%Title: Higgs boson mass in supersymmetry to three loops
%Author: R.V. Harlander, P. Kant, L. Mihaila, M. Steinhauser
%Published: Phys.Rev.Lett. 100 (2008) 191602.
%/
%arxiv/0803.0672
%- {{{ Header:
\documentclass[twocolumn,amssymb,amsmath,nobibnotes,
preprintnumbers,showpacs,prl]{revtex4}
% prl -> no section numbers !!??
\usepackage{epsfig,scalefnt}
%\usepackage{a4wide,epsfig,amsmath,amssymb,cite,scalefnt}
%\usepackage{psfrag}
%\draft
%\twocolumn
\renewcommand{\textfraction}{0}
\renewcommand{\topfraction}{1}
\renewcommand{\bottomfraction}{1}
\newcommand{\dd}{{\rm d}}
\newcommand{\order}[1]{{\cal O}\!\left(#1\right)}
\newcommand{\eqn}[1]{Eq.\,(\ref{#1})}
\newcommand{\eqs}[1]{Eqs.\,(\ref{#1})}
\newcommand{\api}{\frac{\alpha_s}{\pi}}
\newcommand{\lM}{L_{\tilde M}}
\newcommand{\lt}{}
\newcommand{\lgl}{}
\newcommand{\ltM}{L_{t\tilde M}}
\newcommand{\grtsim}{\mbox{\raisebox{-3pt}{$\stackrel{>}{\sim}$}}}
\newcommand{\lessim}{\mbox{\raisebox{-3pt}{$\stackrel{<}{\sim}$}}}
\newcommand{\mudec}{\mu_{\rm dec}}
\newcommand{\mugut}{\mu_{\abbrev\rm GUT}}
\newcommand{\asfull}{\alpha_s^{\rm (full)}}
\newcommand{\asA}{\alpha_s^{(\tilde{t},\tilde{g},6)}}
\newcommand{\asB}{\alpha_s^{(\tilde{g},6)}}
\newcommand{\asC}{\alpha_s^{(6)}}
\newcommand{\abbrev}{\scalefont{.9}}
\newcommand{\drbar}{$\overline{\mbox{\abbrev DR}}$}
\newcommand{\msbar}{$\overline{\mbox{\abbrev MS}}$}
\newcommand{\drbarmath}{\overline{\rm\abbrev DR}}
\newcommand{\msbarmath}{\overline{\rm\abbrev MS}}
\newcommand{\mbDR}[1]{m_b^{\drbarmath{}#1}}
\newcommand{\mbMS}[1]{m_b^{\msbarmath{}#1}}
\newcommand{\asDR}[1]{\alpha_s^{\drbarmath{}#1}}
\newcommand{\asMS}[1]{\alpha_s^{\msbarmath{}#1}}
\newcommand{\betaDRbar}{\beta^{\drbar{}}}
\newcommand{\betaMSbar}{\beta^{\msbar{}}}
\newcommand{\gammaDRbar}{\gamma^{\overline{\rm DR}}}
\newcommand{\gammaMSbar}{\gamma^{\overline{\rm MS}}}
\newcommand{\ZsDRbar}{Z_s^{\overline{\rm DR}}}
\newcommand{\ZsMSbar}{Z_s^{\overline{\rm MS}}}
\newcommand{\ZmDRbar}{Z_m^{\overline{\rm DR}}}
\newcommand{\ZmMSbar}{Z_m^{\overline{\rm MS}}}
\newcommand{\apiDR}{\frac{\asDRbar}{\pi}}
\newcommand{\apiMS}{\frac{\asMSbar}{\pi}}
\newcommand{\aepi}{\frac{\alpha_e}{\pi}}
\newcommand{\lhc}{{\abbrev LHC}}
\newcommand{\qcd}{{\abbrev QCD}}
\newcommand{\mssm}{{\abbrev MSSM}}
\newcommand{\susy}{{\abbrev SUSY}}
\newcommand{\dreg}{{\abbrev DREG}}
\newcommand{\dred}{{\abbrev DRED}}
\newcommand{\mDRbar}{m^{\overline{\rm DR}}}
\newcommand{\mMSbar}{m^{\overline{\rm MS}}}
\newcommand{\bare}{{\rm B}}
\newcommand{\ep}{\epsilon}
\newcommand{\muR}{\mu_{\rm R}}
\newcommand{\Ltop}{L_t}
\newcommand{\Lgluino}{L_{\tilde g}}
\newcommand{\Lstop}[1]{L_{\tilde t#1}}
\newcommand{\mtop}{M_t}
\newcommand{\mstop}[1]{m_{\tilde t_{#1}}}
\newcommand{\mstopbar}[1]{\bar{m}_{\tilde t_{#1}}}
\newcommand{\Mstop}[1]{M_{\tilde t_{#1}}}
\newcommand{\mgluino}{m_{\tilde g}}
\newcommand{\Mgluino}{M_{\tilde g}}
\newcommand{\muSUSY}{\mu_{\rm \susy{}}}
\newcommand{\lmMtMS}{L_{tS}}
\newcommand{\lmMtMst}{L_{t\tilde{t}}}
\newcommand{\lmMtMsq}{L_{t\tilde{q}}}
\newcommand{\lmuMt}{L_{{\mu}t}}
\newcommand{\lmumt}{l_{{\mu}t}}
\newcommand{\lmmtms}{l_{tS}}
\newcommand{\note}[1]{{\tiny (note)}\marginpar {\scriptsize #1}}
\sloppy
%- }}}
\begin{document}
\preprint{SFB/CPP-08-14, TTP08-08, WUB08-02}
%- {{{ title + abstract:
%\wideabs{
\title{
Higgs boson mass in supersymmetry to three loops
}
\author{R.V. Harlander$^{1}$, P. Kant$^{2}$,
L. Mihaila$^{2}$, M. Steinhauser$^{2}$}
\affiliation{$^1$ Fachbereich C, Theoretische Physik,
Universit{\"a}t Wuppertal,
42097 Wuppertal, Germany\\
$^2$ Institut f{\"u}r Theoretische Teilchenphysik,
Universit{\"a}t Karlsruhe,
76128 Karlsruhe, Germany}
\date{\today}
\begin{abstract}
Within the minimal supersymmetric extension of the Standard Model, the
mass of the light CP-even Higgs boson is computed to three-loop
accuracy, taking into account the next-to-next-to-leading order
effects from supersymmetric Quantum Chromodynamics. We consider two
different scenarios for the mass hierarchies of the supersymmetric
spectrum. Our numerical results amount to corrections of about
500~MeV which is of the same order as the experimental
accuracy expected at the CERN Large Hadron Collider (LHC).
\end{abstract}
\pacs{11.30.Pb, 12.38.-t, 14.80.Cp}
\maketitle
%}
%- }}}
%{\bf TODO:\\
%??? check numbers (in text) in Section III
%\\
%??? numerical relevance of ``leading'' terms?
%\\
%??? We have used DRED under the assumption that supersymmetry is not
% violated; there is no formal proof yet.
%}
\section{I. Introduction}
%- {{{ Introduction:
Supersymmetry is currently the most-studied extension of the Standard
Model~(see, e.g., Ref.\,\cite{Nilles:1983ge}). It provides solutions to
some profound theoretical problems of the Standard Model: the fine
tuning of the Higgs mass, the (non-)unification of gauge couplings, a
mechanism for spontaneous symmetry breaking, and a Cold Dark Matter
candidate.
The minimal supersymmetric extension of the Standard Model (\mssm{}) is
based on a two-Higgs-doublet model ({\abbrev 2HDM}) with five physical
Higgs bosons: two {\abbrev CP}-even $h/H$, one {\abbrev CP}-odd $A$
(also named the ``pseudo-scalar'' Higgs), and two charged scalars
$H^\pm$. Each particle of this {\abbrev 2HDM} receives a \susy{} partner
of opposite spin-statistics, where left- and right-handed components of
a Standard Model Dirac fermion are attributed with separate scalars
$\tilde f_{\rm L/R}$ which mix to the physical mass eigenstates $\tilde
f_{1/2}$.
Compared to the Standard Model, the \mssm{} Higgs sector is described by
two additional parameters, usually chosen to be the pseudo-scalar mass
$M_A$ and the ratio of the vacuum expectation values of the two Higgs
doublets, $\tan\beta=v_2/v_1$. The masses of the other Higgs bosons are
then fixed by \susy{} constraints. In particular, the mass of the light
{\abbrev CP}-even Higgs boson, $M_h$, is bounded from above. At
tree-level, it is $M_h