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\title{IMPACT OF TEV-SCALE STERILE NEUTRINOS\\ ON PRECISION LOW-ENERGY OBSERVABLES}
\author{E. AKHMEDOV$^{\dagger}$, A. KARTAVTSEV$^{\ddagger}\,$\footnote{Presenting author. Talk based on arXiv:1302.1872.},
M. LINDNER$^{\dagger}$, L. MICHAELS$^{\dagger}$, J. SMIRNOV$^{\dagger}$}
\address{$^{\dagger}$Max-Planck-Institut f\"ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany\\
$^{\ddagger}$Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, 80805 Munich, Germany}
\maketitle\abstracts{
We study the impact of TeV-scale sterile neutrinos on electroweak precision observables and lepton
number and flavour violating decays in the framework of a type-I see-saw extension of the Standard Model.
At tree level sterile neutrinos manifest themselves via non-unitarity of the PMNS matrix and at one-loop
level they modify the oblique radiative corrections. We perform a numerical fit to the electroweak observables
and find regions of parameter space with a sizable active-sterile mixing which provide a better over-all
fit compared to the case where the mixing is negligible. Specifically we find improvements of the invisible
$Z$-decay width, the charged-to-neutral-current ratio for neutrino scattering experiments and of the
deviation of the $W$ boson mass from the theoretical expectation.}
\section{Introduction}
The Standard Model (SM) is extremely successful and has passed numerous experimental tests. Moreover the
last missing piece, the Higgs particle, has recently likely been seen by the ATLAS and CMS collaborations.
On the other hand, the SM is incomplete as it fails to explain the tiny active neutrino masses, the baryon
asymmetry of the Universe and the existence of dark matter. A simple yet elegant way to solve the first
two or even all of these problems is to supplement the SM by three heavy sterile neutrinos:
\begin{align}
\label{lagrangian}
\mathscr{L}=\mathscr{L}_{SM}&+
{\textstyle\frac12}\bar N_i \bigl(i\slashed{\partial} - M_i\bigr)N_i
- h_{\alpha i}\bar \ell_\alpha {\tilde \phi} N_i
-h^\dagger_{i\alpha} \bar N_i {\tilde \phi}^\dagger \ell_\alpha\,.
\end{align}
The existence of Majorana neutrinos has well known consequences on the phenomenology below the electroweak
scale. In particular, the new states can contribute to the amplitude of the neutrinoless double-beta decay
\cite{Rodejohann:2012xd,LopezPavon:2012zg,Mitra:2012qz,Mitra:2011qr,Blennow:2010th} and induce rare charged
lepton decays \cite{Antusch:2006vwa,Dinh:2012bp}. Furthermore, they can affect the electroweak precision
observables (EWPOs) via tree-level as well as loop contributions and thus provide an explanation for
anomalies in the experimental data. In particular, the tree-level effects result in non-unitarity of the
active neutrino mixing matrix \cite{Antusch:2006vwa} and lead to a suppression of the invisible $Z$-decay
width. This is in agreement with the long standing fact that the LEP measurement of the invisible $Z$-decay
width is two sigma below the value expected in the SM \cite{Beringer:1900zz}. Furthermore the
neutral-to-charged-current ratio in neutrino scattering experiments can be changed thus providing an
explanation for the NuTeV anomaly \cite{NuTeV}. Also a slight shift of the $W$ boson mass from the
value derived from other SM parameters is induced, reducing the tension between the input parameters of
the electroweak fit and the experimentally observed value \cite{Ferroglia:2012ir}.
Encouraged by the fact that sterile neutrinos are very well motivated we study their phenomenological
impact. Specifically we consider TeV-scale sterile neutrinos with a sizable active-sterile mixing and
determine their over-all contributions to the EWPOs and to indirect detection experiments in the framework of
the see-saw type-I extension of the SM.
\section{Impact on low-energy observables}
After the electroweak symmetry breaking the active and sterile \emph{flavor} eigenstates mix. In other words,
the light \emph{mass} eigenstates acquire a small sterile component. Simultaneously, the heavy \emph{mass}
eigenstates acquire a small active component and couple to the $W$- and $Z$-bosons,
\begin{align}
\label{gaugelagrangian}
\mathscr{L}_\text{int}=&-\frac{e}{2c_ws_w}Z_\mu{\textstyle\sum_{i,j=1}^{3+n}}{\textstyle\sum}_{\alpha=e,\m,\t}
\bar{\nu}_i \mathbf{U}^\dagger_{i\alpha}\gamma^\mu P_L \mathbf{U}_{\alpha j} \nu_j\nonumber\\
&-\frac{e}{\sqrt{2}s_w} W_\mu{\textstyle\sum_{i=1}^{3+n}}{\textstyle\sum}_{\alpha=e,\m,\t}\bar{\nu}_i
\mathbf{U}^\dagger_{i\alpha}\gamma^\mu P_L e_\alpha + {\rm h.c.}\,.
\end{align}
This affects the low-energy observables in two ways. First, this introduces additional processes with the
heavy neutrinos in the intermediate state. Second, because the light mass eigenstates acquire a small sterile
component their couplings to the $W$- and $Z$-bosons are smaller than assumed in the Standard Model.
\emph{Lepton universality constraints.}
The second effect has an immediate impact of the probability of the $W$-decay into
a charged lepton of one of the three generations and a light neutrino. First, one can
expect that the decay width is smaller than in the Standard Model, just because the
coupling is smaller. Second, the decay probabilities for the electron, muon and tau
leptons are now in principle different. This goes under the name “lepton universality
violation”. There are relatively stringent experimental constraints on the violation
of lepton universality \cite{Takeuchi:A},
\begin{subequations}
\label{NonUniversConstr}
\begin{align}
\e_e-\e_\m=0.0022 \pm 0.0025\,, \\
\e_\m-\e_\t=0.0017 \pm 0.0038\,,\\
\e_e-\e_\t=0.0039 \pm 0.0040\,,
\end{align}
\end{subequations}
where $\e_\alpha\equiv {\textstyle\sum_{i\geq 4}} |\mathbf{U}_{\alpha i}|^2$.
Note that $\epsilon_\alpha\neq 0$ implies non-unitarity of the PMNS matrix,
i.e. of the $3\times 3$ mixing matrix of the light eigenstates.
\emph{$W$-boson mass.}
The modification of the couplings to the $W$-boson also affects another very
important observable. Because of the non-unitarity the Fermi constant measured
in the muon decay differs from the Fermi constant measured in experiments with
semi-leptonic processes. And because the muon decay width is used as input in
the Standard Model fit, this modification influences many observables. In particular
it changes the theoretical expectation for the $W$-bo\-son mass whose experimental value,
$M_W=80.385\pm0.015$, is roughly one sigma away from the Standard Model
expectation, $80.359\pm0.011$ GeV.
\emph{Invisible Z-decay width.}
The existence of the heavy neutrinos also affects couplings to the $Z$-boson.
Because we now have two neutrino lines, the effect is roughly speaking twice as
strong as for the $W$-boson. Typically, adding new particles to a theory means larger
decay widths, simply because there are more states to decay into. Surprisingly, this is
not what happens with the invisible $Z$-decay width once we add heavy neutrinos. It becomes
smaller instead. Here is the reason. On the one hand, the non-unitarity of the PMNS
matrix makes couplings of the light neutrinos to the $Z$-boson smaller. This automatically
makes the decay width into these states smaller. On the other hand, because the $Z$-boson
is lighter than the heavy neutrinos, it simple cannot decay into the new states for
kinematical reasons. As a result, the invisible $Z$-decay width is smaller than
expected in the Standard Model. Put in another way, this means that the effective number
of neutrinos measured by LEP should be slightly smaller than three. This is in qualitative
agreement with the experimental result
$\Gamma_{\rm inv}/\Gamma_{\rm lept}=5.942\pm0.016$, which is roughly two sigma away
from the Standard Model expectation $5.9721\pm0.0002$.
\emph{Charged to neutral current ratio.}
The existence of the heavy neutrinos makes coupling of the light ones to the $W$-
and $Z$-bosons smaller. The coupling to the $Z$-boson is affected roughly twice as
strong as the coupling to the $W$-boson. The immediate implication is that the
neutral current is suppressed stronger than the charged current. This conclusion
is qualitatively consistent with the results of the NuTeV experiment. After including
a recent NNLO analysis \cite{NuTeV,NNLO} the experimental values for $g_L$ and
$g_R$ are given by $g^2_{\text{L}}=0.3026\pm0.0012$ and $g_{\text{R}}^{2}=0.0303
\pm0.0010$, whereas the Standard Model expectations are $0.3040\pm0.0002$ and
$0.0300\pm0.0002$ respectively.
\emph{Lepton flavor violating decays.}
Recall now that the heavy neutrinos affect the low energy observables
not only because of non-unitarity, but also because they appear as intermediate states
in the Feynman diagrams. The prime example where both effects play a role is a lepton
flavor violating decay $\mu \rightarrow e \gamma$. The contribution of the light
neutrinos is completely negligible. Taking into account the unitarity of the full
mixing matrix $\mathbf{U}$ we find for the contribution of the heavy states,
$\delta_{\nu}=2\, \textstyle{\sum_{i=4}^{3+n}} \mathbf{U}_{e i}^{\ast}
\mathbf{U}_{\mu i}\left[g\left(m_{i}^{2}/M_{W}^{2}\right)-5/3\right]$,
where the second term in the square brackets comes from non-unitarity of the
PMNS matrix, whereas the first one is induced by the intermediate heavy
neutrinos. The recent limit on this branching ratio obtained by the MEG
collaboration \cite{MEG} is
$\text{BR}(\mu^{+}\rightarrow e^{+}\gamma)\leq 5.7\cdot10^{-13}$
at $90\%$ confidence level.
\emph{Neutrinoless double beta decay.}
Another example is neutrinoless double beta decay. The effective electron
neutrino mass is given by $|\langle m_{ee}\rangle|\approx\bigl|
\textstyle{\sum_{i=1}^3}\mathbf{U}_{ei}^{2}m_i
-\textstyle{\sum_{i=4}^{3+n}} F(A,M_i)\mathbf{U}_{ei}^{2}m_i\bigr|$.
Typically, one takes into account only the contributions of the light
neutrinos, this is the first term, but the heavy neutrinos can also contribute,
this is the second term. The experimental bound has also been recently updated
\cite{Macolino:2013ifa} by the GERDA collaboration,
$|\langle m_{ee}\rangle|< 0.2\,-\,0.4\,\,{\rm eV}$.
\emph{STU parameters.}
Last but no definitely not least, the heavy neutrinos can also appear in the self-energy
loops of the $W$- and $Z$-bosons and affect theoretical predictions for the low-energy
observables we have discussed so far. These loop corrections can be taken into account
using the STU parameters developed by Peskin and Takeuchi.
\emph{Combination of the tree-level and loop corrections.}
Explicit expressions for the corrections to the electroweak observables read,
\begin{subequations}
\label{EWobservables}
\begin{align}
\label{Glept}
\frac{\G_{\text{lept}}}{\left[\G_{\text{lept}}\right]_{\text{SM}}}&=1
+0.6\,(\e_{e}+\e_\m+0.0145\,T)-0.0021\, S\,, \\
\label{Ginv}
\frac{\G_{\text{inv}}/\G_{\text{lept}}}{\left[\G_{\text{inv}}/\G_{\text{lept}}\right]_{\text{SM}}}&=1
-0.67\,(\e_{e}+\e_{\m}+\e_{\t})+0.0021\, S-0.0015\, T \,, \\
\label{sinTw}
\frac{\sin^{2}\theta_{\text{w}}^{\text{lept}}}{\bigl[\sin^{2}\theta_{\text{w}}^{\text{lept}}\bigr]_{\text{SM}}}&=1
-0.72\,(\e_{e}+\e_\m+0.0145\,T)+0.0016\, S\,,\\
\label{gL}
\frac{g_{L}^{2}}{\left[g_{L}^{2}\right]_{\text{SM}}}&=1
+0.41\,\e_{e}-0.59\,\e_{\m}-0.0090\, S+0.0022\, T\,, \\
\label{gR}
\frac{g_{R}^{2}}{\left[g_{R}^{2}\right]_{\text{SM}}}&=1
-1.4\,\e_{e}-2.4\,\e_{\m}+0.031\, S-0.0067\, T\,,\\
\label{Mw}
\frac{M_{\text{W}}}{\left[M_{\text{W}}\right]_{\text{SM}}} &= 1
+ 0.11\,\e_{e}+0.11\,\e_{\m}-0.0036\, S+0.0056\, T+ 0.0042\, U \,,
\end{align}
\end{subequations}
where S, T, and U are the STU parameters which encode the loop corrections, and $\epsilon_e$,
$\epsilon_\mu$ and $\epsilon_\tau$ encode the tree-level non-unitarity effects. Importantly,
the loop corrections can be as large as the tree-level ones and therefore we can have partial
cancellation of the tree-level and loop corrections. If this cancellation happens or not of
course depends on the values of the model parameters and this is where we approach the question
of parameter scan.
\emph{Parameter scan.}
The contribution of the heavy neutrinos to most of the observables may be small, but it is decisive
as far as masses and mixing angles of the light neutrinos are concerned. Within the past fifteen
years there has been an enormous progress in this field. On the one hand most of the neutrino
parameters have been measured experimentally. On the other hand, and this is very important for
the parameter scan, Casas and Ibarra have developed a very handy parametrization \cite{Casas:Parameter}
of the full six-by-six neutrino mixing matrix in terms of the experimentally measurable quantities
and a few unknown parameters,
\begin{subequations}
\label{active-sterile-mixing}
\begin{align}
\mathscr{R}&=-i\,{\cal U}\,\hat{m}_{\rm light}^{\frac12}\,O^{*}\hat{m}_{\rm heavy}^{-\frac12} \, ,\\
\mathscr{U}&=\left(1-\mathscr{R}\, \mathscr{R}^{\dagger}\right)^\frac12 {\cal U}\,,
\end{align}
\end{subequations}
with $O$ being an arbitrary complex orthogonal matrix, ${\cal U}$ the \textit{unitary} matrix
diagonalizing $\hat m_\nu$, $\hat m_{\rm heavy}$
the diagonal mass matrix of the heavy neutrinos and $\hat m_{\rm light}$ that of the light neutrinos.
Now comes the question of the number of degrees of freedom. In principle, we start with nine variables:
three masses of the heavy neutrinos plus three complex angles in the matrix $O$. However, the
corrections are expressed in terms of only six quantities: the three parameters of non-unitarity plus
the three STU parameters. In other words, the initial nine degrees of freedom map to six. Out of the
STU parameters, the S and U parameters are negligibly small. So effectively, the initial degrees of
freedom map to four parameters: $\epsilon_e$, $\epsilon_\mu$, $\epsilon_\tau$ and $T$. I would like
to emphasize that our goal was not to perform a full parameter scan, but rather to find examples of
regions in the parameter space where the fit is improved with respect to the Standard Model. For
every considered point in the parameter space we have checked that it is compatible with the
$\mu \rightarrow e \gamma$ and $0\nu\beta\beta$ constraints.
\emph{Rare processes.}
If the point passed this test, as is the case for all points in figure \ref{NH_Smu_RP}, then we
computed the values of the other low-energy observables and used them to calculate $\chi^2$ using
\begin{align}
\label{chisqEWPOdef}
\chi^2_{\text{EWPO}}=\sum_i \frac{(O_i-O_{i,\text{SM}})^2}{(\delta O_i)^2+(\delta O_{i,\text{SM}})^2}\,,
\end{align}
where $O_{i,SM}$ denotes the predictions of the SM, $\delta O_{i,SM}$ are the theoretical
errors and $\delta O_i$ are the experimental errors.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{NH_Smu_RP}
\end{center}
\caption{\label{NH_Smu_RP} $\chi^2$ for four d.o.f. as a function of the ratios of the $\mu\rightarrow e\gamma$
branching ratio and $|\langle m_{ee} \rangle |$ to the corresponding experimental bounds (NH). Here $\e_\mu$ is
suppressed.}
\end{figure}
If the heavy neutrinos had absolutely
no impact on the low-energy observables, then $\chi^2\approx 7.5$. Such points are color-coded by
red on this plot. For the best fit points we get a moderate decrease of $\chi^2$ to 4. These points
are color-coded by green. The improvement of $\chi^2$ per new degree of freedom is roughly one.
Which of the electroweak observables are responsible for this improvement?
\emph{Electroweak fit.}
The answer can be inferred from figure \ref{DevNormalSmallEmu}. The main improvement is due to the
charged-to-neutral current ratio and due to the W-mass. The improvement of the invisible $Z$-decay width
is rather minor. Here the tree-level corrections are largely compensated by the loop ones.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth]{DevNormalSmallEmu}
\end{center}
\caption{\label{DevNormalSmallEmu} EWPOs calculated at the best-fit point for NH and suppressed
$\e_\mu$ (green dots) compared to the experimentally observed values, denoted by the zero line.
The colored lines stand for the respective experimental sigma deviations, thus the displacement
of the predicted values form the observations is presented in units of the experimental error. Note
that for the $0\nu\b\b$ and $\m \rightarrow e \gamma$ constraints we present only the one sigma
exclusion limits. The theoretical predictions of the SM with their theoretical uncertainties
are displayed as well (blue bars). (The best-fit point is at $M_1=20.3$ TeV, $M_2=14.1$ TeV,
$M_3=21.0$ TeV, $\e_e=2.1\cdot 10^{-3}$, $\e_\m=3.0\cdot 10^{-6}$ and $\e_\t=4.5\cdot 10^{-3}$.)}
\end{figure}
Note that figure \ref{DevNormalSmallEmu} assumes normal hierarchy of the light neutrino masses.
For the inverted and quasidegenerate mas hierarchies $\chi^2$ at the best-fit points reduces to
roughly $\chi^2\approx 5.5$ and $\chi^2\approx 5$ respectively \cite{Akhmedov:2013hec}.
\section{\label{Summary}Summary}
To summarize, sterile neutrinos with masses at the TeV-scale and a sizable
active-sterile mixing affect electroweak precision observables as well as
$\mu\rightarrow e\gamma$ and $0\nu\beta\beta$ processes. The effect is twofold.
On the one hand, the coupling of the light mass eigenstates to the $W$- and
$Z$-bosons is smaller than expected in the Standard Model. On the other hand,
the heavy mass eigenstates also couple to the $W$- and $Z$-bosons and can
contribute as intermediate states. Given that there some discrepancies between
the predictions of the Standard Model and the experimental data, corrections
induced by the sterile neutrinos are more than welcome. Accepting some fine-tuning
we can improve the fit of the neutral-to-charged current ratio, of the W-mass,
and to a lesser extent of the invisible decay width.
\section*{Acknowledgments}
The work of A.K. has been supported by the German Science Foundation (DFG) under Grant
KA-3274/1-1 ``Systematic analysis of baryogenesis in non-equilibrium quantum field theory''.
The authors would like to thank M. Blennow, T. Schwetz , J. Heeck, J. Barry and S. Antusch
for helpful discussions.
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\end{document}