Electroweak Symmetry Breaking without the \boldmath{$\mu^2$
Florian Goertz
EWSB_mu=0.tex
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\preprint{CERN-PH-TH-2015-071}
\author{Florian Goertz}
\affiliation{Theory Division,\\
CERN, 1211 Geneva 23, Switzerland}
\emailAdd{florian.goertz@cern.ch}
\title{Electroweak Symmetry Breaking without the \boldmath{$\mu^2$} Term}

\abstract{
We demonstrate that from a low energy perspective a viable electroweak 
symmetry breaking can be achieved
without the (negative sign) $\mu^2$ mass term in the Higgs potential,
thereby avoiding completely the appearance of relevant operators.
We show that such a setup is self consistent and not 
ruled out by Higgs physics. In particular, we point out that it is the 
lightness of the Higgs boson that allows for the electroweak symmetry
to be broken dynamically via operators of $D\geq 4$, consistent with
the power expansion. Beyond that, we entertain how
this scenario might even be preferred phenomenologically
compared to the ordinary mechanism of electroweak symmetry
breaking, as realized in the Standard Model, and argue that
it can be fully tested at the LHC.
In an appendix, we classify UV completions that could lead to such a setup, 
considering also the option of generating all scales dynamically.}
\date{\today}
\begin{document}

\maketitle
\section{Introduction}

A common lore, in particular after the discovery of the Higgs boson, is that electroweak symmetry breaking (EWSB) is 
triggered by a negative sign mass term for a Higgs doublet, either introduced by hand or generated 
dynamically. The ultra-violet~(UV) completion of the Standard Model~(SM) is expected to generate (approximately) a Higgs 
potential of such a form. However, as we will entertain in this article, a more fundamental theory of nature 
could also have an opposite low energy limit, where the appearance of relevant operators is fully avoided. As we will show, in such 
a scenario an 'irrelevant' $D=6$ operator of the type ${\cal O}_6= |H|^6$ could induce a non-trivial vacuum for the scalar 
sector.\footnote{Such operators are expected to be present on general grounds since the SM is not anticipated to be the final theory of nature.} We will point 
out that it is the lightness of the Higgs boson that allows to consider this special setup, where the operators in the Higgs potential
are not only deformed in a sub-leading way, as a phenomenologically viable alternative 
to the SM-form of the potential. It will turn out that this lightness, $m_h^2\ll v^2$,
makes it consistent with the effective field theory (EFT) expansion to fully trade the $(D=2)$ $\mu^2$-operator~for~${\cal O}_6$. 

After having demonstrated  the self-consistency of the setup, we will turn to the phenomenology of the model. 
The most important collider observable is Higgs-boson pair production, where we will show that the LHC is capable 
of fully testing the pure version of the idea. 
Beyond that, the model has intriguing consequences for cosmology. We will see that it just lies in the correct ballpark 
such as to allow for a strong first order phase transition, as required by electroweak baryogenesis.
While in the main part of the paper we just treat the $\mu=0$ Higgs potential as a distinct and interesting boundary condition 
that a potential UV model could fulfill, thereby opening up a new direction in model building, in the appendix we will present and classify 
possible ideas for UV completions of our setup.


\section{The Form of the Higgs Potential}

We consider the SM without relevant operators and instead augment the Higgs potential with a dimension-6 term 
$c_6/\Lambda^2\, {\cal O}_6$ such that it takes the simple form
	\begin{equation}
	\label{eq:pot}
	V(H)= \lambda |H|^4 + \frac{c_6}{\Lambda^2}  |H|^6\,,
	\end{equation}
where all dimensionfull parameters are either zero or at the cutoff of the theory. 

Inspecting the form of the potential, a first observation is that a stable and non-trivial minimum at $|H|^2>0$ 
should be possible if $\lambda < 0$ and $c_6 >0$. In the following, we will check if such a minimum is also viable 
phenomenologically. For any given cutoff scale $\Lambda$, we can first calculate the position of the minimum,
{\it i.e.}, the vacuum expectation value (vev), denoted as $\langle |H|^2\rangle \equiv v^2/2$, via $\partial V/ \partial 
|H|^2=0$.
We find 
	\begin{equation}
	\label{eq:v}
	v^2 = -\frac 4 3 \frac\lambda{c_6} \Lambda^2\,.
	\end{equation}
Clearly, the minimization condition only fixes the relative size of the coefficients $\lambda$ and $c_6/\Lambda^2$. %\footnote{We
%keep the cutoff scale $\Lambda$ explicit here, since it is related to a physical mass scale that we want to discuss later.}
In turn, an electroweak-scale vev can be obtained without the need for a large coefficient of the $D=6$ operator
${\cal O}_6$. The size of the latter will however get fixed by the mass of the physical Higgs excitation around the vev, $h$, 
where in unitary gauge  $H = 1/\sqrt 2 ( 0,  v + h)^T$. This is given as $m_h^2= \partial^2 V/ 
\partial h^2\large{|}_{h=0}$, leading to
	\begin{equation}
	\label{eq:mh}
	m_h^2 =3 v^2 \lambda  + \frac{15}{4} \frac{c_6}{\Lambda^2} v^4\,.
	\end{equation}
The consequences of these relations will be scrutinized in the next section.



\section{Self Consistency of the Setup}

We will now examine quantitatively, if it is possible to generate the vev and the Higgs mass at the
correct values, keeping the parameters in the range of the validity of the EFT,
and then discuss the stability of the low energy setup.
To that extent, we first solve eqs.~(\ref{eq:v}) and (\ref{eq:mh})  for the two free parameters in the 
potential, $\lambda$ and $c_6$, expressing them in terms of the vev, fixed by the Fermi constant as 
$v=246$\,GeV, and the Higgs mass $m_h\approx125$\,GeV. We obtain 
	\begin{equation} 
	\lambda= - \frac{m_h^2}{2 v^2}\approx-0.13\,,\quad c_6= \frac{2 m_h^2}{3 v^2} 
	\frac{\Lambda^2}{v^2}\approx 2.8\, \frac{\Lambda^2}{{\rm TeV}^2}.
	\end{equation}
We inspect that, since $m_h^2/v^2\approx 1/4 \ll 1$, a large cutoff $\Lambda^2 \gg v^2$ is possible while
keeping $c_6 \sim {\cal O}(1)$. We can thus see explicitly that it is the lightness of the Higgs boson which allows for the mechanism to work naturally. 
% Indeed, eqs. (\ref{eq:v}) and (\ref{eq:mh}) taken together lead to $m_h^2 \ll v^2$
% for $\Lambda^2/c_6 \gg v^2$. 
The required $c_6$ versus the cutoff $\Lambda$ is visualized in Figure~\ref{fig:c6lam}, where
$c_6 = 4\pi$ is given by the horizontal dashed line for illustration.
	\begin{figure}[!t]
	\begin{center}
	\includegraphics[height=1.8in]{c6L.pdf}
	\caption{\label{fig:c6lam} Required value of the Wilson coefficient $c_6$ in dependence on the cutoff $\Lambda$. 
	See text for details.}
	\end{center}
	\end{figure}
In particular, setting $\Lambda=0.8\,$TeV ($\Lambda=1$\,TeV) requires only $c_6=1.8$ 
($c_6=2.8$) while even $\Lambda=2\,$TeV is still possible in a rather strongly coupled setup 
with $c_6=11.4 < 4\pi$.\footnote{Note that UV completions could induce even larger couplings,
such as $c_6\sim 16 \pi^2$. Note also that here and in the following we neglect small 
running effects in $\lambda$ and $c_6$ from the cutoff down to the electroweak scale $v$, 
where the potential (\ref{eq:pot}) is considered.}
On the other hand, around $\cal{O}({\rm several})$ TeV, at the latest, new physics (NP) would be expected.
If the new states are uncolored (which we will assume in the following), such mass scales clearly 
introduce no tension with 
current LHC limits. Moreover, we have checked that for all values of the cutoff considered above, the inclusion 
of a $D=8$ operator with ${\cal O}(1)$ coefficient alters the numerical results by only a few per cent 
or less. It is thus save to neglect such operators.

We will now study more detailed the correlation between the needed size of the coefficient $c_6/\Lambda^2$ 
and the physical parameters in the Higgs sector, $m_h$ and $v$, stressing that only a limited part of the 
larger parameter space, considered before the discovery of the Higgs boson, is viable in 
our model.
To finally remove the redundancy of parameters in the low energy theory, allowing for an easier
discussion, we redefine
	\begin{equation}
	c_6 \equiv \bar c_6 \Lambda^2/v^2\,,
	\end{equation}
thereby eliminating the explicit reference to the new scale $\Lambda$ 
(which we shall leave open).
	\begin{figure}[!t]
	\begin{center}
	\includegraphics[height=1.75in]{mh.pdf}\qquad \includegraphics[height=1.75in]{v.pdf}
	\caption{\label{fig:mh} Required value of the Wilson coefficient $\bar c_6$ in dependence on
	the mass of the Higgs boson (left plot) and versus the value of the vev (right plot). See
	text for details.}
	\end{center}
	\end{figure}

In the left panel of Figure~\ref{fig:mh} we depict the required value of $\bar c_6$ versus the Higgs-boson mass. The red dashed line $\bar c_6 \equiv 1$ roughly corresponds to the upper boundary of the viable parameter 
space, not in conflict with the validity of the perturbative EFT. %\footnote{A careful assessment 
%of perturbativity in the EFT leads to the parametric limit $\bar c_6 < 4 \pi m_h/v$, where the numerical
%factor is however eaten up by a combinatorial factor, such that in the end really $\bar c_6<1$.}
We can see clearly that a heavy Higgs boson of only $m_h \gtrsim 300$\,GeV would have already basically 
invalidated our approach. The same would have been true for a smaller vev of $v<100$\, GeV (keeping
$m_h=125$\,GeV), which we show for completeness in the right panel of Figure~\ref{fig:mh}. The 
experimental values $m_h=125$\, GeV and $v=246$\, GeV, visualized by green vertical lines, are however in 
perfect agreement with a reasonable value of $\bar c_6\approx0.17$. Finally, the potential~(\ref{eq:pot}), employing 
these values, is plotted as a solid blue line in Figure~\ref{fig:V}.
It exposes the expected mexican-hat form, featuring a stable minimum at a non-trivial field value.
We conclude that, while it would have been easily possible that the numerical values of the mass
scales generated in nature after EWSB would have excluded our setup, the actual values just
lie in a range that allows for EWSB to be triggered by a single $D=6$ operator instead of a negative mass
squared term.

In order for the strict $\mu^2=0$ setup to be self-consistent, the $\mu^2$ term should also not 
be generated again from loop effects within the effective theory, involving physical scales. 
This is however guaranteed, since the only physical suppression scale~$\Lambda$ can always be 
factored out of loop integrals and never enters dynamically. Nevertheless, note that it is the UV 
completion that eventually needs to explain the absence of this term.


	\begin{figure}[!t]
	\begin{center}
	\includegraphics[height=1.8in]{potCW.pdf}
	\caption{\label{fig:V} Blue curve: The Higgs potential (\ref{eq:pot}), employing the physical values for 
				$m_h$ and~$v$. Red dashed curve: The Higgs potential, including the SM one-loop 
				corrections (\ref{eq:DV}), leading to the shifts (\ref{eq:Dc}).}
	\end{center}
	\end{figure}
Finally, we show that the inclusion of the SM quantum corrections to the potential, generating a
term of the form $|H|^4 \log(H^2/\mu_r^2)$ \cite{Coleman:1973jx}, only corresponds to a small 
perturbation of our setup, leading to little changes in the parameters $\lambda$ and $\bar c_6$. 
Neglecting the tiny impact of light quarks, the SM contributions to the one-loop Coleman-Weinberg potential 
are given by (see, {\it e.g.}, \cite{Quiros:1999jp})
	\begin{equation}
	\label{eq:DV}
	\Delta V =  \frac{1}{64 \pi^2} \sum_{i=W,Z,h,\chi,t} n_i\, M_i^4(H) 
	\left[ \log \frac{M_i^2(H)}{\mu_r^2} - C_i \right] \,.
	\end{equation}
Here, the tree-level field-dependent mass terms read
	\begin{equation}
	\begin{split}
	m_W^2(H)&=\frac{g^2}{2} H^2,\ m_Z^2(H)=\frac{g^2+{g^\prime}^2}{2} H^2,\\
	m_h^2(H)&=6 \lambda H^2,\ m_\chi^2(H)=2 \lambda H^2,\\[1mm]
	m_t^2(H)&=y_t^2 H^2\,,
	\end{split}
	\end{equation}
where we have dropped contributions suppressed by $\Lambda^2$,
the numbers of degrees of freedom are
	\begin{equation}
	n_W=6,\ n_Z=3,\ n_h=1,\ n_\chi=3,\ n_t=-12,
	\end{equation}
and the constants $C_i$ are given by
	\begin{equation}
	C_W=C_Z=5/6,\ C_h=C_\chi=C_t=3/2\,.
	\end{equation} 
In the end, the top quark furnishes the dominant correction. 
Adding (\ref{eq:DV}) to (\ref{eq:pot}), setting the renormalization scale to $\mu_r=v/\sqrt 2$, and solving 
for $c_6$ and $\lambda$ that reproduce correctly $v$ and $m_h$, leads to the shifts
	\begin{equation}
	\label{eq:Dc}
	\Delta \lambda \approx -0.033,\,\quad \Delta \bar c_6 \approx 0.022\,,
	\end{equation}
which is a ${\cal O}(10\%)$ effect. We show the resulting potential as a red dashed line in 
Figure~\ref{fig:V}. It becomes a little bit flatter before the zero of the undisturbed potential and a 
bit steeper afterwards. Moreover, there arises a tiny maximum at low values of $|H|$, such that the origin is
a minimum - however it corresponds to $(V+\Delta V)=0$ and thus lies much higher than the global minimum at 
$|H|=v/\sqrt 2$. The overall changes are very modest. 

\section{Phenomenology}
Beyond the potential direct discovery of new states at (or slightly above) the TeV scale, our model offers distinct signatures in 
Higgs-pair production and cosmology that we want to discuss in the following. 

First of all, the sizable coefficient $\bar c_6 \approx 0.2$ leads to a notable change in the production cross section of 
Higgs pairs, since ${\cal O}_6$ contributes to the trilinear Higgs-self interaction after EWSB. In fact, it increases the cross 
section by a factor of~$\sim (2-3)$. This is in a range that should be possible to exclude at the LHC with 
a luminosity of ${\cal L} \gtrsim$~600\,fb$^{-1}$, see \cite{Goertz:2014qta}.\footnote{Note that $\bar c_6\approx 0.2$ 
corresponds to $c_6\approx -1.2$ in the conventions used to present the final results in \cite{Goertz:2014qta}.}

	\begin{figure}[!t]
	\begin{center}
	\includegraphics[height=1.9in]{barg1.pdf}
	\caption{\label{fig:baryogen} The solid line depicts the required value of the Wilson coefficient $c_6$ in 
	dependence on the cutoff $\Lambda$ in our setup, while the blue region allows for an appropriate first order 
	electroweak phase transition such as to trigger electroweak baryogenesis. See text for details.}
	\end{center}
	\end{figure}
Beyond that, the presence of the operator ${\cal O}_6$ also modifies the electroweak phase transition. Without this 
operator, the phase transition is of second order for \mbox{$m_h=125$\, GeV} (see, {\it e.g.}, \cite{Quiros:1999jp}). This 
excludes the possibility of electroweak baryogenesis within the SM as there is no out-of equilibrium dynamics at the phase 
transition. On the other hand, a sizable contribution of ${\cal O}_6$  changes the Higgs potential such that a first order 
phase transition becomes possible for the physical Higgs mass~\cite{Grojean:2004xa}, allowing for electroweak baryogenesis
(if enough CP violation is present).
In Figure~\ref{fig:baryogen} we show again $c_6$ versus the cutoff $\Lambda$, where now the blue region corresponds
to a first order phase transition that leads to a stable $T=0$ minimum, while in addition sphaleron processes 
are sufficiently suppressed in the broken phase such as to not wash out the generated baryon asymmetry 
\cite{Grojean:2004xa}. The latter requirement leads to the condition $\langle h (T_c) \rangle /T_c \gtrsim 1$ at the critical 
temperature $T_c$.
Very interestingly, our $\mu^2=0$ solution just lies in the middle of the preferred region,
while the SM ({\it i.e.},~$c_6=0$) does not allow for electroweak baryogenesis.

We conclude that the required value of $\bar c_6$ leads to a very interesting phenomenology, allowing for pronounced
effects in Higgs-pair production as well as opening the possibility of the creation of our current universe via electroweak 
baryogenesis. This makes the setup avoiding relevant operators attractive on its own. Beyond that, it calls for an 
examination of how the effective potential (\ref{eq:pot}) could be generated - approximately or exactly - from a UV 
theory. This will be discussed in the Appendix.


\section{Conclusions}

As we know very little about the dynamics of EWSB or how the hierarchy problem is eventually solved in nature, various approaches
to EWSB should be examined and tested, in particular also from the low energy perspective, even if they might not
be the most obvious ones.
In this article, we have demonstrated that setting the notorious relevant operator $|H|^2$ in the Higgs potential to zero
and adding instead an operator ${\cal O}_6=|H|^6$ can lead to a viable electroweak symmetry breaking.
We pointed out that it is the lightness of the Higgs-boson that - perhaps unexpectedly -
 allows for this setup to be self-consistent, in a sense that it is in agreement with perturbativity, 
allowing a reasonably small $c_6$ and a natural NP scale of
 $\Lambda \sim \mbox{$(1-2)$}$\,TeV. The latter is needed in order to create a mass gap that allows to neglect operators with 
$D=8$ and beyond. Moreover, once set to zero, the $\mu^2$ term is not re-generated
in the low energy theory. Eliminating this parameter and adding instead the $D=6$ coefficient $c_6$ keeps the theory
very predictive, since the number of parameters stays the same. 
In particular, the setup is fully testable in experiments currently under way, since relatively large changes in the Higgs-pair 
production cross section are predicted.

The main purpose of our analysis was to study the phenomenological viability of this distinct IR limit, allowing
new directions in model building. However, in the appendix we will also provide already a first classification of UV 
completions that could lead to such a setup at least approximately, or exactly, 
depending on further structures of the UV theory. There, we will also entertain the possibility 
that, while the Higgs boson could be a fundamental scalar, all scales are generated dynamically. 

As it is a distinct theoretical limit, which also opens the possibility of generating 
our universe via baryogenesis at the electroweak scale and interestingly enough is not ruled out by measurements in 
the Higgs sector yet, the $\mu^2 \to 0$  model examined here should be considered as an alternative mechanism of 
breaking electroweak symmetry dynamically.


\paragraph{Acknowledgments}

I am grateful to Roberto Contino,  Adam Falkowski, Gino Isidori, Andrey Katz, Valya Khoze, Andreas Weiler, 
and Jos{\'e} Zurita for useful discussions and comments.
The research of the author is supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (grant no. PIEF-GA-2013-628224).


\begin{appendix}
\section{Possible UV Completions}
\label{sec:App}

So far, the form of the Higgs potential  (\ref{eq:pot}) was considered as a matching 
condition on the unspecified UV completion.
Now, we are going to discuss classes of UV models that could generate such a potential at the tree level. 
The general picture will be that the SM-like theory (featuring $\mu^2=0$) possesses no scale at the 
classical level and is then coupled to a sector that breaks scale invariance, potentially dynamically. 
%as depicted in Figure~\ref{fig:sectors}
Note that such an additional breaking is needed in the first place, since it was found that the breaking of scale invariance 
within the SM by dimensional transmutation is not sufficient to generate the Higgs mass of $m_h=125$\,GeV (see, {\it e.g.}, 
\cite{Hempfling:1996ht,Englert:2013gz}). 
The NP might itself respect scale invariance at the classical level, generating all masses dynamically.
However in the first part of the following discussion we will leave open how the scales in the new sector emerge.


%There, we will in particular provide 
%the connection to models that address the hierarchy problem by invoking the notion of classical scale invariance (CSI)
%\cite{Bardeen:1995kv,Meissner:2006zh,Meissner:2007xv,Englert:2013gz}, showing how in our model all scales could 
%be generated dynamically, in an orthogonal way compared to the usual models of CSI (see, {\it e.g.}, \cite{Hempfling:1996ht, 
%Chang:2007ki,Englert:2013gz} and references therein).
%
%	\begin{figure}[!t]
%	\begin{center}
%	\raisebox{-2mm}{\includegraphics[height=1.2in]{CSI.pdf}}
%	\caption{\label{fig:sectors} Pictorial representation of the breaking of scale invariance via interactions with
%	a new physics sector.}
%	\end{center}
%	\end{figure}

 Let us nevertheless already stress a difference of our approach from the usual one,
often used in models employing classical scale invariance (CSI) as a building principle. In the latter, the $\mu^2$ term is 
forbidden at the tree level, but then regenerated spontaneously, usually via the (loop-induced) vev of an additional scalar 
singlet in a Higgs portal term, mimicking the usual SM Higgs potential  (see, {\it e.g.}, \cite{Hempfling:1996ht,Englert:2013gz},
as well as in general \cite{multi} on models that generate all scales dynamically).
Thus, the low energy 
phenomenology can be quite similar to the one of the SM. In our approach, however, no relevant operator needs to 
be generated in the electroweak-scale theory at all. The breaking of scale invariance is induced in an orthogonal - 
possibly also spontaneous/dynamical - way, via an {\it irrelevant} operator, introduced by integrating out a heavy field 
that couples to the SM. This leads to a distinct low energy phenomenology and full testability of our 
setup. It provides a new minimal way of allowing for a viable EWSB in the presence of the scale-invariant tree-level 
SM Lagrangian, that interestingly features $m_h \to 0$ in the decoupling limit of $\Lambda \to \infty$.
\footnote{Also a mixture of both approaches, generating a very small (potentially even positive) $\mu^2$, while assisting with ${\cal O}_6$ 
to trigger EWSB in a theory respecting scale invariance at the tree level, might be interesting to consider.} 

\subsection{General Discussion}
\label{sec:Gen}
We will start with the option of not worrying weather CSI is a good building principle for the full Lagrangian. 
Depending on how the NP scale is generated, the full models might possess CSI or not. 

A scalar field $S$ with mass $M_S$, transforming as a singlet of $SU(2)_L$, can generate the 
operator ${\cal O}_6$ at the tree level, allowing the NP to be not too light, while a potential contribution to the $|H|^2$ operator could be deferred to the loop level (or beyond, in the presence of additional structures).
The singlet can in general feature two interaction structures with the Higgs boson, $M_S  \lambda_S S |H|^2$ 
and $\lambda_p S^2 |H|^2$, and both are needed to induce ${\cal O}_6$ at the tree level, as shown in 
Figure~\ref{fig:intout}, leading to $c_6/\Lambda^2 \sim \lambda_p|\lambda_S|^2 /M_S^2$. 
Note in that context that generating $c_6$ at the loop level would be less straightforward, since unless the perturbativity bound of the underlying theory is saturated, it would in general tend to be too small.
If one wants to insist on generating all scales dynamically, the dimensionfull coefficient 
in front of $\lambda_S$ could be thought of as a vev of a new field, or arise from a compositeness scale,
see below. Since at the tree level only ${\cal O}_6$ is generated, one could entertain the possibility that
quadratically cutoff-dependent quantum corrections to $\mu^2$ are canceled in an extended singlet NP sector,
e.g. by invoking (partial) supersymmetry or a twin Higgs mechanism,
such as to approximate, or fully satisfy, (\ref{eq:pot}). 
A related discussion on a complete cancellation of UV effects on the $|H|^2$ operator - 
which however there is generated again spontaneously - 
is given in \cite{Englert:2013gz} (see also \cite{Hempfling:1996ht}). 
Alternatively, one could imagine the interaction terms to be cut off by a strong form factor, such that $\lambda_i(p)$ are rapidly vanishing for large momentum $p$ flowing through the vertex. While this would tame loops, it would again not prevent a sizable tree level contribution to ${\cal O}_6$ via integrating out $S$ 
at zero momentum. Finally, one could just enforce the vanishing of the (renormalized) relevant operator by hand at the matching scale. In any case, the scalar $S$ allows to entertain a UV completion where 
EWSB could be driven by ${\cal O}_6$ and not by a negative sign $\mu^2$ term.
If the new singlet does not develop a vev and has a mass $M_S \gg v$, with a significant fraction
not stemming from the SM-Higgs sector, it should be save from current limits. 
A survey of the impact of the new fields from the UV 
completion on Higgs phenomenology (and precision observables) is left for future work. 
For an overview of constraints on scalar extensions of the SM see, {\it e.g.}, 
\cite{deBlas:2014mba}.\footnote{Extension of the vector-boson sector could also generate ${\cal O}_6$ 
at the tree level, see \cite{delAguila:2010mx}.}
	\begin{figure}[!t]
	\begin{center}
	\includegraphics[height=1in]{singlet.jpg}
	\caption{\label{fig:intout} Generation of the operator ${\cal O}_6$ by integrating out the 
	new scalar singlet $S$ at the tree level. See text for details.}
	\end{center}
	\end{figure}

\subsection{Dynamical Generation of all Scales}

Now we are also going to address how the scale in the UV completions emerges, showing concrete possibilities to generate 
all scales dynamically. It will thus not be necessary to assume a relevant operator at all, neither in the shortest-distance 
UV theory, nor in the IR limit $E\sim v$.
 We will first consider compositeness as a rather natural mechanism to obtain
a mass from binding energy, without the need of a relevant operator in the UV theory.
Then we will discuss the option of generating a mass from the Coleman-Weinberg mechanism.

\subsubsection{Compositeness}

One elegant option to obtain the mass for the particles to be integrated out is to consider them to be composites, bound together by a new strong interaction.
We assume the condensation scale, beyond which one could resolve the fundamental constituents, to lie
well above 1\,TeV, such that at currently accessible energies one would never see them directly.
In particular, a set of two fermions (or scalars) could form a massive scalar singlet of $SU(2)_L$,
given that they transform both in the same representation.
These fields could be massless before they condense - 
however the UV constituents could, depending on their quantum numbers, possibly also get a mass from the Higgs 
sector after EWSB. 
Beyond the mass of the composite state, as discussed above, one needs a scale to be present in the
$S |H|^2$ interaction. This in turn could also be generated from the composite sector or arise 
via the vev of a further singlet from the Coleman-Weinberg mechanism (see below).

\subsubsection{Coleman-Weinberg Mechanism}

Before concluding, let us also entertain the option to generate all scales via the Coleman-Weinberg mechanism.
The simplest Coleman-Weinberg setups always rely on a non-negligible vev for a scalar
in order to generate its mass \cite{Hempfling:1996ht,Englert:2013gz}. To avoid potential issues 
with precision tests or Higgs physics from $SU(2)_L$ breaking as well as the direct 
generation of the $\mu^2$ term, in our case we need to assume the presence of one more scalar particle 
compared to the general discussion at the beginning of the section, which however could be elementary. 
In practice, another singlet $S_2$ could obtain a vev from the Coleman-Weinberg mechanism as 
explained before (see also \cite{Englert:2013gz}). In turn, portal interactions with $S$ 
would generate its mass dynamically. On the other hand, direct portals to the SM Higgs need to be strongly suppressed in the thought setup. This could {\it e.g.} be realized geometrically in an 
extra-dimensional framework, with $S$ being
the messengers between the SM and the new singlet $S_2$, residing on different branes.

Finally, nature might have chosen a  completely different way to generate ${\cal O}_6$, while avoiding the 
$\mu^2$ term, still to be found. A further analysis of the potential UV completions, including the examination of dark matter candidates, will be deferred to future work.

\end{appendix}
\begin{thebibliography}{99}

%\bibitem{Bardeen:1995kv}
%  W.~A.~Bardeen,
 % %``On naturalness in the standard model,''
 % FERMILAB-CONF-95-391-T.
  %%CITATION = FERMILAB-CONF-95-391-T;%%

%\bibitem{Meissner:2006zh}
%  K.~A.~Meissner and H.~Nicolai,
%  %``Conformal Symmetry and the Standard Model,''
%  Phys.\ Lett.\ B {\bf 648} (2007) 312
%  [hep-th/0612165].
%  %%CITATION = HEP-TH/0612165;%%
%  %151 citations counted in INSPIRE as of 18 Mar 2015
%
%\bibitem{Meissner:2007xv}
%  K.~A.~Meissner and H.~Nicolai,
%  %``Effective action, conformal anomaly and the issue of quadratic divergences,''
%  Phys.\ Lett.\ B {\bf 660} (2008) 260
%  [arXiv:0710.2840 [hep-th]].
%  %%CITATION = ARXIV:0710.2840;%%
%  %64 citations counted in INSPIRE as of 18 Mar 2015

  %%CITATION = HEP-PH/9604278;%%
%
%\bibitem{Chang:2007ki}
%  W.~-F.~Chang, J.~N.~Ng and J.~M.~S.~Wu,
%  %``Shadow Higgs from a scale-invariant hidden U(1)(s) model,''
%  Phys.\ Rev.\ D {\bf 75} (2007) 115016
%  [hep-ph/0701254 [hep-ph]].
%  %%CITATION = HEP-PH/0701254;%%



\bibitem{Coleman:1973jx}
  S.~R.~Coleman and E.~J.~Weinberg,
  %``Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,''
  Phys.\ Rev.\ D {\bf 7} (1973) 1888.
  %%CITATION = PHRVA,D7,1888;%%

\bibitem{Quiros:1999jp}
  M.~Quiros,
  %``Finite temperature field theory and phase transitions,''
  hep-ph/9901312.
  %%CITATION = HEP-PH/9901312;%%
  %120 citations counted in INSPIRE as of 15 Mar 2015

\bibitem{Goertz:2014qta}
  F.~Goertz, A.~Papaefstathiou, L.~L.~Yang and J.~Zurita,
  %``Higgs boson pair production in the D=6 extension of the SM,''
  arXiv:1410.3471 [hep-ph].
  %%CITATION = ARXIV:1410.3471;%%
  %4 citations counted in INSPIRE as of 28 Mar 2015

\bibitem{Grojean:2004xa}
  C.~Grojean, G.~Servant and J.~D.~Wells,
  %``First-order electroweak phase transition in the standard model with a low cutoff,''
  Phys.\ Rev.\ D {\bf 71} (2005) 036001
  [hep-ph/0407019].
  %%CITATION = HEP-PH/0407019;%%
  %124 citations counted in INSPIRE as of 11 Mar 2015

\bibitem{Hempfling:1996ht}
  R.~Hempfling,
  %``The Next-to-minimal Coleman-Weinberg model,''
  Phys.\ Lett.\ B {\bf 379} (1996) 153
  [hep-ph/9604278].

\bibitem{Englert:2013gz}
  C.~Englert, J.~Jaeckel, V.~V.~Khoze and M.~Spannowsky,
  %``Emergence of the Electroweak Scale through the Higgs Portal,''
  JHEP {\bf 1304} (2013) 060
  [arXiv:1301.4224 [hep-ph]].
  %%CITATION = ARXIV:1301.4224;%%
  %64 citations counted in INSPIRE as of 11 mar 2015

\bibitem{multi}
  W.~A.~Bardeen,
  % %``On naturalness in the standard model,''
   FERMILAB-CONF-95-391-T.
   J.~P.~Fatelo, J.~M.~Gerard, T.~Hambye and J.~Weyers,
  %``Symmetry breaking induced by top loops,''
  Phys.\ Rev.\ Lett.\  {\bf 74} (1995) 492.
  T.~Hambye,
  %``Symmetry breaking induced by top quark loops from a model without scalar mass,''
  Phys.\ Lett.\ B {\bf 371} (1996) 87
  [hep-ph/9510266].
    P.~H.~Frampton and C.~Vafa,
  %``Conformal approach to particle phenomenology,''
  hep-th/9903226.
    K.~A.~Meissner and H.~Nicolai,
  %``Conformal Symmetry and the Standard Model,''
  Phys.\ Lett.\ B {\bf 648} (2007) 312
  [hep-th/0612165].
  W.~-F.~Chang, J.~N.~Ng and J.~M.~S.~Wu,
  %``Shadow Higgs from a scale-invariant hidden U(1)(s) model,''
  Phys.\ Rev.\ D {\bf 75} (2007) 115016
  [hep-ph/0701254].
  R.~Foot, A.~Kobakhidze and R.~R.~Volkas,
  %``electro-weak Higgs as a pseudo-Goldstone boson of broken scale invariance,''
  Phys.\ Lett.\ B {\bf 655} (2007) 156
  [arXiv:0704.1165].
  K.~A.~Meissner and H.~Nicolai,
  %``Effective action, conformal anomaly and the issue of quadratic divergences,''
  Phys.\ Lett.\ B {\bf 660} (2008) 260
  [arXiv:0710.2840].
    S.~Iso, N.~Okada, Y.~Orikasa,
  %``The minimal B-L model naturally realized at TeV scale,''
  Phys.\ Rev.\ D {\bf 80} (2009) 115007
  [arXiv:0909.0128].
    M.~Shaposhnikov and C.~Wetterich,
  %``Asymptotic safety of gravity and the Higgs boson mass,''
  Phys.\ Lett.\ B {\bf 683} (2010) 196
  [arXiv:0912.0208].
   L.~Alexander-Nunneley and A.~Pilaftsis,
  %``The Minimal Scale Invariant Extension of the Standard Model,''
  JHEP {\bf 1009} (2010) 021
  [arXiv:1006.5916].
  T.~Hur and P.~Ko,
  %``Scale invariant extension of the standard model with strongly interacting hidden sector,''
  Phys.\ Rev.\ Lett.\  {\bf 106}, 141802 (2011)
  [arXiv:1103.2571].  
  S.~Iso and Y.~Orikasa,
  %``TeV Scale B-L model with a flat Higgs potential at the Planck scale - in view of the hierarchy problem -,''
  PTEP {\bf 2013} (2013) 023B08
  [arXiv:1210.2848].
   E.~J.~Chun, S.~Jung and H.~M.~Lee,
  %``Radiative generation of the Higgs potential,''
  Phys.\ Lett.\ B {\bf 725} (2013) 158
   [Erratum-ibid.\ B {\bf 730} (2014) 357]
  [arXiv:1304.5815].
    M.~Heikinheimo, A.~Racioppi, M.~Raidal, C.~Spethmann and K.~Tuominen,
  %``Physical Naturalness and Dynamical Breaking of Classical Scale Invariance,''
  Mod.\ Phys.\ Lett.\ A {\bf 29} (2014) 1450077
  [arXiv:1304.7006]. 
T.~Hambye and A.~Strumia,
  %``Dynamical generation of the weak and Dark Matter scale,''
  Phys.\ Rev.\ D {\bf  88} (2013) 055022
  [arXiv:1306.2329].
  C.~D.~Carone and R.~Ramos,
  %``Classical scale-invariance, the electro-weak scale and vector dark matter,''
  Phys.\ Rev.\ D {\bf 88} (2013) 055020
  [arXiv:1307.8428].
     A.~Farzinnia, H.~J.~He and J.~Ren,
  %``Natural Electroweak Symmetry Breaking from Scale Invariant Higgs Mechanism,''
  Phys.\ Lett.\ B {\bf 727} (2013) 141
  [arXiv:1308.0295].
  E.~Gabrielli, M.~Heikinheimo, K.~Kannike, A.~Racioppi, M.~Raidal and C.~Spethmann,
  %``Towards Completing the Standard Model: Vacuum Stability, EWSB and Dark Matter,''
  Phys.\ Rev.\ D {\bf 89} (2014) 1,  015017
  [arXiv:1309.6632].
   R.~Foot, A.~Kobakhidze, K.~L.~McDonald and R.~R.~Volkas,
  %``Poincare Protection for a Natural electro-weak Scale,''
  Phys.\ Rev.\ D {\bf 89} (2014) 115018
  [arXiv:1310.0223].
    M.~Holthausen, J.~Kubo, K.~S.~Lim and M.~Lindner,
  %``Electroweak and Conformal Symmetry Breaking by a Strongly Coupled Hidden Sector,''
  JHEP {\bf 1312} (2013) 076
  [arXiv:1310.4423].
  C.~T.~Hill,
  %``Is the Higgs Boson Associated with Coleman-Weinberg Dynamical Symmetry Breaking?,''
  Phys.\ Rev.\ D {\bf 89} (2014) 073003
  [arXiv:1401.4185].
  J.~Guo and Z.~Kang,
  %``Higgs Naturalness and Dark Matter Stability by Scale Invariance,''
  arXiv:1401.5609.
  S.~Benic and B.~Radovcic,
  %``Electroweak breaking and Dark Matter from the common scale,''
  Phys.\ Lett.\ B {\bf 732} (2014) 91
  [arXiv:1401.8183].
 P.~H.~Chankowski, A.~Lewandowski, K.~A.~Meissner and H.~Nicolai,
  %``Softly broken conformal symmetry and the stability of the electroweak scale,''
  Mod.\ Phys.\ Lett.\ A {\bf 30} (2015) 02,  1550006
  [arXiv:1404.0548].
   K.~Allison, C.~T.~Hill and G.~G.~Ross,
  %``Ultra-weak sector, Higgs boson mass, and the dilaton,''
  Phys.\ Lett.\ B {\bf 738} (2014) 191
  [arXiv:1404.6268].
  W.~Altmannshofer, W.~A.~Bardeen, M.~Bauer, M.~Carena and J.~D.~Lykken,
  %``Light Dark Matter, Naturalness, and the Radiative Origin of the Electroweak Scale,''
  JHEP {\bf 1501} (2015) 032
  [arXiv:1408.3429].
O.~Antipin, M.~Redi and A.~Strumia,
  %``Dynamical generation of the weak and Dark Matter scales from strong interactions,''
  JHEP {\bf 1501} (2015) 157
  [arXiv:1410.1817].
  G.~F.~Giudice, G.~Isidori, A.~Salvio and A.~Strumia,
  %``Softened Gravity and the Extension of the Standard Model up to Infinite Energy,''
  JHEP {\bf 1502} (2015) 137
  [arXiv:1412.2769].

\bibitem{deBlas:2014mba}
  J.~de Blas, M.~Chala, M.~Perez-Victoria and J.~Santiago,
  %``Observable Effects of General New Scalar Particles,''
  arXiv:1412.8480 [hep-ph].
  %%CITATION = ARXIV:1412.8480;%%
  %1 citations counted in INSPIRE as of 31 Mar 2015

\bibitem{delAguila:2010mx}
  F.~del Aguila, J.~de Blas and M.~Perez-Victoria,
  %``Electroweak Limits on General New Vector Bosons,''
  JHEP {\bf 1009} (2010) 033
  [arXiv:1005.3998 [hep-ph]].
  %%CITATION = ARXIV:1005.3998;%%
  %67 citations counted in INSPIRE as of 22 Apr 2015

\end{thebibliography}



\end{document}

1504.00355.extracted_bib
{Bardeen:1995kv % W.~A.~Bardeen, % %``On naturalness in the standard model,'' % FERMILAB-CONF-95-391-T. %%CITATION = FERMILAB-CONF-95-391-T;%% %
1504.00355.extracted_bib
{Meissner:2006zh % K.~A.~Meissner and H.~Nicolai, % %``Conformal Symmetry and the Standard Model,'' % Phys.\ Lett.\ B {648 (2007) 312 % [hep-th/0612165]. % %%CITATION = HEP-TH/0612165;%% % %151 citations counted in INSPIRE as of 18 Mar 2015 % %
1504.00355.extracted_bib
{Meissner:2007xv % K.~A.~Meissner and H.~Nicolai, % %``Effective action, conformal anomaly and the issue of quadratic divergences,'' % Phys.\ Lett.\ B {660 (2008) 260 % [arXiv:0710.2840 [hep-th]]. % %%CITATION = ARXIV:0710.2840;%% % %64 citations counted in INSPIRE as of 18 Mar 2015 %%CITATION = HEP-PH/9604278;%% % %
1504.00355.extracted_bib
{Chang:2007ki % W.~-F.~Chang, J.~N.~Ng and J.~M.~S.~Wu, % %``Shadow Higgs from a scale-invariant hidden U(1)(s) model,'' % Phys.\ Rev.\ D {75 (2007) 115016 % [hep-ph/0701254 [hep-ph]]. % %%CITATION = HEP-PH/0701254;%%
1504.00355.extracted_bib
{Coleman:1973jx S.~R.~Coleman and E.~J.~Weinberg, %``Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,'' Phys.\ Rev.\ D {7 (1973) 1888. %%CITATION = PHRVA,D7,1888;%%
1504.00355.extracted_bib
{Quiros:1999jp M.~Quiros, %``Finite temperature field theory and phase transitions,'' hep-ph/9901312. %%CITATION = HEP-PH/9901312;%% %120 citations counted in INSPIRE as of 15 Mar 2015
1504.00355.extracted_bib
{Goertz:2014qta F.~Goertz, A.~Papaefstathiou, L.~L.~Yang and J.~Zurita, %``Higgs boson pair production in the D=6 extension of the SM,'' arXiv:1410.3471 [hep-ph]. %%CITATION = ARXIV:1410.3471;%% %4 citations counted in INSPIRE as of 28 Mar 2015
1504.00355.extracted_bib
{Grojean:2004xa C.~Grojean, G.~Servant and J.~D.~Wells, %``First-order electroweak phase transition in the standard model with a low cutoff,'' Phys.\ Rev.\ D {71 (2005) 036001 [hep-ph/0407019]. %%CITATION = HEP-PH/0407019;%% %124 citations counted in INSPIRE as of 11 Mar 2015
1504.00355.extracted_bib
{Hempfling:1996ht R.~Hempfling, %``The Next-to-minimal Coleman-Weinberg model,'' Phys.\ Lett.\ B {379 (1996) 153 [hep-ph/9604278].
1504.00355.extracted_bib
{Englert:2013gz C.~Englert, J.~Jaeckel, V.~V.~Khoze and M.~Spannowsky, %``Emergence of the Electroweak Scale through the Higgs Portal,'' JHEP {1304 (2013) 060 [arXiv:1301.4224 [hep-ph]]. %%CITATION = ARXIV:1301.4224;%% %64 citations counted in INSPIRE as of 11 mar 2015
1504.00355.extracted_bib
{multi W.~A.~Bardeen, % %``On naturalness in the standard model,'' FERMILAB-CONF-95-391-T. J.~P.~Fatelo, J.~M.~Gerard, T.~Hambye and J.~Weyers, %``Symmetry breaking induced by top loops,'' Phys.\ Rev.\ Lett.\ {74 (1995) 492. T.~Hambye, %``Symmetry breaking induced by top quark loops from a model without scalar mass,'' Phys.\ Lett.\ B {371 (1996) 87 [hep-ph/9510266]. P.~H.~Frampton and C.~Vafa, %``Conformal approach to particle phenomenology,'' hep-th/9903226. K.~A.~Meissner and H.~Nicolai, %``Conformal Symmetry and the Standard Model,'' Phys.\ Lett.\ B {648 (2007) 312 [hep-th/0612165]. W.~-F.~Chang, J.~N.~Ng and J.~M.~S.~Wu, %``Shadow Higgs from a scale-invariant hidden U(1)(s) model,'' Phys.\ Rev.\ D {75 (2007) 115016 [hep-ph/0701254]. R.~Foot, A.~Kobakhidze and R.~R.~Volkas, %``electro-weak Higgs as a pseudo-Goldstone boson of broken scale invariance,'' Phys.\ Lett.\ B {655 (2007) 156 [arXiv:0704.1165]. K.~A.~Meissner and H.~Nicolai, %``Effective action, conformal anomaly and the issue of quadratic divergences,'' Phys.\ Lett.\ B {660 (2008) 260 [arXiv:0710.2840]. S.~Iso, N.~Okada, Y.~Orikasa, %``The minimal B-L model naturally realized at TeV scale,'' Phys.\ Rev.\ D {80 (2009) 115007 [arXiv:0909.0128]. M.~Shaposhnikov and C.~Wetterich, %``Asymptotic safety of gravity and the Higgs boson mass,'' Phys.\ Lett.\ B {683 (2010) 196 [arXiv:0912.0208]. L.~Alexander-Nunneley and A.~Pilaftsis, %``The Minimal Scale Invariant Extension of the Standard Model,'' JHEP {1009 (2010) 021 [arXiv:1006.5916]. T.~Hur and P.~Ko, %``Scale invariant extension of the standard model with strongly interacting hidden sector,'' Phys.\ Rev.\ Lett.\ {106, 141802 (2011) [arXiv:1103.2571]. S.~Iso and Y.~Orikasa, %``TeV Scale B-L model with a flat Higgs potential at the Planck scale - in view of the hierarchy problem -,'' PTEP {2013 (2013) 023B08 [arXiv:1210.2848]. E.~J.~Chun, S.~Jung and H.~M.~Lee, %``Radiative generation of the Higgs potential,'' Phys.\ Lett.\ B {725 (2013) 158 [Erratum-ibid.\ B {730 (2014) 357] [arXiv:1304.5815]. M.~Heikinheimo, A.~Racioppi, M.~Raidal, C.~Spethmann and K.~Tuominen, %``Physical Naturalness and Dynamical Breaking of Classical Scale Invariance,'' Mod.\ Phys.\ Lett.\ A {29 (2014) 1450077 [arXiv:1304.7006]. T.~Hambye and A.~Strumia, %``Dynamical generation of the weak and Dark Matter scale,'' Phys.\ Rev.\ D {88 (2013) 055022 [arXiv:1306.2329]. C.~D.~Carone and R.~Ramos, %``Classical scale-invariance, the electro-weak scale and vector dark matter,'' Phys.\ Rev.\ D {88 (2013) 055020 [arXiv:1307.8428]. A.~Farzinnia, H.~J.~He and J.~Ren, %``Natural Electroweak Symmetry Breaking from Scale Invariant Higgs Mechanism,'' Phys.\ Lett.\ B {727 (2013) 141 [arXiv:1308.0295]. E.~Gabrielli, M.~Heikinheimo, K.~Kannike, A.~Racioppi, M.~Raidal and C.~Spethmann, %``Towards Completing the Standard Model: Vacuum Stability, EWSB and Dark Matter,'' Phys.\ Rev.\ D {89 (2014) 1, 015017 [arXiv:1309.6632]. R.~Foot, A.~Kobakhidze, K.~L.~McDonald and R.~R.~Volkas, %``Poincare Protection for a Natural electro-weak Scale,'' Phys.\ Rev.\ D {89 (2014) 115018 [arXiv:1310.0223]. M.~Holthausen, J.~Kubo, K.~S.~Lim and M.~Lindner, %``Electroweak and Conformal Symmetry Breaking by a Strongly Coupled Hidden Sector,'' JHEP {1312 (2013) 076 [arXiv:1310.4423]. C.~T.~Hill, %``Is the Higgs Boson Associated with Coleman-Weinberg Dynamical Symmetry Breaking?,'' Phys.\ Rev.\ D {89 (2014) 073003 [arXiv:1401.4185]. J.~Guo and Z.~Kang, %``Higgs Naturalness and Dark Matter Stability by Scale Invariance,'' arXiv:1401.5609. S.~Benic and B.~Radovcic, %``Electroweak breaking and Dark Matter from the common scale,'' Phys.\ Lett.\ B {732 (2014) 91 [arXiv:1401.8183]. P.~H.~Chankowski, A.~Lewandowski, K.~A.~Meissner and H.~Nicolai, %``Softly broken conformal symmetry and the stability of the electroweak scale,'' Mod.\ Phys.\ Lett.\ A {30 (2015) 02, 1550006 [arXiv:1404.0548]. K.~Allison, C.~T.~Hill and G.~G.~Ross, %``Ultra-weak sector, Higgs boson mass, and the dilaton,'' Phys.\ Lett.\ B {738 (2014) 191 [arXiv:1404.6268]. W.~Altmannshofer, W.~A.~Bardeen, M.~Bauer, M.~Carena and J.~D.~Lykken, %``Light Dark Matter, Naturalness, and the Radiative Origin of the Electroweak Scale,'' JHEP {1501 (2015) 032 [arXiv:1408.3429]. O.~Antipin, M.~Redi and A.~Strumia, %``Dynamical generation of the weak and Dark Matter scales from strong interactions,'' JHEP {1501 (2015) 157 [arXiv:1410.1817]. G.~F.~Giudice, G.~Isidori, A.~Salvio and A.~Strumia, %``Softened Gravity and the Extension of the Standard Model up to Infinite Energy,'' JHEP {1502 (2015) 137 [arXiv:1412.2769].
1504.00355.extracted_bib
{deBlas:2014mba J.~de Blas, M.~Chala, M.~Perez-Victoria and J.~Santiago, %``Observable Effects of General New Scalar Particles,'' arXiv:1412.8480 [hep-ph]. %%CITATION = ARXIV:1412.8480;%% %1 citations counted in INSPIRE as of 31 Mar 2015
1504.00355.extracted_bib
{delAguila:2010mx F.~del Aguila, J.~de Blas and M.~Perez-Victoria, %``Electroweak Limits on General New Vector Bosons,'' JHEP {1009 (2010) 033 [arXiv:1005.3998 [hep-ph]]. %%CITATION = ARXIV:1005.3998;%% %67 citations counted in INSPIRE as of 22 Apr 2015