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%- {{{ title and abstract
\title{\vskip-3cm{\baselineskip14pt
\begin{flushleft}
\normalsize SFB/CPP-14-61 \\
\normalsize TTP/TTP14-025 \\
\normalsize Alberta Thy 14-14
\end{flushleft}}
\vskip1.5cm
${\cal O}(\alpha_s^2)$ corrections to the running top-Yukawa
coupling and the mass of the lightest Higgs boson in the MSSM}
\author{ \small
D.~Kunz$^{(a)}$, L.~Mihaila$^{(a)}$, N.~Zerf$^{(b)}$\\
{\small\it (a) Institut f{\"u}r Theoretische Teilchenphysik,
Karlsruhe Institute of Technology (KIT),}\\
{\small\it 76128 Karlsruhe, Germany}\\
{\small\it (b) Department of Physics,
University of Alberta,}\\
{\small\it 4-181 CCIS, Edmonton AB T6G 2E1, Canada}\\
{\small\it E-mail: david.kunz@kit.edu, luminita-nicoletta.mihaila@kit.edu, zerf@ualberta.ca}\\
}
\date{}
\maketitle
\thispagestyle{empty}
\begin{abstract}
In this paper we propose a method to compute the running top-Yukawa coupling
in supersymmetric models with heavy mass spectrum based on the ``running''
and ``decoupling'' procedure. In order to enable this approach
we compute the two-loop SUSY-QCD radiative corrections required in the decoupling process.
The method has the advantage that large logarithmic corrections are automatically resummed through the Renormalization Group Equations.
As phenomenological application we study the effects of this approach on the
SUSY-QCD corrections to the prediction of the
lightest Higgs boson mass at three-loop accuracy. We observe a significant
reduction of the renormalization scale dependence as compared to the direct
method, that is based on the conversion relation between the running and pole mass for the
top quark. The effect of resummation of large logarithmic contributions
consists in an increased prediction for the Higgs boson mass, an observation
in agreement with the previous analyses.
\noindent
PACS numbers: 11.30.Pb, 12.38.-t, 12.38.Bx, 12.10.Kt
\end{abstract}
%- }}}
%- {{{ Introduction:
\section{Introduction}
The recent discovery of the Higgs boson at the Large Hadron Collider (LHC) was a
major milestone not just in particle physics but in the history of
science. It bears the knowledge how the mass comes about at quantum level by
the Higgs mechanism. With the Higgs boson discovery, particle physics entered
a new era of tremendous
intensity of detailed and careful study of its properties. Hopefully,
accurate understanding of the
Higgs phenomenology together with new information from experiments at the LHC
will provide us a tool for exploring new physics.\\
Great interest is currently devoted to the study of the Higgs boson couplings
to the electroweak gauge bosons $W$ and $Z$ and to the top- and bottom-quarks or
tau-leptons. Deviations in these couplings could possibly be observed
once the currently large uncertainties will be improved as part of the LHC
program and at a future Higgs factory. It has been shown that the Higgs couplings
will be sensitive to new physics at the multi-TeV scale once percent
level precision on coupling measurements will be achieved (for a recent review
see \cite{Englert:2014uua}).
The aim of this paper is to propose a method for the computation of the top-Yukawa coupling
within the Minimal Supersymmetric Standard Model (MSSM) taking into account
radiative corrections at ${\cal
O}(\alpha_s^2)$ accuracy. In general, the
relationships between the running couplings and the physical observables
like particle masses are affected by large radiative
corrections~\cite{Sirlin:1985ux,Hempfling:1994ar}. Within the SM, the
relationship between the running top-Yukawa coupling and the physical top-quark
pole mass receives three type of radiative corrections: i) higher order
corrections to the running Yukawa
coupling, ii) contribution to
the fermion pole mass, and
iii) corrections to the relation between the Fermi constant and the SM
parameters. The first contribution is known at three-loop
accuracy~\cite{Chetyrkin:2012rz,Bednyakov:2012en} taking into account
corrections from all sectors of the SM.
The radiative corrections to the top-quark pole mass are known in QCD up to three-loop
order~\cite{Chetyrkin:1999ys,Chetyrkin:1999qi,Melnikov:2000qh,Marquard:2007uj}
and in the electroweak sector up to
two loops~\cite{Faisst:2003px,Jegerlehner:2003sp,Jegerlehner:2003py,Faisst:2004gn,Kniehl:2014yia,Eiras:2005yt}.
The third contribution is known in the SM with
two- and three-loop
accuracy for the genuine electroweak~\cite{Fanchiotti:1992tu,Awramik:2002wn,Faisst:2003px} and
mixed QCD-electroweak~\cite{Chetyrkin:1995js} sectors, respectively.
As was explicitly shown there are two sources for the large radiative
corrections: the
pure QCD contributions to the top-quark pole mass of about $10$~GeV, $2$~GeV
and $0.5$~GeV~\cite{Chetyrkin:1999ys} at one-, two- and three-loop order, respectively; and the
tadpole diagrams when they are taken into account for a
gauge-independent definition of the running-mass~\cite{Faisst:2004gn}. Their
magnitude is comparable with that from QCD sector at
the one-loop order and amounts to about $0.5$~GeV at two loops. As can be
concluded from the numerical values cited above the radiative corrections are
very important for the QCD sector, and even the third-order terms
in the
perturbative series are necessary in order to cope with the current
experimental accuracy on the top-quark mass~\cite{ATLAS:2014wva}.
However, the situation is
going to change if the International Linear Collider (ILC) is built, where a
precision of ${\cal O} (100)$~MeV is expected.\\
When physics beyond the SM is considered, the radiative
corrections to the top-quark pole mass might receive much larger contributions than in
the SM and even diagrams beyond the three-loop order have to be taken into
account to reach the current experimental accuracy. By now, the radiative
corrections to the fermion pole mass are known at
two-loop order for a general theory with massless gauge
bosons~\cite{Martin:2005ch}. The numerical evaluation of the two-loop
self-energy diagrams has been implemented in the code {\tt
TSIL}~\cite{Martin:2005qm}. For supersymmetric models with masses
at the few TeV scale~\footnote{In accordance with the lower bound from the direct
SUSY searches at the LHC (for a review see~\cite{Craig:2013cxa}).} the radiative corrections to the
top-quark pole mass increase by about a factor of four as compared to the
SM results as we show in section~\ref{sec:numerics}.
Thus, the two-loop contributions might become one order of magnitude
larger than the experimental uncertainties and the effects of higher order corrections have
to be considered. The computation of on-shell self-energy diagrams with
several mass scales at the three-loop order is, for the
time being, feasible only using asymptotic expansion techniques and is
computationally very involved. In this paper, we propose an alternative method
that can be applied as long as the top-quark mass is much smaller than the
masses of supersymmetric particles. In the present paper, we focus on the
dominant SUSY-QCD corrections to the running Yukawa coupling in the MSSM and
postpone the study of the contributions originating from Yukawa and electroweak
interactions for a later publication. The SUSY-QCD
corrections to the relation between the running Yukawa coupling and the
top-quark pole mass reduce to the corrections to the ratio between the
pole and the running masses. Explicitly, the running top-quark mass
is determined in the SM with the highest available precision from the
experimentally measured pole mass~\footnote{For a detailed discussion about
the distinction between the pole and the measured top-quark mass we refer
to~\cite{Moch:2014tta}.}. Then, the
running mass is evolved up to
the SUSY scale using the Renormalization Group Equations (RGEs) of the
SM. Afterwards, the running top-quark
mass in the SM is converted to its value in the MSSM. In this step, threshold corrections
at the SUSY-scale are required. In the last step, the running top-quark mass
in the MSSM is evolved at the desired energy scale with the help of MSSM RGEs.
As the RGEs in the SM~\cite{Chetyrkin:2012rz,Bednyakov:2012en} and the
MSSM~\cite{Ferreira:1996ug,Harlander:2009mn} are known to three-loop order,
the threshold corrections are
required at the two-loop order. They are known for light quark masses (e.g. bottom
quark) in the SM to three-loop
accuracy~\cite{Chetyrkin:1997un,Steinhauser:1998cm} and in the MSSM to
two loops~\cite{Bednyakov:2007vm,Bauer:2008bj}. Great interest was devoted to
the determination of the effective bottom-Yukawa coupling in SUSY-models with a large
$\tan \beta$ parameter~\cite{Carena:1999py,Mihaila:2010mp,Noth:2010jy}. For
these models the resummation of $\tan \beta$ enhanced contributions is considered
on top of the two-loop order calculation.
One goal of this paper is
to present the computation of the two-loop SUSY-QCD
threshold corrections to the running top-quark mass. The calculation is similar
with the ones performed in~\cite{Bednyakov:2007vm,Bauer:2008bj}. The advantage
of the method presented here as compared to the direct calculation of on-shell
self-energy diagrams in the MSSM is that the occurring large logarithms of the form
$\ln(M_{\rm top}/M_{\rm SUSY})$ are automatically resummed by the use of
RGEs. The result is a much better convergence of the perturbative series as
will be explicitly shown in the next sections.\\
The second
aim is to study the phenomenological effects of the above
calculation. Obviously, the top-Yukawa coupling is an essential ingredient in
all processes involving interactions between the Higgs boson and top-quark and
top-squarks. However, the most prominent example is probably the effects on the
lightest Higgs boson mass, that receives radiative corrections enhanced by the
fourth power in top-Yukawa coupling. A detailed numerical analysis of the
effects of the proposed method for the determination of the running Yukawa
coupling on the SUSY-QCD corrections to the lightest Higgs boson mass will be discussed in section~\ref{sect:mhphen}.
The paper is structured as follows: in section~\ref{sec:frm} we present our
computational framework; explicit analytical results are discussed in
section~\ref{sec:res} together with their numerical implementation; in section~\ref{sect:mhphen}
we perform a phenomenological analysis of the effects of the two-loop SUSY-QCD
corrections determined in the previous section on the SUSY-QCD prediction of the
lightest Higgs boson mass with three-loop accuracy; in section~\ref{sec:concl}
we summarise our conclusions.
% For the MSSM, the radiative corrections to the
%top-quark pole mass increase by about a factor four for supersymmetric masses
%at the few TeV scale in acc from the
\section{\label{sec:frm} Framework}
An elegant approach to get rid of unwanted large logarithms occurring in the
predictions for observables in multi-scale processes
is to formulate an effective theory (ET) (for a review see Ref.~\cite{Steinhauser:2002rq}).
The parameters of the ET must be modified in
order to take into account the effects of the heavy fields. The ET parameters are related to the
parameters of the full theory by the so-called matching or decoupling
relations. These take into account threshold corrections generated by the heavy
degrees of freedom that are integrated out when the ET is constructed. In
the following, we concentrate on the calculation of the decoupling
coefficients for the strong
coupling and the top-quark mass within SUSY-QCD. They are defined through the
following relations between the bare quantities\\
\begin{eqnarray}
\alpha_s^{0,\prime}&=&\zeta_{\alpha_s}^0 \alpha_s^0\nonumber\\
m_t^{0,\prime}&=& \zeta_{m_t}^0 m_{t}^0\,,
\end{eqnarray}
where the primes label the quantities in the effective theory.
The decoupling coefficients have been computed in QCD including
corrections up to the four-loop
order for the strong coupling~\cite{Schroder:2005hy,Chetyrkin:2005ia} and
three-loop order for quark masses~\cite{Chetyrkin:1997un,Steinhauser:1998cm}.
In the MSSM the two-loop
SUSY-QCD~\cite{Harlander:2005wm, Bednyakov:2007vm,Bauer:2008bj}
and SUSY-EW~\cite{Bednyakov:2007vm,Noth:2010jy} expressions are known. Very
recently, even the three-loop SUSY-QCD corrections to decoupling
coefficient of the strong coupling were computed~\cite{Kurz:2012ff}.
We consider SUSY-QCD with $n_f=6$ active quark and squark flavours and $n_{\tilde{g}}= 1$
gluinos. Furthermore, we assume that all SUSY-particles including squarks and the gluino are much
heavier than the SM particles. Integrating out the heavy
fields from the full Lagrange density, we obtain the Lagrange density corresponding to
the ``effective'' SM with $n_f$ quarks plus non-renormalizable interactions. The latter are
suppressed by negative powers of the heavy masses and will be neglected here.\\
Since the decoupling coefficients are universal quantities, they are
independent of the momenta carried by the incoming and outgoing
particles. The authors of
Refs.~\cite{Chetyrkin:1997un} showed
that the bare decoupling coefficients for the quark mass
$\zeta_m^0$ and for the strong coupling constant $\zeta_s^0$ can be derived
via the relations
\begin{eqnarray}
\zeta_{\alpha_s}^0 &=&
\left[\frac{1+\Gamma^{0,h}_{gcc}(0,0)}{(1+\Pi^{0,h}_c(0))\sqrt{1+\Pi^{0,h}_g(0)}}\right]^{\!2}\,,\nonumber\\
\zeta_{m_t}^0 &=& \frac{1-\Sigma_s(0)}{\sqrt{\zeta_L
\zeta_R}}\quad\mbox{with}\nonumber\\
\zeta_L&=& 1+\Sigma_v(0)-\Sigma_A(0)\quad\mbox{and }\quad \zeta_R\,\,=\,\,1+\Sigma_v(0)+\Sigma_A(0)\,,
\end{eqnarray}
where $\Sigma_s(p^2)$, $\Sigma_v(p^2)$ and $\Sigma_A(p^2)$ are the scalar, the
vector and the axial-vector components of the top-quark self-energy defined
through
\begin{eqnarray}
\Sigma(p^2)= /\!\!\! p (\Sigma_v(p^2) +\gamma_5 \Sigma_A(p^2)) +m_t \Sigma_s(p^2)\,.
\end{eqnarray}
%\label{fig::mthsmuren}
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath sqcd_diags.eps,width=.61\textwidth}
\end{center}
\caption[]{\label{fig::diagrams} Sample Feynman diagrams that
contribute to the two-loop corrections to $\zeta_{m_t}$.
}
\end{figure}
Note that the axial-vector component starts contributing at the two-loop
order. $\Pi^{0,h}_c(p^2)$ and $\Pi^{0,h}_g(p^2)$ are the ghost and gluon vacuum
polarizations and $\Gamma^{0,h}_{gcc}(p,q)$ denotes the amputated Green function
contributing to the gluon-ghost-ghost vertex.
The
superscript $h$ indicates that in the framework of Dimensional Regularization
(DREG) or Dimensional Reduction (DRED) only diagrams
containing at least one heavy particle inside the loops contribute and that
only the hard regions in the asymptotic expansion of the diagrams are
taken into account.\\
In Fig.~\ref{fig::diagrams}
some sample Feynman diagrams contributing to the
decoupling coefficients for the top-quark mass are shown.
The Feynman diagrams were generated with
{\tt QGRAF}~\cite{Nogueira:1991ex}
and further processed with {\tt q2e} and {\tt exp}
~\cite{Harlander:1997zb,Seidensticker:1999bb}. The
reduction of various
vacuum integrals to the master
integral was performed with a self written {\tt
FORM}~\cite{Vermaseren:2000nd} routine~\cite{js:2011}. The reduction of
topologies with two different massive and one massless lines
requires a careful treatment. The related master integral
can be easily derived from its general expression valid for massive
lines, given in
Ref.~\cite{Davydychev:1992mt}.
The finite decoupling coefficients are obtained upon the
renormalization of the bare parameters. They are
given by
\begin{eqnarray}
\zeta_s = \frac{Z_s}{Z_s^\prime} \zeta_s^0
\,,\quad \zeta_m = \frac{Z_m}{Z_m^\prime} \zeta_m^0\,,
\label{eq::zetagren}
\end{eqnarray}
where $Z_s^\prime$ and $Z_m^\prime$ correspond to the renormalization
constants in the
effective theory, and $Z_s$ and $Z_m$ denote the same quantities in the
full theory. Since we are interested in the two-loop results for
$\zeta_i\,,\, i=s,m$, the corresponding renormalization constants for
SUSY-QCD and QCD have to be implemented with the same accuracy. Analytical
results for the latter up to the three-loop order can be found in {\it e.g.}
Refs.~\cite{ Jack:1994kd, Steinhauser:2002rq, Bednyakov:2002sf}.
Apart from the renormalization constants of the external fields, also
parameter renormalization is required. For
the renormalization of the gluino and
squark masses and the squark mixing angle we choose the on-shell
scheme. This scheme allows us to directly use
the physical parameters in the running analyses making the implementation
very simple. The explicit formulae at the one-loop order can be found in
Refs.~\cite{Pierce:1996zz,Harlander:2004tp}. The two-loop counterterms
are known analytically only for specific mass hierarchies~\cite{Kant:2010tf}
and numerically for arbitrary masses~\cite{Martin:2005qm}.\\
For the calculation of $\zeta_{\alpha_s}$ the simultaneous renormalization
of up- and down-type squarks is required. We follow the prescription proposed
in Ref.~\cite{Heinemeyer:2004xw} and fixed the counterterms for the up-squarks and the heavier
down-squark to be on-shell
and derive the counterterm for the lighter down-squark accordingly. In our
limit of neglecting the top-quark mass as compared to the SUSY mass scale it
holds
\begin{equation}
\delta m _{\tilde{d}_1}= \delta m _{\tilde{u}_1}^{\rm
os}\,,\quad\mbox{with}\quad u=u,s,t\quad\mbox{and}\quad d=d,c,b\,,
\end{equation}
where $\delta m _{\tilde{d}_1}$ stands for the mass counterterm of the
light down-squark and $\delta m _{\tilde{u}_1}^{\rm os}$ denotes the on-shell
mass
counterterm of the light up-squark.
For the computation of the decoupling coefficient of the top-quark
mass at order ${\cal O}(\alpha_s^2)$ one needs to renormalize in
addition the top-quark mass and
the $\epsilon$-scalar mass.
As the top-quark mass is neglected w.r.t. SUSY particle masses, an
explicit dependence of the radiative corrections on $m_t$ can occur only
through top-Yukawa coupling. In order to avoid the occurrence of
large logarithms of the form
$\alpha_s^2\log(\mu^2/m_t^2)$ with $\mu\simeq \tilde{M}$, where $\tilde{M}$
stands for the SUSY scale, one has to
renormalize
the top-Yukawa coupling in the \drbar{} scheme. In this way, the large
logarithms are absorbed into the running mass and the higher
order corrections are maintained small. As a consequence, the top-squark
mixing parameter $X_t=A_t-\mu_{\rm SUSY}/\tan\beta$ will be renormalized in a
mixed scheme. Its counterterm is derived from the relation
between the top-quark and squark masses and the mixing angle and mixing parameter.
Finally, the last parameter to be renormalized is the $\epsilon$-scalar
mass~\footnote{For a review on their role in multi-loop calculation
see~\cite{Mihaila:2013wma}.}. To obtain decoupling coefficients
independent of the unphysical parameter $m_\epsilon$, one has to
modify the top squark
masses by finite quantities~\cite{Jack:1994rk,Martin:2001vx}. We adopt in the
present calculation the method proposed in Ref.~\cite{Bednyakov:2007vm} to
choose the $\epsilon$-scalars massive and integrate them out together with the
SUSY particles. In this way we achieve conversion from \msbar{}
to \drbar{} schemes and decoupling of the heavy masses in a single step.
\section{\label{sec:res} Two-loop threshold corrections}
The two-loop results for the decoupling coefficient for the strong coupling is
very compact and we reproduce them below.
%\begin{eqnarray}
\begin{align}
\zeta_{\alpha_s}&=-\frac{\alpha_s}{4\pi}
\frac{1}{3}\bigg[C_A\left(1+2 L_{\tilde{g}}\right)
+T \sum_{\tilde{q}}\sum_{i=1,2} L_{\tilde{q}_i}\bigg]
+ \left(\frac{\alpha_s}{4\pi} \right)^2\bigg\{
C_A^2\left(-\frac{125}{18}-\frac{44}{9} L_{\tilde{g}}+\frac{4}{9}
L_{\tilde{g}}^2 \right)
\nonumber\displaybreak[1]\\
&
+ C_A T
\frac{2}{9}\bigg[30+\sum_{\tilde{q}}\sum_{i=1,2} \bigg(6\frac{
M^2_{\tilde{q}_i}}{ M^2_{\tilde{g}}}
+6 \frac{ M^2_{\tilde{g}}-M^2_{\tilde{q}_i}}{ M^2_{\tilde{g}}} B_{0,{\rm fin}}(M^2_{\tilde{g}},M_{\tilde{q}_i},0)+2 L_{\tilde{q}_i} L_{\tilde{g}}
\nonumber\displaybreak[1]\\
&
-\frac{2 M^4_{\tilde{g}}-5 M^2_{\tilde{g}}
M^2_{\tilde{q}_i}+6 M^4_{\tilde{q}_i}}{M^2_{\tilde{g}}(M^2_{\tilde{g}}-M^2_{\tilde{q}_i})}L_{\tilde{q}_i}
+3 \frac{M^2_{\tilde{g}}}{M^2_{\tilde{g}}-M^2_{\tilde{q}_i}}L_{\tilde{g}}\bigg)
\bigg]
+ T^2\left(\frac{1}{3} \sum_{\tilde{q}}\sum_{i=1,2} L_{\tilde{q}_i}\right)^2
\nonumber\displaybreak[1]\\
&
+ C_F T
\frac{2}{3}\bigg[+\sum_{\tilde{q}}\sum_{i=1,2}\bigg(1+\frac{M^2_{\tilde{g}}}{M^2_{\tilde{q}_i}}-\frac{M^2_{\tilde{g}}-M^2_{\tilde{q}_i}}{M^2_{\tilde{q}_i}}B_{0,{\rm
fin}}(M^2_{\tilde{q}_i},M_{\tilde{g}},0)-2\frac{3 M^2_{\tilde{g}}-2 M^2_{\tilde{q}_i}}{M^2_{\tilde{g}}-M^2_{\tilde{q}_i}} L_{\tilde{q}_i}
\nonumber\displaybreak[1]\\
&
+\left(4+ \frac{M^2_{\tilde{g}}}{M^2_{\tilde{q}_i}}+\frac{2 M^2_{\tilde{q}_i}}{M^2_{\tilde{g}}- M^2_{\tilde{q}_i}}\right)L_{\tilde{g}}
\bigg)
+ \sum_{\rm gen}\bigg(-3\frac{M^2_{\tilde{q}_{u1}}}{M^2_{\tilde{q}_{d1}}}
-\frac{M^2_{\tilde{g}}-M^2_{\tilde{q}_{u1}}}{M^2_{\tilde{q}_{d1}}} B_{0,{\rm
fin}}(M^2_{\tilde{q}_{u1}},M_{\tilde{g}},0)
\nonumber\\
&
+
\frac{M^2_{\tilde{g}}-M^2_{\tilde{q}_{d1}}}{M^2_{\tilde{q}_{d1}}} B_{0,{\rm
fin}}(M^2_{\tilde{q}_{d1}},M_{\tilde{g}},0)+
L_{\tilde{q}_{d1}}-\frac{M^2_{\tilde{q}_{u1}}}{M^2_{\tilde{q}_{d1}}}
L_{\tilde{q}_{u1}} +\frac{1}{2} \frac{M^2_{\tilde{q}_{d1}}}{M^2_{\tilde{g}}-
M^2_{\tilde{q}_i}} L_{\tilde{g}}
\bigg)
\bigg]
\bigg\}
%\end{eqnarray}
\end{align}
In the formula above the sum $\sum_q$ runs over all quark flavours and
$\sum_{\rm gen}$ over the number of generations. $C_A,C_F$ are the quadratic Casimir invariants for the
adjoint and fundamental representations, $T=1/2$ is the Dynkin
index and $L_x=\ln(\mu^2/m_x^2)$, with $x=\tilde{g},\tilde{q}$.
$B_{0,{\rm fin}}(p^2, m_1,m_2)$ denotes the fine part of the $B_0$-function~\cite{Denner:1991kt}. The asymmetry
w.r.t. up- and down-type quarks originate from the special renormalization
scheme of down-type squarks relative to the up-type squarks. Here, $\alpha_s=\alpha_s^{\rm{ (SQCD)}}$
denotes the strong coupling constant in the full theory.
\subsection{Limits}
The final results for two-loop threshold corrections for $\zeta_{m_t}$ are too lengthy to be
displayed here. They are available in
{\tt Mathematica} and {\tt Fortran}
format from http://www.ttp.kit.edu/Progdata/ttp14/ttp14-025. Instead, we present them
for two special mass hierarchies.
\subsubsection{Scenario A}
We consider first the case of all supersymmetric particles having masses
of the same order of magnitude and being much heavier than the top-quark.
\[
\begin{split}
&m_{\tilde{u}} = \ldots = m_{\tilde{b}} = m_{\tilde{t}} =
m_{\tilde{g}}=\tilde{M}\quad \gg\quad
m_t\nonumber\\
& \alpha_s^{(6)} = \zeta_{s}^{\tilde{M}}\, \alpha_s^{\rm{ (SQCD)}} \,,\qquad
m_t^{(6)} = \zeta_{m_t}^{\tilde{M}}\, m_t^{\rm{ (SQCD)}}\,.
\end{split}
\]
$\zeta_{s}^{\tilde{M}},\,\zeta_{m_t}^{\tilde{M}} $ are functions of the
supersymmetric mass $\tilde{M}$, the
soft SUSY breaking
parameters $X_t=A_t-\mu_{\rm SUSY}/\tan\beta$,
the strong coupling constant in the full theory $ \alpha_s^{\rm{ (SQCD)}}$, and the
decoupling scale $\mu$. The superscript $(6)$ indicates that the
parameters are defined in QCD with $n_f=6$ quarks.
%Hierarchie h3:
\begin{eqnarray}
\zeta_{m_t}^{\tilde{M}}&=&-\frac{\alpha_s}{4\pi} C_F \left(-1
+\frac{X_t}{\tilde{M}}+L_{\tilde{M}} \right)
\nonumber\\
&&
+\left(\frac{\alpha_s}{4\pi}
\right)^2\bigg\{
C_F^2\bigg[-\frac{71}{8}-\frac{13}{2} L_{\tilde{M}}+\frac{1}{2}
L_{\tilde{M}}^2
+\frac{X_t}{\tilde{M}}\left(-5+3 L_{\tilde{M}}\right)\bigg]
\nonumber\\
&& + C_F T
\bigg[\frac{109}{3} - 16 L_{\tilde{M}}+12 L_{\tilde{M}}^2 +12 \frac{X_t}{\tilde{M}}\left(-1+L_{\tilde{M}}\right)
\bigg]
\nonumber\\
&&
+ C_F C_A \bigg[-\frac{23}{72}-\frac{37}{6} L_{\tilde{M}}-\frac{1}{2}
L_{\tilde{M}}^2 - \frac{X_t}{\tilde{M}}\left(1+3 L_{\tilde{M}}\right)
\bigg]
\bigg\}
\end{eqnarray}
\subsubsection{Scenario B}
The second scenario we consider is the so called "split-SUSY'' one, with
squarks much heavier than all the other particles:
\[
\begin{split}
&m_{\tilde{u}} = \ldots = m_{\tilde{b}} = m_{\tilde{t}} =\tilde{M}\quad \gg\quad m_{\tilde{g}}\quad \gg\quad
m_t\nonumber\\
& \alpha_s^{(6)} = \zeta_{s}^{\tilde{q}}\, \alpha_s^{\rm{ (SQCD)}} \,,\qquad
m_t^{(6)} = \zeta_{m_t}^{\tilde{q}}\, m_t^{\rm{ (SQCD)}}\,.
\end{split}
\]
The result reads:
%Hierarchy h9:\\
\begin{align}
\zeta_{m_t}^{\tilde{q}} =& \frac{\alpha_s}{4\pi} C_F \bigg\{ \frac{1}{2} - L_{\tilde{M}} + \frac{1}{\tilde{M}^2}(M_{\tilde{g}}^2 - 2 M_{\tilde{g}} X_t) \nonumber\\
&+ \frac{1}{\tilde{M}^4}\bigg[ M_{\tilde{g}}^4(1 + L_{\tilde{M}}
- L_{\tilde{g}}) + 2 X_t M_{\tilde{g}}^3 (-1 + L_{\tilde{g}} - L_{\tilde{M}}) \bigg]\bigg\}\nonumber \displaybreak[1] \\
& + \left(\frac{\alpha_s}{4\pi}\right)^2 C_F\bigg\{-\frac{C_A}{72 \tilde{M}^4}\bigg[\tilde{M}^4\big(-481 + 432 L_{\tilde{M}} + 108 L_{\tilde{M}}^2 \nonumber \\
& + 120 L_{\tilde{g}} - 72 L_{\tilde{g}}^2 + 576 \zeta(2)\big) - 36 M_{\tilde{g}}^3\big(M_{\tilde{g}}(38 + 4 L_{\tilde{M}} \nonumber \\
& + 13 L_{\tilde{M}}^2 + 2 L_{\tilde{g}} - 20 L_{\tilde{M}}L_{\tilde{g}} + 7L_{\tilde{g}}^2 - 20 \zeta(2)) \nonumber \\
& - 4 X_t(5 + 19 L_{\tilde{M}} + 3 L_{\tilde{M}}^2 - 16L_{\tilde{g}} - 3 L_{\tilde{M}} L_{\tilde{g}} + 2 \zeta(2))\big) \nonumber \\
& + 72 \tilde{M}^2 M_{\tilde{g}}\big(2 X_t(7 + 6L_{\tilde{M}} - 3 L_{\tilde{g}} + 2 \zeta(2))\nonumber\\
& + M_{\tilde{g}}(-15 + 3 L_{\tilde{M}} - 6 L_{\tilde{g}} + 10 \zeta(2))\big)\bigg]\nonumber \displaybreak[1]\\
& +\frac{C_F}{8\tilde{M}^4}\bigg[ -8 \tilde{M}^2 M_{\tilde{g}}\big(M_{\tilde{g}}(21 + L_{\tilde{M}} - 20\zeta(2)) \nonumber \\
& + X_t(5- 6L_{\tilde{M}} - 8\zeta(2))\big) - 2 M_{\tilde{g}}^3 \big(M_{\tilde{g}}(175 + 60 L_{\tilde{M}}^2 \nonumber \\
& + 90 L_{\tilde{g}} + 56 L_{\tilde{g}}^2 - 2L_{\tilde{M}}(43 + 58 L_{\tilde{g}}) - 104 \zeta(2)) \nonumber\\
& + 4 X_t (15 - 6 L_{\tilde{M}}^2 + 11 L_{\tilde{g}} + L_{\tilde{M}}(-17 + 6L_{\tilde{g}}) - 8 \zeta(2))\big)\nonumber \\
& + \tilde{M}^4\big(-189 - 48 L_{\tilde{M}} + 4 L_{\tilde{M}}^2 + 120 \zeta(2)\big)\bigg] \nonumber\displaybreak[1] \\
& + \frac{T}{3 \tilde{M}^4}\bigg[-36 \tilde{M}^2 M_{\tilde{g}} \big((-1 + L_{\tilde{M}})M_{\tilde{g}} + X_t - 2 L_{\tilde{M}} X_t\big) \nonumber \\
& + 3 M_{\tilde{g}}^3\big((-13 - 12 L_{\tilde{M}}^2 + 6L_{\tilde{g}} + 6L_{\tilde{M}}(-3 + 2 L_{\tilde{g}}))M_{\tilde{g}} \nonumber \\
& + 4 (7 + 6 L_{\tilde{M}}^2 + L_{\tilde{M}}(9 - 6L_{\tilde{g}}) - 3 L_{\tilde{g}})X_t\big) \nonumber \\
& + \tilde{M}^4\big(127 - 30 L_{\tilde{M}} + 36 L_{\tilde{M}}^2 - 36 \zeta(2)\big)\bigg]\bigg\}
\end{align}
We have checked the formulae above against the exact calculation both
analytically and numerically. First, we have computed the
two-loop decoupling coefficient $\zeta_{m_t}$ for scenarios A and B making use
of asymptotic expansion method (explicitly Large Mass expansion) that is
available
in the code {\tt exp} and compared with the expansion of the exact
result. For scenario A we obtain agreement for the dominant term ({\it i.e.}
neglecting corrections proportional with the mass differences between the SUSY
particles.) For scenario B, we verified the agreement for the first three
terms of the expansion in mass ratio $M_{\tilde g}^2/\tilde{M}^2$. Also, the
direct numerical comparison of the exact and asymptotically expanded
results gives very good agreement.
%\section{Results}
\subsection{\label{sec:numerics} Numerical results}
In this section we present the phenomenological effects of the two-loop SUSY-QCD
threshold corrections on the prediction of the running top-quark mass at the
SUSY mass scale. We also present the comparison with the direct prediction
obtained from the ratio between the running and the pole mass within SUSY-QCD as
described in the code {\tt TSIL}~\cite{Martin:2005qm}.\\
Our method can be summarised in the following sequence:
\begin{equation}
\begin{split}
M_t^{\rm OS}\,\stackrel{(i)}{\to}\,
m_t^{\msbarmath}(M_t)\,\stackrel{(ii)}{\to}\,
m_t^{\msbarmath}(\mu_{\rm dec})\,\stackrel{(iii)}{\to}\,
m_t^{\rm{ SQCD},\overline{\rm DR}}(\mu_{\rm dec})\,\stackrel{(iv)}{\to}\,
m_t^{\rm{ SQCD},\overline{\rm DR}}(\mu)\,,
\label{eq::mbrun}
\end{split}
\end{equation}
where $M_t^{\rm OS}$ denotes the top-quark pole mass and $m_t^{\msbarmath}$
and $m_t^{\rm{ SQCD},\overline{\rm DR}}$ stand for the running top-quark mass in the
SM and SUSY-QCD in the \msbar{} and \drbar{} schemes, respectively. $\mu_{\rm
dec}$ is the scale at which the decoupling is performed
and it is usually chosen comparable with SUSY masses.
If not stated otherwise, we fix it to be the arithmetic average over the squarks and gluino on-shell masses:
\begin{equation}
\mu_{\rm{dec}}=\frac{1}{13}\big[M_{\tilde{g}}+\sum_{\tilde{q}}M_{\tilde{q}}\big]\,.
\end{equation}
But of course, $\mu_{\rm{dec}}$ can be chosen arbitrarily and the dependence
of the running top-quark on it is a measure of the theoretical
uncertainties (for details, see Fig.~\ref{fig:mudec} and its discussion).
Also in the numerical setup, we implemented it as a free
parameter that can be varied. In the step $(i)$ the relation between the top-quark pole
and running masses in the SM is required. We implemented the
three-loop results~\cite{Chetyrkin:1999ys,Chetyrkin:1999qi,Melnikov:2000qh,Marquard:2007uj}. The
RGEs for the SM and the MSSM that are necessary
in the steps $(ii)$ and $(iv)$ are known to
three-loop
accuracy~\cite{Chetyrkin:2012rz,Bednyakov:2012en,Ferreira:1996ug,Harlander:2009mn}
as well. Let us mention, that in QCD the quark anomalous dimension was recently computed even at
the five loop order~\cite{Baikov:2014qja}. For consistency, the threshold
corrections evaluated in the step $(iii)$ are necessary at two loops.
For the explicit numerical evaluation we use for the SM parameters their
values cited in Ref.~\cite{pdg}. For the MSSM parameters we employ two
scenarios that we call "heavy Higgs'' and "heavy sfermions'', respectively. We
obtain the numerical values with the help of the spectrum generator {\tt
SOFTSUSY}~\cite{Allanach:2001kg}. The input parameters for the spectrum
generator are as follows:
i) In the "heavy sfermions'' scenario we define all \drbar{} breaking parameters at the input scale following the Supersymmetry Les Houches Accord (SLHA)~\cite{Skands:2003cj,Allanach:2008qq} \verb|EXTPAR 0| $Q_{\rm{in}}=\tilde{m}_t$.
Where $\tilde{m}_t$ is an alias for the right handed top squark mass breaking parameter \verb|EXTPAR 46| and kept as free parameter.
Further we identify the third generation doublet mass breaking $\tilde{m}_{Q_3}$ alias \verb|EXTPAR 43| with $\tilde{m}_t$.
All other sfermion mass breaking parameters have a common value $\tilde{m}_f=\tilde{m}_t+1\rm{TeV}$.
The trilinear couplings (\verb|EXTPAR 11-13|) are given by $A_{t}=20\rm{GeV}$ and $A_{\tau}=A_b=4\rm{TeV}$.
The gaugino mass parameters (\verb|EXTPAR 1-3|) are set to $M_1=M_2=M_3=1.5 \rm{TeV}$.
For the bilinear coupling of $H_u$ and $H_d$ (\verb|EXTPAR 23|) we chose $\mu_{\rm{SUSY}}=200\rm{GeV}$.
The mass of the pseudo scalar Higgs boson (\verb|EXTPAR 23|) is set to $M_A=1\rm{TeV}$.
Finally, the ratio of the two vacuum expectation values (\verb|MINPAR 3|) is set to $\tan\beta=10$.
The given parameters choice results in very weakly mixing top squarks,
which are about $1\rm{TeV}$ lighter than the other sfermions and thus have largest impact in the decoupling process.
By increasing the value of $\tilde{m}_t$ one automatically pushes the squark mass spectrum to higher values.
However, in order to successfully describe the currently measured mass for the lightest Higgs boson,
one is forced to use multi TeV range values for $\tilde{m}_t$, because of the
very weak mixing between the top squarks which have nearly equal masses.
ii) In the "heavy Higgs'' scenario we define all \drbar{} breaking parameters at the input scale $Q_{\rm{in}}=\sqrt{M_{\rm{SUSY}}^2+M_t^{\rm{OS}\,2}}$.
Here $M_{\rm{SUSY}}=1\rm{TeV}$ is a common breaking mass parameter for all sfermions except $\tilde{m}_t$ (\verb|EXTPAR 46|), which we keep as free parameter.
The remaining input parameters are given by:
\begin{align}
A_{\tau}&=A_{b}=2469.49{\rm{GeV}}, & A_t&=1.5{\rm{TeV}}, &\mu_{\rm{SUSY}}&=200\rm{GeV},\nonumber\\
M_1&= 5 s^2_w/(3c_w^2)M_2,& M_2 &=200\rm{GeV},& M_3 &=800\rm{GeV},\nonumber\\
M_A&=1{\rm{TeV}}, & \tan\beta&=20\,. & &
\end{align}
Here $s_w$ and $c_w$ are the sine and cosine of the Weinberg mixing angle $\theta_W$.
In contrast to the first scenario we do have light Higgs masses for sub TeV values of $\tilde{m}_t$ due to the stop mixing.
Moreover one can have a very light stop of order $300{\rm{GeV}}$ for $\tilde{m}_t$ values having nearly same size.
Please note that for the pure SUSY-QCD analysis done in this paper,
the effect of changing breaking parameters of particles transforming as QCD
singlets is very weak with respect to SUSY-QCD decoupling effects. We provide
the SLHA input files for the two scenarios considered here in electronic
format on the web page http://www.ttp.kit.edu/Progdata/ttp14/ttp14-025.
\begin{figure}[t]
\begin{center}
\epsfig{figure=\figurepath spectrum_ModMhMax_MtDRbarOfmuDec_1.eps,width=.9\textwidth}
\caption[]{\label{fig:mudec} Decoupling scale dependence of
the running top quark mass for the ``heavy Higgs'' scenario. The
curves show the results obtained within the decoupling method at
one (dotted line), two (dashed line) and three (full line) loops. The
renormalization scale was fixed to $\mu_{\rm ren}=\sqrt{M_{\tilde{t}_1}M_{\tilde{t}_2}}$.
}
\end{center}
\end{figure}
In the following, we denote as leading order (LO) value for the running
top-quark mass in the decoupling method
(DEC), the value obtained with one-loop RGEs and without threshold
corrections. The next-to-leading (NLO) and the next-to-next-to-leading order
(NNLO) values in the decoupling method are calculated employing two- and
three-loop RGEs and one- and two-loop threshold corrections, respectively. The
NLO top-quark mass computed in the direct method ( with the code {\tt TSIL}) takes
into account the one-loop relation between the running- and pole-quark mass,
whereas the NNLO prediction is based on the two-loop relation.
In a first step, we study the dependence of the running top-quark mass on the
unphysical parameter $\mu_{\rm{dec}}$. As this parameter is not fixed by the
theory, the dependence of the physical quantities on it provides us a measure
of the accuracy of the method itself. We display in Fig.~\ref{fig:mudec} such
a dependence for the ``heavy Higgs'' scenario. It is particularly important
to perform the study in this scenario as one of the supersymmetric particles (
the light top-squark) has the mass at an intermediate scale of about
$300$~GeV, whereas the rest of the particles have masses around $1$~TeV. The natural question
to be addressed in this case is whether the one-step decoupling approach, where
all supersymmetric particles are integrated out at once, is still appropriate or a
multi-step procedure is required. In the Fig.~\ref{fig:mudec} the dotted,
dashed and full lines depict the running top quark mass evaluated at scale
$\mu_{\rm ren}=\sqrt{M_{\tilde{t}_1}M_{\tilde{t}_2}}$ at LO, NLO and NNLO and the decoupling scale is
varied in the range from $M_{\rm top}$ to $2 M_{\rm SUSY}$. As expected from
theoretical consideration and clearly illustrated in the
plot, a substantial improvement of the stability of the predictions
w.r.t. the variation of $\mu_{\rm{dec}}$ is obtained when going from one-
to three-loop accuracy. While at the two-loop level a variation of the
top-quark mass of about $2$~GeV is still present, at three loops the
variation amounts to less than $100$~MeV. Since the expected experimental
accuracy for the top-quark mass measurement even at the future ILC does not go below
$100$~MeV, we can conclude that the method proposed here is well suited also
for scenarios with lighter supersymmetric particles with masses around $300$~GeV.
Furthermore, in Fig.~\ref{fig:mtrenhsf} we present the running top-quark mass in the full
theory (in the ``heavy sfermion'' scenario) as a function of the scale $\mu_{\rm ren}$ at which it is evaluated.
The black (middle) curves display the results obtained with the method
proposed in this paper (that we refer at as the decoupling method
and is denoted as ``DEC'' in the legend of the
plot)
at one (dotted line) two (dashed line) and three
(full line) loops. The blue curves show the predictions obtained directly via the
ratio between the top-quark pole and the running masses in the SUSY-QCD,
using the code {\tt TSIL}. The dashed line corresponds to the
one-loop results and the full line displays the two-loop contributions. As
can be seen from the figure, the radiative corrections for the decoupling method
are very small (tenth of MeV between one- and two-loop order contributions and
negligible at the three loops) as
compared with the current experimental uncertainty on the
top quark pole mass of about $1$~GeV. The perturbative series is very
well converging and the contributions from the unknown higher order
corrections are negligible for all renormalization scales. The
radiative corrections obtained via the direct method are much larger than
the experimental uncertainty. Even at the two-loop order, they amount to
$10$~GeV at the electro-weak scale and increase further at renormalization
scales comparable with the squark masses. In this case higher order
contributions are necessary to bring the theoretical precision at the same
level as the experimental one. One observes also that the predictions obtained
in the two methods at the two-loop order agree very well for small renormalization scales below
$400$~GeV. This can be explained by the fact that in this case the
logarithmic contributions (of the form $\ln(M_{\rm top}/\mu)$) are small and
the resummation gives only minor
corrections. When the running top-quark mass is evaluated at high energy
scales the resummation of the large logarithms becomes important and the
difference between the two predictions can reach about $10$~GeV. Let us also
point out that in the domain where the resummation is expected to bring only small
effects the differences between the two methods decrease considerably when
going from one to two loops as is expected in perturbation theory.
In Fig.~\ref{fig:mtmrhsf} the running top-quark mass is shown as a function of
the squark mass breaking parameter $\tilde{m}_{t}$, that can be interpreted
in the ``heavy sfermion'' scenario as a scale for the SUSY mass spectrum. In this case the renormalization scale
is chosen as geometric mean value of the top-squark masses. One notices that
the radiative corrections calculated with the decoupling approach are very small for all
SUSY scales. The direct computation deliver even at the two-loop order
radiative corrections of about ${\cal O}(10)$~GeV. As expected, the two
methods provide results in very good agreement at the two-loop order for low
SUSY mass scales, where no large logarithmic corrections are present.
%Plots description... Comparison with TSIL and convergence of the series.....\\
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath spectrum_HeavySfermions_MtDRbarOfmuRen_1.eps,width=.91\textwidth}
\caption[]{\label{fig:mtrenhsf} Renormalization scale dependence of
the running top quark mass for the ``heavy sfermion'' scenario. The
black curves show the results obtained within the decoupling method at
one (dotted line), two (dashed line) and three (full line) loops. The
blue lines display the results obtained with the code {\tt TSIL} at
one (dashed line) and two (full line) loops.
}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath spectrum_HeavySfermions_MtDRbarOfMtildet_1.eps,width=.91\textwidth}
\caption[]{\label{fig:mtmrhsf} Dependence of
the running top-quark mass on the SUSY scale $\tilde{m}_{t}$
for the ``heavy sfermion'' scenario. The convention for the curves is
the same as in Fig.~\ref{fig:mtrenhsf}.
}
\end{center}
\end{figure}
Fig.~\ref{fig:mthh} shows the running top-quark mass as a function of
the renormalization scale in the "heavy Higgs'' scenario. For the chosen
input parameters, the predictions of the two methods differ at the one-loop
level by more than $10$~GeV. At the two-loop level the predictions of the two
approaches agree well within the present experimental uncertainty on the
top-quark pole mass for low-energy scales, whereas for large renormalization scales the difference
amounts to few GeV. Let us also point out that the differences between one- and
two-loop contributions within the decoupling method amounts to about $4$~GeV,
whereas the genuine three-loop contributions are very small, below
$100$~MeV. This observation proves the good convergence of the perturbative
methods in the decoupling approach. The direct calculation based on the code
{\tt TSIL} provides similarly
large radiative corrections at the two-loop level. However, in order to
reduce the theoretical uncertainty at a similar level with the experimental
one, we need higher order radiative corrections that are
currently not available.
In summary, we conclude that the two methods provide results in good agreement
for low SUSY mass scales or renormalization scales, but they differ
significantly when the SUSY particles become heavy, in the multi TeV
range. Also, the rapid convergence of the perturbative series for the decoupling
method allows us to reduce the genuine theoretical uncertainties due to unknown
higher order corrections well below the present experimental uncertainty on the
top-quark pole mass. The discrepancies between the predictions obtained within
the two methods can have important phenomenological implications, depending on
the process and observables under consideration.
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath spectrum_ModMhMax_MtDRbarOfmuRen_1.eps,width=.91\textwidth}
\caption[]{\label{fig:mthh} Dependence of the running top quark mass
on the renormalization scale within the ``heavy Higgs'' scenario. The convention for the curves is
the same as in Fig.~\ref{fig:mtrenhsf}.
}
\end{center}
\end{figure}
\section{\label{sect:mhphen} The mass of the lightest Higgs boson}
The Higgs boson mass measurement by ATLAS $125.5\pm 0.2\pm
0.6$~GeV~\cite{ATLAS:2013mma} and
CMS $125.7\pm 0.3\pm 0.3 $~GeV~\cite{CMS:yva} already reached an
amazing precision. In the MSSM, the
lightest Higgs boson mass is predicted.
Beyond the tree-level approximation, it is a function of the top-squark
masses and mixing parameters. It grows logarithmically with the
top-squark masses and can be used to determine an upper bound for the
supersymmetric (SUSY) mass scale from the measured Higgs boson mass, once the mixing
parameters are fixed. This approach has received considerable attention
recently~\cite{Feng:2013tvd,Hahn:2013ria,Draper:2013oza}, partially because
the direct searches for SUSY particles
at the LHC remained unsuccessful, indicating a possible lower bound for their mass
scale in the TeV range.\\
Since the dependence of the Higgs boson mass on the SUSY masses is
logarithmic, high-precision measurements and theoretical predictions are
required. The genuine theoretical
uncertainties due to unknown higher order corrections are expected to
grow with the SUSY mass scale. For top-squark masses in the multi TeV
range they were estimated to be of about
$5-7$~GeV~\cite{Hahn:2013ria,Draper:2013oza}.
By now, the complete one-loop~\cite{Ellis:1990nz,Brignole:1992uf} and
dominant two-loop~\cite{Heinemeyer:1998np,Degrassi:2001yf}
corrections are implemented
in the numerical programs FeynHiggs~\cite{Heinemeyer:1998yj} and
CPsuperH~\cite{Lee:2003nta} using on-shell
particle masses, and in SOFTSUSY~\cite{Allanach:2001kg} and SPheno~\cite{Porod:2003um} using
running parameters. The
dominant SUSY-QCD three-loop corrections are taken into account in the code
{\tt H3m}~\cite{Kant:2010tf}, for which a mixed renormalization scheme was
employed. However, the three-loop contributions are known only for specific
mass hierarchies between the SUSY particles.
The dominant contributions to the leading (LL) and
next-to-leading (NLL)
logarithmic terms in the ratio between the top quark mass and
the typical scale of SUSY particle masses $\ln(M_{\rm
top}/M_{\rm SUSY})$ have been obtained in
Ref\cite{Martin:2007pg}. Very recently, the generalization of the LL and NLL
approximation has been derived~\cite{Hahn:2013ria} and implemented
in the code {\tt FeynHiggs}, up
to the seventh loop-order. Furthermore, in
Ref.~\cite{Draper:2013oza} the recent calculations of the
three-loop beta-functions for the SM
coupling
and the two-loop corrections to the Higgs boson mass in
the SM
have been used to derive (presumably) the dominant NNLL corrections at the four-loop order.
In this section we focus on the numerical effects that the new prescription
for the determination of the running top-quark mass and top-Yukawa coupling
has on the predictions of the lightest Higgs boson mass taking into account
SUSY-QCD radiative corrections. We implemented the
resummation of the large logarithms of the form $\ln(M_{\rm top}/M_{\rm SUSY})$ contained in
the running of the top-Yukawa coupling (as discussed in the previous
section) on top of the three-loop SUSY-QCD corrections to the lightest
Higgs boson mass encoded in the
program~{\tt H3m}. As will be shown, this type of resummation is necessary for SUSY
masses in the multi TeV range and enable us to reduce the effects
of the unknown higher order contributions. In the following, we evaluate the
running top-quark mass and couplings with the highest accuracy both for the
decoupling and the direct methods and use them further for the calculation of
the lightest Higgs boson mass at one-, two- and three-loop accuracy.
In order to allow a convenient evaluation of general MSSM scenarios
including the stated low scale ones,
the {\tt Mathematica} package {\tt SLAM}~\cite{Marquard:2013ita} has been implemented in {\tt H3m}.
It provides an easy to use interface for calling and reading SUSY
spectrum generator output full automatically using the SLHA.
Moreover it enables the ability to save and recall SUSY spectra in and from a data base.
Besides reading in user provided SLHA spectrum generator output files,
it is now possible to enter the SLHA input file for the spectrum generators
defining the SUSY scenario directly in {\tt Mathematica}
relaxing the restriction to predefined scenarios in earlier versions of {\tt
H3m}.\\
The MSSM parameters derived in this way are stored and further used to compute the running
top-quark mass, the running top-Yukawa coupling and the strong coupling
constant in the MSSM using the decoupling method described in
section~\ref{sec:frm}. This step is realized through stand-alone routines as
is also explained in the flowchart plot in Fig.~\ref{fig:flow}. Afterwards, the
stored values for the input MSSM parameters together with the \drbar{}
couplings just derived are delivered to the code {\tt H3m}. The computation
of the lightest Higgs boson mass follows then the steps described in
Ref.~\cite{Kant:2010tf}. The user has also the possibility to choose the way
the running top-quark mass is computed.
The command\\ {\tt SetOptions[H3GetSLHA, calcmt->\{``MtTSIL''\}]} allows to use
at this stage the code {\tt TSIL}, whereas the decoupling method is
implemented as default option. A
direct comparison between the predictions obtained with the two methods will
be presented below.\\
The new version of the code {\tt H3m} together with few simple example
programs are available from
{\tt https://www.ttp.kit.edu/Progdata/ttp14/ttp14-025/}. Apart from the
implementation of the interface program {\tt SLAM} and the routine for the
computation of the running top-quark mass and the strong coupling constant
through the decoupling method, we improved on the determination of the
mixing angle and reduced oscillations in the Higgs mass by setting the W-boson mass
fixed to is on-shell value as given by the PDG~\cite{pdg}.
%\tikzstyle{int}=[draw, minimum height=0.5cm, minimum width=4cm]
%\tikzstyle{init} = [pin edge={to-,thin,black}]
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath flowchart.eps,width=.91\textwidth}
%\vspace*{-30em}
\caption[]{\label{fig:flow} Flowchart of the new version of {\tt H3m}. First, the user calls
the code {\tt SLAM} to set the MSSM parameters. A subsequent call of {\tt H3m}
computes $M_h$.}
\end{center}
\end{figure}
In Fig.~\ref{fig:mhrenhsf} it is shown the dependence of the lightest Higgs
boson mass $M_h$ in the ``heavy sfermion'' scenario on the renormalization
scale, taking into account beyond
one-loop only SUSY-QCD radiative corrections. The black curves display the
one-(dotted line), two-(dashed line) and
three-loop (full line) contributions obtained with the running
top-quark mass in the decoupling method. The blue curves present the
same predictions using the code {\tt TSIL} for the
calculation of the running top-quark mass. The
explicit value of the running-top quark mass can be read from
Fig.~\ref{fig:mtrenhsf}. It is
known that the renormalization scale dependence of an observable gives
an estimation for the magnitude of unknown higher order corrections.
This enables us to use it for the determination of the theoretical uncertainty.
As expected the renormalization scale dependence is reduced when going
from one- to two- and to three-loop order corrections in both
schemes. However, the direct determination of the running top-quark mass
(blue curves) is affected by large logarithmic corrections that in turn
induces large radiative corrections to the Higgs boson mass. Even at
the three-loop order, the scale variation of $M_h$
amounts to about $5$~GeV, more than an order of magnitude larger than
the current
experimental accuracy on $M_h$ and few times bigger than the parametric
uncertainties. In contrast, the resummation of the logarithmic
corrections to the running top-Yukawa coupling through the decoupling
method renders the scale dependence of $M_h$ at three-loop order very
mild about tens of MeV. One can also observe, that low values for the
renormalization scale around the top-quark pole mass are not well
suited for the present scenario, especially when the three-loop order
contributions are not taken into account. In this case, radiative
corrections even beyond the three-loop order are required in order to
cope with the experimental precision. The difference between the
predictions for $M_h$ obtained with the two methods is sizeable and can
amount to few GeV for large values of the renormalization
scale.
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath spectrum_HeavySfermions_MhOfmuRen_SUSYQCD_1.eps,width=.9\textwidth}
\caption[]{\label{fig:mhrenhsf} Dependence of the Higgs boson mass
on the renormalization scale within the ``heavy sfermion'' scenario.
The black curves display the one-(dotted line), two-(dashed line) and
three-loop (full line)
contributions to the lightest Higgs boson mass, when the running
top-quark mass is determined using the decoupling method. For all
three curves the running top-quark mass is evaluated with three-loop
accuracy. The blue curves present the same predictions using for the
calculation of the running top-quark mass the code {\tt TSIL}. Beyond
one-loop, only the SUSY-QCD contributions are taken into consideration.
}
\end{center}
\end{figure}
The dependence of $M_h$ on the SUSY breaking parameter $\tilde{m}_{t}$ is
shown in Fig.~\ref{fig:mhmrhsf}. As described in section~\ref{sec:numerics},
$\tilde{m}_{t}$ can be interpreted as an estimation of the SUSY mass
parameter. The convention for the line style is the same as in the previous
figure. The renormalization scale is fixed as the geometric mean value of the top-squark
masses, thus in the TeV range. The radiative
corrections to the Higgs boson mass increase with the SUSY mass scale as
expected. The predictions obtained using the two
methods for the derivation of the top-Yukawa coupling are in good agreement
for low SUSY scales of about $500$~GeV, but they differ significantly
for heavy SUSY spectrum in the multi TeV range. The Higgs boson mass predicted
through the decoupling method is always heavier and has a steeper dependence
on the SUSY spectrum as compared to the one obtained through
the direct method. This difference can be explained by the effects of
resumming large logarithms within the first approach. Since we use the RGEs at
the three-loop order the next-to-next-to-leading-logarithms are resummed. A
similar behaviour of predictions
for $M_h$ based on resummation was observed in the previous
works~\cite{Hahn:2013ria,Draper:2013oza}. Let us point out the big impact of
the resummation procedure for constraining the SUSY parameter space. While the
prediction of $M_h$ through the decoupling method allows SUSY mass scales of
about $4$~TeV, the present scenario is already excluded when the direct method
is employed. This observation explains also the need for very precise theoretical
predictions for the Higgs boson mass.
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath spectrum_HeavySfermions_MhOfMtildet_SUSYQCD_1.eps,width=.91\textwidth}
\caption[]{\label{fig:mhmrhsf} Dependence of
the lightest Higgs boson mass on the SUSY scale $\tilde{m}_{t}$
for the ``heavy sfermion'' scenario. The same convention for the line
style is used as for the Fig.~\ref{fig:mhrenhsf}.
}
\end{center}
\end{figure}
In Fig.~\ref{fig:mhrenhh} the dependence of $M_h$ on the renormalization scale
is shown for the ``heavy Higgs'' scenario. As can be read from the figure,
the scale dependence is reduced when higher order radiative corrections are
taken into account. However, even at the three-loop order the variation of
$M_h$ with the renormalization scale can amount to few GeV for both
methods. It is also important to notice that low values of the
renormalization scale are characterized by large radiative contributions. A
better alternative is the choice of the renormalization scales above $500$~GeV. In the
decoupling method this choice will reduce the scale variation to about
$200$~MeV. As for the previous scenario, we observe a milder scale variation
when the decoupling method is employed, that can be associated with smaller
higher order corrections. The difference between the predictions obtained in
the two frameworks amounts to few GeV for renormalization scales in the TeV
range. The two methods provide same values for $M_h$ for renormalization scale
of about $400$~MeV, for which also the predictions for the running top-quark mass coincide.
\begin{figure}
\begin{center}
\epsfig{figure=\figurepath spectrum_ModMhMax_MhOfmuRen_SUSYQCD_1.eps,width=.91\textwidth}
\caption[]{\label{fig:mhrenhh} Dependence of the lightest Higgs boson mass
on the renormalization scale within the ``heavy Higgs'' scenario.
The same convention for the line style is used as in Fig.~\ref{fig:mhrenhsf}.
}
\end{center}
\end{figure}
\section{\label{sec:concl} Conclusions}
In this paper, we consider the calculation of ${\cal
O}(\alpha_s^2)$ radiative corrections to the running top-quark mass and
Yukawa coupling with the MSSM. Our method is based on the ``running and
decoupling'' technique, that has the advantage to resumm the large logarithms
by the use of RGEs. Our numerical analysis performed in
section~\ref{sec:numerics} showed that the method is very stable upon higher
order radiative corrections. The remaining theoretical uncertainty is estimated
by half of the magnitude of genuine three-loop order contributions and amounts
to about $100$~MeV. The method proposed here provides results for the running
top-quark mass in good agreement with the
predictions of the code {\tt TSIL} for light SUSY spectra and for low-energy
scales at which the top-quark mass is computed. For heavy SUSY particles and
for high renormalization scales the differences between the two methods can
easily reach few GeV. However, the relation between the running and the on-shell
top-quark mass in the MSSM, on which the code {\tt TSIL} rely, is affected by
large radiative corrections in the range of GeV. One also observes that the
two methods agree better when
going from LO to NLO and NNLO.\\
The exact analytical results of our method are available in electronic
format. In addition, we provide in the paper the results for two specific
mass hierarchies.
The second part of the paper presents the
effects that the new determination of the running top-Yukawa
coupling has on the prediction of the lightest Higgs boson mass in the SUSY-QCD.
We observe a much milder dependence on the renormalization scale of the Higgs
boson mass predicted with three-loop accuracy. The renormalization scale
dependence is usually interpreted as a measure for the missing higher
order corrections. Thus, this improvement is very welcomed given the present
difficulty to achieve radiative corrections to the Higgs boson mass beyond
the two-loop level. We also notice that the predictions obtained through the
decoupling method are higher than the one derived in the direct
method. The difference can amount to several GeV for SUSY masses in the TeV
range. This behaviour can be explained by the effect of resumming the large
logarithms through the use of RGEs in the determination of the running
top-Yukawa coupling.\\
Furthermore, we implemented the decoupling method described above together
with the code {\tt SLAM}, that provides an
interface to spectrum generators, in the existing code {\tt H3m}. In this
way, the code {\tt H3m} computes the
three-loop SUSY-QCD corrections to the Higgs boson mass, taking into account
the resummation of the large logarithms of the form $\ln(M_{\rm top}/M_{\rm SUSY})$.
Finally, we want to stress that for the final prediction of the lightest Higgs
boson mass within the MSSM, in addition to the SUSY-QCD corrections discussed
in this paper also higher order corrections induced by the top- and
bottom-Yukawa couplings have to be considered. A similar analysis for the
Yukawa sector is not yet feasible because the two-loop contributions to the
decoupling coefficient of the Yukawa couplings induced by mixed QCD and top- or
bottom-Yukawa corrections are not known in the literature. This analysis is
beyond the scope of the present paper and we postpone it to a future project.
However, it is important to emphasize that the contributions that we do not
consider here (including
two-loop top- and bottom-Yukawa and electro-weak corrections) can
range from $0.5$~GeV to $2.5$~GeV without resummation effects, depending
on the parameters of the two
scenarios analysed. These numbers have to be
compared with the sum of the one- and two- and three-loop SUSY-QCD
contributions that can reach from $20$~GeV up to $35$~GeV when no
resummation effects are included. Thus,
the effects of resummation discussed above that account only for the
SUSY-QCD sector,
although not complete, provide an essential contribution to the prediction of
the lightest Higgs boson mass for supersymmetric models with heavy particles.
%- {{{ Acknowledgements:
\bigskip
\noindent
{\large\bf Acknowledgments}\\
We are grateful to Matthias Steinhauser for continuous support and for
enlightening conversations. We thank Philipp Kant for providing
us assistance with the code {\tt H3m} and for many useful discussions.
This work was supported by the DFG through SFB/TR~9 ``Computational Particle
Physics''.
%- }}}
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\end{document}