrinomassmatrix}
\end{eqnarray}
where  $M_{1}$ e $M_{2}$ are the standard  soft breaking terms of the gauginos. We remark that  on considering the hierarchy $v_N< v_\nu < v_d< v_u<v_S$  the bottom right $3 \times 3$ block of this matrix, which   involves only  the neutrinos, decouples from the  gauginos and Higgsinos sector leaving the neutrinos with the following mass matrix in the basis $(\nu,N^{c},S)$
\begin{eqnarray}
M_\nu \approx  \left(\begin{array}{ccc}
 0 & Y_{\nu} v_{u} & 0 \\
 Y_{\nu} v_{u} & 2 \lambda_{v} v_{N} & \lambda_{s} v_{S} \\
 0 & \lambda_{s} v_{S} & \lambda_{s} v_{N}
 \end{array}\right).
 \label{neutrinomassmatrix}
\end{eqnarray}

For this decoupling to be effective we must have
$v_\nu$ of order MeV or less. Diagonalization of this mass matrix implies
that the lightest neutrino, which is predominantly the standard one, $\nu$, get the following mass expression,
\begin{equation}
m_\nu \approx \frac{Y_{\nu}^2}{\lambda_s}\frac{ v_{u}^2}{ v_{S}^2}  v_{N} .
\label{ISSmass}
\end{equation}

This is exactly  the mass expression of the ISS mechanism.  For $v_S$ around
TeV and $v_N$ around keV we obtain neutrinos at eV scale for  $v_u$ at
electroweak scale. In the case of three generations the pattern of the neutrino
masses will be determined by $Y^{ij}_\nu$ .

To demonstrate the validity of
these aproximations we can compute the mass eigenvalues from the full matrix in
(\ref{generalneutrinomassmatrix}). For typical values of the supersymmetric parameters and $v_S \sim 10$ TeV, $v_N \sim 10$ keV, $v_\nu \sim 1$ MeV and  $Y_\nu \sim \lambda_s=0.21$, we have the following order of magnitude for the  mass
eigenvalues ($\sim$ TeV, $\sim$ TeV,
$\sim 10^2$ GeV, $\sim 10^2$ GeV, $\sim 10^{-1}$ eV, $\sim$ TeV,
$\sim$ TeV), where the lightest particle is  exclusively
the standard neutrino. This result is encouraging and indicates that RPV is an interesting framework to realize the SUSYISS mechanism.

We end this section making a comparison of the SUSYISS developed here with the $\mu\nu$SSM in Ref. {\cite{munucase}}. This model resorts to R-parity violation to solve the $\mu$ problem. However neutrino masses at sub-eV scale require considerable amount of fine tuning of the Yukawa couplings. We stress that, in spite of the fact that  the SUSYISS model contains the particle content of the $\mu\nu$SSM, unfortunately it  does not realize the $\mu\nu$SSM.  This is so because if we allow a term like $\hat S \hat H_u \hat H_d$ in the superpotential in Eq. (\ref{superpotential}), as consequence the entries  $\psi_{h^{0}_{d}} S$ and $\psi_{h^{0}_{u}} S$ in the mass matrix in Eq. (\ref{generalneutrinomassmatrix}) would develop robust values which jeopardize the realization of the ISS mechanism.

\section{The mass of the Higgs}

Now, let us focus on the scalar sector of the model. We restrict our interest   in checking if the model may accommodates   a $125$ GeV Higgs mass  without resorting to  tight loop contributions.  For the case of  one generation the model involves five neutral scalars whose mass terms compose a  $5\times 5 $  mass matrix that we consider in the basis $(H_u\,,\,H_d\,, \tilde \nu \,,\, \tilde N \,,\, \tilde S)$. We do not show such a mass matrix here because of the complexity of  their entries.  Instead of dealing with a $5 \times 5$ mass matrix, which is  very difficult to handle analytically, we make use of a result that says that an upper bound on the mass of the lightest scalar,  which we consider as the Higgs, can be obtained by computing the eigenvalues of the $2 \times 2$ submatrix in the upper left corner of this $5 \times 5$ mass matrix\cite{upperbound}. This is a common procedure adopted in such cases which  give us an idea of the potential of the model to generate the 125 GeV Higgs  mass.

The dominant terms of this $2 \times 2$ submatrix  are given by,
\begin{eqnarray}
M^{2}_{2\times 2}\approx \left(\begin{array}{cc} B\mu\cot(\beta)+
M^{2}_{Z}\sin^{2}(\beta)-\frac{Y_{\nu}\lambda_{s}v_\nu}{2v_u}
v^2_{S} & -B\mu-M^{2}_{Z}\sin(\beta)\cos(\beta)\\
-B\mu-M^{2}_{Z}\sin(\beta)\cos(\beta) & B\mu \tan(\beta) + M^{2}_{Z}\cos^{2}(\beta)  \end{array}\right).
\label{2x2massmatrix}
\end{eqnarray}
We made use of the hierarchy among the VEVs, as discussed above, to obtain such a $2 \times 2$ submatrix.  On diagonalizing this $2 \times 2$  submatrix we obtain  the following upper bound on the mass of the Higgs,
\begin{equation}
m^{2}_{h}\leq
M^{2}_{Z}\cos^2(2\beta)-\frac{Y_{\nu}\lambda_{s}v_\nu}{2v_u}v^2_{S}.
\label{upperbound}
\end{equation}
Note also  that  Eq. ({\ref{vS}}) imposes that either  $M^2_{\tilde
S}$ or $v^2_{S}$ is negative. In order to the second term in Eq. (\ref{upperbound})  gives  a positive contribution to the Higgs mass we  take $M^2_S$ negative and $Y_\nu$ and $\lambda_s$ with opposite sign.

What is remarkable in the mass expression in Eq. (\ref{upperbound}) is that the second term provides a robust correction to the Higgs mass even  involving the parameters that dictate the neutrino masses as the
couplings $Y_\nu$ and $\lambda_s$ and the VEV $v_S$. This  suggest an interesting connection between
neutrino and Higgs mass.  For illustrative proposals, perceive that for
$Y_\nu$ of the same order of $\lambda_s$, $v_\nu$ around MeV, $v_u$ around
$10^2$ GeV and $v_{S}$ of order tens of TeV, the second term provides a contribution of tens of
GeV to the Higgs mass. This contribution is enough to alleviate the pressure on
the stop masses and their mixing in order to keep valid the principle of naturalness.

In order to check the range of values the stop mass and the $A_t$ term may
develop in this model, we add to  $m^{2}_{h}$ given above  the leading 1-$\textit{loop}$ corrections
coming from the MSSM stop terms\cite{loopcorrection},
\begin{equation}
\Delta m^{2}_{h}=
\frac{3m^{4}_{t}}{4\pi^{2}v^{2}}\left(log\left(\frac{m^{2}_{s}}{m^{2}_{t}}\right)
+
\frac{X^{2}_{t}}{m^{2}_{s}}\left(1-\frac{X^{2}_{t}}{12m^{2}_{s}}\right)\right),
\label{Limh2}
\end{equation}
where $m_{t}=173.2$ GeV is the top mass, $v=\sqrt{v^2_u + v^2_d}=174$ GeV is the VEV
of the standard model, $X_{t}\equiv A_t-\mu cot (\beta)$ is the stop mixing parameter and $m_{s} \equiv
(m_{\tilde{t}_{1}}m_{\tilde{t}_{2}})^{1/2}$ is the SUSY scale (scale of
superpartners masses) where $m_{\tilde t}$ is the stop mass. In the analysis done below, we work with
degenerated stops and, in all plots, we take $v_\nu=1$ MeV and
$v_{S}=4 \times 10^4$ GeV.

 Figure 1 shows  possible values for the magnitude of $Y_\nu$ and $\lambda_s$
 that provide a Higgs with a mass of $125$ GeV. Note that the plot tells us that
 such a mass requires $Y_\nu$ and $\lambda_s$ around
 $10^{-1}$. This range of values for $Y_\nu$ and $\lambda_s$ provides, through
 Eq.(\ref{ISSmass}),   $m_\nu \approx 0.1$ eV for $v_S=10$ TeV and $v_N=10$ keV.
 Thus neutrino at sub-eV scale is compatible with $m_h=125$ GeV effortlessly.

 Figure 2  tell us  that the model yields the desired Higgs mass for stop mass
 and mixing parameters below the TeV scale. Finally, Figure 3  shows that  a
 Higgs of mass of $125$ GeV is obtained for a broad range of values of $tan (\beta)$.

Let us discuss a little some phenomenological aspects of the SUSYISS mechanism
developed here. First of all, observe that the aspects of RPV concerning the
mixing among neutralinos and neutrinos, as well as charginos and charged
leptons, are dictated by the VEVs $v_\nu$ and $v_{N}$ and the couplings $Y_\nu$
and $\lambda_s$, which are both small. The squarks sector is practically
unaffected. Thus, with relation to these sectors, the phenomenology of the
SUSYISS mechanism is practically similar to the case  of the supersymmetric
version of the ISS mechanism\cite{ISSSUSY1, ISSpheno}. The signature of the
SUSYISS mechanism developed here should manifest mainly in the  scalar sector of
the model due to the mixing of the neutral scalars with the sneutrinos which
will generate Higgs decay channel with lepton flavor violation $h \rightarrow
l_i l_j$.

In general, as far as we know, this is the first time the ISS mechanism is
developed in the framework of RPV. Thus many theoretical, as well
phenomenological aspects of the model proposed here must be addressed such as
experimental constraints from RPV, accelerator physics, analysis of the
renormalization group equation, spontaneously SUSY breaking schemas,  etc.,
which we postpone to a future paper\cite{future}. Moreover, needless to say that
in SUSY models with RPV the lightest supersymmetric particle is not stable which
means that neither the neutralino nor sneutrino are  candidates for dark
matter\cite{DMcandidate} any longer. We would like to remark that
because of the $Z_3$ symmetry used in the superpotential above cosmological
domain wall problems are expected\cite{DWproblem}. However, the solution of this
problem  in the NMSSM as well in  the $\mu\nu$SSM\cite{munucase} cases may be
applied to our case, too\cite{DWsolution}.

Finally, concerning the stability of the vacuum, we have to impose that the potential be bounded from below when the scalar fields become  large in any direction of the space  fields and that the potential does not present charge and color breaking  minima. Concerning the latter condition, we do not have to worry about this condition here because the new scalar fields associated to the superfield singlets, $\hat S$ and $\hat N^C$,  are neutrals under electric and color charges. Concerning the former issue, the worry arises because at large values of the fields the quartic terms dominate the potential. Thus we have to guarantee  that  at large values of the fields the potential be positive. Thus we have to worry with the quartic couplings, only.  The negative value of $\lambda_s$  leads to two negative quartic  terms. Considering this, on analyzing the potential above, we did not find any direction in the field space in which $\lambda_s$ negative leads to a negative potential. All direction we find involves a set of condition where it is always possible to guarantee that the potential be positive at large value of the fields\cite{casas}.  Moreover, we took $\lambda_s$ negative for convenience. We may arrange the things such that all couplings be positive. For example, on taking $\lambda_s$ positive, $v_\nu$  in Eq. (\ref{vnu}) get negative, which guarantee a positive contribution to the Higgs masses and that all quartic couplings be positive. However, a complete analysis of the stability of the potential is necessary. This  will be done in \cite{future}.

\section{ Conclusions}

In this work we proposed the realization of the SUSYISS model in the framework
of RPV. The main advantage of such framework is that it allows  the realization
of the SUSYISS model with a minimal set of superfield content where the
superfields $\hat S$ and $\hat N^C$ of the minimal implementation are sufficient
to realize the model. To grasp the important features of the SUSYISS,  we
restricted our work to the case of one generation of superfields.   As nice
result, the canonical mass parameters $M_N$ and $\mu_N$ of the SUSYISS mechanism
are recognized as the VEVs of the scalars $\tilde S$ and $\tilde N$ that compose
the superfields $\hat S$ and $\hat N^C$. There is no way to fix the values of
the VEVs $v_S$ and $v_N$. However, it seems plausible that $v_S$ and $v_N$
develop values around TeV and keV scale, respectively. Thus, we conclude that
RPV seems to be an interesting framework for the realization of the SUSYISS
mechanism. We recognize that in order to establish the model a lot of work have
to be done, yet. For example, we have to  find the spontaneous SUSY breaking
scheme that better accommodates the mechanism, develop the phenomenology of
the model and its embedding in GUT schemes. We end by saying  that the main
results of this work are that  the model proposed here realizes minimally the
SUSYISS mechanism and provides a 125 GeV Higgs mass respecting the naturalness principle.

\acknowledgments
This work was supported by Conselho Nacional de Pesquisa e
Desenvolvimento Cient\'{i}fico- CNPq (C.A.S.P and  P.S.R.S ) and Coordena\c c\~ao de Aperfei\c coamento de Pessoal de N\'{i}vel Superior - CAPES (J.G.R).