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adding eqs. to calculate the Yukawa couplings y_f(MZ)
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Expander authored and Expander committed Feb 22, 2014
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\newcommand{\ESSM}{E$_6$SSM\xspace}
\newcommand{\code}[1]{\lstinline|#1|} % inline source code
\newcommand{\textoverline}[1]{$\overline{\mbox{#1}}$}
\newcommand{\DRbar}{\textoverline{DR}} % inline source code
\newcommand{\DRbar}{\textoverline{DR}\xspace}
\newcommand{\MSbar}{\textoverline{MS}\xspace}
\newcommand{\unit}[1]{\,\text{#1}} % units
\newcommand{\userinput}{\text{<input>}}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\def\at{\alpha_t}
\def\ab{\alpha_b}
\def\as{\alpha_s}
Expand Down Expand Up @@ -446,11 +450,191 @@ \subsection{Boundary conditions}

\subsubsection{Calculation of the gauge couplings $g_i(M_Z)$}

{\color{red} copy gauge couplings section from the manual.}
The low-scale constraint calculates the gauge couplings $g_i(M_Z)$.
It does so by first calculating $e_{\text{susy}}^{\text{\DRbar}}(M_Z)$
and $g_{3,\text{susy}}^{\text{\DRbar}}(M_Z)$ via
%
\begin{align}
\alpha_{\text{e.m.},\text{susy}}^{\text{\DRbar}}(M_Z) &=
\frac{\alpha_{\text{e.m.},\text{SM}}^{(5),\text{\MSbar}}(M_Z)}{1 -
\Delta\alpha_{\text{e.m.},\text{SM}}(M_Z) -
\Delta\alpha_{\text{e.m.},\text{susy}}(M_Z)} ,\\
\Delta\alpha_{\text{e.m.},\text{SM}}(\mu) &=
\frac{\alpha_\text{e.m.}}{2\pi} \left[\frac{1}{3}
- \frac{16}{9} \log{\frac{m_t}{\mu}} \right],\\
\Delta\alpha_{\text{e.m.},\text{susy}}(\mu) &=
\frac{\alpha_\text{e.m.}}{2\pi} \left[ -\sum_{\text{susy particle }
i}
C_i \log{\frac{m_i}{\mu}} \right],\\
e_{\text{susy}}^{\text{\DRbar}}(M_Z) &=
\sqrt{4\pi\alpha_{\text{e.m.},\text{susy}}^{\text{\DRbar}}(M_Z)}
\end{align}
%
\begin{align}
\alpha_{\text{s},\text{susy}}^{\text{\DRbar}}(M_Z) &=
\frac{\alpha_{\text{s},\text{SM}}^{(5),\text{\MSbar}}(M_Z)}{1 -
\Delta\alpha_{\text{s},\text{SM}}(M_Z)
- \Delta\alpha_{\text{s},\text{susy}}(M_Z)} ,\\
\Delta\alpha_{\text{s},\text{SM}}(\mu) &=
\frac{\alpha_\text{s}}{2\pi} \left[
-\frac{2}{3} \log{\frac{m_t}{\mu}} \right],\\
\Delta\alpha_{\text{s},\text{susy}}(\mu) &=
\frac{\alpha_\text{s}}{2\pi}\left[ \frac{1}{2}-\sum_{\text{susy
particle } i} C_i \log{\frac{m_i}{\mu}} \right] ,\\
g_{3,\text{susy}}^{\text{\DRbar}}(M_Z) &=
\sqrt{4\pi\alpha_{\text{s},\text{susy}}^{\text{\DRbar}}(M_Z)} ,
\end{align}
%
where
%
\begin{align}
\alpha_{\text{e.m.},\text{SM}}^{(5),\text{\MSbar}}(M_Z) &= 1/127.944,\\
\alpha_{\text{s},\text{SM}}^{(5),\text{\MSbar}}(M_Z) &= 0.1185,
\end{align}
%
are the \MSbar electromagnetic and strong cougling constants in the
Standard Model including only $5$ quark flavours
\cite{Beringer:1900zz}. Afterwards, SARAH's tree-level expressions
for the Weinberg angle and the $U(1)_Y$ and $SU(2)_L$ gauge couplings
$g_Y$ and $g_2$ in terms of $M_{W,\text{susy}}^{\text{\DRbar}}(M_Z)$,
$M_{Z,\text{susy}}^{\text{\DRbar}}(M_Z)$ and
$e_{\text{susy}}^{\text{\DRbar}}(M_Z)$ are used to calculate
$g_{1,\text{susy}}^{\text{\DRbar}}(M_Z)$ and
$g_{2,\text{susy}}^{\text{\DRbar}}(M_Z)$. In the MSSM, for example,
one has
%
\begin{align}
\theta_{W,\text{susy}}^{\text{\DRbar}}(M_Z) &=
\arcsin\sqrt{1 - \left(\frac{M_{W,\text{susy}}^{\text{\DRbar}}(M_Z)}{M_{Z,\text{susy}}^{\text{\DRbar}}(M_Z)}\right)^2} ,\\
g_{1,\text{susy}}^{\text{\DRbar}}(M_Z) &=
\sqrt{\frac{5}{3}} \frac{e_{\text{susy}}^{\text{\DRbar}}(M_Z)}{\cos\theta_{W,\text{susy}}^{\text{\DRbar}}(M_Z)} ,\\
g_{2,\text{susy}}^{\text{\DRbar}}(M_Z) &=
\frac{e_{\text{susy}}^{\text{\DRbar}}(M_Z)}{\sin\theta_{W,\text{susy}}^{\text{\DRbar}}(M_Z)} ,
\end{align}
%
where
%
\begin{align}
\left(M_{W,\text{susy}}^{\text{\DRbar}}(M_Z)\right)^2 &=
\left(M_W^{\text{pole}}\right)^2 + \re \Pi_{WW}^T(p^2 = (M_W^{\text{pole}})^2) ,\\
\left(M_{Z,\text{susy}}^{\text{\DRbar}}(M_Z)\right)^2 &=
\left(M_Z^{\text{pole}}\right)^2 + \re \Pi_{ZZ}^T(p^2 = (M_Z^{\text{pole}})^2)
\end{align}
%
and $M_W^{\text{pole}} = 80.404\unit{GeV}$, $M_Z^{\text{pole}} =
91.1876\unit{GeV}$ \cite{Beringer:1900zz}.

\subsubsection{Calculation of the Yukawa couplings $y_f(M_Z)$}

{\color{red} copy Yukawa couplings section from the manual.}
At the low-scale constraint the expressions for the Yukawa coupling
matrices $y_f(M_Z)$ are calculated from the tree-level mass matrices
of the SM-fermions $m_u$, $m_d$, $m_e$. In the MSSM, for example,
these relations read
%
\begin{align}
y_u^{\text{\DRbar}}(M_Z) &= \frac{\sqrt{2} m_{u}^T}{v_u} , &
y_d^{\text{\DRbar}}(M_Z) &= \frac{\sqrt{2} m_{d}^T}{v_d} , &
y_e^{\text{\DRbar}}(M_Z) &= \frac{\sqrt{2} m_{e}^T}{v_d}.
\end{align}
%
At the low-scale the fermion mass matrices are composed as
%
\begin{align}
m_u =
\begin{pmatrix}
m_{u}^{\userinput} & 0 & 0 \\
0 & m_{c}^{\userinput} & 0 \\
0 & 0 & m_{t,\text{susy}}^{\text{\DRbar}}(M_Z)
\end{pmatrix} ,\\
m_d =
\begin{pmatrix}
m_{d}^{\userinput} & 0 & 0 \\
0 & m_{s}^{\userinput} & 0 \\
0 & 0 & m_{b,\text{susy}}^{\text{\DRbar}}(M_Z)
\end{pmatrix} ,\\
m_e =
\begin{pmatrix}
m_{e}^{\userinput} & 0 & \\
0 & m_{\mu}^{\userinput} & 0 \\
0 & 0 & m_{\tau,\text{susy}}^{\text{\DRbar}}(M_Z)
\end{pmatrix},
\end{align}
%
where $m_{u,c,d,s,e,\mu}^{\userinput}$ are read from the \code{Block
SMINPUTS} of the SLHA input file \cite{Skands:2003cj}. The 3rd
generation quark masses are calculated in the \DRbar scheme from the
SLHA user input quantities $m_t^\text{pole}$,
$m_{b,\text{SM}}^{\text{\MSbar}}(M_Z)$,
$m_{\tau,\text{SM}}^{\text{\MSbar}}(M_Z)$ \cite{Skands:2003cj}. In
detail, the top quark \DRbar mass is calculated via
%
\begin{align}
\begin{split}
m_{t,\text{susy}}^{\text{\DRbar}}(\mu) &= m_t^\text{pole} +
\re\Sigma_{tt}^{S,\text{heavy}}(m_t^\text{pole}) \\
&\phantom{=\;} + m_t^\text{pole}
\left[ \re\Sigma_{tt}^{L,\text{heavy}}(m_t^\text{pole}) +
\re\Sigma_{tt}^{R,\text{heavy}}(m_t^\text{pole}) + \Delta
m_t^{(1),\text{qcd}} + \Delta m_t^{(2),\text{qcd}} \right] ,
\end{split}
\end{align}
%
where the $\Sigma_{ff}^\text{heavy}$ denotes the self-energy of
fermion $f$ without the Standard Model contributions. The appearing
QCD corrections $\Delta m_t^{(1),\text{qcd}}$ and $\Delta
m_t^{(2),\text{qcd}}$ are taken from
\cite[Eq.\ (58),(61)]{Bednyakov:2002sf} and read
%
\begin{align}
\Delta m_t^{(1),\text{qcd}} &= -\frac{g_3^2 \left(5-3 \log\left(\frac{m_t^2}{\mu^2}\right)\right)}{12 \pi^2},\\
\begin{split}
\Delta m_t^{(2),\text{qcd}} &= \left(\Delta
m_t^{(1),\text{qcd}}\right)^2 \\
&\phantom{=\;} - \frac{g_3^4 \left[396
\log^2\left(\frac{m_t^2}{\mu^2}\right)-1476
\log\left(\frac{m_t^2}{\mu^2}\right)-48 \zeta(3)+2011+16 \pi
^2 (1+\log 4)\right]}{4608 \pi^4}.
\end{split}
\end{align}
%
The \DRbar mass of the bottom quark is calculated via
\cite{Baer:2002ek,Skands:2003cj}
%
\begin{align}
m_{b,\text{susy}}^{\text{\DRbar}}(\mu) &=
\frac{m_{b,\text{SM}}^{\text{\DRbar}}(\mu)}{1 -
\re\Sigma_{bb}^{S,\text{heavy}}(m_{b,\text{SM}}^\text{\MSbar})/m_b
- \re\Sigma_{bb}^{L,\text{heavy}}(m_{b,\text{SM}}^\text{\MSbar}) -
\re\Sigma_{bb}^{R,\text{heavy}}(m_{b,\text{SM}}^\text{\MSbar})} ,\\
m_{b,\text{SM}}^{\text{\DRbar}}(\mu) &=
m_{b,\text{SM}}^{\text{\MSbar}}(\mu) \left(1 - \frac{\alpha_s}{3
\pi} - \frac{23}{72} \frac{\alpha_s^2}{\pi^2} + \frac{3
g_2^2}{128 \pi^2} + \frac{13 g_Y^2}{1152 \pi^2}\right) ,
\end{align}
%
where $m_{b,\text{SM}}^{\text{\MSbar}}(M_Z)$ is user input. The
\DRbar mass of the $\tau$ is calculated via
%
\begin{align}
\begin{split}
m_{\tau,\text{susy}}^{\text{\DRbar}}(\mu) &=
m_{\tau,\text{SM}}^{\text{\DRbar}}(\mu) +
\re\Sigma_{\tau\tau}^{S,\text{heavy}}(m_{\tau,\text{SM}}^\text{\MSbar}) \\
&\phantom{=\;} + m_{\tau,\text{SM}}^{\text{\DRbar}}(\mu) \left[
\re\Sigma_{\tau\tau}^{L,\text{heavy}}(m_{\tau,\text{SM}}^\text{\MSbar})
+
\re\Sigma_{\tau\tau}^{R,\text{heavy}}(m_{\tau,\text{SM}}^\text{\MSbar})
\right] ,
\end{split}\\
m_{\tau,\text{SM}}^{\text{\DRbar}}(\mu) &= m_{\tau,\text{SM}}^{\text{\MSbar}}(\mu)
\left(1 - 3 \frac{g_Y^2 - g_2^2}{128 \pi^2}\right),
\end{align}
%
where $m_{\tau,\text{SM}}^{\text{\MSbar}}(M_Z)$ is user input. To
convert the fermion masses from the \MSbar to the \DRbar scheme the
Yukawa coupling conversion from \cite[Eq.~(19)]{Skands:2003cj} was
used and it was assumed that the VEV is already given in the \DRbar
scheme.

\subsubsection{Electroweak symmetry breaking}
In \fs the one-loop electroweak symmetry breaking conditions are
Expand Down Expand Up @@ -1308,6 +1492,16 @@ \section{Tests}
[arXiv:0907.4682 [hep-ph]].
%%CITATION = ARXIV:0907.4682;%%
%35 citations counted in INSPIRE as of 21 Sep 2013

%\cite{Skands:2003cj}
\bibitem{Skands:2003cj}
P.~Z.~Skands, B.~C.~Allanach, H.~Baer, C.~Balazs, G.~Belanger, F.~Boudjema, A.~Djouadi and R.~Godbole {\it et al.},
%``SUSY Les Houches accord: Interfacing SUSY spectrum calculators, decay packages, and event generators,''
JHEP {\bf 0407} (2004) 036
[hep-ph/0311123].
%%CITATION = HEP-PH/0311123;%%
%394 citations counted in INSPIRE as of 22 Feb 2014

%\cite{Allanach:2008qq}
\bibitem{Allanach:2008qq}
B.~C.~Allanach, C.~Balazs, G.~Belanger, M.~Bernhardt, F.~Boudjema, D.~Choudhury, K.~Desch and U.~Ellwanger {\it et al.},
Expand Down Expand Up @@ -1444,7 +1638,31 @@ \section{Tests}
%%CITATION = HEP-PH/0510419;%%
%130 citations counted in INSPIRE as of 03 Nov 2013

%
%\cite{Beringer:1900zz}
\bibitem{Beringer:1900zz}
J.~Beringer {\it et al.} [Particle Data Group Collaboration],
%``Review of Particle Physics (RPP),''
Phys.\ Rev.\ D {\bf 86} (2012) 010001.
%%CITATION = PHRVA,D86,010001;%%
%3229 citations counted in INSPIRE as of 22 Feb 2014

%\cite{Bednyakov:2002sf}
\bibitem{Bednyakov:2002sf}
A.~Bednyakov, A.~Onishchenko, V.~Velizhanin and O.~Veretin,
%``Two loop O(alpha-s**2) MSSM corrections to the pole masses of heavy quarks,''
Eur.\ Phys.\ J.\ C {\bf 29} (2003) 87
[hep-ph/0210258].
%%CITATION = HEP-PH/0210258;%%
%35 citations counted in INSPIRE as of 22 Feb 2014

%\cite{Baer:2002ek}
\bibitem{Baer:2002ek}
H.~Baer, J.~Ferrandis, K.~Melnikov and X.~Tata,
%``Relating bottom quark mass in DR-BAR and MS-BAR regularization schemes,''
Phys.\ Rev.\ D {\bf 66} (2002) 074007
[hep-ph/0207126].
%%CITATION = HEP-PH/0207126;%%
%55 citations counted in INSPIRE as of 22 Feb 2014

\end{thebibliography}
\end{document}

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