text:
Q: How do you show that an $L^p$ entire (holomorphic on the complex plane) function is $0$? Just to clarify, I want to show that:
If $f$ is entire and $\int_{\mathbb{C}} |f|^p dxdy <\infty$, then $f=0$.
I think I can show that this is the case for $p=2$, but I'm not sure about other values of $p$...

A: Use Hölder's inequality and Cauchy's integral formula to show the function and its derivatives all vanishes at zero.

A: $\newcommand\RR{\mathbb{R}}$
$\newcommand\NN{\mathbb{N}}$
$\newcommand\e{\varepsilon}$
$\newcommand\lbg{\lambda}$
If $f: \RR \to \RR$ is an analytic unbounded Lebesgue integrable function, then it must in particular satisfy this property (because it is continuous):
\[
   \forall \e,M > 0 \; \exists x_0 : |x| > |x_0| \Rightarrow
   \lbg(\{x \in \RR : |f(x)| > M\}) < \e,
\]
where $\lbg$ is the usual Lebesgue measure.
Then you just use that $f$ is analytic iff $\forall \text{ compact } K \subset \RR \; \exists C > 0 : x \in K, n \in \NN \Rightarrow \left\vert\frac{\partial^n f}{\partial x^n}(x)\right\vert
\leq C^{n+1}n!$
But it is easy to see that this is impossible since $|f'|$ will have to be arbitrarily large - or in other words that you can go far enough out to the right on the x-axis that $|f|$ has to go from being larger than some big $M$ to smaller than some small $m$ on a very small interval.
This is missing quite a few details, but it should be pretty easy to fill out.

A: Right! Thank you, I didn't think about using Hölder's.
In short,
\begin{equation}
|f(0)| = \left|\int_{|z|=R} \frac{f(z)dz}{z} \right| <  \left(\int_{|z|=R} |f(z)|^p |dz|\right)^{1/p} \left(\int_{|z|=R} |z|^{-q} |dz|\right)^{1/q}
\end{equation}
And $\left(\int_{|z|=R} |z|^{-q} |dz|\right)^{1/q} = R^{-1+1/q} (2\pi)^{1/q} = R^{-1/p} (2\pi)^{1/q}$
\begin{equation}
\int_0^\infty \left((2\pi)^{1/q} |f(0)| R^{1/p}\right)^p dR < || f ||_p^p <\infty
\end{equation}
And the only way for this to fly is for $f(0)=0$, for $p\le 1$.


text:
Birthdate: 23rd July, 1989
Birthplace: Hammersmith, London, UK
From Wikipedia, the free encyclopedia Daniel Jacob Radcliffe (born 23 July 1989) is an English actor who rose to prominence playing the titular character in the Harry Potter film series. His work on the series has earned him several awards and more than £60 million. Radcliffe made his acting debut at age ten in BBC One's television movie David Copperfield (1999), followed by his film debut in 2001's The Tailor of Panama. Cast as Harry at the age of eleven, Radcliffe has starred in seven Harry Potter films since 2001, with the final installment releasing in July 2011. In 2007 Radcliffe began to branch out from the series, starring in the London and New York productions of the play Equus, and the 2011 Broadway revival of the musical How to Succeed in Business Without Really Trying. The Woman in Black (2012) will be his first film project following the final Harry Potter movie. Radcliffe has contributed to many charities, including Demelza House Children's Hospice and The Trevor Project. He has also made public service announcements for the latter. In 2011 the actor was awarded the Trevor Project's "Hero Award".
Daniel Jacob Radcliffe (born 23 July 1989) is an English actor who rose to prominence playing the titular character in the Harry Potter film series. His work on the series has earned him several awards and more than £60 million.
Radcliffe made his acting debut at age ten in BBC One's television movie David Copperfield (1999), followed by his film debut in 2001's The Tailor of Panama. Cast as Harry at the age of eleven, Radcliffe has starred in seven Harry Potter films since 2001, with the final installment releasing in July 2011. In 2007 Radcliffe began to branch out from the series, starring in the London and New York productions of the play Equus, and the 2011 Broadway revival of the musical How to Succeed in Business Without Really Trying. The Woman in Black (2012) will be his first film project following the final Harry Potter movie.
Radcliffe has contributed to many charities, including Demelza House Children's Hospice and The Trevor Project. He has also made public service announcements for the latter. In 2011 the actor was awarded the Trevor Project's "Hero Award".
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The story follows a young lawyer, Arthur Kipps (Radcliffe), who is ordered to travel to a remote village and sort out a recently...
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redpajama_set_name: RedPajamaCommonCrawl


Weathers (suspension) will transfer to Nicholls State, Jon Rothstein of CBS Sports reports.
Weathers was dismissed from the Oklahoma State basketball program in mid-January due to a violation of team rules. However, it didn't take long for the guard to find a new landing spot. He will have a couple more years of eligibility for Nicholls State, presumably after he sits out the first half of the 2019-20 season.
Weathers was dismissed from the Oklahoma State basketball team Wednesday, Nathan Ruiz of The Oklahoman reports.
Weathers (ankle) had five points (2-4 FG, 0-1 3Pt, 1-3 FT), six rebounds and three assists across 18 minutes off the bench in Monday's loss to Baylor.
Weathers (ankle) is a game-time decision for Monday's game against Baylor, Jon Rothstein of CBS Sports reports.
Weathers was taken to the locker room during Saturday's game against West Virginia due to an ankle sprain, Nathan Ruiz of The Oklahoman reports.
Weathers put up 13 points (3-12 FG, 0-1 3Pt, 7-10 FT), seven rebounds, six assists, and a steal in the loss to Oklahoma on Saturday.


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Home / Multi-Media / Audio / Podcast with Elana Meyer at Prague Marathon 2014
Podcast with Elana Meyer at Prague Marathon 2014
Yomog Meje
@YSMediaLtd
The memory of Elana Meyer celebrating with Derartu Tulu after the 1992 Olympic 10,000 m final in Barcelona rightly has its place in history, not just for athletics but as a symbol of the sport creating a bridge across what had once seemed an impossible divide.
Tulu of Ethiopia had just won the gold medal while Meyer had taken the silver for South Africa, the first of her country's medallists on its return to the Olympic fold as the apartheid era came to an end.
Meyer remains intent on achievement but now her goal is to help raise standards in South African marathon running at the elite level. Playing a leading role in the management of the Cape Town Marathon, she hopes the cream of the nation's young distance talent will turn their attentions to this classic distance.
[LISTEN] Podcast with Elana Meyer - Prague Marathon
During the past weekend Meyer was a guest of honour at the Volkswagen Prague Marathon. She competed in a marathon relay event on Sunday, teaming up with former marathon world record holder Ingrid Kristiansen of Norway and Czech long distance stars Robert Stefko and Anezka Drahotova.
"It means so much to me to be running in Prague, the city of Emil Zatopek. He was an idol of mine when I began to run, I wanted to emulate him and run all the distance events, from 5,000 to the marathon," said Elana Meyer.
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Lawrence O'Donnell took time out on his Monday program to further explain why he calls himself a "socialist."
O'Donnell has repeatedly used the term to define his political leanings.
On Monday, he said that he has called himself a socialist since he "first read the definition of socialism in the first economics class I took in college." However, he stressed that he didn't believe in pure socialism. Being a socialist, he said, didn't mean "that we choose the socialist option every time, but we do consider socialism a reasonable option under certain circumstances." Those circumstances, he said, included things like Medicare or Social Security.
O'Donnell said he disliked "bad socialism"--such as agriculture subsidies--as much as he disliked "bad capitalism"--such as for-profit health care--and that he was "as much a capitalist as I am a socialist." However, he explained that, since America was the only advanced country where socialist is "considered a dirty word, I feel more compelled to stand up for the socialist side of me."
O'Donnell then played an extended clip of Bill Maher trying to sell socialism to skeptical Americans.


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Daily News and F1 Comment: Friday 16th October, 2015
October 16, 2015 · by thejudge13	· in Daily F1 News and Comment.	·
Another GP2 champion sidelined for Formula One
It used to be the case that if a driver won the GP2 title, he got a full F1 race seat the following year in Formula One. That trend ceased with the 2012 GP2 champion – Davide Valsecchi – and is set to continue with the latest series winner, Stoffel Vandoorne.
McLaren are still hoping he may get a drive for Manor or whatever form Lotus may appear for 2016. However, the Woking team have contingency plans for their protege, to ensure he does not spend a season sidelined from track action. With Kevin Magnussen leaving McLaren at the end of this year, Vandoorne will assume the majority of the simulator duties at the MTC.
Further, the young Belgium is being lined up for a competitive seat in the Japanese 'Super Formula' series, which of course would be appreciated by Honda. Clearly Button and Alonso will not be asked to give up FP1 practice sessions, so Vandoorne will ply his trade in the next best series where downforce and the speed of the cars will prepare him for an F1 debut in 2017.
The question to be asked for drivers entering GP2 has to be whether the expense is really worth it anymore. The teams in this series are predominantly funded by drivers and their sponsors and with a price tag in excess of £1million – it appears rather an expensive entry on the road to nowhere.
Red Bull will reunite with Renault
Following the Singapore GP, TJ13 exclusively revealed to much derision that Red Bull had opened a conversation with Renault about receiving the French engine manufacturers power units in 2016. Having won four consecutive Formula One WCC and WDC titles, Red Bull finished second in the constructors race in 2014 and are likely to finish fourth this year. Niki Lauda says, "They should stay, the most important thing is they must stay."
Surprisingly – or not, Franz Tost now talks positively about the potential for Renault to catch up with Ferrari and Mercedes in during 2016. And Daniel Kvyat is positive about the Renault power unit too. F1.com
Kaltenborn says Ecclestone does understand the EU complaint
When asked in Sochi by Martin Brundle why Sauber and Force India have complained to the EU, Bernie Ecclestone claimed he had 'no idea' of the details of the complaint. The team principal of Sauber, Monisha Kalternborn says Ecclestone knows exactly what their complaint is about.
Deduction = Bernie is lying. Crash
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Ecclestone F1 engine regualtions dismissed out of hand →
2 responses to "Daily News and F1 Comment: Friday 16th October, 2015"
mikecloud54 October 16, 2015 at 00:41 · · Reply →
I had a thought. It might not be a brilliant thought, but it is a thought nonetheless. What if each driver was allocated an extra and complete powertrain for the season? This powertrain could be used either for races or for testing, but not for both. If it was used for testing then neither of the main drivers could drive it. It would have to be the reserve driver, test driver, or any up and coming driver who wanted to showcase their talents. The poorer teams may not be able to afford to test, so they would benefit from the extra engine during the race season. The richer teams could afford the test and any improvements made would trickle down to the teams they supplied, so everyone would benefit. Not only that, but Pirelli could test their tyres, the race tracks could earn some extra money and the tv stations could get some cheap tv. Okay, shoot me down now. 🙂
Lloyd October 16, 2015 at 01:34 · · Reply →
You idiot. Your suggestion is far too brilliant and altogether sensible for F1 to take it seriously 😉


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Prada Goes Fur-Free
The Italian brand vows to stop using animal fur, starting with its Spring/Summer 2020 collection.
by Pameyla Cambe (Singapore)
It looks like the domino effect of fashion's fur-free movement is still in place, because it has just earned itself a new member in fashion powerhouse, Prada. In collaboration with the Fur Free Alliance (FFA), which comprises 50 animal protection organizations in over 40 countries, the Prada Group has announced that it will no longer use animal fur in its future products.
The new policy applies to all of the brands under the Prada Group, including Italian fashion labels Prada and Miu Miu. It will be in effect starting from the brands' Spring/Summer 2020 Women's collections, while remaining fur products will be sold until quantities are exhausted. Additionally, both brands will continue to use leather in future collections.
In a statement, Miuccia Prada shared, "Focusing on innovative materials will allow the company to explore new boundaries of creative design while meeting the demand for ethical products."
In the past year, several other brands have gone fur-free to meet that same demand, such as Gucci, Versace and Burberry. Other brands like Chanel chose to abandon the use of exotic skins instead. Having Prada, one of fashion's biggest names, join them is certainly a big bonus to the sustainability movement, and will hopefully cause a ripple effect on a global scale.
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Ever kick yourself for not investing in one of Mary Katrantzou's most iconic designs? Chastise yourself no longer. The multi-award-winning British fashion designer has just reissued a capsule collection of her most famous pieces exclusively on Matchesfashion.com. Ready, steady….shop!
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text:
\section{Introduction}

Lepton flavour is an accidental symmetry of the Standard Model (SM), and there are many extensions of the SM,
like the seesaw models, supersymmetric SM, flavour-changing $Z'$ or scalars, leptoquarks,
or left-right symmetric models, that can
naturally break this symmetry. Even within the ambit of SM, neutrino mixing provides a
source of leptonic flavour violation (LFV), but the rates are too small to be observed in any near future
\cite{LFVrates}. Thus, observation of any LFV decay is a smoking gun signal of New Physics (NP).
In general four types of LFV processes have been looked for: (i) leptonic decays ($\tau\to 3e$, $\tau\to 3\mu$,
$\mu\to 3e$, $\tau\to 1e+2\mu$, $\tau\to 1\mu+2e$), (ii) radiative decays ($\tau\to e\gamma$, $\tau\to\mu\gamma$,
$\mu\to e\gamma$), (iii) semileptonic decays ($\ell_1\to \ell_2M$, where $M$ is some meson), and (iv)
conversion (like $\mu \to e$). They are not all
independent, {\em e.g.}, a flavour-changing electromagnetic penguin can also give rise to leptonic LFV
decays. Most of the decays, however,
have very stringent limits \cite{LFVrates}, the branching ratios (BR) being typically of the order of $10^{-8}$
or even smaller.

The interest in such LFV decays have recently been rekindled from the observation that some of the
semileptonic $B$ decay modes show anomalous deviations from the SM expectations, which may
possibly be explained by lepton flavour non-universality (LFNU)
as well as LFV. As an example, let us refer the reader
to a recent attempt in Refs.\ \cite{CKMS,CKMS2}, where the authors have shown that
both $R_K, R_{K^*}$ and $R(D), R(D^*)$ anomalies can be explained satisfactorily with only two new
operators, if the weak and mass bases of the charged leptons $\{\mu,\tau\}$ are related by a field rotation
\cite{GGL}. The apparent excess in the LFV decay channel of the Higgs boson,
$h\to\mu\tau$, as was once reported by the CMS Collaboration \cite{1502.07400}, coupled with such $B$ decay
anomalies, could lead to some well-motivated and fairly constrained models of LFV
\cite{Choudhury:2016ulr}.
The LFV operators with only leptonic fields cam also be induced by Renormalisation Group (RG) running
of semileptonic operators \cite{feruglio}.
Thus, there is enough motivation to seriously look into such LFV channels; they
might be observed at the Large Hadron Collider (LHC) \cite{lhc-mutau},
or dedicated super-B factories like Belle-II. Implications of such LFV decays may also be found in
Ref.\ \cite{request}.

In this paper, we will focus solely on the leptonic channel $\tau\to 3\mu$.
The channel has already been studied in detail in the literature; there are both model-independent
\cite{0711.0792,1506.07786,1511.07434} as well as partially model-dependent \cite{0802.0049,1701.00870}
studies. The Belle Collaboration has an upper bound on the BR \cite{1001.3221}
\begin{equation}
{\rm BR}(\tau\to 3\mu) < 2.1\times 10^{-8}
\label{e:bound}
\end{equation}
at 90\% confidence level (CL). One reason for looking at this particular channel is the possibility
of leptonic rotation in the $\{\mu,\tau\}$ sector as mentioned above, which invariably leads to such LFV
channels out of lepton flavour conserving operators in the weak basis. Another reason, of course, is
the relative ease with which the final state muons can be detected in both hadronic and $e^+e^-$ colliders.

Here we would like to push the studies on LFV in $\tau\to 3\mu$ further by asking and answering a few
questions. Observation of even a single $\tau\to 3\mu$ event is a definite signal for NP.
Assuming that one observes, possibly at a super-B factory like Belle-II,
a few events for the LFV decay in question, will one be able to unearth the nature of the
possible operators that can lead to such a decay?
It has been shown \cite{0711.0792,0802.0049} that there can be six independent
LFV operators in the chiral
basis that lead to $\tau\to 3\mu$. If the final state muon polarisations are not measured,
all the operators are {\em a priori} equally probable, and obviously only the number of events will not tell
us anything about the presence or absence of any of these operators; it can only yield some estimate on
the respective Wilson coefficients (WC).
So, are there observables which will help us to differentiate between these operators?
We will show that this is indeed possible, without using higher-order
differential cross-sections like Ref.\ \cite{0711.0792} which may have very few number of events in each bin.
At this point, we also note that some more operators can be generated through Fierz reordering,
but obviously they are not independent of the first six, and therefore we will not consider them any further.

The second question that we would like to ask is whether the existence of more than one such operators can be
disentangled from the data. Here the answer will be partially positive, unless, again, the muon polarisations are
measured. If one can have a sizeable number of events, and measure the muon polarisations too, one
may have in principle further observables, but we would like to be conservative. Anyway, as we will show,
one does not expect more than 70 events or so at the most with an integrated luminosity ${\cal L}_{\rm int} =
50$ ab$^{-1}$ at Belle-II.

We will use the method of Optimal Observables (OO), which has already been used in different areas of
particle physics \cite{gunion,atwood}, and in particular, for flavour physics \cite{S3,zc}.
This method displays the amount of significance level (``how many sigma" in standard parlance)
by which one point in the allowed parameter
space can be separated from another point. This is the only way to approach the question of
model differentiation before the arrival of the data. Once one has the data, other methods, like the
unbinned multivariate maximum likelihood, may be employed.

A related question is  the number of events with which one can have a successful differentiation among models,
where a model is specified by its operator structure and WCs. As expected, if the number of events is too small,
it will be harder to differentiate among various models, or in other words, the significance level will be lower.
We will quantify this statement subsequently.

The paper is arranged as follows. In Section 2, we enlist all the possible NP operators that can give rise to
the $\tau\to 3\mu$ decay, and the observables that we deal with are discussed in Section 3.
In Section 4, we show the differentiation among models with only one NP
operator. Section 5 deals with models with two such NP operators, and we discuss how well the presence
of the second operator can be found out from various observables. Section 6 summarizes and concludes
the paper.

\section{The New Physics Operators}

For this section, we will follow the notation and convention of Ref.\ \cite{0802.0049}. The most general LFV
Lagrangian can be written as
\begin{eqnarray}
{\cal L} &=& \frac{1}{\Lambda^2}
\bigg[g^S_{LL} (\overline\mu_L\mu_R)(\overline\mu_R\tau_L) + g^S_{LR} (\overline\mu_L\mu_R)(\overline\mu_L\tau_R) +
g^S_{RL} (\overline\mu_R\mu_L)(\overline\mu_R\tau_L) + g^S_{RR} (\overline\mu_R\mu_L)(\overline\mu_L\tau_R)  \nonumber\\
&& \left. +  g^V_{LL} (\overline\mu_R\gamma^\alpha\mu_R)(\overline\mu_L\gamma_\alpha\tau_L) +
g^V_{LR} (\overline\mu_R\gamma^\alpha\mu_R)(\overline\mu_R\gamma_\alpha\tau_R) \right.\nonumber\\
&& \left. +  g^V_{RL} (\overline\mu_L\gamma^\alpha\mu_L)(\overline\mu_L\gamma_\alpha\tau_L) +
g^V_{RR} (\overline\mu_L\gamma^\alpha\mu_L)(\overline\mu_R\gamma_\alpha\tau_R) \right.\nonumber\\
&&  +  \frac12 g^T_{LR} (\overline\mu_L\sigma^{\alpha\beta}\mu_R)(\overline\mu_L\sigma_{\alpha\beta}\tau_R) +
\frac12 g^T_{RL} (\overline\mu_R\sigma^{\alpha\beta}\mu_L)(\overline\mu_R\sigma_{\alpha\beta}\tau_L) \bigg]\,,
\label{lfvlag}
\end{eqnarray}
and we will denote the operator accompanying $g^X_{IJ}$ ($X=S,V,T$, and $I,J=L,R$) as $O^X_{IJ}$.
$\Lambda$ is the cutoff scale, which we will set at 5 TeV for our analysis.
We separate the operators into three major classes: ${\sf S}$ (operators of the form
$O^S_{IJ}$), ${\sf V}$ (the $O^V_{IJ}$ operators) and ${\sf T}$ (the tensor
operators $O^T_{IJ}$). Thus, the effective Lagrangian is of the form
\begin{equation}
{\cal L} = \frac{1}{\Lambda^2} \left[ \sum_{I,J=L,R} \left(g^S_{IJ} O^S_{IJ} + g^V_{IJ} O^V_{IJ}\right)
+ \sum_{I\not=J} g^T_{IJ} O^T_{IJ}\right]\,.
\label{e:lag-conc}
\end{equation}
In the above mentioned basis, not all the ten operators are independent; Fierz transformation relates
the two tensor operators with the rest, and the pairs $O^S_{LL}$-$O^V_{LL}$ and
$O^S_{RR}$-$O^V_{RR}$ are also related \cite{0711.0792}. Thus, only four scalar and two vector
operators are enough to span the operator basis.
However, we keep all of them for the time being, as the mediator that has been integrated out may give rise
to operators that are linear combinations of the six independent ones, like
the tensor operators that can be generated from some hypothetical spin-2 mediators.

Writing the BR in terms of the new WCs \cite{0802.0049}, one may easily show that the decay
$\tau \to 3\mu$ has maximal sensitivity to {\sf V} or {\sf T} operators.
As an example, the present bound on ${\rm BR}(\tau\to 3\mu) < 2.1\times 10^{-8}$ translates to
$|g_{RL}^S| \approx 1$, while $|g_{RL}^T|, |g_{RL}^V| \sim {\cal O}(0.1)$.
Thus, for a given number of events, the reach for {\sf V} or {\sf T} WCs is better than that for the {\sf S}
ones.

The left-chiral fields being $SU(2)$ doublets, one can also get a neutrino-antineutrino pair out of the operators
$O^S_{RR}$ and $O^V_{RR}$, which technically gives an extra contribution to the SM $\tau$ decay channel
$\tau\to \mu\overline\nu_\mu \nu_\tau$. However, the couplings will turn out to be so constrained as not to
affect this channel in any significant amount. Similar LFV operators for $\mu\to 3e$ may affect the
extraction of the Fermi coupling $G_F$ from the muon lifetime in a measurable way.

One may try to look for $\tau\to\mu\gamma$ by contracting a pair of muons and taking the photon
with momentum $q^\mu$ out from the loop. For the scalar operators, the contribution vanishes in the $q^2=0$
limit, and for the vector operators, this amounts to charge and not the transition magnetic moment
renormalisation.

To begin with, we will consider the presence of one operator at a time. This generates six independent
models spanning over the ${\sf S}$ and ${\sf V}$ classes. Next, we will consider the presence of two
operators at a time, which include the well-motivated combinations like
\begin{eqnarray}
O^S_{9L} = O^S_{LL} + O^S_{RL}\,,  && O^S_{10L} = O^S_{LL} - O^S_{RL}\,,  \nonumber\\
O^S_{9R} = O^S_{LR} + O^S_{RR}\,, &&  O^S_{10R} = O^S_{LR} - O^S_{RR}\,,
\end{eqnarray}
and similarly for the ${\sf V}$ class. Our goal will be to pinpoint whether or not these two coupling scenarios
can be differentiated from those involving only one coupling at a time.

\section{Observables}

In this section, we will define the observables which we have used in our analysis to differentiate the effects of different NP operators.
As an example, we consider the ${\sf S}$ class of operators, $O^S_{IJ}$, taken from Eq.\ (\ref{e:lag-conc}), and
consider the decay $\tau^-\to \mu^-\mu^+\mu^-$.
The double differential cross-section for the antimuon is given, after integrating out the
phase space for the two muons, by
\begin{align}
\frac{d B_\tau}{dx\, d(\cos\theta)} &= \frac{T_\tau\, m_\tau^5}{ 128 \times 48\pi^3 \Lambda^4}\left[
3x^2 (|g_{RL}^S|^2 +|g_{LL}^S|^2+|g_{RR}^S|^2+|g_{LR}^S|^2)\right.\nonumber\\
&\left.-x^3 (3|g_{RL}^S|^2+2|g_{LL}^S|^2+2|g_{RR}^S|^2 + 3 |g_{LR}^S|^2)\right. \nonumber \\
&\left.+\,x^2 \cos\theta\,(3 |g_{RL}^S|^2 + |g_{LL}^S|^2 - |g_{RR}^S|^2 - 3|g_{LR}^S|^2 )\right.\nonumber\\
&\left.- x^3 \cos\theta\,(3|g_{RL}^S|^2+ 2|g_{LL}^S|^2- 2|g_{RR}^S|^2- 3 |g_{LR}^S|^2)\right]\,,
\label{e:dgdx}
\end{align}
where we take the muons to be massless, and use the notation
\begin{equation}
B_\tau \equiv {\rm BR} (\tau\to 3\mu)\,.
\end{equation}
Here, $T_\tau = 1/\Gamma$ is the lifetime of the $\tau$ lepton,
$x = 2E_{\overline\mu}/m_\tau$ is the reduced energy of the
antimuon, and $\theta$ is angle between the polarization of the $\tau$ and the momentum of the antimuon,
following the convention of Ref.\ \cite{1506.07786}. For further discussion, let us define
\begin{eqnarray}
g_1 &\equiv& |g_{RL}^S|^2 +|g_{LL}^S|^2+|g_{RR}^S|^2+|g_{LR}^S|^2\,,\nonumber\\
g_2 &\equiv& 3|g_{RL}^S|^2+2|g_{LL}^S|^2+2|g_{RR}^S|^2 + 3 |g_{LR}^S|^2\,,\nonumber\\
g_3 &\equiv& 3 |g_{RL}^S|^2 + |g_{LL}^S|^2 - |g_{RR}^S|^2 - 3|g_{LR}^S|^2\,, \nonumber\\
g_4 &\equiv& 3|g_{RL}^S|^2+ 2|g_{LL}^S|^2- 2|g_{RR}^S|^2- 3 |g_{LR}^S|^2\,.
\end{eqnarray}
Thus
\begin{equation}
B_\tau = \frac{T_\tau\, m_\tau^5}{ 128 \times 24\pi^3 \Lambda^4}\left[g_1 - \frac14 g_2\right]\,.
\label{brscalar}
\end{equation}
The number of events gives the information only on the combination $g_1-\frac14 g_2$.

Another observable is the observed integrated forward-backward asymmetry $A_{FB}$, defined as
\begin{equation}
A_{FB} = \frac{N_F - N_B}{N_F + N_B}\,,
\label{e:afb}
\end{equation}
with
\begin{eqnarray}
N_F &=& \sigma_{\rm Prod} {\cal L}_{\rm int}\epsilon\, \int_{0}^{1}\, dx\, \int_{0}^{1}\, d(\cos\theta)
\frac{dB_\tau}{dx\, d(\cos\theta)}\,,\nonumber\\
N_B &=& \sigma_{\rm Prod} {\cal L}_{\rm int} \epsilon\, \int_{0}^{1}\, dx\, \int_{-1}^{0}\, d(\cos\theta)
\frac{dB_\tau}{dx\, d(\cos\theta)}\,,
\end{eqnarray}
where $\sigma_{\rm Prod}$ is the $\tau$ production cross-section, ${\cal L}_{\rm int}$ is the integrated luminosity, and
$\epsilon$ is the combined detection efficiency in the $\tau\to 3\mu$ channel.

We will also define the $x$-dependent asymmetry, normalised to the total decay width, as
\begin{equation}
\frac{dA_{FB}}{dx} \equiv A'_{FB}(x) = \sigma_{\rm Prod} {\cal L}_{\rm int} \epsilon\,
\frac{\int_{0}^{1}\, d(\cos\theta)\,  \frac{dB_\tau}{dx\, d(\cos\theta)} -
\int_{-1}^{0}\, d(\cos\theta)\, \frac{dB_\tau}{dx\, d(\cos\theta)}}{N_B + N_F}
\equiv \frac{N_F(x) - N_B(x)}{N}\,,
\label{afbx1}
\end{equation}
where $N=N_F + N_B$ gives the total number of signal events.

Instead of the antimuon, one can play an identical game with one of the same-sign muons
({\em i.e.}, the one with the same sign as the decaying $\tau$), say the more
energetic of the two. Let us define
\begin{eqnarray}
g'_1 &\equiv& |g_{RL}^S|^2 + 3 |g_{LL}^S|^2+ 3 |g_{RR}^S|^2+ |g_{LR}^S|^2\,,\nonumber\\
g'_2 &\equiv& |g_{RL}^S|^2+ 4 |g_{LL}^S|^2+ 4 |g_{RR}^S|^2 + |g_{LR}^S|^2\,,\nonumber\\
g'_3 &\equiv& |g_{RL}^S|^2 + 7 |g_{LL}^S|^2 - 7  |g_{RR}^S|^2 - |g_{LR}^S|^2\,, \nonumber\\
g'_4 &\equiv& |g_{RL}^S|^2+ 4 |g_{LL}^S|^2- 4 |g_{RR}^S|^2-  |g_{LR}^S|^2\,,
\end{eqnarray}
so that
\begin{equation}
\frac{dB_\tau}{dy\, d(\cos\alpha)} =
\frac{T_\tau\, m_\tau^5}{128 \times 96\pi^3 \Lambda^4}\left[
3y^2 g'_1 - 2y^3 g'_2 + y^2 \cos\alpha\, g'_3 -2y^3\cos\alpha\, g'_4\right]\,,
\label{e:dgdy}
\end{equation}
where
$y = 2E_{\mu}/m_\tau$ is the reduced energy of the more energetic same-sign muon,
and $\alpha$ is angle between its direction and the polarisation of the $\tau$ lepton.

In an analogous way to Eq.\ (\ref{afbx1}), one can define
\begin{equation}
{\cal A}'_{FB}(y) =
\frac{N_F(y)-N_B(y)}{N}=
\sigma_{\rm Prod} {\cal L}_{\rm int} \epsilon\,
\frac{\int_{0}^{1}\, d(\cos\alpha)\,  \frac{dB_\tau}{dy\, d(\cos\alpha)} -
\int_{-1}^{0}\, d(\cos\alpha)\, \frac{dB_\tau}{dy\, d(\cos\alpha)}}{N}.
\label{afby1}
\end{equation}
As will be shown later, the observables $A_{FB}^{\prime}$ and ${\cal A}_{FB}^{\prime}$
are useful to differentiate the sensitivities of the subtypes of operators within a particular type,
say {\sf S} or {\sf V}.

Similarly, for the {\sf V} class of models, one obtains, in an analogous way to Eq.\ (\ref{e:dgdx}),
\begin{align}
\frac{d B_\tau}{dx\, d(\cos\theta)} &= \frac{T_\tau\, m_\tau^5}{ 128 \times 12\pi^3 \Lambda^4}\left[
3x^2 (4|g_{RL}^V|^2 +|g_{LL}^V|^2+|g_{RR}^V|^2+4|g_{LR}^V|^2)\right.\nonumber\\
&\left.-2x^3 (6|g_{RL}^V|^2+|g_{LL}^V|^2+|g_{RR}^V|^2 + 6 |g_{LR}^V|^2)\right. \nonumber \\
&\left.+ x^2 \cos\theta(12 |g_{RL}^V|^2 + |g_{LL}^V|^2 - |g_{RR}^V|^2 - 12|g_{LR}^V|^2 )\right.\nonumber\\
&\left.- 2x^3 \cos\theta\,(6|g_{RL}^V|^2+ |g_{LL}^V|^2- |g_{RR}^V|^2- 6 |g_{LR}^V|^2)\right]\,.
\label{e:dgdxv}
\end{align}
The corresponding BR is
\begin{equation}
B_\tau = \frac{T_\tau\, m_\tau^5}{ 128 \times 12\pi^3 \Lambda^4}\left[2 g_1 -  g_2\right]\,,
\label{brvector}
\end{equation}
where,
\begin{eqnarray}
g_1 &\equiv& 4|g_{RL}^V|^2 +|g_{LL}^V|^2+|g_{RR}^V|^2+4|g_{LR}^V|^2\,,\nonumber\\
g_2 &\equiv& 6|g_{RL}^V|^2+|g_{LL}^V|^2+|g_{RR}^V|^2 + 6 |g_{LR}^V|^2\,,\nonumber\\
g_3 &\equiv& 12 |g_{RL}^V|^2 + |g_{LL}^V|^2 - |g_{RR}^V|^2 - 12|g_{LR}^V|^2\,, \nonumber\\
g_4 &\equiv& 6|g_{RL}^V|^2+ |g_{LL}^V|^2- |g_{RR}^V|^2- 6 |g_{LR}^V|^2\,.
\end{eqnarray}
From Eqs.\ (\ref{e:dgdx}), (\ref{brscalar}), (\ref{e:dgdxv}), and (\ref{brvector}), one finds that {\sf V}-type
operators generate more events than {\sf S}-type operators, if the orders of magnitude of their WCs are similar.
The number of events as well as the angular distribution depend on the model subtype. We refer the
reader to Fig.\ \ref{f:brvswc}, which shows this explicitly.

\begin{figure}[htbp]
\centering
   \subfloat[\label{brvsgrlzoomed}]
  {\includegraphics[height=5.5cm]{Fig1a.png}}
  \subfloat[\label{brvsgrrzoomed}]
  {\includegraphics[height=5.5cm]{Fig1b.png}}
  \caption{ (a) Variation of ${\rm BR}(\tau \to 3\mu)$ with the WCs $g_{RL}^I$ ($I = S,V,T$).
  (b) The same for $g_{RR}^I$ ($I = S,V$). The horizontal line shows the present limit. The results for $g^I_{LR}$
  are identical to those for $g^I_{RL}$, and the results for $g^I_{LL}$ are identical to those for $g^I_{RR}$.}
\label{f:brvswc}
\end{figure}

For the same-sign muon, one gets
\begin{equation}
\frac{dB_\tau}{dy\, d(\cos\alpha)} =
\frac{T_\tau\, m_\tau^5}{128 \times 24\pi^3 \Lambda^4}\left[
3y^2 g'_1 - 8y^3 g'_2 + y^2 \cos\alpha\, g'_3 - 8y^3\cos\alpha\, g'_4\right]\,,
\end{equation}
with
\begin{eqnarray}
g'_1 &\equiv& 4|g_{RL}^V|^2 + 3|g_{LL}^V|^2+ 3|g_{RR}^V|^2+ 4|g_{LR}^V|^2\,,\nonumber\\
g'_2 &\equiv& |g_{RL}^V|^2+ |g_{LL}^V|^2 + |g_{RR}^V|^2 + |g_{LR}^V|^2\,,\nonumber\\
g'_3 &\equiv& 4 |g_{RL}^V|^2 + 7|g_{LL}^V|^2 -7 |g_{RR}^V|^2 - 4|g_{LR}^V|^2\,, \nonumber\\
g'_4 &\equiv& |g_{RL}^V|^2+ |g_{LL}^V|^2- |g_{RR}^V|^2-  |g_{LR}^V|^2\,.
\end{eqnarray}
In an analogous way to Eqs.\ (\ref{afbx1}) and (\ref{afby1}), one can define $A'_{FB}(x)$ and
${\cal A}'_{FB}(y)$.

While we will not discuss the {\sf T}-type operators separately,
the double differential decay distribution is given by,
\begin{equation}
\frac{d B_\tau}{dx\, d(\cos\theta)} = \frac{T_\tau\, m_\tau^5}{ 128 \times 4\pi^3 \Lambda^4}
\, 9 x^2(1-x)\, \left[g_1
+ g_2 \cos\theta\right]\,,
\end{equation}
where,
\begin{equation}
g_1 \equiv |g_{RL}^T|^2+ |g_{LR}^T|^2\,,\ \
g_2 \equiv |g_{RL}^T|^2- |g_{LR}^T|^2\,.
\end{equation}
Thus
\begin{equation}
B_\tau = \frac{3 T_\tau\, m_\tau^5}{ 128 \times 8\pi^3 \Lambda^4}\, g_1\,.
\end{equation}

\section{Analysis}

In this section, we discuss the current and future sensitivities of the {\sf S}, {\sf V}, and {\sf T}
operators on the observables $B_{\tau}$, $A'_{FB}$. and ${\cal A}'_{FB}$. The next subsection deals
with the simplified cases where only one operator is considered to be present at a time. While this
is instructive and sheds a lot of light on the differentiating power of the observables, a more realistic
scenario might involve more than one operators. Thus, in the next Section, we discuss the cases
where two operators are simultaneously present, and try to see whether those two-operator models can
be separated from the single-operator ones.

\subsection{One Operator Models}

Let us assume, to start with, that only one out of the ten possible operators shown in Eq.\ (\ref{lfvlag}) is
present, notwithstanding the fact that not all of them are mutually independent, some being related to the
others through Fierz rearrangement.
In Fig.\ \ref{f:brvswc}, we show how the branching ratio $B_{\tau}$ depends on the WCs $g_{RL}^{S,V,T}$ and
$g_{RR}^{S,V}$. Identical plots are obtained if one replaces $LR$ with $RL$, and $RR$ with $LL$. This
$R\leftrightarrow L$ symmetry is true for all subsequent observables and their distributions, which reduces
possible independent cases worth discussing by a factor of 2.

Given the combination $IJ$, if theory tells us the approximate magnitude of the WC $g^X_{IJ}$, even with
the number of events as the sole observable, one can almost immediately differentiate $X=S$ case with $X=V$
or $T$. With higher statistics even a differentiation between {\sf V} and {\sf T} may be possible. The present
limit on $B_\tau$ is translated to
$\vert g^T_{RL}\vert \leq 0.13$, $\vert g_{RL}^{V}\vert \leq 0.20$, $\vert g^S_{RL}\vert \leq 0.80$,
$\vert g_{RR}^{V}\vert \leq 0.28$, and $\vert g^S_{RR}\vert \leq 0.57$. However, as we do not have any
{\em a priori} knowledge of the magnitudes of the WCs, we have to look for some other observables and
use the number of events as a normalisation. In other words, we will assume that the total number of events
has been given to the community by the experimentalists and see how much extra information we can
extract.

In Fig.\ \ref{f:dbdx}, we show how the differential rate $dB_\tau/dx$ varies with the muon energy variable $x$
for a fixed value of the WC, set at $0.1$.
With the normalisation included, the area under the curve gives the total number of events in different
$x$-bins. Note that due to possible paucity of events, one may have only a few bins, 2 or 3, before the
data starts thinning out too much to have any statistical significance. Thus, the continuous distribution
showed in Fig.\ \ref{f:dbdx} is an idealised scenario. Even then, what we find is that the number of events
will be markedly different for different classes of operators if the WCs are of the same order, which is
very much along the expected line. On the other hand, the asymmetry variables $A'_{FB}(x)$ or
${\cal A}'_{FB}(y)$ must show identical pattern for all operators, {\sf S}, {\sf V},
or {\sf T}, with a fixed chirality structure, because the overall normalisation cancels in the ratio.

\begin{figure}[t]
\begin{center}
 \includegraphics[height=4.5cm]{Fig2a.png}~~~
 \includegraphics[height=4.5cm]{Fig2b.png}
\end{center}
\caption{The decay rate distribution $dB_\tau/dx$ for different operators, with the relevant
$g^{S,V,T}_{IJ} \approx 0.1$. }
\label{f:dbdx}
 \end{figure}

The next task would be to differentiate among the various chiral subclasses of a particular class of model.
For illustration, we will take the {\sf S} class of models, and consider the presence of one {\sf S}-type
operators at one time. As the sensitivities are higher for {\sf V} and {\sf T} classes, whatever results one has
for {\sf S} will only be more enhanced and pronounced for other classes. At the same time, if the underlying
theory predicts $g_{IJ}$ values of the order of unity, very small number of events will be harder to sustain
under {\sf V} or {\sf T} classes.

\begin{figure}[t]
\centering
  \subfloat[\label{Ax_{fb}(binned) for 50 events}]
  {\includegraphics[height=4.5cm]{Fig3a.png}}
  \subfloat[\label{Ax_{fb}(binned) for 20 events}]
  {\includegraphics[height=4.5cm]{Fig3b.png}}

   \subfloat[\label{Ay_{fb}(binned) for 50 events}]
  {\includegraphics[height=4.5cm]{Fig3c.png}}
  \subfloat[\label{Ay_{fb}(binned) for 20 events}]
  {\includegraphics[height=4.5cm]{Fig3d.png}}
  \caption{ $A'_{FB}(x)$ for the antimuon with (a) 50 and (b) 20 events for the
  four single coupling {\sf S} class of models. The same for the more energetic of the two muons,
  ${\cal A}'_{FB}(y)$, with $($c$)$ 50 and (d) 20 events.
}
\label{f:single}
\end{figure}

In the single-coupling scheme, we consider four different models, depending upon which operator contributes, and denote them as
\begin{equation}
{\rm Model~1}: O^S_{RL}\,,\ \ {\rm Model~2}: O^S_{LL}\,,\ \
{\rm Model~3}: O^S_{RR}\,,\ \ {\rm Model~4}: O^S_{LR}\,.
\end{equation}
If only one operator contributes, $A'_{FB}(x)$ becomes a function of $x$ only and does not depend
on the magnitude of the WCs:
\begin{equation}
A'_{FB}(x)_{1} = -A'_{FB}(x)_{4} = 6(x^2-x^3)\,,\ \
A'_{FB}(x)_{2} = -A'_{FB}(x)_{3} = x^2-2x^3\,.
\end{equation}
The integrated asymmetry $A_{FB}(i)$ for the $i$-th model can be obtained by integrating $x\in [0:1]$, and the values are
\begin{equation}
A_{FB}(1) = -A_{FB}(4) = \frac12\,,\ \
A_{FB}(2) = -A_{FB}(3) = -\frac16\,.
\end{equation}
There is a zero crossing only for models 2 and 3 at $x=\frac12$.

Similarly, for the forward-backward asymmetry ${\cal A}'_{FB}(y)$, we find
\begin{equation}
{\cal A}'_{FB}(y)_{1} = -{\cal A}'_{FB}(y)_{4} = y^2 - 2y^3\,,\ \
{\cal A}'_{FB}(y)_{2} = -{\cal A}'_{FB}(y)_{3} = \frac12 (7y^2-8y^3)\,.
\end{equation}
While all of them show zero-crossing, for the last two models such crossing occurs almost at the end of
the kinematic range at $y=7/8$. The integrated asymmetries are (for $y\in [0:1]$)
\begin{equation}
{\cal A}_{FB}(1) = {\cal A}_{FB}(3) = - {\cal A}_{FB}(2) = -{\cal A}_{FB}(4) = -\frac16\,.
\end{equation}

The $A'_{FB}(x)$s for different models are shown in Figs.\
\ref{Ax_{fb}(binned) for 50 events} and \ref{Ax_{fb}(binned) for 20 events}. Similarly, ${\cal A}'_{FB}(y)$s
are shown in Figs.\ \ref{Ay_{fb}(binned) for 50 events} and \ref{Ay_{fb}(binned) for 20 events}.
In these figures, every theoretical line has
broadened out to a thick band, often overlapping with each other. This happens because the number of the events is
limited. For every $x$ ($y$), the error margin in $A'_{FB}(x)$ is approximately given by
\begin{equation}
\delta A'_{FB} = \sqrt{\left(\frac{\partial A'_{FB}(x)}{\partial N_F(x) }\right)^2 (\delta N_F(x))^2 +
\left(\frac{\partial A'_{FB}(x)}{\partial N_B(x) }\right)^2 (\delta N_B(x))^2
+ \left(\frac{\partial A'_{FB}(x)}{\partial N}\right)^2 (\delta N)^2}
\end{equation}
where
\begin{equation}
\delta N_F(x) = \sqrt{N_F(x)},\ \  \delta N_B(x) = \sqrt{N_B(x)}\,,
\end{equation}
are the statistical errors in the number of events in the forward and backward directions respectively,
and $\delta N = \sqrt{N}$.

The expression for ${\cal A}'_{FB}(y)$ is analogous.
We have not considered the correlation between $N_F(x)$ and $N_B(x)$; depending on the sign
of the correlation, the expression can be an overestimation or underestimation,
but as we do not have any {\em a priori}
knowledge of the distribution, it is better to stick to zero correlation.
The bands in Fig.\ \ref{f:single} indicate the $1\sigma$ error margins.
Clearly, the resolving power is much less for 20 events than with 50 events.

Because of the probable paucity of events, the asymmetries may be measured
 only with a limited number of bins. But even with two bins, low-$x$ ($0<x<0.5$) and high-$x$ ($0.5<x<1$),
 one should be able to differentiate between competing models.

The existing bound on $\tau\to 3\mu$ comes from the analysis of 782 fb$^{-1}$ data from the Belle
collaboration \cite{1001.3221}, and 468 fb$^{-1}$ data from the BaBar collaboration \cite{1002.4550}.
With a production cross-section of $0.919$ nb for the $\tau^+\tau^-$ pairs, one gets 720 million such pairs
at Belle and 420 million at BaBar. For 50 ab$^{-1}$ of integrated luminosity at Belle-II, one expects
$N_P=4.6\times 10^{10}$ $\tau^+\tau^-$ pairs. With a detection efficiency of $7.6\%$ \cite{1001.3221},
and using the present bound given in Eq.\ (\ref{e:bound}), the maximum number of such events is about
73. For our discussion, we will use two scenarios: one with $N=20$ and the other with $N=50$.
Note that the errors are only statistical in nature. There may be other uncertainties, like fixing the direction
of the $\tau$ polarisation, which will widen up the bands, but that effect is expected to be small with the
$\tau$ detection ability of Belle-II.

\begin{figure}[t]
\centering
  \subfloat[\label{dbrdx for 50 events}]
  {\includegraphics[height=4.5cm]{Fig4a.png}}
  \subfloat[\label{dbrdy for 50 events}]
  {\includegraphics[height=4.5cm]{Fig4b.png}}

  \caption{ (a) $dB_\tau/dx$ and (b) $dB_\tau/dy$ for the four
  single coupling {\sf S} class of models.
}
\label{f:single2}
\end{figure}

If $A_{FB}$ turns out to be positive (negative), the viable models are 1 or 3 (2 or 4). Similarly, the
positive (negative)-${\cal A}_{FB}$ models are 2 and 4 (1 and 3). Thus, measurement of only the sign of
these asymmetries leave us with a twofold ambiguity. However, $x(y)$-dependent asymmetry measurements
have the ability to resolve the same; {\sf S} and {\sf V}-type models behave identically.
It is enough to measure the asymmetry in two bins: low $x(y)$ bin for $0\leq x(y) < \frac12$ and
high $x(y)$ bin for $\frac12\leq x(y) \leq 1$. If these measurements were precise enough, it would have
been sufficient not only to pinpoint the model but also to explore whether more than one operators are
contributing \footnote{We, however, show a continuous $x(y)$-distribution of the asymmetries; one just needs
to integrate over the respective bin to have an idea of the relative magnitudes.}.
Unfortunately, with a limited number of events, the measurements cannot be that precise.

Our results $dB_{\tau}/dx$ and $dB_{\tau}/dy$ in the single-coupling schemes are shown in
Fig.\ \ref{f:single2}. The lines broaden out into bands if we take the errors and uncertainties into account.
Such broadening, in all probability, will make the lines indistinguishable from one another.
However, all these models can be separated from each other
from the asymmetry measurements, particularly in the high-$x(y)$ bin.

\section{Two Operator Models}

Once we establish that given enough events ($\sim 50$), it will be straightforward to differentiate between
several one-operator models, the next question is: what if the data is not compatible with any of them? Note that
for the single-operator scheme to stand good, all the observables, and not only a few of them,
 have to be in the right ballpark specified by that model.

However, even the principle of Occam's Razor may not be enough reason for not invoking the
double-operator scheme.
We will, as before, be confined within the {\sf S} class of models,
and consider the cases where two WCs are nonzero at a time. As we have shown, this will be the case
if the underlying theory forces the muon current to be pure $S$ or $P$ \footnote{And pure $V$ or $A$
for the {\sf V} class of models.}.
Thus, the question we ask is: If the new physics is described by two
$\tau\to 3\mu$ operators, with what confidence level can we differentiate that from those cases where only
one of them is present? Note that the number of events will act as the tightest constraint on the parameter space.
We will try to differentiate these models, hereafter called $O2$ for two effective operators,
from that with only one operator, which we call the `seed' model.
For example, we consider the following $O2$ models:
\begin{eqnarray}\label{modlst1}
{\rm Model~A} &:& O_{RL}^S~{\rm and}~O_{LL}^S\,,\nonumber\\
{\rm Model~B} &:& O_{RL}^S~{\rm and}~O_{RR}^S\,,\nonumber\\
{\rm Model~C} &:& O_{RL}^S~{\rm and}~O_{LR}^S\, .
\end{eqnarray}
We will compare the differentiability of these models (A, B and C) with the seed model with one operator
 $O^S_{RL}$. To achieve this goal, we need to find the parameter space spanned by $g^S_{RL}$ and
another $g^S_{IJ}$, which depends on what $O2$ model we consider.
We further check with what confidence level
the allowed regions for models A, B and C can be separated from the single-operator model given by
$O^S_{RL}$. To complete the study, we include three further $O2$ models, namely,
\begin{eqnarray}\label{modlst2}
{\rm Model~D} &:& O_{RR}^S~{\rm and}~O_{RL}^S\,,\nonumber\\
{\rm Model~E} &:& O_{RR}^S~{\rm and}~O_{LL}^S\,,\nonumber\\
{\rm Model~F} &:& O_{RR}^S~{\rm and}~O_{LR}^S\,,
\end{eqnarray}
where the first operator is treated as the seed.

Models B and D are different, because of different seeds. The seed models are chosen in such
a way as to have positive $A_{FB}$ for the opposite-sign muon for them;
the negative $A_{FB}$ models will have a corresponding
relationship, which can be obtained by flipping $L$ and $R$, $L \leftrightarrow R$ \footnote{
If we consider the asymmetry based on the more energetic like-sign muon, models A-F have negative
asymmetry, while the corresponding $L\leftrightarrow R$ models have positive asymmetry.}.
Let us mention here that the confidence interval contours will depend on the type of seed operators being considered.

For this part of the analysis, we will use the Optimal Observable (OO) technique. For a detailed discussion
on this technique, we refer the reader to Refs.\ \cite{gunion,atwood}. In the context of $B$ decays, this
method has been applied in Refs.\ \cite{S3,zc}. The essential point of the OO technique is that this gives the
optimal set of observables (which are in general functions of experimental observables) with which two
points in the parameter space of different models can be differentiated with maximum efficiency. In other
words, this gives the maximum possible separation, in terms of confidence level, between two points
in the parameter space as a function of experimental observables. In practice, the systematic errors
reduce the confidence intervals.

As has been shown in Refs.\ \cite{S3,zc}, this method is all the more useful when one does not have
any experimental data; in the presence of data, one can do a maximum likelihood analysis. This also means
that not all the systematic uncertainties are taken into account. Thus, OO acts more as a motivational tool
to the experimentalists than as an instrument for detailed quantitative theoretical studies.

Even with only the {\sf S} class of operators, the parameter space of WCs is four-dimensional. A complete
analysis is not only cumbersome but also of very little help in the real-life scenario where the number of
events will definitely be below 100 and therefore a fine scan of the parameter space, with a two-dimensional
binning on $x$($y$) and $\cos\theta(\cos\alpha)$, will have so few events per bin as to make the analysis
meaningless. The only constraint on the WCs comes from the non-observability of the decay.

In the OO technique, one writes any observable ${\cal O}$, depending on a variable $\phi$, as
\begin{equation}
{\cal O}(\phi) = \sum_i C_i f_i(\phi)\,,
\label{e:def-oo}
\end{equation}
which can be generalised to a set of variables denoted by $\bm{\phi}$. Here, all the $f_i$s are independent, and $C_i$s are some constants.
The major goal of this technique is to extract $C_i$s. In our case, $C_i$s will be functions of $g_i$s and $g'_i$s defined earlier.
Our analysis can be done by defining a quantity analogous to $\chi^2$, such as
\begin{align}\label{chidef}
 \chi^2 &= \sum_{i,j} (C_i - C_i^0)  V_{ij}^{-1}(C_j - C_j^0) .
\end{align}
The $C_i^0$s are called the seed values, which can be considered as model inputs.
The covariance matrix $V_{ij}$s are defined as
\begin{equation}
V_{ij} = \langle \Delta C_i \Delta C_j\rangle = \frac{ (M^{-1})_{ij} \sigma_T}{N}\,, \text{with}\ \
M_{ij} = \int \frac{f_i(\phi)f_j(\phi) }{O(\phi)} \, d\phi.
\label{vijdef}
\end{equation}
In the above expression of $V_{ij}$, $\sigma_T = \int O(\phi)\, d\phi$ and $N = \sigma_{\rm Prod}{\cal L}_{\rm int}
\epsilon B_\tau$, as defined earlier. For a specific model, $\chi^2$ gives the
confidence level separation between the seed value $C_i^0$ (seed model) and the model under consideration,
parametrized by $C_i$.

Looking at Eq. (\ref{chidef}), it is clear that the shape of the fixed $\chi^2$ hypersurface depends on $V^{-1}_{i j}$,
and the centroid of that (where $\chi^2 =\chi^2_{min} = 0$) changes with the seed values. These fixed $\chi^2$
surfaces are what determines the separation between models essentially.
Thus separation between any two models 1 and 2, with seed at 1, will in general be not equal to the separation
when the seed is at 2. This is the reason for treating Models B and D separately in
Eqs.\ (\ref{modlst1}) and (\ref{modlst2}).

In the case of single operator model, the seed values of the WCs corresponding to the 50 events are obtained as
\begin{equation}
|g^S_{RL}|^2 = 0.44~{\rm (A,B,C)}\,,\ \ \
|g^S_{RR}|^2 = 0.22~{\rm (D,E,F)}\,.
\end{equation}
For the negative $A_{FB}$ models, one may take $|g^S_{LR}|^2=0.44$ and $|g^S_{LL}|^2 =
0.22$. Our additional inputs are
\begin{equation}
m_\tau = 1.78~{\rm GeV}\,,\ \ T_\tau= 290.3~{\rm fs}\,, \ \ \Lambda = 5~{\rm TeV}\,.
\end{equation}

We show our results for the ${\sf S}$ class of models; ${\sf V}$ class of models will show identical
results. The observables that we use are $A'_{FB}(x)$, ${\cal A}'_{FB}(y)$ (both defined in the previous section), as well as
$dB_\tau/dx$ and $dB_\tau/dy$, the expressions for which can be obtained from Eqs.\ (\ref{e:dgdx}) and (\ref{e:dgdy}):
\begin{eqnarray}
\frac{dB_\tau}{dx} &=&
\frac{m_\tau^5\, T_\tau}{ 128 \times 24\pi^3 \Lambda^4}(3x^2 g_1 - x^3 g_2)\,,\nonumber\\
\frac{dB_\tau}{dy} &=&
\frac{m_\tau^5 T_\tau}{ 128 \times 48\pi^3 \Lambda^4}(3y^2 g'_1 - 2y^3 g'_2)\,.
\end{eqnarray}
From Eq.\ (\ref{afbx1}), one can write
\begin{equation}
A'_{FB}(x) = \frac{1}{B_\tau}\times T_\tau \frac{m_\tau^5}{ 128 \times 48\pi^3 \Lambda^4}(x^2 g_3 - x^3 g_4)\,.
\end{equation}
For 50 events, $B_\tau$ = $1.43\times 10^{-8}$.
Similarly,
\begin{equation}
{\cal A}'_{FB}(y) = \frac{1}{B_\tau}\times T_\tau \frac{m_\tau^5}{ 128 \times 96\pi^3 \Lambda^4}
(y^2 g'_3 - 2y^3 g'_4)\,.
\label{afby}
\end{equation}

Before we show our results for all the 6 models, let us mention a few important points here.
\begin{enumerate}

 \item The determination of $\chi^2$ involves an integration over the variable $\phi$ of Eq.\
 (\ref{e:def-oo}). If over the region of integration the observable for the seed model becomes zero
 for any value of $\phi$, the integration diverges. Thus, one has to cut off such badly behaving regions.
 For example, if the observable for the seed model becomes zero at the end points, say $a$ and $b$,
 one has to perform the integral between $a+\epsilon$ and $b-\epsilon$, where $\epsilon$ is taken to be so small
 as not to affect the observable (like, say, the number of events). More concrete examples are given
 below.

 \item One may ask why we do not use a two-variable analysis and use the double differential cross-section
 as the observable. This would have certainly been useful, and more powerful as an analytical tool, if we could
 manage a large number of events so that even the two-dimensional bins have enough number of events.
 With a small number of events, such an analysis would not give much useful information.
 \end{enumerate}

 All our observables depend on only two functions, $f_1$ and $f_2$, with the argument being $x$ for
 the opposite-sign muon and $y$ for the like-sign more energetic muon.
 Depending on the observables, the combinations $C_1$ and $C_2$ are as follows.

\subsection{Observable: $A'_{FB}(x)$}

\begin{figure}[t]
\centering
  \subfloat[\label{RL-LL}]
  {\includegraphics[height=5cm]{Fig5a.png}}
  \subfloat[\label{RL-RR}]
  {\includegraphics[height=5cm]{Fig5b.png}}
  \subfloat[\label{RL-LR}]
  {\includegraphics[height=5cm]{Fig5c.png}}  \\
  \subfloat[\label{RR-RL}]
  {\includegraphics[height=5cm]{Fig5d.png}}
  \subfloat[\label{RR-LL}]
  {\includegraphics[height=5cm]{Fig5e.png}}
  \subfloat[\label{RR-LR}]
  {\includegraphics[height=5cm]{Fig5f.png}}
  \caption{The differentiability of the models A-F, shown in (a)-(f) respectively, from the
  `seed' model, with $A'_{FB}(x)$ as the observable.}
\label{f:afbx}
\end{figure}

\begin{table}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|}
 \hline
 Model & Seed & Second operator & $C_1$ & $C_2$\\
 \hline
 A & $RL$ & $LL$ & $3 |g_{RL}^S|^2 + |g_{LL}^S|^2$ & $3 |g_{RL}^S|^2 + 2 |g_{LL}^S|^2$\\
 B & $RL$ & $RR$ & $3 |g_{RL}^S|^2 - |g_{RR}^S|^2$ & $3 |g_{RL}^S|^2 - 2 |g_{RR}^S|^2$\\
 C & $RL$ & $LR$ & $3 |g_{RL}^S|^2 - 3 |g_{LR}^S|^2$ & $3 |g_{RL}^S|^2 - 3 |g_{LR}^S|^2$\\
 D & $RR$ & $RL$ & $ - |g_{RR}^S|^2 + 3 |g_{RL}^S|^2$ & $-2 |g_{RR}^S|^2 + 3 |g_{RL}^S|^2$\\
 E & $RR$ & $LL$ & $ - |g_{RR}^S|^2 + |g_{LL}^S|^2$ & $-2 |g_{RR}^S|^2 + 2 |g_{LL}^S|^2$\\
 F & $RR$ & $LR$ & $ - |g_{RR}^S|^2 - 3 |g_{LR}^S|^2$ & $-2 |g_{RR}^S|^2 - 3 |g_{LR}^S|^2$\\
 \hline
  \end{tabular}
  \caption{$C_1$ and $C_2$ for different models. The observable is $A'_{FB}(x)$.}
 \label{tab-afbx}
 \end{center}
 \end{table}

We show the coefficients $C_1$ and $C_2$ in Table \ref{tab-afbx}, and
Fig.\ \ref{f:afbx} displays our results for Models A-F. As the plots are not self-explanatory,
let us clearly specify what they mean. The diagonal band with negative slope, in each of the plots,
represents the allowed region in the parameter space of the various $O2$ models. Only the two relevant
WCs are taken to be nonzero, keeping the others fixed at zero.
Once the experimentalists obtain a certain number of events, this will specify a line in the
two-dimensional parameter space over which the allowed models, each of them specified by some WCs,
may lie. The exact position of the line will depend on what model one chooses,
but the analysis must take into account the constraint imposed by this line \footnote{That is why
even for a two-parameter model the degree of freedom is only 1.}. The uncertainties in the data
will broaden the line to a band, whose width will ultimately depend on the number of events as well as the
detector parameters. As a very rough guess, we take $\sqrt{N}$ to be the width for the line with $N$
events. The plots are drawn for $N=50$; thus, the band includes all the points for which the number of events
lie approximately between 43 and 57. The separation contours are drawn on these bands only. We expect the
bands to be narrower in actual experiments.

Let us consider Fig.\ \ref{RL-LL}. This takes $|g^S_{RL}|^2 = 0.435$ as the seed value. The plot
tells us that this one coupling model can be differentiated from the one with $|g^S_{RL}|^2 = 0.2$ and $|g^S_{LL}|^2 = 0.1$
at more than $3\sigma$ if we have approximately 50 events and use $A'_{FB}(x)$ as our observable.
Similarly, the model with $|g^S_{RL}|^2 = 0.35$ and
$|g^S_{LL}|^2 = 0.05$ can be separated from the above mentioned seed model by less than $2\sigma$.
The actual numbers should be even worse as the
systematic uncertainties will also creep in. Similar conclusions hold for Models B, C, D, E and F, for which
the results are shown from Figs. \ref{RL-RR} to \ref{RR-LR} respectively.
As we mentioned before, contours for Models B and D are not the same, although they involve the same
set of operators. This is because the seed is different, which ultimately control the correlation matrix.

For models D-F, the seed model has a zero crossing for $A'_{FB}(x)$ at $x=\frac12$. Unlike in the case
of the differential decay distribution and observables proportional to it, the integrated observable in this case
may become negative in different parts of the parameter space. This makes the covariance matrix $V_{i j}$
not positive definite. We note here that for our purpose, {\em i.e.} to construct $\chi^2$, the integrated
observables serve only as the normalization of $V^{-1}_{i j} (= M_{i j})$. We have taken the modulus of the
integrand for each value of $x (y)$ for this reason.
This, while keeping the nature of the error ellipsoids intact, will always keep the covariance matrix
positive definite. On the flip-side, this makes the integral diverge at the zero crossing point. Thus, to evaluate
the correlation matrix $V_{ij}^{-1}$ and to cancel this divergence, one has to remove the tiny
patch $0.495 < x < 0.505$ from integration. This
has only a negligible effect on the number of events, but keeps the necessary integrations convergent.

\begin{figure}[t]
\centering
  \subfloat[\label{2coupdist1}]
  {\includegraphics[height=5cm]{Fig6a.png}}
  \subfloat[\label{2coupdist2}]
  {\includegraphics[height=5cm]{Fig6b.png}}
  \caption{Comparison of $A'_{FB}(x)$ for the $O2$ Models A-F with the
  respective `seed' model. The relevant WCs are taken from Fig.\ \ref{f:afbx}.}
\label{2cdafbx}
\end{figure}

\begin{table}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|}
  \hline
 Model  &  $|g_{RL}^S|^2$  & $|g_{LL}^S|^2$  & $|g_{RR}^S|^2$ & $|g_{LR}^S|^2$ & $A_{FB}$ (int.) \\
  \hline
1(Seed) &      0.435       &      -          & 	   -          &     -          & 0.5     \\
 \hline
      A &      0.265       &     0.085       & 	   -          &     -          & 0.239    \\
 \hline
      B &      0.186       &        -        & 	   0.125      &     -          & 0.309    \\
 \hline
      C &      0.305       &        -        & 	    -         &   0.131        & 0.2    \\
 \hline
3(Seed) &        -         &        -        & 	   0.218      &     -          & 0.167    \\
 \hline
      D &      0.063       &        -        & 	   0.186      &     -          & 0.215    \\
 \hline
      E &        -         &      0.110     & 	   0.108      &     -          & -0.001    \\
 \hline
      F &        -         &         -       & 	   0.188     &     0.061      & 0.073    \\
 \hline
  \end{tabular}
 \end{center}
 \caption{The WCs, as obtained from Fig.\ \ref{f:afbx},
 for which Models A-F are separable from the
 respective seed models at $3\sigma$ level. The integrated asymmetries are also shown.}
 \label{t:2cdafbx}
 \end{table}

In Fig. \ref{2cdafbx}, we show, as an illustration, the behaviour of $A'_{FB}(x)$ for Models A-C
vis-a-vis the seed Model 1 and Models D-F with seed Model 2, for which the differentiability is at the
$3\sigma$ level. The corresponding WCs, extracted from Fig.\ \ref{f:afbx}, are displayed in Table \ref{t:2cdafbx}.
We note that Models A-C can be differentiated from the seed Model 1 with only $|g^S_{RL}|^2$
for all values of $x$. On the other hand, Models D and F can be differentiated from seed Model 2
with only $|g^S_{RR}|^2$ (Model 3) only for medium values of $x$, and zero-crossing of $A'_{FB}(x)$ plays a crucial role.

\subsection{Observable: ${\cal A}'_{FB}(y)$ }

In an analogous way, one can use the more energetic of the like-sign muons, and the corresponding
asymmetry ${\cal A}'_{FB}(y)$. The coefficients $C_1$ and $C_2$, from Eqs.\ (\ref{e:dgdy})
and (\ref{afby1}), are shown in Table \ref{tab-afby}.

\begin{table}[htbp]
\begin{center}
 \begin{tabular}{|c|c|c|c|c|}
 \hline
 Model & Seed & Second operator & $C_1$ & $C_2$\\
\hline
 A & $RL$ & $LL$ & $ |g_{RL}^S|^2 + 7 |g_{LL}^S|^2$ & $ |g_{RL}^S|^2 + 4  |g_{LL}^S|^2$\\
 B & $RL$ & $RR$ & $|g_{RL}^S|^2 - 7 |g_{RR}^S|^2$ & $|g_{RL}^S|^2 - 4 |g_{RR}^S|^2$\\
 C & $RL$ & $LR$ & $|g_{RL}^S|^2 - |g_{LR}^S|^2$ & $|g_{RL}^S|^2 - |g_{LR}^S|^2$\\
 D & $RR$ & $RL$ & $ -7 |g_{RR}^S|^2 + |g_{RL}^S|^2$ & $ -4 |g_{RR}^S|^2 +  |g_{RL}^S|^2$\\
 E & $RR$ & $LL$ & $ -7 |g_{RR}^S|^2 + 7 |g_{LL}^S|^2$ & $ -4 |g_{RR}^S|^2 + 4 |g_{LL}^S|^2$\\
 F & $RR$ & $LR$ & $ - 7 |g_{RR}^S|^2 - |g_{LR}^S|^2$ & $ - 4 |g_{RR}^S|^2 -  |g_{LR}^S|^2$\\
 \hline
 \end{tabular}
 \caption{$C_1$ and $C_2$ for different models, with ${\cal A}'_{FB}(y)$ as the observable.}
 \label{tab-afby}
 \end{center}
 \end{table}

\begin{table}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|}
  \hline
 Model  &  $|g_{RL}^S|^2$  & $|g_{LL}^S|^2$  & $|g_{RR}^S|^2$ & $|g_{LR}^S|^2$ & $A_{FB} (int.)$ \\
  \hline
1(Seed) &      0.435       &      -          & 	   -          &     -          & -0.167     \\
 \hline
      A &      0.314       &     0.061       & 	   -          &     -          & -0.074    \\
 \hline
      B &      0.313       &        -        & 	   0.061      &     -          & -0.167    \\
 \hline
      C &      0.216       &        -        & 	    -         &   0.219        &  0.001    \\
 \hline
 3(Seed) &        -         &        -        & 	   0.218      &     -          & -0.167    \\
 \hline
      D &      0.187       &        -        & 	   0.124      &     -          & -0.167    \\
 \hline
      E &        -         &      0.099     & 	   0.119      &     -          & -0.016    \\
 \hline
      F &        -         &         -       & 	   0.121      &     0.194      & -0.018    \\
 \hline
   \end{tabular}
 \end{center}
 \caption{The WCs, as obtained from Fig.\ \ref{f:afby},
 for which Models A-F are separable from the
 respective seed models at $3\sigma$ level. The integrated asymmetries are also shown.}
   \label{t:2cdafbxy}
 \end{table}

 \begin{figure}[htbp]
\centering
  \subfloat[\label{RL-LL3}]
  {\includegraphics[height=5cm]{Fig7a.png}}
  \subfloat[\label{RL-RR3}]
  {\includegraphics[height=5cm]{Fig7b.png}}
  \subfloat[\label{RL-LR3}]
  {\includegraphics[height=5cm]{Fig7c.png}}\\
  \subfloat[\label{RR-RL3}]
  {\includegraphics[height=5cm]{Fig7d.png}}
  \subfloat[\label{RR-LL3}]
  {\includegraphics[height=5cm]{Fig7e.png}}
  \subfloat[\label{RR-LR3}]
  {\includegraphics[height=5cm]{Fig7f.png}}

  \caption{The differentiability of the models A-F, shown in (a)-(f) respectively, from the
  `seed' model, with ${\cal A}'_{FB}(y)$ as the observable.}
\label{f:afby}
\end{figure}

\begin{figure}[t]
\centering
  \subfloat[\label{2coupdist3}]
  {\includegraphics[height=5cm]{Fig8a.png}}
  \subfloat[\label{2coupdist4}]
  {\includegraphics[height=5cm]{Fig8b.png}}
  \caption{Comparison of the distributions of the asymmetry ${\cal A}'_{FB}(y)$ for the models A-F from the
  respective `seed' model. Here the relevant WCs are taken from the plots in figure \ref{f:afby}.}
\label{f:2cdafby}
\end{figure}

In Fig.\ \ref{f:2cdafby}, we show the distribution of ${\cal A}'_{FB}(y)$ for Models A-F, comparing
A-C with Model 1 as seed and D-F with Model 3 as seed respectively. The corresponding WCs are given in
Table \ref{t:2cdafbxy}. While a $3\sigma$ separation between the models is possible, one notes that
the differentiation works best in the middle-$y$ region, rather than at the endpoints.

\subsection{Observable: $dB_\tau/dx$ and $dB_\tau/dy$}

Study of the differential BRs is instructive. First, let us refer the reader to Tables \ref{t:2wcdbrdx} and
\ref{t:2wcdbrdy} respectively for the coefficients $C_1$ and $C_2$ in all the models considered. For
both $dB_\tau/dx$ and $dB_\tau/dy$,
this shows immediately that Models A and B must yield identical distributions; same is true
for the pair D and F. This is because the BR does not depend on the change $R \leftrightarrow L$.
Models C and E are very poorly differentiable from their respective seeds (at less than $1\sigma$) and so
we do not discuss them any further, neither do we show the corresponding separation plots.

Even though the pattern seems similar, there is an important difference. With $dB_\tau/dx$ as the
observable, we can separate Models A(B) or D(F) from the corresponding seed models at $3\sigma$
or more, depending on the respective WCs. This can be seen from Figure \ref{f:dbrdx}, as well as Table
\ref{t:2cddbx}. With $dB_\tau/dy$ as the observable, there is no available parameter space with 50
events where any model can be separated at more than $2\sigma$ from the seed models. This is why
we do not show the corresponding plots for $dB_\tau/dy$. Thus, as far as the measurement of the number
of events in different energy bins goes, it is preferable to detect the unlike-sign muon, than one of the
like-sign muons.

As the {\sf V} class of models show an identical behaviour, we conclude that based on only the data and
without any {\em a priori} knowledge of the WCs, it is impossible to differentiate between the classes,
but within a particular class, it is possible to differentiate among the various Lorentz structures of the
effective operators.
With enough events, one should be able to differentiate single-operator models
from the double-operator models, like those with pure $S$ or $V$ ($O_9$),  or with pure $P$ or
$A$ ($O_{10}$) muon current.
If we have approximately 50 events, $A'_{FB}(x)$ may help us differentiate
$O_{9L}$ or $O_{10L}$ models from $RL$ by about $5\sigma$, and $O_{9R}$ and $O_{10R}$ models
from $RR$ by about $7\sigma$. With $dB_\tau/dx$,
the former set is differentiable at about $3\sigma$, while the latter is at less than $2\sigma$. (The $9(10)L
(R )$ models are specified by equal magnitudes of the two WCs.)
As $C_1$ and $C_2$ involve $\vert g_{IJ}\vert^2$, and hence it is insensitive to the sign or
phase of the WCs.

\begin{table}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|}
 \hline
 Model & Seed & Second operator & $C_1$ & $C_2$\\
 \hline
 A & $RL$ & $LL$ & $|g_{RL}^S|^2 + |g_{LL}^S|^2$ & $3 |g_{RL}^S|^2 + 2 |g_{LL}^S|^2$\\
 B & $RL$ & $RR$ & $|g_{RL}^S|^2 + |g_{RR}^S|^2$ & $3 |g_{RL}^S|^2 + 2 |g_{RR}^S|^2$\\
 C & $RL$ & $LR$ & $|g_{RL}^S|^2 + |g_{LR}^S|^2$ & $3 |g_{RL}^S|^2 + 3 |g_{LR}^S|^2$\\
 D & $RR$ & $RL$ & $ |g_{RR}^S|^2 +  |g_{RL}^S|^2$ & $2 |g_{RR}^S|^2 + 3 |g_{RL}^S|^2$\\
 E & $RR$ & $LL$ & $ |g_{RR}^S|^2 + |g_{LL}^S|^2$ & $ 2 |g_{RR}^S|^2 + 2 |g_{LL}^S|^2$\\
 F & $RR$ & $LR$ & $ |g_{RR}^S|^2 + |g_{LR}^S|^2$ & $2 |g_{RR}^S|^2 + 3 |g_{LR}^S|^2$\\
 \hline
  \end{tabular}
    \caption{$C_1$ and $C_2$ for different models. The observable is $dB_\tau/dx$.}
    \label{t:2wcdbrdx}
 \end{center}
 \end{table}

  \begin{table}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|}
 \hline
 Model & Seed & Second operator & $C_1$ & $C_2$\\
\hline
 A & $RL$ & $LL$ & $ |g_{RL}^S|^2 +3 |g_{LL}^S|^2$ & $  |g_{RL}^S|^2 +4  |g_{LL}^S|^2$\\
 B & $RL$ & $RR$ & $|g_{RL}^S|^2 +3 |g_{RR}^S|^2$ & $ |g_{RL}^S|^2 +4 |g_{RR}^S|^2$\\
 C & $RL$ & $LR$ & $ |g_{RL}^S|^2 + |g_{LR}^S|^2$ & $|g_{RL}^S|^2 + |g_{LR}^S|^2$\\
 D & $RR$ & $RL$ & $ 3 |g_{RR}^S|^2 + |g_{RL}^S|^2$ & $ 4 |g_{RR}^S|^2 +  |g_{RL}^S|^2$\\
 E & $RR$ & $LL$ & $ 3 |g_{RR}^S|^2 + 3 |g_{LL}^S|^2$ & $ 4 |g_{RR}^S|^2 + 4 |g_{LL}^S|^2$\\
 F & $RR$ & $LR$ & $ 3 |g_{RR}^S|^2 +  |g_{LR}^S|^2$ & $ 4 |g_{RR}^S|^2 +  |g_{LR}^S|^2$\\
 \hline
 \end{tabular}
     \caption{$C_1$ and $C_2$ for different models. The observable is $dB_\tau/dy$.}
     \label{t:2wcdbrdy}
 \end{center}
 \end{table}

\begin{figure}[t]
\centering
  \subfloat[\label{RL-LL2}]
  {\includegraphics[height=3.9cm]{Fig9a.png}}
  \subfloat[\label{RR-RL2}]
  {\includegraphics[height=3.9cm]{Fig9b.png}}
   \caption{The differentiability of the Models A and D, shown in (a) and (b) respectively, from the
  `seed' model, with $dB_\tau/dx$ as the observable. Model B is identical with Model A, and Model F is
  identical with Model D
 }
\label{f:dbrdx}
\end{figure}

\begin{table}[htbp]
 \begin{center}
\begin{tabular}{|c|c|c|c|c|}
  \hline
 Model  &  $|g_{RL}^S|^2$  & $|g_{LL}^S|^2$  & $|g_{RR}^S|^2$ & $|g_{LR}^S|^2$  \\
  \hline
1(Seed) &      0.435       &      -          & 	   -          &     -           \\
 \hline
      A &      0.202       &     0.117       & 	   -          &     -           \\
  \hline
3(Seed) &        -         &        -        & 	   0.218      &     -           \\
 \hline
      D &      0.372      &        -        & 	   0.032      &     -           \\
 \hline
   \end{tabular}
\end{center}
 \caption{The representative values of the WCs, obtained from Fig.\ref{f:dbrdx}, for which Models A and D
 can be differentiated from the
 respective seed models at $3\sigma$ level.}
  \label{t:2cddbx}
\end{table}

In general, LFV models can also involve electrons in the final state, from operators leading to
$\tau^-\to e^- e^+ e^-$ or $\tau^- \to e^- e^+ \mu^-$. The BRs of these channels have bounds comparable to
that of $\tau\to 3\mu$, so we may expect a similar number of events at Belle-II, and a similar analysis
will work. However, detection of final state electrons in an $e^-e^+$ machine will have lesser efficiency than
that of final state muons.

 \section{Conclusion}

In this paper, we focus on the LFV decay $\tau \to 3\mu$. This is of crucial importance in the light of
semileptonic $B$-decay anomalies, which hint at some new physics involving second and third generation
leptons, probably a mixing among the charged leptons. The present limit on this mode translates to $\sim 70$
events at the most at Belle-II with 50 ab$^{-1}$ integrated luminosity. While even a single event will
unequivocally indicate new physics, we try to answer a more ambitious question: is it possible to say
anything about the underlying operators from the observables? Needless to say, the answer will be vital
for model builders.

The answer to this question would have been much easier if final state muon polarisations could have been
measured. As far as the present technologies go, this is not easily attainable. However, as we show, one can
form other observables, which are relatively clean and at the same time can yield significant information.
One of the observables is the asymmetry of either the unlike-sign muon, or the like-sign more energetic
muon, which is to be measured with respect to the initial $\tau$-polarisation direction. If one can measure
the asymmetries, even with the associated error margins, in different energy bins, this can differentiate
between the different types of operators in a particular class (scalar, vector, or tensor).

Another important observable, as expected, is the number of events in different energy bins, either the
unlike-sign or the like-sign muons. Just like the asymmetries, it can
potentially differentiate among the different chiral structures of the operators, although to a lesser extent.
Given the total number of events, one can also have an idea of the magnitude of the relevant WCs. We
expect more events for {\sf V} or {\sf T} type operators, so their WCs, $g^V_{IJ}$ or $g^T_{IJ}$, can be
probed better.

It may so happen that there are more than one NP operators. A typical case is when the muon current
is purely vector or axial-vector in nature. If we have enough number of events ($\sim 50$), we should be able
to say whether there is only one underlying operator or two. Asymmetries are the better observables,
but the distribution of the number of events can also help and act as complementary ones.

One must, however, remember that such an analysis involves the risk of underestimating the errors
by neglecting the systematic uncertainties. Thus, this is to be seen more as a motivation to the experimentalists.
Once the data is available, other powerful analysis methods, like the maximum likelihood, can be applied.

{\bf Acknowledgements}:
AK thanks the Science and Engineering Research Board (SERB), Government
of India, for a research grant. SKP is supported by the grants IFA12-PH-34 and
SERB/PHY/2016348.

redpajama_set_name: RedPajamaArXiv


For long, people have been for the wonder drug that will stop ageing or increase life expectancy for the masses. It seems like a Harvard doctor is closer to developing this remarkable drug that all desire. But instead of reversing ageing, this drug would make the human body healthier by replenishing all vital organs. With the help of a perfected medicine of this sort, cardiovascular maladies and Alzheimer's can be avoided, while the hair & skin would regain the splendor of the youth. Another great effect of this treatment will be that the fertility of the consumer would increase and people of both sexes would be able to sire children at ripe ages.
Dr. Ronald DePinho, a cancer specialist at Harvard University, has tested the miracle drug in mice and got results that even he did not expect. In the detailed coverage of the achievement published in Nature magazine, it was reported that the skin, brains, guts as well as other vital organs of the mice in question could be likened to those of an 80year old human. But, in less than two months into the research, Dr. DePinho observed that there was remarkable change in the appearance of the mice, and they seemed to be completely renewed. Moreover, they were absolutely infertile before the tests began, but had brought forth large litters at the time of the recording of results.
The unexpected effects emanated from the action of the drug on telomeres, which are a sort of caps for chromosomes, with function being the protection of the chromosomes. As one ages, the telomeres' size reduces, thereby increasing the instances of Alzheimer's and allied ailments. In due course of time, their size reduces to the limits and the cells die. This cell-death can be avoided if a naturally-occurring enzyme, called telomerase, can be activated, which is exactly what the drug in question did. Dr. DePinho's drug shocked the enzyme in the mice, thereby altering the ageing process of the test subjects, which had replicated that of humans.
According to the doctor, the basic effects can be termed as giving an 80year old the body of a 50year old. If administered in middle age, this medicine would easily slow down the onset of Alzheimer's, heart diseases & diabetes, and even increase lifespan. The doctor is now going to direct his energies into developing such a drug for mass usage, but he told the Daily Mail that there is a lot of work to be done, since factors other than telomeres would contribute to the success of the medication. His contemporaries have hailed the discovery as 'beautiful', but commented that the perfected anti-ageing drug is a decade away.
« A Glass Of Red Wine Can Keep A Check On Your Blood Glucose Levels!


Even at night it's warm and inviting!
Hot coffee and tea available around the clock!
Located between Binghamton, Endicott and Johnson City, our Vestal, NY hotel is ideally situated in the southern tier of New York with easy access to Rte. 17, I-81 and I-88. The Holiday Inn Express & Suites, Vestal, NY hotel is near SUNY Binghamton University.
Leisure travelers will love using our Vestal accommodations as a home base for sightseeing within the region. Whether visiting one of the local horse farms or taking advantage of the wealth of outdoor adventures nearby, there is plenty to do in Vestal. This hotel is within easy driving distance of Binghamton Senators Hockey, the Binghamton Zoo, Roberson Museum of Science Center & Link Planetarium and the Conklin Players Club.
Business travelers can appreciate our convenient lodging in Vestal, New York near BAE Systems, IBM, Frito-Lay and Link Flight Simulations. There are three medical centers and hospitals within three miles of the hotel. Being in Vestal means being close to the Greater Binghamton Regional Airport. Our choice Vestal, New York accommodations also boasts an on-site conference room and free, high-speed Internet access.
Our hotel in Vestal, NY offers all the comforts of home. We hope that you take advantage of the complimentary Express Start breakfast or enjoy a dip in the indoor pool. Call our Vestal, New York hotel today, and enjoy our great deals and rates for the Tri-Cities.
Our hotel rooms in Vestal, New York offer a 32" flat screen TV with HBO and an easy chair to help you relax after a busy day of travel. The new bath amenities and Kohler showerhead will rejuvenate you, and every bed is triple-sheeted for comfort. Free high-speed wireless Internet access is provided.
Pool hours are from 8am to 10pm. There is no life guard on duty. Parents must accompany all children.
Two Treadmills, two Elipticals, free weight dumbell set, and workout bench. Equipped with air conditioning, flat screen TV, and water fountain.
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Currently renovating rooms, exterior is complete. Public space complete.
We are excited to welcome the family and friends of those graduating from Binghamton University this year! Enjoy your time with us in Vestal!


text:
Ovrture
Chris Snavely
Chris believes that the best outcomes occur as a result of meaningful collaboration. Having worked with more than 50 educational and health care organizations of all shapes and sizes, he leads the creative exchanges and helps design project workflows that steadily build consensus.
Prior to his civilian life, Chris was an officer in the U.S. Navy and had the privilege of serving five years as a Surface Warfare Officer. While not in the office, Chris loves being with his young family and trying to find enough time to keep fit.
Education: U.S. Naval Academy, BS; U.S. Naval Postgraduate School, Masters in Information Systems and Operations
Maria Barton2018-01-25T14:11:34-05:00July 30th, 2015|
Post Office Box 1139 | State College, PA 16804-1139
t 814.234.3672 f 814.234.3786 | contact@snavelyassociates.com
Copyright © 2021 Snavely Associates | DEI Statement


text:
Q: JUnit : when testing method with annotation, test ignores annotation I have really weird issue. Heres my method
@SomeAnnotation(value = "test")
public void doSomethink() {
   ..
}

When I'm using my application everythink works fine (when calling method doSomethink() in debug it also goes inside annotation), but when I run test like below
@Test
public void testDoSomethink() {
   service.doSomethink();
}

My test completly ignores annotation and goes straight into doSomethink method. Am i missing somethink? I think that piece of code is enough but let me know if you need some more.
SomeAnnotation
package org.springframework.security.access.prepost;

import java.lang.annotation.Documented;
import java.lang.annotation.ElementType;
import java.lang.annotation.Inherited;
import java.lang.annotation.Retention;
import java.lang.annotation.RetentionPolicy;
import java.lang.annotation.Target;

@Target({ElementType.METHOD, ElementType.TYPE})
@Retention(RetentionPolicy.RUNTIME)
@Inherited
@Documented
public @interface SomeAnnotation{
    String value();
}

By ignore i mean that when running through test it simply bypass annotation like its not even there

A: Annotations are not code which gets executed, but it is meta-information which needs an annotation processor to do something with it.
In case of Java EE for example, the annotations are processed by the container (e.g. the app server you use) and result in doing additional things like setting up a transaction or do a mapping between entity and table.
So in your case, it seems that you expect Spring to do something with the annotation (which you experience when debugging your application). When running the test case, there is no framework or container doing it, so you have to either mock or simulate that, or use something like Arquillian to run your test within the needed environment (e.g. a container).

A: See this later answer -> https://stackoverflow.com/a/60299629/3661748
Essentially, you have to create a validator.  Then use that validator to validate the class.
That validator will trigger the annotations and collect the violations into an iterable collection.
You can extract those violations to validate individual messages if you want some thorough testing.

redpajama_set_name: RedPajamaStackExchange


From time to time I have a couple of drinks at Van Moll, one of the best cafés on the inner city ring of Eindhoven. Just a ten minute walk from the trainstation you will find this bar that is hidden from the average crowd in Eindhoven. People having drinks here are not passing-by, but go here for the atmosphere… An atmosphere that is awesome! The café is laid-back, chill, different and the crowd is the same. The crowd is mixed with well-informed habitants of Eindhoven, expats and visitors who have found this spot.
This bar slash cafe slash brewery is one of the best places to hang out with friends and enjoy a good conversation. The music is toned down unlike other bars so it's easy to understand one another and enjoy the company.The interior of this bar is nice. Lots of wood combined in a modern fashion and lots of open space and light topped of by a NEON-sign and lots of space to sit and chat. Some people come here to play some of the games that are available. But most all people come here because this spot is very chill, open-minded and there are lots of beers to choose from. I stopped counting how much there are, but it can be said they are specialised in serving beer and offer a HUGE range. Van Moll does not only offer beers, but brews them theirself. Through a clear glass window you can see the brewery downstairs while sipping on your ale. This is the first brewery in Eindhoven and they do this because they love and breathe beer and believe they can add value by providing top-beers and inventing new flavours.
Do NOT worry if you're not a beer-lover, Van Moll offers some great kinds of rum, whisky and other drinks too, not you're average stuff, but top of the line. The rum or whisky can be mixed with coke (although they don't advise it), but not the usual kind (you'll see).
If you're ever in Eindhoven or if you haven't had drinks at Van Moll yet (shame on you) be sure to visit this great drink hotspot! For other Drink Hotspots in Eindhoven check out the Drinks page. For the latest hotspots or events follow me on Facebook, Twitter or Instagram.
22 november 2014 | Filed under Drink spots.


Locality: Sofiya Mine, Nikitovka Deposit, Novyy Donbass, Gorlovka, Doneck Oblast', Ukraine Link to MinDat.org Location Data.
Name Origin: Named for the composition, FERROus iron and HEXAHYDrate in allusion to the six water molecules in the crystal structure.
Comments: White to light brownish crust of ferrohexahydrite on matrix.
Location: Sohland, Bautzen, Oberlausitz, Saxony, Germany. Scale: Picture size 4 mm.
note: Specific Gravity of Ferrohexahydrite =1.93 gm/cc.
U=PEFerrohexahydrite x rElectron Density= 13.82 barns/cc.


text:
Home › Malware
Over 70 Rutter's Stores Hit by Payment Card Breach
By Eduard Kovacs on February 14, 2020
Convenience store and gas station chain Rutter's informed customers this week that their payment card information may have been stolen by cybercriminals.
Rutter's has over 70 locations in Central Pennsylvania, West Virginia and Maryland, and a vast majority of them are affected. One of the impacted stores is in West Virginia and the other 70 are in Pennsylvania.
The company said it discovered the breach recently, after being notified by a third-party that "there may have been unauthorized access to data from payment cards that were used at some Rutter's locations."
Rutter's investigation revealed on January 14 that hackers had planted malware on payment processing systems, allowing them to obtain information from credit and debit cards used at point-of-sale (PoS) devices at fuel pumps and convenience stores.
The malware was designed to look for card track data, which includes cardholder name, card number, expiration date and internal verification code. However, the company says its convenience stores are equipped with PoS terminals that accept EMV cards. In the case of customers who inserted their EMV cards into the terminal's chip reader, the malware may have only stolen the card number and expiration date.
Rutter's also noted that the malware did not appear to have copied data from all the payment cards processed by the compromised systems.
"There is no indication that other customer information was accessed. Please note this incident is not the result of a handheld 'skimmer' being placed on a Rutter's fuel pump," Rutter's said.
The company also pointed out that car washes, ATMs and lottery machines were not impacted.
The malware was mostly present between October 1, 2018, and May 29, 2019, but in some cases the theft of payment card data may have started on August 30, 2018, or September 20, 2018. The list of affected stores and the timeframe when the malware was present are listed on the Rutter's website.
Visa warned in December that sophisticated cybercrime groups had been targeting North American gas stations with PoS malware.
Related: Over 50 Islands Restaurants Hit by Payment Card Breach
Related: Wawa Facing Lawsuits Over Data Breach at All of its Stores
Related: Landry's Discloses Payment Card Incident


Flash memory specialist SanDisk has just unveiled its smallest USB Flash drive to date, the Cruzer Blade. About the size of a paper clip and weighing in at just 2.5 grams, the drives come in storage capacities starting from 2GB right up to a 4,000 MP3-capacity 16GB.
The SanDisk Cruzer Blade's sturdy little frame measures just 1.63 x 0.69 x 0.29 inches and weighs about the same as a penny. The company's Kent Perry explained that "at half the size of some other USB drives, our compact Cruzer Blade provides consumers with highly portable storage at a great value. The drive lets users transfer their digital data quickly between computers, and delivers a storage boost to devices such as netbooks that often need additional memory but are difficult to upgrade easily."
Although SanDisk's smallest USB flash drive, the Cruzer Blade is not of course the smallest available. Sony has been offering its 1.26 x 0.57 x 0.1 inch Sony MicroVault Tiny in 1GB to 8GB capacities for some time and Kingmax also has a range of teeny drives, the company's smallest being the 1.24 x 0.47 x 0.07 inch, 8GB or 16GB Super Stick Mini.
The Cruzer Blade is available now in the U.S. and Canada, with the 2GB version costing US$14.99, 4GB coming in at US$21.99, the 8GB drive priced at US$38.99 and the 16GB variety topping the price tree at US$77.99.


Diy Modern Farmhouse Murphy Bed - How To Build The Bed And . New Build Diy Modern Bed 2018 .. A Murphy Bed is a great way to turn any room into a guest room with the pull of a handle. This DIY Murphy bed project is easy to follow with the step-by-step tutorial and free printable plans. Easy To Build Diy Wall Bed For $150 - Your Modern Family We love to build DIY things in our house (like our DIY mudroom bench). Our house has one room downstairs and all of the other bedrooms are upstairs.


Menu's will no longer lose focus when swapping between gamepad and mouse navigation.
There have been significant changes to the Alchemist Brawler ranging from ability speed increases as well as added mechanics. Throw flask now has a short duration and changes to Turtle Shell Tonic allow even more defensive capabilities including reflecting or absorbing knock backs. The Alchemist continues to be a strong choice when played defensively and is a great "area denial" brawler.


The act is administered by the Wage and Hour Division and requires employers to pay covered employees who are not otherwise exempt at least the federal minimum wage and overtime pay of one-and-one-half-times the regular rate of pay.
While the minimum wage applies to most workers, exemptions apply to individuals in "exempt" positions such as certain executives, administrative and professional employees, as well as other groups, such as tipped employees and child labor.
The Wage and Hour Division enforces the labor standards provisions of the Immigration and Nationality Act (INA) that apply to aliens authorized to work in the U.S. under certain non-immigrant visa programs (H-1B, H-1B1, H-1C, H2A).
The multitude of state and federal rules for fringe benefits, travel pay, garnishments, escheatment, multi-state taxation, recordkeeping, and more.
HRTrainingCenter.com provides training courses on all of these topics. Simply click on an applicable title below to see specifics about, or to order, a training session.