\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
 -.2ex}{2.3ex plus .2ex}{\large\sc}}
\def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus
 -.2ex}{1.5ex plus .2ex}{\normalsize\sc}}
\makeatother

\newcommand{\newcommand}{\newcommand}
\newcommand{\renewcommand}{\renewcommand}

\newcommand{\subs}[1]{{\vspace*{0.5cm}}%
{\noindent\underline{\small\sc #1}}{\addcontentsline{toc}{subsubsection}{#1}}%
{\vspace*{0.3cm}}}

\newcommand{\subss}[1]{{\vspace*{0.5cm}}%
{\noindent\underline{\small\sc #1}}%
{\vspace*{0.3cm}}}

\newcommand{\chap}[1]{{\clearpage}%
\begin{center}%
{\noindent\underline{\large\sc #1}}{\addcontentsline{toc}{section}{#1}}%
\end{center}%
{\vspace*{0.3cm}}}

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}

\newcommand{\trac}[2]{{\textstyle\frac{#1}{#2}}}

\newcommand{\ex}[1]{\mbox{e}^{\,\textstyle#1}}

\newcommand{\CC}{\Bbb{C}}
\newcommand{\HH}{\Bbb{H}}
\newcommand{\PP}{\Bbb{P}}
\newcommand{\RR}{\Bbb{R}}
\newcommand{\ZZ}{\Bbb{Z}}
\newcommand{\II}{\Bbb{I}}
\newcommand{\EE}{\Bbb{E}}
\newcommand{\TT}{\Bbb{T}}
\newcommand{\DD}{\mathrm{I}\!\mathrm{D}}

\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\d}{\delta}
\newcommand{\ga}{\gamma}
\newcommand{\la}{\lambda}
\newcommand{\f}{\phi}
\newcommand{\phi}{\psi}
\newcommand{\e}{\eta}
\renewcommand{\c}{\chi}
\newcommand{\eps}{\epsilon}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}

\newcommand{\symx}{{\mathhexbox@\msafam@73}}
\newsymbol\smallsmile 1360
\newsymbol\smallfrown 1361
\newcommand{\ad}{\mathop{\mbox{ad}}\nolimits}
\newcommand{\tr}{\mathop{\mbox{tr}}\nolimits}
\newcommand{\Tr}{\mathop{\mbox{Tr}}\nolimits}
\newcommand{\Det}{\mathop{\mbox{Det}}\nolimits}
\renewcommand{\det}{\mathop{\mbox{det}}\nolimits}
\newcommand{\rk}{\mathop{\mbox{rk}}\nolimits}
\newcommand{\del}{\partial}
\newcommand{\diag}{\mathop{\mbox{diag}}\nolimits}
\newcommand{\ra}{\rightarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\LRa}{\Leftrightarrow}
\newcommand{\lra}{\leftrightarrow}
\newcommand{\ot}{\otimes}
\renewcommand{\ss}{\subset}
\newcommand{\nul}{\noindent\underline}
\newcommand{\non}{\nonumber\\}
\newcommand{\mat}[4]{\left(\begin{array}{cc}#1&#2\\#3&#4\end{array}\right)}
\renewcommand{\lg}{\frak{g}}
\newcommand{\G}[3]{\Gamma^{#1}_{\;{#2}{#3}}}
\newcommand{\nam}{\nabla_{\mu}}
\newcommand{\nan}{\nabla_{\nu}}
\newcommand{\dx}{\dot{x}}
\newcommand{\dxl}{\dot{x}^{\la}}
\newcommand{\dxm}{\dot{x}^{\mu}}
\newcommand{\dxn}{\dot{x}^{\nu}}
\newcommand{\ddx}{\ddot{x}}
\newcommand{\ddxm}{\ddot{x}^{\mu}}
\newcommand{\ddxn}{\ddot{x}^{\nu}}
\newcommand{\dxi}{\dot{\xi}}
\newcommand{\ddxi}{\ddot{\xi}}

\def\cite{\cite}
\def\over{\over}
\def{ 1\ov 2}{{ 1\over 2}}
\def{1 \ov 4}{{1 \over 4}}
\def\tilde{\tilde}
\def{\rm f}{{\rm f}}
\def{\rm h}{{\rm h}}
\def{\rm k}{{\rm k}}
\def{\rm x}{{\rm x}}
\def{\rm y}{{\rm y}}
\def{\rm z}{{\rm z}}
\def\rm \theta{\rm \theta}
\def{\rm const}{{\rm const}}
\def\epsilon{\epsilon}
\def{light-cone}{{light-cone}}
\def\ZZ{\ZZ}
\def\RR{\RR}
\def\RR{\RR}
\def\EE{\EE}
\def\gamma{\gamma}
\def\footnote{\footnote}
\newcommand{\rf}[1]{(\ref{#1})}
\def \phi {\phi}
\def \bibitem{\bibitem}
\def \sigma{\sigma}
\def{\cal E}{{\cal E}}
\def{\cal H}{{\cal H}}
\def{D}{{\rm D}}
\def \mu {\mu}
\def {\rm a} {{\rm a}}

\setlength{\parskip}{6pt plus 2pt}%
\addtolength{\hoffset}{-1.5cm} \addtolength{\textwidth}{3cm}

\begin{document}

\begin{center}
{\Large\sc Brown-York Energy and Radial Geodesics}
\end{center}
\vspace{0.2cm}

\begin{center}
{\large\sc Matthias Blau} and {\large\sc Blaise Rollier}
\end{center}

\vskip 0.2 cm
\centerline{\it Institut de Physique, Universit\'e de Neuch\^atel}
\centerline{\it Rue Breguet 1, CH-2000 Neuch\^atel, Switzerland}

\vspace{1cm}

We compare the Brown-York (BY) and the standard Misner-Sharp (MS)
quasilocal energies for round spheres in spherically symmetric space-times
from the point of view of radial geodesics. In particular, we show that
the relation between the BY and MS energies is precisely analogous to that
between the (relativistic) energy $E$ of a geodesic and the effective
(Newtonian) energy $E_{\text{eff}}$
appearing in the geodesic equation, thus
shedding some light on the relation between the two. Moreover, for
Schwarzschild-like metrics we establish a general relationship between
the BY energy and the geodesic effective potential which explains and
generalises the recently observed connection between negative BY energy
and the repulsive behaviour of geodesics in the Reissner-Nordstr\o m
metric. We also comment on the extension of this connection between
geodesics and the quasilocal BY energy to regions inside a horizon.

\section{Introduction}

It is a consequence of the fundamental general covariance of general
relativity that there is no well-defined covariant notion of the local
energy density of the gravitational field. The next best thing is perhaps
the notion of a quasilocal energy (QLE), i.e.\ the energy contained
in a two-dimensional surface.  Numerous definitions of QLE have been
proposed in the literature (for a detailed and up-to-date review with many
references see \cite{szab}), and these tend to be mutually inequivalent
even in simple cases such as the Kerr metric \cite{berg}.

There is at least one case, however, in which there appears to be
\textit{almost} universal agreement as to what the QLE should be,
namely for round spheres (i.e.\ orbits of the rotational isometry
group) in spherically symmetric space-times. In that case, the
classical Misner-Sharp (MS) energy \cite{ms} (see e.g.\ \cite{szab} or
\cite{hayward} for recent discussions) is widely considered to be the
``standard'' definition of the energy for round spheres.

One serious contender to this definition is based on the Brown-York (BY)
QLE \cite{by1}. The definition of the BY energy is based on the covariant
Hamilton-Jacobi formulation of general relativity, and this makes it a
natural object to consider in a variety of contexts, with numerous
attractive features. However, the standard
BY energy for round spheres does not
agree with the standard MS energy (even for the Schwarzschild metric), and
this fact has occasionally been used as an argument against the BY energy
as a ``good'' definition of a QLE (see e.g.\ the discussions in
\cite{szab,hayward}).

In this article we will look at the relationship and differences between
the MS and BY energies for round spheres from the point of view of
geodesics and
their associated energy concepts like the relativistic geodesic energy
and the effective Newtonian potential. In general, one would not expect
point-like objects to be able to probe something not quite local like a
QLE. However, the situation is different for round spheres for which the
QLE is independent of the angular coordinates. In such a situation it is
fair to ask whether there is a relation between the gravitational energy
as felt by a point-like observer (geodesic) and that defined according
to some QLE prescription.

Originally, our investigation of these issues was prompted by an
observation and a remark in \cite{lsy}. There it was observed that
for the Reissner-Nordstr\o m metric the BY energy becomes negative
for sufficiently small radius. In \cite{lsy} it was suggested that
this negative energy is strictly related to the well-known repulsive
behaviour exhibited by the geodesics of massive neutral particles in
the Reissner-Nordstr\o m metric.

What supports this point of
view is the fact that the energy indeed becomes negative at precisely
the radius where radial geodesics begin to experience the repulsive
behaviour of the Reissner-Nordstr\o m core. This clearly hints at a
deeper connection between geodesic and quasilocal energy, or, in the words of
\cite{lsy}:``\textit{The turnaround radius agrees with the radius where
the quasilocal energy becomes negative, so it seems that the two effects
are very likely connected.}''
We will indeed be able to establish a general relationship between the BY
energy and the geodesic effective potential (for radial geodesics) and,
in particular, a relation between negative BY energy and a repulsive
behaviour for geodesics.

Our results also shed some light on the difference between the MS and
BY energies for round spheres.  In particular, for Schwarzschild-like
metrics $ds^2 = -f(r)^2 dt^2 + f(r)^{-2} dr^2 + r^2 d\Omega^2$, for
which geodesics are conveniently described in terms of an effective
Newtonian potential, we observe that the MS energy is directly related
to the effective potential $V_{\text{eff}}(r)$ for radial goedesics, and that
the relation between the MS and BY energies is strictly analogous to the
relation $E_{\text{eff}} = \trac{1}{2}(E^2 -1)$ between the energy appearing in
the effective potential equation and the relativistic geodesic energy $E$
of the particle.  Therefore, inasmuch as $E$ is a relativistic energy and
$E_{\text{eff}}$ an effective Newtonian energy, perhaps one interpretation of
the difference between the BY and MS energies for round spheres is to say
that the former provides one with a relativistic notion of gravitational
energy while the MS energy is more like an effective Newtonian quantity.

We also briefly discuss the extension of these results to regions inside
a horizon. An extension of the BY energy to this case was proposed in
\cite{lsy}. However, it has been remarked\footnote{e.g.\ by one of the
referees, and by Ruth Durrer (private communication)} that the proposal of
\cite{lsy} should perhaps better be thought of as a quasi-local momentum.
Our geodesic perspective is compatible with this
point of view since, as we will show, the relation between the BY
``energy'' of \cite{lsy} and the effective potential inside the horizon
is identical to that between $E$ and $E_{\text{eff}}$ provided that one
considers geodesics that are \textit{spacelike} (outside the horizon).

We believe that the message of this work is two-fold: First of all,
it shows that there are situations where geodesic test particles can be
useful to probe candidate definitions of QLE.  Moreover, these results
also illuminate the difference between the BY and MS energies
and provide further evidence that the BY definition of a QLE provides
a good (relativistic) measure of the gravitational energy even though
(or even precisely because) it does not agree with the standard (and
perhaps somewhat more Newtonian) MS energy for round spheres.

\section{Brown-York and Misner-Sharp Energy for Spherical Symmetry}

We briefly recall the definition of the BY and MS energies
for round spheres, referring to the original literature (e.g.\
\cite{by1,by2,lsy} and \cite{ms})
and the review article \cite{szab} for details.
We will consider a general
spherically symmetric metric written in the form
\be
ds^2 = -N(t,r)^2 dt^2 + f(t,r)^{-2}dr^2 + r^2 d\Omega^2
\label{met2}
\ee
with $d\Omega^2= d\theta^2 + \sin^2\theta d\phi^2$
the standard line-element on the unit 2-sphere.
Even though we will
only consider 4-dimensional space-times in this article, the extension to
higher dimensions is rather straightforward.
In contrast to \cite{by2} we prefer to work directly
with the area radius $r$ as the radial coordinate.
The \textit{round spheres} in this space-time (the orbits of the
rotational isometry group) are the 2-spheres $t=\text{const.},
r=\text{const}$.
It was shown in \cite{by1,by2} that,
in any region in which $\del_t$ is timelike and $\del_r$ is
spacelike, the standard
BY quasilocal energy $E_{BY}(t,r)$ associated to a round sphere
of radius $r$, and calculated with respect to the standard static
observers associated to the spatial slicing $t=\text{const.}$
is given by
\be
E_{BY}(t,r) = \frac{r}{G_N}(1-f(t,r))\;\;,
\label{eby2}
\ee
where $G_N$ is Newton's constant.

This BY energy differs from the ``standard''
Misner-Sharp (MS) energy \cite{ms} for round spheres which, for a metric of
the type (\ref{met2}) and for any $(t,r)$, is given by
\be
E_{MS}(t,r) = \frac{r}{2G_N}\left(1-f(t,r)^2\right)\;\;.
\label{ems1}
\ee
For example, for the Reissner-Nordstr\o m metric
$N(r)^2 = f(r)^2 = 1 - \frac{2m}{r}  + \frac{e^2}{r^2}$
one has
\be
\begin{aligned}
E_{BY}(r) &= \frac{r}{G_N}
\left(1-\sqrt{1 - \frac{2m}{r}+\frac{e^2}{r^2}}\right)\\
E_{MS}(r) &= \frac{1}{G_N}\left(m-\frac{e^2}{2r}\right)\;\;.
\end{aligned}
\ee
Both reduce to the ADM mass $M=m/G_N$ asymptotically,
$\lim_{r\ra\infty} E_{BY}(r) =
\lim_{r\ra\infty} E_{MS}(r) = M$
(and for the Schwarzschild metric one evidently
has $E_{MS}(r)=M$ for all $r$).
Moreover, for sufficiently small values of $r$, $r<r_0 = e^2/2m$
both the MS and the BY energy are negative (note that the expression for the
BY energy is also valid inside the inner horizon $r_-$ and that $r_0 < r_-$).
The qualitative (and not just quantitative) difference bewteen the
BY and MS energies is e.g.\ illustrated by the fact that,
unlike the MS energy, the  BY energy is finite at $r=0$,
$E_{BY}(0)=-|e|/G_N$ \cite{lsy}.

\section{Brown-York and geodesic energy for Schwarzschild-like metrics}

In order to analyse the BY energy (and its relationship with the MS energy)
from the point of view of geodesics, we now specialise to Schwarzschild-like
metrics, i.e.\ static spherically symmetric metrics with $N(r)=f(r)$ (the
extension to time-dependent Schwarzschild-like metrics with $f=f(t,r)$ is
straightforward),
\be
ds^2 = -f(r)^2 dt^2 + f(r)^{-2} dr^2 + r^2 d\Omega^2\;\;.
\label{met4}
\ee
In this case the behaviour of timelike radial
geodesics is governed by the effective potential equation
\be
\trac{1}{2}\dot{r}^2 + V_{\text{eff}}(r) = E_{\text{eff}}\;\;,
\label{veff2}
\ee
where the effective (and effectively Newtonian) potential
$V_{\text{eff}}(r)$ is related to $f(r)^2$ by
\be
f(r)^2 = 1 + 2 V_{\text{eff}}(r)\;\;,
\label{fv}
\ee
and the effective energy $E_{\text{eff}}$ is given in terms of the
relativistic geodesic energy per unit rest mass
$E=f(r)^2 \dot{t}$ of the particle by
\be
E_{\text{eff}} = \trac{1}{2}(E^2 - 1)\;\;.
\label{eeff}
\ee
For later we note that $E=f(r_m)$ where $r_m$ (the index could indicate
a minimum or maximum) is a turning point, $\dot{r}_m=0$, of the trajectory,
and that
in the asymptotically flat case (which we take here to simply mean
$\lim_{r\ra\infty} f(r)=1$) for scattering trajectories that reach
(or start out at) $r\ra\infty$ one also has the relation
$E^2 = 1 + \dot{r}_{\infty}^2\geq 1$ between $E$ and
the velocity at infinity.
In particular, for scattering trajectories in
the Reissner-Nordstr\o m metric, for the minimal radius $r_m=r_m(E)$ one has
$r_m(E) \leq r_m(E=1) = e^2/2m$,
which, as noted in \cite{lsy}, agrees with the radius
$r_0$ where the BY (and MS) energy becomes negative.

In order to now
study the relations among the Brown-York energy, the Misner-Sharp
energy, and the geodesic effective potential, it turns out to be convenient to
introduce the corresponding potentials
\be
V_{BY}(r) := -G_{N}\frac{E_{BY}(r)}{r}\qquad\qquad
V_{MS}(r) := -G_{N}\frac{E_{MS}(r)}{r}\;\;.
\ee
Using the definition (\ref{ems1}) of the MS energy and (\ref{fv}),
one immediately sees that
\be
V_{MS}(r) = -\trac{1}{2}(1-f(r)^2) = V_{\text{eff}}(r)\;\;.
\label{mseff}
\ee
Thus the MS potential agrees on the nose with the effective potential
and has a clear physical interpretation in the present context. In
particular, negative MS energy is strictly correlated with a repulsive
behaviour of the effective potential for radial geodesics.

What about the BY potential? Given the above relation between the
MS energy and radial geodesics, one's first thought may perhaps
be\footnote{This was not our first thought, but we are grateful to one
of the referees for reminding us that it should perhaps have been.}
that to establish a link with the BY energy one should calculate the
latter for freely falling (geodesic) rather than static observers,
or for static observers in comoving (Novikov) coordinates.
The Schwarzschild BY energy for geodesic obervers was first determined
in \cite{bm} and more recently, also motivated by the appearance of
the first version of the present article on the arXiv, in \cite{yc}
(with a slightly different prescription).
For example, the result of \cite{bm} (for an observer initially at rest at
infinity) is
\be
E_{BY}^{\text{freefall}}(r) = \frac{r}{G_N}
\left(\sqrt{1+ \frac{2m}{r}}-1\right)\;\;.
\ee
In Novikov coordinates $(\tau,R)$, where $\tau$ is the proper time of a
radially infalling observer and $R$ is related to the maximal radius
$r_m$ of the geodesic by $R= \sqrt{\frac{r_m}{2m}-1}$,
the Schwarzschild metric reads (see e.g.\ \cite[\S 31.4]{mtw})
\be
ds^2 = -d\tau^2 + \frac{R^2+1}{R^2} \left(\frac{\del r}{\del R}\right)^2 dR^2
+ r(\tau,R)^2 d\Omega^2\;\;.
\ee
Calculating the BY energy for ``static'' observers in this space-time, one
finds
\be
\label{enov}
E_{BY}^{\text{Novikov}}(r) = \frac{r}{G_N}
\left(1 - \frac{R}{\sqrt{R^2 +1}}\right)
= \frac{r}{G_N}
\left(1 - \sqrt{1-\frac{2m}{r_m}}\right)\;\;.
\ee
This can e.g.\ be seen to agree with the result of
\cite{yc}, based on the calculation of the freefall BY energy in Kruskal
coordinates. Thus, neither do the above results reproduce the MS
energy, nor do they appear to be related to the effective geodesic potential
in any other particularly useful or illuminating way.

In this context it is perhaps also worth pointing out that in
\cite{by2} a change of coordinates (foliation) $t\ra T(t,r)$ for the
Schwarzschild metric was exhibited with respect to which the standard
BY energy takes the MS value $E_{BY}(r) = M$. Such a foliation, giving
$E_{BY}(r)=E_{MS}(r)$, can also readily be constructed for the general
Schwarzschild-like metric \eqref{met4}. It suffices to choose $T(t,r)$ such
that
\be
\label{T}
dT = dt + \frac{1-f^2}{f^2(1+f^2)} dr
\;\;.
\ee
To see this note that for a general spherically symmetric metric of the
form
\be
ds^2 = -N(t,r)^2 dt^2 + F(t,r)^{-2}(dr + A(t,r)dt)^2 + r^2 d\Omega^2
\ee
the BY energy is still given by \eqref{eby2} (with $f\ra F$),
and that with the choice \eqref{T} one has
$1-F = \trac{1}{2}(1-f^2)$, so that indeed $E_{BY}(r) = E_{MS}(r)$.
However, the physical significance of this choice of foliation escapes us, and
this construction does not appear to shed any light on the relationship
between the BY energy and geodesic notions of energy.

Thus we now return to the task of relating the standard BY energy
\eqref{eby2} to the geodesic effective potential.
Substituting (\ref{fv}) in (\ref{eby2}), one finds
\be
E_{BY}(r) = \frac{r}{G_N}(1-\sqrt{1+2 V_{\text{eff}}(r)})\;\;,
\ee
or
\be
1+ V_{BY}(r) = \sqrt{1+2V_{\text{eff}}(r)}\;\;.
\label{vbyveff}
\ee
While this relation, which we may also read as the relation between the MS
energy and the BY energy, may appear to be somewhat obscure, it reveals several
interesting features of the BY potential $V_{BY}(r)$
and its relation to $V_{\text{eff}}(r)=V_{MS}(r)$:

\begin{enumerate}

\item First of all
we observe that (\ref{vbyveff}) and (\ref{veff2})
allow us to express the BY potential in terms of geodesic quantities as
\be
\label{vbyr}
1+ V_{BY}(r) = \sqrt{E^2 -\dot{r}^2}\leq E\;\;.
\ee
In other words, the relation between the Brown-York potential and the
relativistic energy $E$
(per unit rest mass) of the particle can be phrased as

\textit{The energy $E$ of the geodesic particle is greater or equal to
the sum of its rest mass and the gravitational potential energy (as measured
by $V_{BY}(r)$), with equality at points where $\dot{r}=0$.}

Thus the BY potential appears to provides a reasonable
measure of the energy of the gravitational field in this context.
The inequality $E\geq 1+ V_{BY}(r)$ should be compared and
contrasted with the analogous equation $E_{\text{eff}} \geq V_{MS}(r)$
for the MS (or effective) potential that follows from (\ref{veff2}).
This suggests a certain
 analogy $E_{BY} \lra E$ and $E_{MS} \lra E_{\text{eff}}$.

\item This analogy is strengthened by the observation that \eqref{vbyveff}
implies
\be
V_{\text{eff}}(r) = \trac{1}{2}\left((1+V_{BY}(r))^2-1\right),
\label{vv2}
\ee
which shows that the relation between $V_{\text{eff}}$ and
$1+V_{BY}$ is identical to the relation $E_{\text{eff}} = \frac{1}{2}(E^2-1)$
\eqref{eeff}
between the effective energy $E_{\text{eff}}$ and the geodesic particle
energy $E$.

Thus, since $E$ is a relativistic energy and $E_{\text{eff}}$ an effective
Newtonian quantity, it is tempting to say that the BY energy provides one
with a relativistic notion of gravitational energy while the MS energy
is really more like an effective Newtonian quantity. So far, however,
this is only a suggestion, based on the geodesic analogy that we have
developed here, and further analysis of this issue, in other settings,
will be required to substantiate (or disprove) this interpretation of
the difference between $E_{MS}$ and $E_{BY}$.

\item
Finally, \eqref{vbyveff} implies that
$V_{\text{eff}}(r)$ and
$V_{BY}(r)$ have the same zeros
and that the BY potential is repulsive/positive whenever (and
whereever) the effective potential is repulsive.
Thus the BY energy is negative if and only if the effective
potential is repulsive. In particular, \eqref{vbyr} leads
to a simple expression for the BY energy at any
turning point $r_m$ ($\dot{r}_m=0$) of the potential, namely
$1+ V_{BY}(r_m) = E$ or
\be
E_{BY}(r_m) = \frac{r_m}{G_N}(1-E)\;\;.
\ee
This also follows directly from the definition \eqref{eby2} and the
previously noted $E=f(r_m)$.
In particular, $E_{BY}(r_m)$ is negative for scattering trajectories with
$E> 1$. Thus non-positive BY energy is necessary for a repulsive
behaviour of radial geodesics. This provides a simple explanation and
proof of a generalisation of the
observation made in \cite{lsy} in the context of the Reissner-Nordstr\o m
metric. Note also that, for the Schwarzschild metric, at $r=r_m$ one has
$E_{BY}(r_m)=E_{BY}^{\text{Novikov}}(r_m)$,
so that static and freely falling
observers can agree on the energy at a turning point of the freely falling
observer, as they should.
\end{enumerate}

All in all this provides us with a coherent picture of the relation
between geodesic notions of energy on the one hand, and the quasilocal
gravitational MS and BY energies for round spheres on the other.

Finally we comment briefly, from the present geodesic point of view,
on the extension $E_{LSY}(r)$
of the BY energy $E_{BY}(r)$ to the interior of a horizon proposed in
\cite{lsy}. Writing the Schwarzschild-like metric as
\be
ds^2 = -\epsilon f(r)^2 dt^2 + \epsilon f(r)^{-2} dr^2 + r^2 d\Omega^2\;\;,
\label{met5}
\ee
with $\epsilon=\pm 1$ corresponding to the exterior (interior) region,
the definition of \cite{lsy} is
\be
E_{LSY}(r) = \frac{r}{G_N}(1-\epsilon f(r))
\ee
($E_{LSY}(r) = E_{BY}(r)$ in the region $\epsilon=+1$).
We now write the effective potential equation for radial timelike
($\lambda=+1$) or spacelike ($\lambda=-1$) geodesics as
\be
\trac{1}{2}\dot{r}^2 + V_{\text{eff}}^{\lambda}(r) = E_{\text{eff}}^{\lambda}
\ee
where
$E_{\text{eff}}^{\lambda} = \trac{1}{2}(E^2-\lambda)$.
Then one easily finds
\be
\epsilon\lambda V_{\text{eff}}^{\lambda}(r) =
\trac{1}{2}\left((1+V_{LSY}(r))^2-\epsilon\right)\;\;.
\label{vv3}
\ee
This relation between the effective potential
$V_{\text{eff}}^{\lambda}(r)$ and
$1+V_{LSY}(r)$ is identical to the relation
between the effective Newtonian energy $E_{\text{eff}}^\lambda$
and the relativistic geodesic
energy $E$ for any $\epsilon$ provided that one
correlates the region of interest (specified by
$\epsilon$) with the character of the geodesic (indicated by $\lambda$)
by making the choice
$\epsilon = \lambda$.
Thus for $\epsilon=-1$
$E_{LSY}(r)$ appears to be naturally associated with spacelike
geodesics.

\subsection*{Acknowledgements}

We are grateful to the referees for their useful comments and suggestions.
This work forms part of a Master thesis project of B.R.\ performed jointly
at the Universit\'e de Gen\`eve and the Universit\'e de Neuch\^atel. M.B.\
acknowledges financial support from the Swiss National Science Foundation
and the EU under contract MRTN-CT-2004-005104.

\renewcommand{\Large}{\normalsize}


\section*{Dynamical Behaviour of $O$ in Lattice Gases}

The dynamical behaviour of the anisotropic order parameter  $m$ [see Eq.~\eqref{eq:def-m} in the Letter] following a quench to the critical point is well described by
the Gaussian theory for all the three lattice gas models studied, $i.e.,$ driven lattice gas with either constant (IDLG) or random (RDLG) infinite drive and equilibrium lattice gas (LG). In other words, in the short-time regime, $m \sim t^{1/2}$ [see Eq. \eqref{eq:mt}] and the Binder cumulant $g$ of the lowest transverse mode [defined in Eq. \eqref{eq:binder}] is zero in this regime. The alternative order parameter $O,$ however, distinguishes between the driven (IDLG, RDLG) and the equilibrium (LG) lattice gases.

In order to understand  this, we first write the phenomenological scaling form for $O$,  analogous to Eq. \eqref{eq:scalingass} in  the Letter,
\begin{eqnarray}
O (t, L_{\parallel} ; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O (t/L_{\parallel}^{z/(1+\Delta)} ; S_\Delta).\quad
\label{eq:Oscalingass}
\end{eqnarray}
We already remarked that, in the LG, this scaling form is not compatible with the prediction $O \sim t^{1/8}  L_{\parallel}^{-1/2}$ of the Gaussian theory.  However, following Ref. \cite{AS2002}, it can be argued that, at short times, the only dependence of $O$ on the system size $L_{\parallel}$ is of the form $O \sim L_\parallel^{-1/2}$ which is very well confirmed by numerical simulations. Accordingly,  the generic behaviour of $O$ can be assumed to be
\begin{eqnarray}
O \sim t^{\alpha} L_\parallel^{-1/2}, \label{eq:O}
\end{eqnarray}
where $\alpha$ is a phenomenological exponent to be determined. This, along with Eq. \eqref{eq:Oscalingass}, implies $\tilde f_O(x) \sim x^{\alpha}.$ Comparing the finite-size behaviour in Eq.~\eqref{eq:O} with Eq.~\eqref{eq:Oscalingass} one actually infers,
\begin{eqnarray}
\alpha &=& \frac{1+ \Delta -2 \beta/\nu}{2 \, (4- \eta)}. \label{eq:alpha}
\end{eqnarray}
This equation, together with the hyperscaling relation $\Delta - 2 \beta/\nu= - \eta$ in two spatial dimensions, shows that the prediction $\alpha = 1/8$ of the Gaussian theory [see Eq. \eqref{eq:Ot}] can be obtained only when $\eta=0,$ which is the case for the IDLG (exactly) and the RDLG (approximately) but not for the LG.

On the other hand,  Eq.~\eqref{eq:alpha} predicts $\alpha = 1/10$ upon substituting the values of the critical exponents corresponding to the Ising  universality class (LG). This is consistent with the numerical simulation results presented in the main text, see Fig. \ref{fig:ising}(b) therein.

\begin{figure}[th]
\vspace*{0.2 cm}
 \centering
 \includegraphics[width=10 cm]{./compare_binder.pdf}

\caption{Comparison between the temporal evolution of the Binder cumulants $g$ corresponding to the $12^{th}$ transverse mode, $i.e.,$ with $n_\perp =12,$ in the LG (lowest curve), IDLG and RDLG (two upper curves) on a $32 \times 32$ lattice. \label{fig:b}}
 \label{fig:binder}
\end{figure}

The emergence of this new value $1/10$ of the exponent $\alpha$ must be traced back to the non-Gaussian nature of higher fluctuating modes in the LG. In fact, even though the lowest mode behaves identically in all the three models we considered,  characterized by the same behaviour of $m$, higher modes show a significant difference in the non-driven case.

To illustrate this, we measured the Binder cumulants of higher modes which is defined  analogously to Eq.~(11), using transverse modes other than the first, i.e., with $\mu=\tilde \sigma(0,2 \pi n_\bot/L_\bot)$ and $n_\bot>1.$
 Figure \ref{fig:b} compares the same for all the three lattice gases for the mode with $n_\perp =12$ on a $32 \times 32$ lattice. Clearly, the curve corresponding to the LG (lowest, blue) departs from Gaussian behaviour $g=0$ (in practice, $e.g.,$ $|g| \lesssim 0.005,$ corresponding to the shaded gray area) much earlier than it does for the IDLG  or RDLG (two upper curves, red and green respectively).

Accordingly, the different dynamical behaviour of $O$, which involves a sum over all modes, can be attributed to the non-Gaussian nature of the higher modes in the LG.
Such a departure is not entirely surprising. In fact, for higher modes, mesoscopic descriptions such as the ones in Eqs. \eqref{eq:L-DLG} or \eqref{eq:g_evol} are not expected to hold, while the anisotropy at the microscopic level could be the mechanism leading to the Gaussianity of higher modes in the driven models.


\section{Introduction}\label{sec:intro}
The ATLAS and CMS collaborations at Large Hadron Collider (LHC) have
recently discovered a new bosonic resonance of mass around 125 GeV
\cite{:2012gk, ATLAS:science,:2012gu,CMS:science, :2012br}. Measuring its coupling
to different  Standard Model (SM) particles and establishing its nature are going to
be leading aims of future LHC runs. Although it is yet to be confirmed
as SM Higgs, in this paper we specify this resonance as Higgs and denote it by $H$.
Any deviations from its SM  nature should exhibit in its coupling to
different particles. Anomalous couplings of Higgs may come in both gauge and Yukawa
sectors. Establishing the nature of the Higgs will require a precise
measurement of its gauge as well as Yukawa couplings. In future LHC runs the coupling of Higgs to $W, Z$ bosons
will be measured in several different channels such as $H \to Z Z^{*}\to 4 \ell$. However, measuring
its coupling to fermions as well as loop induced couplings like $HZ\gamma$ are going to be
relatively more challenging.

In this regard, several studies
\cite{Isidori:2013cla,Gonzalez-Alonso:2014rla,Buchalla:2013mpa,Gao:2014xlv,Bhattacharya:2014rra,Keung:1983ac,Hagiwara:1993sw,Curtin:2013fra,Bodwin:2014bpa,Korchin:2014kha,Delaunay:2013pja,Giudice:2008uua,Kagan:2014ila,Chen:2014gka,
Beneke:2014sba,Manohar:2000dt,Isidori:2013cga,Brod:2013cka,Falkowski:2014ffa,Bergstrom:1985hp,Grinstein:2013vsa,Bodwin:2013nua}
have been
directed towards rare Higgs decays such as $H \to Z V$; $V$ being a vector quarkonium ($J^{\rm{PC}} = 1^{--}$).
Although the branching ratios are small, rare Higgs decays offer
complimentary information about Higgs couplings~\cite{Isidori:2013cla} and can serve as important probe of ``New Physics'' (NP).
Besides, subsequent decays of $Z$ and $V$ into  pair of leptons make them experimentally clean channels.
Moreover, the decay rates are further enhanced by resonant production of $V$ and could be seen in high
luminosity LHC runs or in future colliders. Among rare Higgs decays, the decay to a vector quarkonium
($ J/\psi, \Upsilon$) received considerable attention in recent times.
 Refs.~\cite{Gonzalez-Alonso:2014rla,Gao:2014xlv,Bhattacharya:2014rra} have studied $H \to Z V$ process
 with the aim to probe Higgs couplings and new physics.

There exist three different processes that contribute to the $H \to Z V$
decay. Refs.~\cite{Isidori:2013cla,Gonzalez-Alonso:2014rla} calculate
the decay rates for $H \to Z V$ via $H \to Z^* Z$ with $Z^* \to V$.
Although in SM the process $H \to Z \gamma^{*} \to Z V$  is loop suppressed,
Ref.~\cite{Gao:2014xlv} shows that it can provide a
significant contribution depending on the nature of the vector boson $V$.
There exist a third contribution where $H \to Z V$ is produced via $H \to q \bar{q}\to Z V$
and Ref.~\cite{Bhattacharya:2014rra} studies this channel assuming anomalous coupling
in Yukawa sector. As any of the above three processes could be anomalous, in this paper we
perform a model independent analysis of $H\to ZV$ decay without making any assumption on its
origin.

We first calculate the SM contribution of the three processes and their respective interferences
to the $H\to ZV$ decay.
 We also write down the most general $H  Z V$ vertex and derive corresponding angular asymmetries from
it. These asymmetries have been discussed in Ref.~\cite{Modak:2013sb,Modak:2014zca} in the context of
$H \to Z Z^*\to 4 \ell$ and also in Ref~.\cite{Buchalla:2013mpa} to probe non standard Higss coupling
via angular analysis. They provide powerful tools which can probe SM as well as any anomalous contributions to
the decay. Similar asymmetry has also been discussed in Ref.~\cite{Bhattacharya:2014rra}
to measure $CP$ odd properties of Yukawa sector in $H \to Z V$ decays.
 In our work we construct all possible asymmetries and perform a case by case analysis discussing relative contributions
of different diagrams and their consequences on respective asymmetries.

The plan of the paper goes as follows. In section~\ref{sec:direct} we compute the SM contributions
of the three processes and compare their relative strengths.
Section~\ref{sec:anghvz} is devoted to formalize the angular analysis and construction of angular
asymmetries for $H \to Z V$ with further decays of $V$ and $Z$  into pair of leptons. We also
discuss how to probe different Higgs coupling using these angular asymmetries. In Section~\ref{sec:conclusion} we conclude
our results.

\section{Standard Model contribution of different channels to $H \to Z V $ process}
\label{sec:direct}

\begin{figure*}[hbtp!]
\centering
\includegraphics[width=.9\textwidth]{hvz.eps}
\caption{Feynman diagrams contributing to $H \to Z V$, $V$ being a vector quarkonium resonance.
The diagrams originate from three different couplings: (a)  tree level $HZZ$ coupling,
 (b) loop induced $HZ\gamma$ coupling, (c)  $H q \bar{q}$ Yukawa coupling.}
\label{fig1}
\end{figure*}

We start our discussion by first estimating the relative strength of SM contribution of different channels to
the process $H \to Z V$, where $V$ is a vector quarkonium ($J^{PC}=1^{--}$). In particular we will focus on $J/\psi(1S)$ and $\Upsilon(1S)$ but our analysis is general and can be used for any vector quarkonium. These decays receive contributions from three
different diagrams as shown in Fig.\ref{fig1}. Some of these contributions have been individually studied in previous
works \cite{Gonzalez-Alonso:2014rla, Gao:2014xlv, Bhattacharya:2014rra} but a combined analysis is still lacking.

 The relative strengths of the diagrams and their interference terms
vary depending on the final vector quarkonium resonance. Because of quite different masses of
$J/\psi$ and $\Upsilon$ resonances the relative
strengths of these diagrams differ appreciably in the two cases.
We explicitly calculate the individual contributions for $J/\psi (1S)$ and $\Upsilon(1S)$
to demonstrate this fact.

In SM the first diagram Fig.\ref{fig1}(a), originates from tree level $HZZ$ gauge coupling.
The matrix element for it is given by
\begin{align}
\mathcal{M}_1 & = -\mathcal{K}_1\left(a^{ZZ}_1 g_{\mu \nu}\right)\epsilon_1^{*\mu}\epsilon_2^{*\nu}
\end{align}
where
\begin{align}
 \mathcal{K}_1=\frac{2 g~g^q_V f_V}{\cos\theta_W}\frac{M_V M_Z^2}{M_Z^2-M_V^2}\label{k1},
\end{align}
with $\theta_W$ as Weinberg angle, $g^q_V=(\frac{1}{4}-\frac{2}{3}\sin^2\theta_W)$ for Charm($c$) quark and
 $g^q_V=(-\frac{1}{4}+\frac{1}{3}\sin^2\theta_W)$ for Bottom($b$) quark. Also,  $\epsilon_1^{\mu}(q_1)$ and $\epsilon_2^{\nu}(q_2)$
are the polarization vectors for $Z$ and $V$ having momenta $q_1$ and $q_2$ respectively. Moreover,
$f_V$ is defined by the matrix element $\bracket{0}{\bar{q}\gamma^{\mu}q}{V(q_2,\epsilon_2)}=f_V M_V \epsilon_2^{\mu}$.

Since in SM the $HZ\gamma$ coupling is forbidden at tree level, the second diagram Fig.\ref{fig1}(b),
can only arise via loop processes. One can compute this process by writing down an effective lagrangian
for the $HZ\gamma$ coupling \cite{Korchin:2013ifa,Gao:2014xlv,Bergstrom:1985hp,Hagiwara:1993sw}
 The matrix element for this diagram is given by
\begin{align}
\mathcal{M}_2 &=-\mathcal{K}_2 \left(a^{Z\gamma}_1 ~q_1.q_2 ~g_{\mu \nu}- a^{Z\gamma}_2 q_{1\nu}q_{2\mu}\right)
\epsilon_1^{*\mu}\epsilon_2^{*\nu}
\end{align}
where
\begin{align}
\mathcal{K}_2 = \frac{g\,\alpha\, Q^f f_V}{2 \pi v} \frac{C_{Z\gamma}}{M_V}.
\end{align}
$C_{Z\gamma}$ is the dimensionless effective
coupling constant for the $HZ\gamma$ vertex \cite{Bergstrom:1985hp,Hagiwara:1993sw,Korchin:2013ifa}, $\alpha = \frac{e^2}{4 \pi}$ and
$Q^f = \frac{2}{3}, \frac{-1}{3}$ for $V = J/\psi, \Upsilon$ respectively.

The third contribution Fig.\ref{fig1}(c) comes from   $Hq\bar{q}$ Yukawa coupling  and is given by
\begin{align}
\mathcal{M}_3 &=-\mathcal{K}_3 \left(a^{Zq\bar{q}}_1 ~q_1.q_2 ~g_{\mu \nu}- a^{Zq\bar{q}}_2 q_{2\mu} q_{1\nu}\right)
\epsilon_1^{*\mu}\epsilon_2^{*\nu}
\end{align}
where
\begin{align}
\mathcal{K}_3 = \frac{4\sqrt{3} g g^q_V \phi_0}{\cos \theta_W\left(M_H^2-M_Z^2-M_V^2\right)}\left(\frac{M_V G_F}{2\sqrt{2}}\right)^{\frac{1}{2}},
\end{align}
and $\phi_0$ is the wave function of the
vector quarkonium resonance evaluated at zero three momentum \cite{Bodwin:2013gca,Bodwin:1994jh,Bodwin:2014bpa}.

The total decay width for $H \to Z V$ process is combination of all three contributions
 given by
\begin{align}
\Gamma_{total}& =\Gamma_1+\Gamma_{2}+\Gamma_{3}+\Gamma_{12}+\Gamma_{13}+\Gamma_{23}.
\end{align}
where $\Gamma_i$ are obtained from $|\mathcal{M}_i|^2$ and $\Gamma_{ij}$
are interference terms between $\mathcal{M}_i$ and $\mathcal{M}_j$ with $i,j=1,2,3$. The individual
contributions for both $ J/\psi(1S) $ and $\Upsilon(1S)$ are listed in Table~\ref{tab}.
\begin{table}[htb]
\caption{Contributions to the branching fraction from the three contributing diagrams and their interferences for $J/\psi(1S)$ and $\Upsilon(1S)$ resonances. The total decay width of Higgs is assumed to be $4.07$ MeV. Values  of $f_V=0.405(0.680)$ GeV
\cite{Gonzalez-Alonso:2014rla} and $\phi^2_0=0.073(0.512)$ $\rm{GeV}^3$\cite{Bodwin:2013gca}  for $J/\psi(\Upsilon)$.}

\begin{tabular}{c c c}\hline\hline
$\mathcal{B}r(H \to Z V)$  \hspace{1cm} & $ J/\psi(1S) $                    \hspace{1cm}    & $\Upsilon(1S)$  \\ \hline
$\mathcal{B}r_{\Gamma_1}$          \hspace{1cm}  &  $1.75 \times 10^{-6}$     \hspace{1cm}   & $1.68 \times 10^{-5}$ \\
$\mathcal{B}r_{\Gamma_2}$                       \hspace{1cm} & $1.14 \times 10^{-6}$       \hspace{1cm} &  $8.33 \times 10^{-8}$ \\
$\mathcal{B}r_{\Gamma_3}$                       \hspace{1cm}  & $8.52 \times 10^{-9}$       \hspace{1cm} &  $5.80 \times 10^{-7}$ \\
$\mathcal{B}r_{\Gamma_{12}}$                      \hspace{1cm}  & $4.50 \times 10^{-7}$       \hspace{1cm} & $1.10\times 10^{-6}$ \\
$\mathcal{B}r_{\Gamma_{13}}$                      \hspace{1cm} & $3.89 \times 10^{-8}$       \hspace{1cm} &  $2.89 \times 10^{-6}$ \\
$\mathcal{B}r_{\Gamma_{23}}$                      \hspace{1cm} & $1.97\times 10^{-7}$       \hspace{1cm} & $4.40\times 10^{-7}$ \\ \hline \hline
\end{tabular}
 \label{tab}
\end{table}

From Table\,\ref{tab} it is clear that the relative contributions of the three channels is different
for  $ J/\psi $ and $\Upsilon$ resonances. In case of $ J/\psi$ the dominant contributions
come from $\Gamma_{1}$ and $\Gamma_{2}$ corresponding to $HZZ$  and $HZ\gamma$ couplings respectively.
The subleading contributions come from the interference terms $\Gamma_{12}$ and $\Gamma_{23}$. The
contribution $\Gamma_{3}$ coming from $Hq\bar{q}$ coupling is negligibly small. The
major contribution from Yukawa sector will come from the interference term $\Gamma_{23}$.
Therefore while probing the anomalous Yukawa couplings one should not neglect the contribution of
the interference terms over $\Gamma_{3}$.

However, in case of $\Upsilon$ the situation is quite different. The leading contribution comes only from the $\Gamma_{1}$ term
whereas $\Gamma_{12}$ and $\Gamma_{13}$ provide the subleading contributions.
The contribution of $\Gamma_{3}$ is now larger than $\Gamma_{2}$ but still negligibly small
compared to $\Gamma_{1}$. Again as before while probing anomalous Yukawa coupling the
effect of interference terms can not be neglected.

As discussed above the rare Higgs decays $H \to Z V$ are sensitive not only
to $HZZ$ coupling but also to $HZ\gamma$ and $Hq\bar{q}$ couplings. Moreover,
depending on nature of $V$, the contribution of various Higgs couplings to the decay
widths vary widely. Hence these decay modes have potential to provide information
complimentary to $H \to Z Z^{*} \to 4 \ell$ ``golden channel''.
Also, $H \to Z V$ decays followed by a subsequent
decay of $Z$ and $V$ into pair of leptons will provide a experimentally clean channel
that can be used to probe them in future colliders or high luminosity LHC runs.
In next section we will discuss  the angular analysis technique which provide a powerful tool
for probing such couplings.

\section{Angular analysis and Observables for $H \to Z V \to 4 \ell$ process}
\label{sec:anghvz}

In this section we formalize the necessary technique to probe $HZV$ vertex.
We start with writing down general structure of the vertex and the different helicity amplitudes for
$H \to Z V$ process.
The general Lorentz structure of the $HZV$ vertex can be written as
\begin{equation}\label{eq:VHZZ0}
   V^{\alpha \beta}_{HZV}=
  \bigg( a_1 \, g^{\alpha \beta} + a_2 \, P^{\alpha}P^{\beta}
  + i a_3 \,\epsilon^{\alpha \beta \mu \nu} \; q_{1\mu}\,q_{2\nu} \bigg),
\end{equation}
where $a_1$, $a_2$ and $a_3$ are vertex factors, $P$ is the momentum of
Higgs boson and $q_1$ and $q_2$ are momenta of $Z$ and $V$ respectively. In SM
\begin{align}
a^{\rm{SM}}_1 & = -(\mathcal{K}_1\,a^{ZZ}_1+\mathcal{K}_2\,a^{Z\gamma}_1 q_1.q_2+\mathcal{K}_3\,a^{q\bar{q}}_1 q_1.q_2),
\label{a1} \\
a^{\rm{SM}}_2 & = \left(\mathcal{K}_2~a^{Z\gamma}_2+\mathcal{K}_3 a^{q\bar{q}}_2\right), \label{a2} \\
a^{\rm{SM}}_3 & = 0. \label{a3}
\end{align}
The couplings $a_1$, $a_2$ and $a_3$ can be extracted via angular asymmetries discussed below.
Any deviation from SM values will indicate anomalous nature of $H \to Z V$ decay.

 The decay under consideration can be expressed in terms of three helicity
amplitudes $\mathcal{A}_{L}$, $\mathcal{A}_{\parallel}$ and $\mathcal{A}_{\perp}$ defined
in the transversity basis as
\begin{align}
  \mathcal{A}_{L} &= (M_H^2 - M_Z^2 - M_V^2)\, a_1 + M_H^2\, X^2 \, a_2, \label{eq:AL}\\%[2ex]
  \mathcal{A}_{\parallel} &= \sqrt{2 }M_H M_V \, a_1, \label{eq:AA}\\%[2ex]
  \mathcal{A}_{\perp} &= \sqrt{2}M_H^2 M_V\, X  \, a_3, \label{eq:AP}
\end{align}
where $M_H$, $M_Z$ and $M_V$ are masses of $H$, $Z$ and $V$ respectively with
\begin{eqnarray}\label{eq:X}
  X&=&\displaystyle\frac{\sqrt{\displaystyle\lambda(M_H^2,M_Z^2,M_V^2)}}{\displaystyle 2M_H}
\end{eqnarray}
where
$\lambda(x,y,z)= x^2+y^2+z^2-2\,x\,y-2\,x\,z-2\,y\,z$.
The helicity amplitudes $\mathcal{A}_{L}$, $\mathcal{A}_{\parallel}$ and $\mathcal{A}_{\perp}$ have
definite parity properties. $\mathcal{A}_{L}$, $\mathcal{A}_{\parallel}$ are $CP$ even in nature
where as $\mathcal{A}_{\perp}$ is $CP$ odd.

The full angular distribution for $H \to Z_{(\ell^+ \ell^-)} V_{(\ell^+ \ell^-)}$ is given
by following expression

\begin{widetext}
\begin{align}\label{angdist}
 & \frac{8\pi}{\Gamma}\frac{d^3\Gamma}{ d\cos{\theta_1} \;
    d\cos{\theta_2} \; d\phi} = 1 +\frac{|f_{\parallel}|^2-|f_{\perp}|^2}{4}
  \cos\,2\phi\big(1-P_2(\cos\theta_1)\big)
  \big(1-P_2(\cos\theta_2)\big)
+\frac{1}{2} \text{Im}(f_{\parallel} f_{\perp}^*)\,\sin\,2\phi \nonumber \\
& \times \big(1-P_2(\cos\theta_1)\big) \big(1-P_2(\cos\theta_2)\big)
+\frac{1}{2}(1-3
  \modulus{f_L}^2)\,\big(P_2(\cos\theta_1)+P_2(\cos\theta_2)\big)
+\frac{1}{4}(1+3\modulus{f_L}^2)\,P_2(\cos\theta_1)P_2(\cos\theta_2)\nonumber\\
  & +\frac{9}{8\sqrt{2}} \left( \text{Re}(f_L
    f_{\parallel}^*)\,\cos\phi + \text{Im}(f_L f_{\perp}^*)\,\sin\phi
  \right)
  \sin\,2\theta_1\,\sin\,2\theta_2
+\eta \Bigg(\frac{3}{2} \text{Re}(f_{\parallel}
  f_{\perp}^*)\big(\cos\theta_2 (2 + P_2(\cos\theta_1)) \nonumber\\
 & - \cos\theta_1
  (2 + P_2(\cos\theta_2))\big)
-\frac{9}{2\sqrt{2}}\text{Re}(f_L f_{\perp}^*)
  \cos\theta_2\cos\phi \sin\theta_1\sin\theta_2
+\frac{9}{2\sqrt{2}}\text{Im}(f_L f_{\parallel}^*)
  \cos\theta_2\sin\phi
  \sin\theta_1\sin\theta_2\Bigg),
\end{align}
\end{widetext}
where the angle $\theta_1$($\theta_2$) is the angle between three momenta of $\ell^+$ in $Z$($V$) rest frame
and the direction of three momenta of $Z$($V$) in $H$ rest frame. The angle $\phi$ is defined as the angle between
the normals to the planes defined by  $Z\to \ell^+ \ell^-$ and  $V\to \ell^+ \ell^-$ in $H$ rest frame.
The expressions for {\it{helicity fractions}} $f_L$, $f_{\parallel}$ and $f_{\perp}$ are given in the appendix.

 Integrating Eq.\eqref{angdist} with respect to the angles
$\cos{\theta_1}$ or $\cos{\theta_2}$ or $\phi$, one can obtain following
uniangular distributions:
\begin{align}
  \frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_1} &=
  \frac{1}{2} + t_2\,P_2(\cos\theta_1) - t_1
  \cos\theta_1,  \label{eq:t1dist} \\
  \frac{1}{\Gamma}\frac{d\Gamma}{ d\cos\theta_2} &=
  \frac{1}{2} + t_2\,P_2(\cos\theta_2) , \label{eq:t2dist} \\
  \frac{2\pi}{\Gamma}\frac{d\Gamma}{d\phi} &= 1 +
  u_2\,\cos\,2\phi + v_2\,\sin\,2\phi  \label{eq:phidist}
\end{align}
where  $P_2(\cos\theta_{1,2})$ are second degree Legendre Polynomial and
\begin{align}
\label{eq:t1}
t_1 &=\frac{3}{2} \, \eta \text{Re}(f_{\parallel} f_{\perp}^*),\\
\label{eq:t2}
t_2 &=\frac{1}{4} (1-3\modulus{f_L}^2),\\
\label{eq:v2}
v_2 &=\frac{1}{2}\,\mathrm{Im}(f_{\parallel}f_{\perp}^*),\\
\label{eq:u2}
u_2 &=\frac{1}{4} (|f_{\parallel}|^2-|f_{\perp}|^2).
\end{align}
The uniangular distributions in Eq.\eqref{eq:t1dist}, Eq.\eqref{eq:t2dist} and Eq.\eqref{eq:phidist}
will give us arsenal to probe the $H\to Z V$ coupling.
The observables $t_1$, $t_2$, $u_2$ and $v_2$ can be extracted using following
asymmetries:

\begin{align}
&  t_1 =  \frac{1}{\Gamma} \left( \int_{-1}^{0} - \int_{0}^{+1} \right)
   \frac{d\Gamma}{ d\cos\theta_1} \, d\cos\theta_1,
\label{eq:T10asym}\\%
&  t_2 = \!\frac{4}{3 \Gamma}\! \left( \int_{-1}^{-\frac{1}{2}} -
  \int_{-\frac{1}{2}}^{+\frac{1}{2}} + \int_{+\frac{1}{2}}^{+1} \right)
\frac{d\Gamma}{d\cos\theta_{1,2}}  d\cos\theta_{1,2}
\label{eq:T20asym}\\%
&  v_2 = \frac{\pi}{2\Gamma} \left( \int_{-\pi}^{-\frac{\pi}{2}} -
  \int_{-\frac{\pi}{2}}^{0} + \int_{0}^{+\frac{\pi}{2}} -
  \int_{+\frac{\pi}{2}}^{+\pi} \right) \frac{d\Gamma}{d\phi}d\phi,
   \label{eq:V20asym}\\%
&  u_2 = \frac{\pi}{2 \Gamma} \left( \int_{-\pi}^{-\frac{3\pi}{4}} -
  \int_{-\frac{3\pi}{4}}^{-\frac{\pi}{4}} +
  \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}
  -\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} + \int_{\frac{3\pi}{4}}^{\pi}
  \right)\frac{d\Gamma}{d\phi} d\phi.
\label{eq:U20asym}
\end{align}

The observables $t_1$, $t_2$, $u_2$, $v_2$ are functions of $a_1$, $a_2$,
$a_3$ and their measurements will allow us to probe $H\to Z V$ coupling.
In SM $t_1$, $t_2$, $u_2$, $v_2$ have unique values which can be computed using
the SM values of the couplings $a_1$, $a_2$, $a_3$ given in Eq.\eqref{a1},
Eq.\eqref{a2} and Eq.\eqref{a3}. The anomalous nature, if any, of $a_1$, $a_2$ , $a_3$ will
show up in the observables as deviation from their SM values.

As discussed in Section \ref{sec:direct}, the rare Higgs decays are sensitive to $HZZ$, $HZ\gamma$
and $Hq\bar{q}$ couplings. Therefore, any deviation of the observables $t_1$, $t_2$, $u_2$, $v_2$ from their SM values
can not a priori be attributed to anomalous nature of any one sector. However, when taken in conjugation with
other decays like $H\to Z Z^{*}\to 4\ell$ they can provide complimentary information about $HZ\gamma$
and $Hq\bar{q}$ couplings. For example, if any hint of anomalous nature is found in $H\to Z Z^{*}\to
4\ell$ decay, one expects to see corresponding deviations in the observables of $H \to ZV$ for both $J/\psi$ and $\Upsilon$.
On the other hand if $H\to Z Z^{*}\to 4\ell$ decay observables turn out to be consistent with the SM
values then $HZZ$ contribution in rare decays should also be SM like. In such a scenario any observed anomaly in $H \to Z V$
can only come from either $HZ\gamma$ or $H q\bar{q}$ couplings. As the magnitude of their contributions in $H \to J/\psi$ and $H \to \Upsilon$ are quite different, this fact can be exploited to further narrow down the origin of the anomalous behaviour.
Moreover, for Higgs decay to both $J/\psi$ and $\Upsilon$, the effect of any anomaly in Yukawa sector will manifest itself predominantly
through the interference terms.

In principle Higgs can have anomalous couplings in more than one sector. If so, it will be relatively more difficult to make any definite conclusions about the relative contributions of the three sectors to the anomalous couplings of Higgs in
$H \to Z V$ decays.

\section{Conclusion}\label{sec:conclusion}
We discuss the feasibility of probing the rare Higgs decays $H\to Z V$
followed by further decay of $V$ and $Z$ into pair of leptons. We have shown that the $H\to Z V$ decays
receive contributions from gauge as well as Yukawa sector and depending on the nature of $V$, can be sensitive to anomalous couplings in more than one sector. They can provide complimentary information about the gauge and Yukawa couplings of Higgs.
We then showed how observables extracted as angular asymmetries will allow us to extract information about $HZV$ coupling.
Such asymmetries have the potential to infer the nature of gauge and Yukawa couplings of Higgs in future colliders or
high luminosity LHC runs.

\acknowledgments
We thank Dibyakrupa Sahoo and Rahul Sinha for many fruitful discussions.