New method for precise determination of top quark mass at LHC
Sayaka Kawabata
Current measurements of the top quark mass which have achieved a precision of less than $1$\,GeV involve a theoretical problem that the definition of the measured mass is ambiguous in perturbation theory. As a possible solution to the problem, we present a new method to measure the top quark mass at the LHC. This method uses lepton energy distribution and has a boost-invariant nature. We discuss strategies towards a precise determination of theoretically well-defined top quark masses such as the $\overline{\rm MS$ mass with the method. As a first step in this direction, a simulation analysis at the leading order is performed considering actual experimental circumstances. The result indicates that this method with further improvements is capable of realizing a precision of less than $1$\,GeV at the LHC.
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		Toyama International Workshop on Higgs as a Probe of New Physics 2015, 11--15, February, 2015
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\title{New method for precise determination of top quark mass at LHC}

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\author{Sayaka Kawabata}
\affiliation{Department of Physics, Tohoku University, Sendai 980-8578, JAPAN}
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\begin{abstract}
Current measurements of the top quark mass which have achieved a precision of less than $1$\,GeV involve a theoretical problem that the definition of the measured mass is ambiguous in perturbation theory.
As a possible solution to the problem, we present a new method to measure the top quark mass at the LHC.
This method uses lepton energy distribution and has a boost-invariant nature.
We discuss strategies towards a precise determination of theoretically well-defined top quark masses such as the $\overline{\rm MS}$ mass with the method.
As a first step in this direction, a simulation analysis at the leading order is performed considering actual experimental circumstances.
The result indicates that this method with further improvements is capable of realizing a precision of less than $1$\,GeV at the LHC.
\end{abstract}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:introduction}
The top quark mass is a key parameter in various particle physics.
It is an important input to electroweak precision fits, and % due to its large contributions to radiative corrections.
%As a consequence, 
its precise value is required for the consistency check of the Standard Model (SM) and constraining models of new physics with these fits~\cite{Baak:2012kk,Ciuchini:2013pca,Baak:2014ora}.
%In addition, its role in examination of the SM vacuum stability is getting more decisive~\cite{Degrassi:2012ry,Buttazzo:2013uya} after the discovery of the Higgs boson.
In addition, the top quark mass plays a decisive role in examination of the SM vacuum stability~\cite{Degrassi:2012ry,Buttazzo:2013uya}.
% especially after the discovery of the Higgs boson.
%Furthermore, predictions beyond the SM often depend strongly on the value of the top quark mass.
The top quark mass has been measured at the Tevatron and at Run I of the Large Hadron Collider (LHC).
%Recently the top mass has been measured with a precision of $0.4$\,\% in the Tevatron and LHC experiments.
Their first combined result yields $m_t=173.34\pm 0.76$\,GeV~\cite{ATLAS:2014wva}.
Higher precision is expected to be achieved by the upgraded LHC with the collision energy of $\sqrt{s}=13$-$14$\,TeV~\cite{CMS:2013wfa}.

However, it has been recognized that the definition of the measured mass is ambiguous in perturbation theory.
% involves a problem in its definition in perturbation theory.
The measured mass is often referred to as ``Monte-Carlo (MC) mass" and distinguished from theoretically well-defined top quark masses, e.g. the $\overline{\rm MS}$ mass and the pole mass.
%should not be identified as the top quark pole mass for the following reason.
%The problem arises in non-perturbative aspects of the top-quark decay products.
%In order to use the top mass value as an input to the above theoretical calculations of electroweak precision fits and the vacuum stability and other calculations, the mass should be related with the SM Lagrangian in a definite way.
%The mass to be determined should be related to the SM Lagrangian in a definite way as we use it as an input to perturbation theory.
The reason for this is as follows:
%This is due to non-perturbative effects which involve jets in the top-quark decay products. 
in the above measurements, the top quark mass is basically obtained from measuring kinematic distributions of the top quark final state and comparing the distributions with theoretical predictions.
For the theoretical predictions, MC event generators are used.
%The above measurements are performed by comparing theoretical predictions and experimental data of distributions of top-quark decay products.
Although the predictions ought to be based on reliable perturbation theory to extract theoretically well-defined top quark masses,
%The following problem occurs if we reconstruct the top quark mass naively from momenta of top-quark decay products.
formation of jets which includes hadronization processes cannot be derived from first principles within the framework of perturbative QCD.
For this reason, phenomenological models are used to describe hadronization in the MC generators.
%There are phenomenological models to describe the hadronization and they are used in Monte Carlo simulation to predict the distributions in the above experiments.
As a result, the top quark mass obtained using jet momenta depends on hadronization models which are not based on perturbation theory.
Note that we cannot separate decay products purely from a top quark in the first place.
The top quark inevitably has color connections to other colored particles, and thus, hadronization always involves other particles, too.
Moreover, process dependence of hadronization makes it difficult to quantify uncertainties of the models further.
%(It is often referred to as ``Monte Carlo mass" $m_t^{\rm MC}$.)
%Therefore, $m_t^{\rm MC}$ is not well-defined in perturbation theory despite its high precision.
%As an example, let us consider reconstructing the invariant mass of the top decay products in the top decay $t\rightarrow bW\rightarrow b\ell\nu$.
%In this case, firstly, the neutrino is undetected and thus, to reconstruct the neutrino momentum, information about jet momenta is required.
%Secondly, $b$-jet momentum involves hadronization processes.
%Thirdly, the top quark inevitably has color connections to other colored particles, which makes it impossible in principle to separate only the top-quark decay products at hadron level.
%From these reasons, the measured mass obtained by comparing theoretical predictions and experiments of the top invariant mass distribution is still affected by hadronization effects and thus the Monte Carlo mass.
More detailed discussion about this problem can be found in, e.g. Ref.~\cite{Hoang:2014oea}.

One way to avoid this problem is to use inclusive observables instead of kinematics of top quarks.
The ATLAS and the CMS collaborations have obtained the top quark pole mass as $m_t^{\rm pole}=172.9^{+2.5}_{-2.6}$\,GeV~\cite{Aad:2014kva} and $m_t^{\rm pole}=176.7^{+3.8}_{-3.4}$\,GeV~\cite{Chatrchyan:2013haa}, respectively, from measurements of inclusive $t\overline{t}$ production cross section.
With the same approach, the $\overline{\rm MS}$ mass has also been obtained as $m_t^{\overline{\rm MS}}(m_t^{\overline{\rm MS}})=160.0^{+5.1}_{-4.5}$\,GeV by the D0 collaboration~\cite{Abazov:2011pta}.
Although these masses have clear definitions in perturbation theory, their errors are still considerably large.

If one aims at a high accuracy of less than $1$\,GeV in determination of theoretically well-defined top masses, we must take care of the fact that the quark pole mass suffers from an intrinsic uncertainty of the order of $\Lambda_{\rm QCD}$.
%Moreover, it is known that the concept of a quark pole mass itself contains a problem.
A quark has a color, and therefore, we cannot extract a single quark.
It means that in principle the top quark propagator does not have a pole if it is computed non-perturbatively.
Therefore, we can specify the top quark pole mass only by defining it order-by-order in perturbation theory.
Though the pole mass is useful in some cases, it is sensitive to infrared physics and exhibits poor convergence of the perturbative series when used in perturbative computations.
Avoiding this difficulty requires utilization of the so-called short-distance masses, which improves convergence of perturbation drastically.
Among others, the $\overline{\rm MS}$ mass is commonly used, where improved convergence has been confirmed in various perturbative QCD predictions~\cite{Langenfeld:2009wd,Hoang:1998nz,Dowling:2013baa}. 
In this context, the top quark $\overline{\rm MS}$ mass, among various definitions of the top quark mass, should be determined with high precision.
%it is important to determine a theoretically well-defined top quark mass, especially the $\overline{\rm MS}$ mass with high precision.

In the light of the above situation and theoretical issues, we propose a new method for top quark mass reconstruction aiming a precise determination of the top quark $\overline{\rm MS}$ mass at the LHC~\cite{Kawabataa:2014osa}.
The method utilizes lepton energy distribution.
Since the lepton in the top decay $t\rightarrow  bW\rightarrow b\ell\nu$ is not affected by hadronization effects, our method can determine theoretically well-defined top quark masses.
Moreover, another important feature of our method is a boost invariance.
The lepton energy distribution depends on velocity distribution of parent top quarks, which is difficult to know accurately due to, e.g. uncertainties in parton distribution functions (PDFs).
On the other hand, our method is, in principle, independent of top quark velocities.
%To be specific,
The experimental observable we use in this method is lepton energy distribution in any Lorentz frame, and theoretical prediction compared with it is the lepton energy distribution in the rest frame of the top quark.
In this comparison, the boost dependence of the distribution is canceled, which is a non-trivial point of our method.
%Therefore, in principle we can obtain the top quark mass only from information concerning the leptonic decay process of the top quark.
We call this method the ``weight function method."

In Sec.~\ref{sec:mtDetermination} we present an outline of the weight function method and strategies towards a precise determination of the top quark pole and $\overline{\rm MS}$ masses with the method.
As a first step in the direction, we perform a simulation analysis at the leading order (LO), investigating the experimental viability of the method.
The result of the LO analysis and its consequent prospects are given in Sec.~\ref{sec:LOAnalysis}.
%Conclusions are given in Sec.~\ref{sec:conclusion}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Top mass determination with weight function method}
\label{sec:mtDetermination}
In this short article, we give only the main points of the weight function method.
For further details of the method we refer to Refs.~\cite{Kawabata:2011gz,Kawabata:2013fta,Kawabataa:2014osa}.
In the weight function method, the top quark mass can be obtained by the following three steps:
\begin{enumerate}
	\item Compute a weight function $W(E_\ell,m)$ which is given by
		\begin{equation}
			W(E_\ell,m)=\!\int \!dE \left.\mathcal{D}_0(E;m)\frac{1}{EE_\ell} \,({\rm odd~fn.~of~}\rho)\right|_{e^{\rho}=E_\ell/E}\,,
			\label{eq:WF}
		\end{equation}
		where $\mathcal{D}_0(E;m)$ is the normalized lepton energy distribution in the rest frame of the top quark whose mass is $m$.
	\item Construct a weighted integral $I(m)$, using the weight function $W(E_\ell,m)$ and a measured lepton energy distribution $D(E_\ell)$:
		\begin{equation}
			I(m) ~\equiv~ \int dE_\ell \,D(E_\ell) \,W(E_\ell,m)\,.
		\end{equation}
	\item Then the top quark mass can be obtained as the zero of $I(m)$:
		\begin{equation}
			I(m=m_t^{\rm rec}) ~=~ 0\,.
		\end{equation}
\end{enumerate}
The method is based on an assumption that the lepton angular distribution in the top quark rest frame is flat~\footnote{
For top quarks produced via $t\overline{t}$ at the LHC, this assumption is valid at sub-percent level~\cite{Bernreuther:2006vg}.
The deviation from the flatness should be taken into account as small corrections if needed.
}.
%and is applicable to determinations of other parameters if the assumption is satisfied, namely the lepton angular distribution in the rest frame of the parent particle is flat.
Note that this method does not consider finite-width effects of the top quark.
The effects should be incorporated as small corrections to the method.
%They are not negligible if one aims high precision, and therefore, we should consider them in an additional way.

For realization of a precise top mass determination with the method, experimental viability of the method is vital.
We have confirmed this in Ref.~\cite{Kawabataa:2014osa}, by performing a simulation analysis at LO with consideration of actual experimental circumstances.
The results are summarized in the following section.
The subjects of our future works will be to include theoretical corrections such as higher-order QCD corrections and finite-width effects of the top quark.
In the rest of this section, we discuss strategies towards a precise determination of the top quark pole and $\overline{\rm MS}$ masses considering these theoretical corrections.

%In addition to the above theoretical aspects, experimental viability of the method is certainly vital for realization of a precise top mass determination with the method.
%We investigate this in the following section, by performing a simulation analysis at LO with consideration of actual experimental circumstances.
%The result indicates the possibility of achieving high precision with this method.
%In the following, we present strategies towards a precise determination of theoretically well-defined top quark mass with this method.
%Then we present strategies towards a precise determination of the top quark pole and $\overline{\rm MS}$ masses with this method.


With the weight function method, the top quark pole mass can be obtained by computing the lepton energy distribution in the top quark rest frame $\mathcal{D}_0(E;m)$ in the on-shell scheme.
In the computation of $\mathcal{D}_0$, the pole mass parameter of the top quark is supposed to be set to $m$.
% (where $m$ represents the pole mass parameter of the top quark).
Then the zero of $I(m)$ gives the value of the top quark pole mass.
Higher-order corrections which are relevant to $\mathcal{D}_0$ are only those concerning the top-quark decay process (ignoring effects of the top quark width% such as off-shellness, non-factorizable corrections and, sub- and non-resonant contributions
), and
they are now available to next-to-next-to-leading order (NNLO) in perturbative QCD~\cite{Gao:2012ja,Brucherseifer:2013iv}.
%The QCD higher-order corrections concerning the top quark decay are now available at the next-next-to-leading order (NNLO).
The obtained pole mass can be converted to the $\overline{\rm MS}$ mass using the relation between them which is currently complete to four-loop order~\cite{Marquard:2015qpa}.

As we have mentioned in the previous section, the quark pole mass suffers from intrinsic ambiguities and leads to a bad convergence behaviour of the perturbative series.
Consequently, the determination of the $\overline{\rm MS}$ mass via the pole mass would receive uncertainties.
% when we aim at high precision including high-order corrections.
A naive approach to obtain the $\overline{\rm MS}$ mass directly (namely bypassing the pole mass) with the weight function method is to compute $\mathcal{D}_0$ in the $\overline{\rm MS}$ scheme, setting the $\overline{\rm MS}$ mass parameter of the top quark to $m$.
%A naive approach that the lepton energy distribution in the top-quark rest frame is calculated in the $\overline{\rm MS}$ scheme so that the parameter $m$ in $\mathcal{D}_0$ represents the $\overline{\rm MS}$ mass parameter is not limited by our method.
Though our method is not limited to a particular renormalization scheme, the lepton distribution in the top quark rest frame $\mathcal{D}_0$ is not suited to the $\overline{\rm MS}$ scheme in a certain region of phase space if we naively perform $\alpha_s$ expansion~\footnote{See similar discussions in Refs.~\cite{Beneke:1998rk,Falgari:2013gwa}.}.
%the $\overline{\rm MS}$ scheme is not a natural scheme for distributions in the top quark rest frame in naive $\alpha_s$ expansions.\footnote{
%See similar discussions in Ref.~\cite{Falgari:2013gwa}.
%}
Therefore, it would be helpful to use other short-distance masses which differ from the pole mass by smaller amounts than the $\overline{\rm MS}$ mass as well as being insensitive to infrared physics.
Such masses are, for example, the potential-subtracted (PS) mass~\cite{Beneke:1998rk} and the 1S mass~\cite{Hoang:1999zc}.
Once we obtain these masses with our method, they can be converted to the $\overline{\rm MS}$ mass.
In addition, since the weighted integral $I(m)$ is conceptually close to an inclusive observable, being integrated over the lepton energy, the possibility that $I(m)$ [instead of $\mathcal{D}_0(E;m)$] is a suitable observable for naive $\alpha_s$ expansion in the $\overline{\rm MS}$ scheme is worth investigating.
%Taking this fact into account, several directions toward a direct determination of the top quark $\overline{\rm MS}$ mass are probable.
%One approach is to calculate $\mathcal{D}_0$ in the on-shell scheme but use the resultant weighted integral $I(m)$ in terms of the $\overline{\rm MS}$ mass.
%It is probable that the observable integrated over the lepton energy $E_\ell$ is tamed in the $\alpha_s$ expansions in the $\overline{\rm MS}$ scheme.
%Another possible approach is to utilize other short-distance masses than the $\overline{\rm MS}$ mass.
%Expressing the $\mathcal{D}_0$ distribution in terms of another short-distance mass like the PS mass or 1S mass, the obtained mass as the zero of $I(m)$ (namely, the reconstructed PS mass or 1S mass) is supposed to be converted to the $\overline{\rm MS}$ mass.
%These masses can avoid the renormalon problem yet the concepts are closer to the pole mass than the $\overline{\rm MS}$ mass.
%Thus, they might be suitable to our method.
 
Furthermore, consideration of finite-width effects of the top quark is necessary to achieve a high precision.
%Additionally, to achieve a high precision, considerations of the effects due to the top quark width is necessary.
The weight function method accommodate the so-called factorizable corrections
%, that is, corrections which concern only the production or decay process of top quarks,
in resonant and sub-resonant contributions if there exists an intermediate top quark which emits a lepton.
On the other hand, the method does not consider effects due to off-shellness (propagation corrections), non-factorizable corrections and sub- or non-resonant contributions where no parent top quark of a lepton intermediates.
Corrections required due to these effects can be computed based on such frameworks as the complex-mass scheme~\cite{Denner:1999gp} and the effective field theory (EFT) approach~\cite{Beneke:2003xh,Falgari:2013gwa}.
%need to be treated as corrections.
%To calculate such corrections, schemes to treat finite-width effects in a systematic way such as the complex-mass scheme and effective field theory (EFT) approaches would be helpful.
%Ultimately it is desirable to determine the $\overline{\rm MS}$ mass within the framework of the complex-mass scheme (or the EFT framework) including both QCD and electroweak corrections.

\section{Simulation analysis at LO}
\label{sec:LOAnalysis}

In this section, we outline the results of a simulation analysis of the top mass reconstruction with the weight function method at LO, which is performed in Ref.~\cite{Kawabataa:2014osa}.
%first outline some important issues concerning experimental circumstances and explain a required modification to the method.
%Taking the modification into account, we then perform a simulation analysis of the top mass reconstruction at LO, and present the result.
%We refer to Ref.~\cite{Kawabataa:2014osa} for more details.
%Based on the view stated in the previous section, the first simulation analysis at the leading order has been performed.
%In this article, we first outline important issues concerning real experimental circumstances and required modification to the method.
%Next we present the result of the detailed simulation analysis considering real experimental circumstances and mention the resultant prospects.
In the analysis, we investigate experimental viability of the method, considering various experimental effects.
We study $t\overline{t}$ production events with lepton+jets final states at $\sqrt{s}=14$\,TeV at the LHC.
%In this analysis, top quarks produced via $t\overline{t}$ which decay into a lepton plus jets at the collision energy of $14$\,TeV at the LHC are considered.
%Dilepton channel can also be applied in principle.

Among various sources of experimental effects, such as detector acceptance, event selection cuts and background contributions, lepton cuts affect our method most seriously.
%In real experiments, there are effects of detector acceptance, event selection cuts and background contributions.
%Above all, lepton cuts cause the most serious effect to our method.
Typical lepton cuts at the LHC are, for example, $p_T(\ell)>20$\,GeV and $|\eta (\ell)|<2.4$.
These cuts severely deform lepton distributions, and as a consequence, the obtained mass with our method (the zero of $I(m)$) deviates from the true mass value.
This requires a modification of the method and we cope with it by compensating the lost events by the lepton cuts with MC simulation events.
The true mass can be obtained as the zero of $I(m)$ imposing the consistency condition that the top mass value $m_t^c$ of the compensated MC part coincides with the zero of $I(m)$.
%The compensated MC part has a dependence on the top mass value.
%We call the top mass for the compensated part $m_t^c$.
%The true mass can be obtained by the condition that $m_t^c$ is consistent with the zero of $I(m)$.
The normalization of lepton distributions of the compensated part is determined so that its junction to the data part is smooth.
We confirmed that the compensated MC part is less sensitive to the top mass determination owing to this way of normalization determination.
Uncertainties of the compensated MC part associated with factorization scale dependence and PDF uncertainties are estimated.

In Fig.~\ref{fig:WFn=2-15} we show weight functions $W(E_\ell,m)$ at LO with the odd function of $\rho$ in Eq.~(\ref{eq:WF}) taken as $n\,\tanh (n\rho)/\cosh (n\rho)$.
Using the weight functions and the compensated lepton distributions with various $m_t^c$ after event selection cuts, we construct weighted integrals $I(m)$.
Fig.~\ref{fig:ImWithmtMCAftCuts} shows the weighted integrals with the weight function corresponding to $n=2$.
%Weighted integrals with the weight function corresponding to $n=2$ are shown in Fig.~\ref{fig:ImWithmtMCAftCuts}.
%Figure~\ref{fig:ImWithmtMCAftCuts} shows the weighted integrals using compensated distributions with various $m_t^c$ after event selection cuts.
\begin{figure}[t]
	\begin{center}
		\begin{tabular}{c}
		
		 	\begin{minipage}{0.45\hsize}
					\includegraphics[width=.8\textwidth]{WFn=2-15.eps}
					\caption{\label{fig:WFn=2-15} Weight functions $W(E_\ell,m)$ at LO with the odd function of $\rho$ in Eq.~(\ref{eq:WF}) taken to be $n\,\tanh (n\rho)/\cosh (n\rho)$.}
			\end{minipage}
\hspace{1cm}	
		 	\begin{minipage}{0.45\hsize}
					\includegraphics[width=.8\textwidth]{Im2_VariousmtcNewFit_AftCuts_mt173.eps}
					\caption{\label{fig:ImWithmtMCAftCuts} Weighted integrals $I(m)$ with various $m_t^{\rm c}$. The weight function used corresponds to $n=2$. The input value of the top quark mass is $173$\,GeV.}
			\end{minipage}		
						
		\end{tabular}	
	\end{center}
	\vspace{-.5cm}
\end{figure}
One can see in Fig.~\ref{fig:ImWithmtMCAftCuts} that the variation of the zero of $I(m)$ is much less than the corresponding variation of $m_t^c$.
The input value of the top quark mass to the data part, which corresponds to the true mass in real experiments, is $173$\,GeV in the figure.
From these weighted integrals, we reconstruct the top quark mass as $174.1$\,GeV in this simulation analysis.
%From these $I(m)$s, the top quark mass is reconstructed as the value of $m_t^c$ which coincides with the zero of $I(m)$.
%The reconstructed masses are given in Table~I.
%The input top quark mass, which is the mass for the data part and the true mass in real experiments, is taken as $m_t=167,\,170,\,173,\,176$ and $179$\,GeV, and for each value of the input mass the reconstructed mass with this method is shown.
The estimated MC statistical error due to limitation of the capable number for generated and analyzed events is $+1.0$/$-1.1$\,GeV.
%In this analysis, the capable number for generated and analyzed events is limited, and the MC statistical error due to it on the top mass determination is estimated to be about $1$\,GeV.
In addition, the expected shift due to effects of the top width is $+0.34$\,GeV.
Taking them into account, the reconstructed mass is consistent with the input mass.
Table~\ref{tab:uncertainties} summarizes estimates of uncertainties from several major sources in the top mass determination at LO.
\begin{table}[t]
	\centering
	\caption{Estimates of uncertainties in GeV from several sources in the top mass determination at LO. The weight function used in this evaluation corresponds to $n=2$. The signal and background statistical errors correspond to those with an integrated luminosity of $100$\,fb$^{-1}$.}
	\vspace{.06cm}
	\begin{tabular}{c|c|c|c|c}
		%\hline 
%		\hline
		~Signal stat. error~ & ~~~~~Fac. scale~~~~~ & ~~~~~~~PDF~~~~~~~ & ~Jet energy scale~ & ~Background stat. error~ \\ \hline
%		& (signal) & (signal) & (signal) & \\ \hline
		$0.4$ & $+1.5$/$-1.4$ & $0.6$ & $+0.2$/$-0.0$ & $0.4$\\
%		\parbox[c][.6cm][c]{0cm}{} $0.4$ & $^{+1.5}_{-1.4}$ & $0.6$ & $^{+0.2}_{-0.0}$ & $0.4$\\
%		\hline
	\end{tabular}
	\label{tab:uncertainties} 
	\vspace{-0.1cm}
\end{table}

In ideal experiments, the weight function method does not depend on uncertainties concerning top-quark production processes, such as factorization scale dependence and PDF uncertainties (as stated in Sec.~\ref{sec:introduction}).
In practice, however, it depends on these uncertainties via the compensated part, and as a result, the factorization scale uncertainties are the dominant source in the LO analysis.
These uncertainties are expected to be reduced by using a MC generator with NLO corrections to top production processes for the compensated part.
The result of the LO analysis indicates that with further improvements mentioned in Sec.~\ref{sec:mtDetermination}, this method is capable of realizing a precision of less than $1$\,GeV in the determination of the top quark $\overline{\rm MS}$ mass at the LHC.



%\section{Conclusions}
%\label{sec:conclusion}

% If you have acknowledgments, this puts in the proper section head.
%\bigskip % extra skip inserted
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgments}
This work is based on analyses performed in collaboration with Y. Shimizu, Y. Sumino and H. Yokoya, and I would like to thank all of them.
The work was supported by Grant-in-Aid for JSPS Fellows under the program number 24$\cdot$3439.
\end{acknowledgments}

\bigskip % extra skip inserted
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  D.~Buttazzo, G.~Degrassi, P.~P.~Giardino, G.~F.~Giudice, F.~Sala, A.~Salvio and A.~Strumia,
  %``Investigating the near-criticality of the Higgs boson,''
  JHEP {\bf 1312} (2013) 089
  [arXiv:1307.3536 [hep-ph]].
  %%CITATION = ARXIV:1307.3536;%%
  %275 citations counted in INSPIRE as of 24 Apr 2015

%\cite{ATLAS:2014wva}
\bibitem{ATLAS:2014wva}
  [ATLAS and CDF and CMS and D0 Collaborations],
  %``First combination of Tevatron and LHC measurements of the top-quark mass,''
  arXiv:1403.4427 [hep-ex].
  %%CITATION = ARXIV:1403.4427;%%
  %176 citations counted in INSPIRE as of 24 Apr 2015

%\cite{CMS:2013wfa}
\bibitem{CMS:2013wfa}
  CMS Collaboration [CMS Collaboration],
  %``Projected improvement of the accuracy of top-quark mass measurements at the upgraded LHC,''
  CMS-PAS-FTR-13-017.
  %%CITATION = CMS-PAS-FTR-13-017;%%
  %5 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Hoang:2014oea}
\bibitem{Hoang:2014oea}
  A.~H.~Hoang,
  %``The Top Mass: Interpretation and Theoretical Uncertainties,''
  arXiv:1412.3649 [hep-ph].
  %%CITATION = ARXIV:1412.3649;%%
  %4 citations counted in INSPIRE as of 18 May 2015

%\cite{Aad:2014kva}
\bibitem{Aad:2014kva}
  G.~Aad {\it et al.}  [ATLAS Collaboration],
  %``Measurement of the $t\overline{t}$ production cross-section using $e\mu $ events with $b$ -tagged jets in $pp$ collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS detector,''
  Eur.\ Phys.\ J.\ C {\bf 74} (2014) 10,  3109
  [arXiv:1406.5375 [hep-ex]].
  %%CITATION = ARXIV:1406.5375;%%
  %38 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Chatrchyan:2013haa}
\bibitem{Chatrchyan:2013haa}
  S.~Chatrchyan {\it et al.}  [CMS Collaboration],
  %``Determination of the top-quark pole mass and strong coupling constant from the t t-bar production cross section in pp collisions at $\sqrt{s}$ = 7 TeV,''
  Phys.\ Lett.\ B {\bf 728} (2014) 496
   [Phys.\ Lett.\ B {\bf 728} (2014) 526]
  [arXiv:1307.1907 [hep-ex]].
  %%CITATION = ARXIV:1307.1907;%%
  %59 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Abazov:2011pta}
\bibitem{Abazov:2011pta}
  V.~M.~Abazov {\it et al.}  [D0 Collaboration],
  %``Determination of the pole and MSbar masses of the top quark from the $t\bar{t}$ cross section,''
  Phys.\ Lett.\ B {\bf 703} (2011) 422
  [arXiv:1104.2887 [hep-ex]].
  %%CITATION = ARXIV:1104.2887;%%
  %64 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Langenfeld:2009wd}
\bibitem{Langenfeld:2009wd}
  U.~Langenfeld, S.~Moch and P.~Uwer,
  %``Measuring the running top-quark mass,''
  Phys.\ Rev.\ D {\bf 80} (2009) 054009
  [arXiv:0906.5273 [hep-ph]].
  %%CITATION = ARXIV:0906.5273;%%
  %290 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Hoang:1998nz}
\bibitem{Hoang:1998nz}
  A.~H.~Hoang, M.~C.~Smith, T.~Stelzer and S.~Willenbrock,
  %``Quarkonia and the pole mass,''
  Phys.\ Rev.\ D {\bf 59} (1999) 114014
  [hep-ph/9804227].
  %%CITATION = HEP-PH/9804227;%%
  %135 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Dowling:2013baa}
\bibitem{Dowling:2013baa}
  M.~Dowling and S.~O.~Moch,
  %``Differential distributions for top-quark hadro-production with a running mass,''
  Eur.\ Phys.\ J.\ C {\bf 74} (2014) 11,  3167
  [arXiv:1305.6422 [hep-ph]].
  %%CITATION = ARXIV:1305.6422;%%
  %10 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Kawabataa:2014osa}
\bibitem{Kawabataa:2014osa}
  S.~Kawabata, Y.~Shimizu, Y.~Sumino and H.~Yokoya,
  %``Weight function method for precise determination of top quark mass at Large Hadron Collider,''
  Phys.\ Lett.\ B {\bf 741} (2015) 232
  [arXiv:1405.2395 [hep-ph]].
  %%CITATION = ARXIV:1405.2395;%%
  %6 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Kawabata:2011gz}
\bibitem{Kawabata:2011gz}
  S.~Kawabata, Y.~Shimizu, Y.~Sumino and H.~Yokoya,
  %``Boost-Invariant Leptonic Observables and Reconstruction of Parent Particle Mass,''
  Phys.\ Lett.\ B {\bf 710} (2012) 658
  [arXiv:1107.4460 [hep-ph]].
  %%CITATION = ARXIV:1107.4460;%%
  %8 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Kawabata:2013fta}
\bibitem{Kawabata:2013fta}
  S.~Kawabata, Y.~Shimizu, Y.~Sumino and H.~Yokoya,
  %``Measurement of physical parameters with a weight function method and its application to the Higgs boson mass reconstruction,''
  JHEP {\bf 1308} (2013) 129
  [arXiv:1305.6150 [hep-ph]].
  %%CITATION = ARXIV:1305.6150,;%%
  %4 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Bernreuther:2006vg}
\bibitem{Bernreuther:2006vg}
  W.~Bernreuther, M.~Fuecker and Z.~G.~Si,
  %``Weak interaction corrections to hadronic top quark pair production,''
  Phys.\ Rev.\ D {\bf 74} (2006) 113005
  [hep-ph/0610334].
  %%CITATION = HEP-PH/0610334;%%
  %69 citations counted in INSPIRE as of 25 May 2015

%\cite{Gao:2012ja}
\bibitem{Gao:2012ja}
  J.~Gao, C.~S.~Li and H.~X.~Zhu,
  %``Top Quark Decay at Next-to-Next-to Leading Order in QCD,''
  Phys.\ Rev.\ Lett.\  {\bf 110} (2013) 4,  042001
  [arXiv:1210.2808 [hep-ph]].
  %%CITATION = ARXIV:1210.2808;%%
  %40 citations counted in INSPIRE as of 24 May 2015

%\cite{Brucherseifer:2013iv}
\bibitem{Brucherseifer:2013iv}
  M.~Brucherseifer, F.~Caola and K.~Melnikov,
  %``$\mathcal O(\alpha_s^2)$ corrections to fully-differential top quark decays,''
  JHEP {\bf 1304} (2013) 059
  [arXiv:1301.7133 [hep-ph]].
  %%CITATION = ARXIV:1301.7133;%%
  %31 citations counted in INSPIRE as of 24 May 2015

%\cite{Marquard:2015qpa}
\bibitem{Marquard:2015qpa}
  P.~Marquard, A.~V.~Smirnov, V.~A.~Smirnov and M.~Steinhauser,
  %``Quark Mass Relations to Four-Loop Order in Perturbative QCD,''
  Phys.\ Rev.\ Lett.\  {\bf 114} (2015) 14,  142002
  [arXiv:1502.01030 [hep-ph]].
  %%CITATION = ARXIV:1502.01030;%%
  %2 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Beneke:1998rk}
\bibitem{Beneke:1998rk}
  M.~Beneke,
  %``A Quark mass definition adequate for threshold problems,''
  Phys.\ Lett.\ B {\bf 434} (1998) 115
  [hep-ph/9804241].
  %%CITATION = HEP-PH/9804241;%%
  %239 citations counted in INSPIRE as of 24 May 2015

%\cite{Falgari:2013gwa}
\bibitem{Falgari:2013gwa}
  P.~Falgari, A.~S.~Papanastasiou and A.~Signer,
  %``Finite-width effects in unstable-particle production at hadron colliders,''
  JHEP {\bf 1305} (2013) 156
  [arXiv:1303.5299 [hep-ph]].
  %%CITATION = ARXIV:1303.5299;%%
  %15 citations counted in INSPIRE as of 24 Apr 2015

%\cite{Hoang:1999zc}
\bibitem{Hoang:1999zc}
  A.~H.~Hoang and T.~Teubner,
  %``Top quark pair production close to threshold: Top mass, width and momentum distribution,''
  Phys.\ Rev.\ D {\bf 60} (1999) 114027
  [hep-ph/9904468].
  %%CITATION = HEP-PH/9904468;%%
  %143 citations counted in INSPIRE as of 24 May 2015

%\cite{Denner:1999gp}
\bibitem{Denner:1999gp}
  A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
  %``Predictions for all processes e+ e- ---> 4 fermions + gamma,''
  Nucl.\ Phys.\ B {\bf 560} (1999) 33
  [hep-ph/9904472].
  %%CITATION = HEP-PH/9904472;%%
  %266 citations counted in INSPIRE as of 25 May 2015

%\cite{Beneke:2003xh}
\bibitem{Beneke:2003xh}
  M.~Beneke, A.~P.~Chapovsky, A.~Signer and G.~Zanderighi,
  %``Effective theory approach to unstable particle production,''
  Phys.\ Rev.\ Lett.\  {\bf 93} (2004) 011602
  [hep-ph/0312331].
  %%CITATION = HEP-PH/0312331;%%
  %72 citations counted in INSPIRE as of 25 May 2015

\end{thebibliography}


\end{document}


1505.06933.extracted_bib
{Baak:2012kk M.~Baak, M.~Goebel, J.~Haller, A.~Hoecker, D.~Kennedy, R.~Kogler, K.~Moenig and M.~Schott {et al., %``The Electroweak Fit of the Standard Model after the Discovery of a New Boson at the LHC,'' Eur.\ Phys.\ J.\ C {72, 2205 (2012) [arXiv:1209.2716 [hep-ph]]. %%CITATION = ARXIV:1209.2716;%% %299 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Ciuchini:2013pca
1505.06933.extracted_bib
{Ciuchini:2013pca M.~Ciuchini, E.~Franco, S.~Mishima and L.~Silvestrini, %``Electroweak Precision Observables, New Physics and the Nature of a 126 GeV Higgs Boson,'' JHEP {1308 (2013) 106 [arXiv:1306.4644 [hep-ph]]. %%CITATION = ARXIV:1306.4644;%% %70 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Baak:2014ora
1505.06933.extracted_bib
{Baak:2014ora M.~Baak {et al. [Gfitter Group Collaboration], %``The global electroweak fit at NNLO and prospects for the LHC and ILC,'' Eur.\ Phys.\ J.\ C {74 (2014) 3046 [arXiv:1407.3792 [hep-ph]]. %%CITATION = ARXIV:1407.3792;%% %69 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Degrassi:2012ry
1505.06933.extracted_bib
{Degrassi:2012ry G.~Degrassi, S.~Di Vita, J.~Elias-Miro, J.~R.~Espinosa, G.~F.~Giudice, G.~Isidori and A.~Strumia, %``Higgs mass and vacuum stability in the Standard Model at NNLO,'' JHEP {1208 (2012) 098 [arXiv:1205.6497 [hep-ph]]. %%CITATION = ARXIV:1205.6497;%% %437 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Buttazzo:2013uya
1505.06933.extracted_bib
{Buttazzo:2013uya D.~Buttazzo, G.~Degrassi, P.~P.~Giardino, G.~F.~Giudice, F.~Sala, A.~Salvio and A.~Strumia, %``Investigating the near-criticality of the Higgs boson,'' JHEP {1312 (2013) 089 [arXiv:1307.3536 [hep-ph]]. %%CITATION = ARXIV:1307.3536;%% %275 citations counted in INSPIRE as of 24 Apr 2015 %\cite{ATLAS:2014wva
1505.06933.extracted_bib
{ATLAS:2014wva [ATLAS and CDF and CMS and D0 Collaborations], %``First combination of Tevatron and LHC measurements of the top-quark mass,'' arXiv:1403.4427 [hep-ex]. %%CITATION = ARXIV:1403.4427;%% %176 citations counted in INSPIRE as of 24 Apr 2015 %\cite{CMS:2013wfa
1505.06933.extracted_bib
{CMS:2013wfa CMS Collaboration [CMS Collaboration], %``Projected improvement of the accuracy of top-quark mass measurements at the upgraded LHC,'' CMS-PAS-FTR-13-017. %%CITATION = CMS-PAS-FTR-13-017;%% %5 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Hoang:2014oea
1505.06933.extracted_bib
{Hoang:2014oea A.~H.~Hoang, %``The Top Mass: Interpretation and Theoretical Uncertainties,'' arXiv:1412.3649 [hep-ph]. %%CITATION = ARXIV:1412.3649;%% %4 citations counted in INSPIRE as of 18 May 2015 %\cite{Aad:2014kva
1505.06933.extracted_bib
{Aad:2014kva G.~Aad {et al. [ATLAS Collaboration], %``Measurement of the $t\overline{t$ production cross-section using $e\mu $ events with $b$ -tagged jets in $pp$ collisions at $\sqrt{s=7$ and 8 TeV with the ATLAS detector,'' Eur.\ Phys.\ J.\ C {74 (2014) 10, 3109 [arXiv:1406.5375 [hep-ex]]. %%CITATION = ARXIV:1406.5375;%% %38 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Chatrchyan:2013haa
1505.06933.extracted_bib
{Chatrchyan:2013haa S.~Chatrchyan {et al. [CMS Collaboration], %``Determination of the top-quark pole mass and strong coupling constant from the t t-bar production cross section in pp collisions at $\sqrt{s$ = 7 TeV,'' Phys.\ Lett.\ B {728 (2014) 496 [Phys.\ Lett.\ B {728 (2014) 526] [arXiv:1307.1907 [hep-ex]]. %%CITATION = ARXIV:1307.1907;%% %59 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Abazov:2011pta
1505.06933.extracted_bib
{Abazov:2011pta V.~M.~Abazov {et al. [D0 Collaboration], %``Determination of the pole and MSbar masses of the top quark from the $t\bar{t$ cross section,'' Phys.\ Lett.\ B {703 (2011) 422 [arXiv:1104.2887 [hep-ex]]. %%CITATION = ARXIV:1104.2887;%% %64 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Langenfeld:2009wd
1505.06933.extracted_bib
{Langenfeld:2009wd U.~Langenfeld, S.~Moch and P.~Uwer, %``Measuring the running top-quark mass,'' Phys.\ Rev.\ D {80 (2009) 054009 [arXiv:0906.5273 [hep-ph]]. %%CITATION = ARXIV:0906.5273;%% %290 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Hoang:1998nz
1505.06933.extracted_bib
{Hoang:1998nz A.~H.~Hoang, M.~C.~Smith, T.~Stelzer and S.~Willenbrock, %``Quarkonia and the pole mass,'' Phys.\ Rev.\ D {59 (1999) 114014 [hep-ph/9804227]. %%CITATION = HEP-PH/9804227;%% %135 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Dowling:2013baa
1505.06933.extracted_bib
{Dowling:2013baa M.~Dowling and S.~O.~Moch, %``Differential distributions for top-quark hadro-production with a running mass,'' Eur.\ Phys.\ J.\ C {74 (2014) 11, 3167 [arXiv:1305.6422 [hep-ph]]. %%CITATION = ARXIV:1305.6422;%% %10 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Kawabataa:2014osa
1505.06933.extracted_bib
{Kawabataa:2014osa S.~Kawabata, Y.~Shimizu, Y.~Sumino and H.~Yokoya, %``Weight function method for precise determination of top quark mass at Large Hadron Collider,'' Phys.\ Lett.\ B {741 (2015) 232 [arXiv:1405.2395 [hep-ph]]. %%CITATION = ARXIV:1405.2395;%% %6 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Kawabata:2011gz
1505.06933.extracted_bib
{Kawabata:2011gz S.~Kawabata, Y.~Shimizu, Y.~Sumino and H.~Yokoya, %``Boost-Invariant Leptonic Observables and Reconstruction of Parent Particle Mass,'' Phys.\ Lett.\ B {710 (2012) 658 [arXiv:1107.4460 [hep-ph]]. %%CITATION = ARXIV:1107.4460;%% %8 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Kawabata:2013fta
1505.06933.extracted_bib
{Kawabata:2013fta S.~Kawabata, Y.~Shimizu, Y.~Sumino and H.~Yokoya, %``Measurement of physical parameters with a weight function method and its application to the Higgs boson mass reconstruction,'' JHEP {1308 (2013) 129 [arXiv:1305.6150 [hep-ph]]. %%CITATION = ARXIV:1305.6150,;%% %4 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Bernreuther:2006vg
1505.06933.extracted_bib
{Bernreuther:2006vg W.~Bernreuther, M.~Fuecker and Z.~G.~Si, %``Weak interaction corrections to hadronic top quark pair production,'' Phys.\ Rev.\ D {74 (2006) 113005 [hep-ph/0610334]. %%CITATION = HEP-PH/0610334;%% %69 citations counted in INSPIRE as of 25 May 2015 %\cite{Gao:2012ja
1505.06933.extracted_bib
{Gao:2012ja J.~Gao, C.~S.~Li and H.~X.~Zhu, %``Top Quark Decay at Next-to-Next-to Leading Order in QCD,'' Phys.\ Rev.\ Lett.\ {110 (2013) 4, 042001 [arXiv:1210.2808 [hep-ph]]. %%CITATION = ARXIV:1210.2808;%% %40 citations counted in INSPIRE as of 24 May 2015 %\cite{Brucherseifer:2013iv
1505.06933.extracted_bib
{Brucherseifer:2013iv M.~Brucherseifer, F.~Caola and K.~Melnikov, %``$\mathcal O(\alpha_s^2)$ corrections to fully-differential top quark decays,'' JHEP {1304 (2013) 059 [arXiv:1301.7133 [hep-ph]]. %%CITATION = ARXIV:1301.7133;%% %31 citations counted in INSPIRE as of 24 May 2015 %\cite{Marquard:2015qpa
1505.06933.extracted_bib
{Marquard:2015qpa P.~Marquard, A.~V.~Smirnov, V.~A.~Smirnov and M.~Steinhauser, %``Quark Mass Relations to Four-Loop Order in Perturbative QCD,'' Phys.\ Rev.\ Lett.\ {114 (2015) 14, 142002 [arXiv:1502.01030 [hep-ph]]. %%CITATION = ARXIV:1502.01030;%% %2 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Beneke:1998rk
1505.06933.extracted_bib
{Beneke:1998rk M.~Beneke, %``A Quark mass definition adequate for threshold problems,'' Phys.\ Lett.\ B {434 (1998) 115 [hep-ph/9804241]. %%CITATION = HEP-PH/9804241;%% %239 citations counted in INSPIRE as of 24 May 2015 %\cite{Falgari:2013gwa
1505.06933.extracted_bib
{Falgari:2013gwa P.~Falgari, A.~S.~Papanastasiou and A.~Signer, %``Finite-width effects in unstable-particle production at hadron colliders,'' JHEP {1305 (2013) 156 [arXiv:1303.5299 [hep-ph]]. %%CITATION = ARXIV:1303.5299;%% %15 citations counted in INSPIRE as of 24 Apr 2015 %\cite{Hoang:1999zc
1505.06933.extracted_bib
{Hoang:1999zc A.~H.~Hoang and T.~Teubner, %``Top quark pair production close to threshold: Top mass, width and momentum distribution,'' Phys.\ Rev.\ D {60 (1999) 114027 [hep-ph/9904468]. %%CITATION = HEP-PH/9904468;%% %143 citations counted in INSPIRE as of 24 May 2015 %\cite{Denner:1999gp
1505.06933.extracted_bib
{Denner:1999gp A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth, %``Predictions for all processes e+ e- ---> 4 fermions + gamma,'' Nucl.\ Phys.\ B {560 (1999) 33 [hep-ph/9904472]. %%CITATION = HEP-PH/9904472;%% %266 citations counted in INSPIRE as of 25 May 2015 %\cite{Beneke:2003xh
1505.06933.extracted_bib
{Beneke:2003xh M.~Beneke, A.~P.~Chapovsky, A.~Signer and G.~Zanderighi, %``Effective theory approach to unstable particle production,'' Phys.\ Rev.\ Lett.\ {93 (2004) 011602 [hep-ph/0312331]. %%CITATION = HEP-PH/0312331;%% %72 citations counted in INSPIRE as of 25 May 2015