# Enhancements to cavity quantum electrodynamics system

A. D. Cimmarusti , J. A. Crawford , D. G. Norris and L. A. Orozco  Joint
Quantum Institute, Department of Physics, University of Maryland and National
Institute of Standards and Technology, College Park, MD 20742-4111, U.S.A.

###### Abstract

We show the planned upgrade of a cavity QED experimental apparatus. The system
consists of an optical cavity and an ensemble of ultracold 85Rb atoms coupled
to its mode. We propose enhancements to both. First, we document the building
process for a new cavity, with a planned finesse of $\sim$20000\. We address
problems of maintaining mirror integrity during mounting and improving
vibration isolation. Second, we propose improvements to the cold atom source
in order to achieve better optical pumping and control over the flux of atoms.
We consider a 2-D optical molasses for atomic beam deflection, and show
computer simulation results for evaluating the design. We also examine the
possibility of all-optical atomic beam focusing, but find that it requires
unreasonable experimental parameters.

###### pacs:

42.15.Eq, 42.50.Pq, 32.60.+i

##  I Introduction

The simplest realization of a cavity quantum electrodynamic system consists of
a single material fermion coupled to a boson field. Our work in the optical
regime couples Rb atoms and a finite number of modes of an optical cavity
berman94 . These systems have numerous applications in quantum information
science turchette95b ; cirac97 ; gheri98 ; wilk07b , and also enable the study
of quantum optics effects difficult to observe in free space norris10 ;
norris09a ; terraciano09 ; bishop08 ; guerlin07 ; harochebook ; hennrich05 .

Our current cavity QED experimental setup incorporates two independent vacuum
chamber components: a spherical hexagon and a cube (See Fig. 1). The former
houses an unbalanced Magneto-Optical Trap (MOT), from which we obtain a Low-
Velocity Intense Source (LVIS) of 85Rb atoms lu96 . The MOT uses a pair of
coils to generate a quadrupole magnetic field, with two retro-reflected laser
beams in the horizontal plane, and a third along the axis of the strong
magnetic field gradient (vertical direction). At the bottom of the chamber,
the vertical beam strikes a gold mirror and quarter-wave plate that have a
1.5-mm diameter hole, yielding a region of unbalanced laser intensity that
pushes out a beam of cold atoms. Directly below the hexagon lies a cubic
chamber which contains the optical cavity. Atoms propagate 3-4 cm through the
mirror to the cavity and are optically pumped to a desired $m$ state en route.
A fraction of the atoms pass between the cavity mirrors (2.2 mm spacing) and
couple to the TEM00 mode ($1/e$ field radius 56 $\mu$m) for $\sim$ 5 $\mu$s
before striking the bottom of the vacuum chamber.

Figure 1:  (color online) Current vacuum chamber (not drawn to scale).
Unbalanced MOT (red beams) by hole in mirror creates LVIS.

The current arrangement has room for improvement in several fronts: Light from
the vertical MOT beam propagates collinearly with the atoms, complicating the
desired optical pumping distribution and driving the atoms weakly while they
are coupled to the cavity mode. We are unable to access the regime of
collective strong coupling in this system, due to the small number of atoms
that couple fully to the cavity mode. This is primarily due to the
distribution of transverse velocities of the atoms coming from the LVIS, which
results in a broadening spatial profile and lower atom density as the beam
propagates. To increase the single atom coupling we decrease the mode volume
by a factor of two and keep the decay rate of the cavity increasing the
finesse also by a factor of two. However, the reduction of the volume
complicates the collective strong coupling.

The structure of the paper is as follows: Section 2 talks about the progress
in the construction of the new optical cavity. In section 3, we discuss
deflection of an LVIS by a 2-D optical molasses (OM). We explore the
possibility of focusing an atomic beam in section 4, and we end with
conclusions in section 5.

##  II Building the optical cavity

At the heart of every optical cavity QED apparatus lies a high finesse (low
loss) resonator. As the optical transmission coefficients for the mirrors used
in the resonator may be as low as a few parts per million, reducing the
absorptive losses to this level is a great experimental challenge. Our present
cavity (decay rate $\kappa/2\pi=2.8\times 10^{6}$ s-1 and finesse of 11,000)
suffers from higher than expected losses, reducing its finesse by about a
factor of two from the intended value ( $\sim$25,000, as calculated using the
reflectivity of the mirrors). We believe the losses are due to adhesive
residue inadvertently deposited on the mirror surfaces during construction, as
well as the aging of the high-reflectivity coatings over the nearly two
decades since deposition. We are constructing a new cavity with small and
well-characterized losses, as well as slightly stronger coupling to the atoms
and enhanced vibration isolation.

Two spherical mirrors with Research Electro-Optics (REO) high-reflectivity
coatings form the cavity. For precise alignment, these circular mirrors sit in
a square groove machined into a single piece of MACOR, a low-outgassing
nasa_outgas , machinable ceramic that can withstand high temperatures. The
MACOR is adhered atop two separate piezoelectric transducers (PZTs), bridging
the 2 mm gap between them. With the MACOR affixed to the PZTs and the PZTs
affixed to a stainless steel base, we cut the MACOR piece in two, thereby
allowing for independent translation of each half of the cavity. It is
desireable to cut the MACOR with minimal force, so as to avoid putting stress
on the PZTs or comprimising the alignment of the groove in the MACOR. This
procedure provides good alignment of the MACOR grooves between the two halves,
so that simply resting the mirrors in the grooves before gluing keeps the
reflective surfaces parallel. To maintain high vacuum ($\sim 10^{-8}$ Torr or
lower) with the cavity mount inserted in the chamber, we use only adhesives
that NASA rates as low-outgassing based on percent total mass loss at low
pressure nasa_outgas  such as Loctite Hysol 1C. In place of solder, we use
EPO-TEK H20E, a silver-filled electrically conductive low-outgassing epoxy, to
bond the PZTs to wires and the stainless steel base. Figure 2 shows the
various steps followed.

Figure 2:  (color online) Photographs of the cavity building process. (A) Non-
magnetic steel base, (B) PZTs on base, (C) MACOR with slit in the middle and
(D) high reflectivity mirrors.

The choice of curvature and coating for the two mirrors depends on the desired
cavity properties. We aim to construct a cavity with coupling in the
intermediate regime (i.e. cavity decay rate $\kappa$ of the same order as
single-atom coupling rate $g$ and spontaneous emission rate $\gamma$), and
with a finesse of about 20,000. After nearly two decades of aging, the REO
mirrors require direct measurement of their transmission properties in order
to determine the expected finesse.

We use a 5 mW, 780 nm laser beam and a mounted optical power meter to measure
the intensity transmission coefficient $T$ directly. A 780 nm filter reduces
background light into the detector to below 0.01 nW. After measuring the power
of the unimpeded beam, we move a mirror into the path of the laser such that
it is incident on the coated side, and the mirror reflects the majority of the
power back through the optical isolator before the laser. Finally, we use an
aperture after the mirror to carefully block as much of the scatter as
possible (without clipping the transmitted beam itself), and record the
intensity of transmitted light. Taking the ratio of the transmitted optical
power to the incident optical power (over several independent trials), we
determine the current transmissions of the high-reflectivity mirrors. The
labeled $T$ values for the mirrors did not always agree with our measurements
as Table 1 shows.

# | $R$ | Date | Target $T$ | Range $T$
---|---|---|---|---
6 | 45 cm | 11/98 | 300 ppm | 235 - 297 ppm
3 | 45 cm | 11/98 | 15 ppm | 7.57 - 8.06 ppm
Table 1:  Mirror parameters: # number of mirrors, $R$ radius of curvature,
Target transmission and Range of trasmission.

Choosing mirrors with transmissions of $T_{1}$ = 8 ppm and $T_{2}$= 297 ppm
(so that over 97% of photons in the cavity will exit through one mirror,
making detection simpler), we calculate finesse assuming that the absorption
losses are very small with Eq. (1)

$F=\frac{\pi}{1-\sqrt{(1-T_{1})(1-T_{2})}}.$ (1)

These directly measured transmissions give an expected finesse of
$\sim$20,000.

We use a method of impedance mismatching in which the stainless steel cavity
mount sits atop a stack of materials with very different resonant frequencies,
such that vibrations do not easily propagate through the entire stack, to
improve the mechanical decoupling of the cavity. The materials include lead,
copper, and Sorbothane (a shock absorbing synthetic viscoelastic urethane
polymer). In order to find the optimal configuration of the materials, we
place the cavity base atop a test stack and use a PZT attached to the base as
a microphone, connecting its output to a spectrum analyzer (Stanford Research
Systems SR770 FFT Network Analyzer). We then strike the tabletop with a
hammer, delivering controlled “delta function” impulses, several times over a
span of ten seconds. We collect PZT voltage and average the spectrum over a
range of frequencies from 0 to 100 kHz, and we compare dozens of different
combinations of damping materials.

We obtain a drastic improvement to the effectiveness of the damping stack by
altering the geometry of the Sorbothane layer, based on advice from the
manufacturers of Sorbothane sorbothane . Instead of a solid sheet of
Sorbothane, we cut the material into twelve roughly 0.25-inch squares, and
space the squares out over the area of the damping stack. As shown in Fig. 3,
the small pieces of Sorbothane make this geometry significantly more effective
at damping, as they have more room to deform sideways and dissipate
vibrational energy.

Figure 3:  (color online) Power spectra after impulse excitation. Damping
layers: Lead-Sorbothane-Lead.

##  III Atomic beam deflection

We propose a new atomic beam design to address the problem of the vertical MOT
beam that propagates together with the atoms, inspired by wang10 ; nellessen89
; witte92 ; ashkin70 , in which a 2-D optical molasses deflects the atomic
beam from its initial trajectory and into the cavity mode (see Fig. 4). The
benefits of deflection are threefold: It rids us of the unwanted MOT light, it
allows for better control of the atom number, and it provides a way to lower
the average speed of the atoms traversing the cavity mode. The latter
translates into longer transit times for the atoms in the cavity mode.

A pair of retro-reflected laser beams, intersecting at right angles in the
$x$-$y$ plane, create a 2-D optical molasses (see Fig. 4). The molasses acts
as a viscous damping force of form $\vec{F}=-\beta\vec{v}_{T}$, opposing the
motion of the atoms in the $x$-$y$ plane, and rapidly damping the velocity in
these directions, $\vec{v}_{T}$, to a mean of zero. Aiming the initial atom
beam from the LVIS at some angle with respect to the $x$, $y$, and $z$ axes,
the molasses damps all velocity components except those along $z$, giving an
effective deflection into the $z$ direction while also reducing the mean
longitudinal velocity of the beam.

Figure 4:  (color online) LVIS of atoms exit (a) hole angled to the vertical
axis of our optical cavity, scatter light from a (b) 2-D optical molasses.

Implementing this geometry requires a major modification of our apparatus. We
simulate the beam deflection process to find the optimal experimental
parameters before changing the system. The code for this simulation is in C++
with the aid of Root, an open source object-oriented data analysis framework
developed by CERN and used extensively in particle physics. The code currently
relies on histogramming, graphing, geometry and physics libraries of Root and
can be obtained from the website of the author under the terms of the GNU
General Public License andres_sim_beam

The first stage of the simulation involves the modeling of an atomic beam in
our current LVIS setup. The main goal of the simulation is to give a
reasonable estimate for the ratio of atoms that couple to the cavity mode to
the total number of atoms that exit the unbalanced MOT. The code does not
simulate a MOT and assumes no atom-atom interactions.

We create the beam by randomly assigning $(x,y)$ starting coordinates to each
atom from a uniform distribution across the area of the hole in the LVIS
mirror (see Fig. 1), defining the vertical starting position as $z=0$. In a
similar way, we randomly specify $(v_{x},v_{y})$ initial transverse
velocities, constrained by the geometric collimation mechanism of the LVIS
mirror aperture to a maximum value as in Eq. (2):

$v_{T,max}=\frac{D_{h}}{d_{mh}}v_{0},$ (2)

where $D_{h}$ is the diameter of the hole, $d_{mh}$ is the distance from the
center of the MOT to the center of the hole and $v_{0}$ is the mean
longitudinal speed of the atom beam from the unbalanced MOT. The initial
vertical velocity components $v_{z}$ come from a Gaussian distribution with
mean $v_{0}=14$ m/s and FWHM 2.7 m/s, in accordance with lu96 .

Figure 5 shows 1000 atomic trajectories using parameters from our current LVIS
and cavity setup. Each atom propagates until it reaches within one waist of
the cavity mode center or misses and reaches the chamber wall. The white spot
at the top center of the figure corresponds to the position of the cavity
mode. The simulations show that 7 - 9 % of atoms coming from the LVIS couple
to the cavity mode, where the range corresponds to the uncertainty in the
actual distance between LVIS aperture and cavity mode (between 3 and 4 cm.)
For the planned smaller optical cavity, the range drops to 4 - 6 %, down by
approximately a factor of two as expected from the mode volume.

Figure 5:  (color online) One thousand atomic trajectories a beam exiting a
1.5 mm hole. The optical cavity mirror spacing is 2.2 mm and sits at
$z_{cav}=4$ cm. Colors correspond to different atoms.

We next simulate the atomic beam deflection proposal. We follow the treatment
of metcalf99  for the calculation of the scattering rates in steady-state for
a two-level atom in the presence of a laser. We give each of the four beams a
transverse Gaussian profile as in Eq. (3):

$s_{0,i}^{(g)}=s_{0}\left(\frac{w_{0}}{w(z_{i})}\right)^{2}\exp{\left[-2\left(\frac{r_{i}^{2}}{w^{2}(z_{i})}\right)\right]},$
(3)

where $z_{i}$ and $r_{i}$ are the longitudinal and transverse coordinates with
respect to the $k$-vector of the $i$th beam, $w_{0}$ is the waist of each
beam, $s_{0}$ is the on-resonance saturation parameter at the center of each
waist (i.e., at $z_{i}$=$r_{i}$= 0), and $w(z_{i})$ is the beam spot size as a
function of longitudinal position along each beam. $s_{0,i}^{(g)}$ is the
Gaussian beam correction to the on-resonance saturation parameter. We use it
to calculate the saturation parameter per beam for each atom as Eq. (4):

$s_{i}=\frac{s_{0,i}^{(g)}}{1+\left(2\left(\delta_{i}-\vec{k_{i}}\cdot\vec{v}\right)/\gamma\right)^{2}},$
(4)

where $\delta=\omega_{\ell}-\omega_{a}$ is the laser detuning with respect to
the atomic transition frequency, $\vec{k}$ is the laser wavevector, $\vec{v}$
is the velocity of the atom, $\gamma$ is the transition linewidth. We make the
approximation that the molasses beams do not interfere, which can be
accomplished experimentally by using orthogonal polarizations and spatially
separating the $x$ and $y$ beam pairs. In the simplified model we add the
contribution from each beam independently and find the total saturation
parameter $s_{T}=\sum_{i=1}^{4}s_{i}$. We then calculate the total scattering
rate $\gamma_{p}$ as in Eq. (5):

$\gamma_{p}=\frac{\gamma}{2}\frac{s_{T}}{1+s_{T}},$ (5)

When an atom enters the molasses region (defined within the $1/e^{2}$
intensity region of the beams), it immediately begins scattering at a rate
$\gamma_{p}$. A scattering event involves a recoil velocity of magnitude
$\hbar k/m$ in the direction of the absorbed beam, and a recoil of the same
magnitude but random direction for the reemission. We determine which beam the
atom absorbs from by comparing a random number to the ratio
$\gamma_{p,i}/\gamma_{p}$, i.e. the relative weight of each beam compared to
the total scattering rate. The beam with the largest weight will be most
likely to give the atom a momentum kick. We then select two more random
numbers to determine the angle of reemission in three dimensions. Updating the
velocity of the atom with the result of the two recoils, we allow the atom to
propagate freely until the next scattering event. To determine this time step,
we pick random numbers from the exponential waiting-time distribution (Eq. 6),

$W(t)=\gamma_{p}e^{-\gamma_{p}t},$ (6)

which governs the probability of the next event occuring in time $t$ for a
Poisson random process. At each time step we record the position and velocity
of the atom. The algorithm repeats until the atom exits the molasses.

We use very weakly focused Gaussian beams with parameters $w_{0}=w(z_{i})=7.5$
mm, for simulating the proposed atomic beam setup, such that the beam
intensity has negligible $z_{i}$ dependence. We choose a slightly smaller
aperture size in the LVIS mirror, 1 mm, to achieve a smaller initial beam, and
we set the angles of the initial LVIS beam trajectory as $\theta=30^{\circ}$
(polar) and $\phi=45^{\circ}$ (azimuthal) with respect to the cavity
coordinate system.

Figure 6 shows the effect of laser detuning on the beam profile. We graph the
average polar angle of the atomic beam velocity at a fixed position just after
it exits the molasses region.

Figure 6:  Average polar angle of atoms after 2-D OM as a function of laser
detuning for beams with $w_{0}=w=7.5$ mm and $s_{0}=3$.

Using the same parameters as before, but choosing the optimal detuning of
$\delta/\gamma=-0.5$, Fig. 7 shows atomic trajectories under the action of the
2-D optical molasses. For this simulation, we placed the cavity at $z_{cav}=1$
cm and obtained 11 % of atoms coupling to the mode.

Figure 7:  (color online) Atomic beam trajectories deflected by 2-D OM with
$\delta/\gamma=-0.5$ and $s_{0}=3$.

Figure 8 shows the $x$ component of the velocity for the atoms. We calculate
the RMS transverse speed of the collimated atomic beam and find 15.3 cm/s,
yielding a Doppler temperature of 120 $\mu$K. This value agrees with a simple
calculation. For a true 1-D system with two beams, we write the heating rate
as $4\gamma_{p}E_{r}$, where $E_{r}=\hbar\omega_{r}$ is the recoil energy
metcalf99 . One factor of two comes from the presence of two beams and the
second because each absorption and re-emission cycle involves two recoils in
that dimension. In three dimensions, the recoil energy from isotropic re-
emission is divided equally among all directions, so this second factor of two
becomes $(1+N/3)$, where $N$ is the number of dimensions with a pair of
molasses beams. When equated to the cooling rate along any one dimension, this
gives Doppler temperature $T=\hbar(4/12,5/12,6/12)\gamma/k_{b}$ for
$N=(1,2,3)$ pairs of molasses beams. In our case $N=2$ and the result is in
agreement with the simulation.

Figure 8:  (color online) Atomic beam $v_{x}$ damped by 2-D OM with
$\delta/\gamma=-0.5$ and $s_{0}=3$.

##  IV Laser lens: Focusing

The 2-D OM induces a deflection in the atomic beam (see Fig. 7), and
collimates the beam down to the Doppler temperature in the transverse
directions, but does not focus the atoms. This requires a position dependent
force. Eq. 3 shows the possibility of introducing spatial dependence through
the factor $w(z_{i})$. By focusing the laser beams tightly at distance $p$
away from the central axis as shown in Fig. 9, we can create an intensity
gradient over the interaction region and induce atomic beam focusing balykin88
.

It is possible to achieve focusing by using a 2-D MOT as a magneto-optical
lens berthoud98 ; labeyrie99 . This gives a position dependent force from the
magnetic field gradient that shifts the atomic resonances, allowing spatial
compression. However, its setup carries undesirable consequences. The
proximity of the 2-D MOT coils to the optical cavity, make accurate control of
the magnetic fields in the cavity very difficult.

Figure 9:  Simplified schematic of a 1-D laser lens.

Figure 9 shows that when $\vec{v}=0$, $F_{1}-F_{2}=-\hbar
k\left(\gamma_{p,2}-\gamma_{p,1}\right)\neq 0$ away from the axis. We estimate
how tightly focused the beams have to be to achieve a usable intensity
imbalance. Fig. 10 shows our results under the following assumptions. We take
a fixed point for the calculation, at $z=1$ mm. We choose this value because
it represents a typical position of an atom in the beam as it exits the 2-D
OM. The beam spot size at the center of the laser lens region is 5 mm and an
intensity per beam at the center of $s_{0}=2$. We set $\vec{v}=0$, to
characterize the position dependent part of the interaction.

Figure 10:  Scattering rate ratio for an atom at $z=1$ mm ($\vec{v}=0$) in a
laser lens with $s_{0}(0,0)=3$ and $w(0)=5$ mm.

These results indicate that a successful laser lens requires very tight
focusing ($\sim$0.5 $\mu$m). We separate the deflection and the focusing, and
elaborate a more sophisticated model for a two-dimensional laser that includes
more of the atomic structure through a six level atom model correction
bouyer94  (See Fig. 11).

Figure 11:  (color online) 6-level atomic structure. Driven by $\sigma^{+}$
and $\sigma^{-}$ polarized light. Decays and excitations take place according
the Clebsch-Gordan coefficients shown metcalf99 .

The algorithm for the laser lens builds upon the one for the 2-D OM. The
6-level atom correction weighs the scattering rates per beam and spontaneous
emission using the Clebsch-Gordan coefficients. We use two different
quantization axes, $x$ and $y$, depending from which beam the atom most
recently absorbed. This approach makes the code simpler.

Results from the simulation are complicated. They show focusing for very small
waists. The refinements in the model (i.e. 6-level correction) do not seem to
loosen such tight requirements. We will continue to explore this, and evaluate
its practical benefits.

##  V Conclusions

We have demonstrated enhancements to increase the quality of our cavity and
allow us to couple more atoms to its modes.

In cavity construction, we expect that careful experimental measurement of the
transmissions of our mirrors have allowed us to make significantly more
accurate predictions of finesse. Additionally, our progress in reducing
mechanical vibrations via impedance mismatching of damping materials will help
to ensure stable operation of the cavity.

We show encouraging simulation results for the implementation of 2-D optical
molasses for the deflection and collimation of a Low Velocity Intense Source
of atoms. Parameter explorations pointed at an optimal molasses laser beam
detuning at $\delta/\gamma=-0.5$. By placing the cavity at a moderately short
distance from the molasses, we showed that we can get up to 11 % of atoms into
the new cavity mode, in contrast with our simulation predictions for our
current system of about 6 %.

Despite initial evidence that the laser lens technique can achieve atomic
focusing, we consider it highly difficult to implement in our experiment, due
primarily to the very small laser beam waists required for its effectiveness
($<1$ $\mu$m).

## Acknowledgments

This work was supported by the National Science Foundation (NSF). We thank J.
R. Ramos for his participation in early stages of the optical cavity building
process. We are grateful to S. L. Rolston, H. J. Carmichael and M. Scholten
for their stimulating discussions and guidance. Thanks to the Root developers
and users for their invaluable advice.

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# Solving the Naturalness Problem with Feeble Coupled Sectors

J. Lorenzo Díaz-Cruz<EMAIL_ADDRESS>CIIEC and CIFFU, BUAP  Facultad de
Ciencias Físico - Matemáticas, BUAP
Apdo. Postal 1364, C.P. 72000, Puebla, Pue. México

###### Abstract

The discovery of a light Higgs boson means that whatever form new physics
takes, it should keep stable the Higgs mass. Besides the well-known solutions
to the naturalness problem (Supersymmetry, Conformal symmetry, Compositeness,
etc), models that include heavy particles with feeble couplings to the
Standard Model (SM) can be considered natural, since the corrections to the
Higgs mass remains of the order of the electroweak (EW) scale. This solution
can be used for model building too, with realizations that include the see-saw
mechanism for neutrino masses and FIMP dark matter models, but it also holds
for generic sectors that have Planck-suppressed couplings with the SM. One can
also incorporate this solution within the SMEFT framework; the corresponding
higher-dimensional operators induce small corrections to both the Higgs mass
and its self-coupling, a prediction that can be tested at a future Higgs
factory. We present a natural extension of the SMEFT that describes
corrections to the SM, while also including a Feeble Coupled Sector aimed to
account for the dark cosmos, with predictions for new signals that can be
tested too.

1\. Introduction. Finding a solution to the hierarchy or naturalness problem,
and exploring the corresponding models, have played an important role for the
development of particle physics. The identification of quadratic divergences
as a potential peril for elementary scalars [1], motivated the search for new
theories where such problem could be eliminated. The importance of those
solutions, such as supersymmetry, composite and technicolor models, should not
be underestimated, both for conceptual and experimental aspects. In the first
place they provided roads to look for and explore models that addressed such
problem; furthermore those models, despite its failure or success, justified
the experimental search for the Higgs boson, which many doubted it actually
existed after the Standard Model (SM) was proposed. The development of
radiative corrections helped to corner the Higgs in a mass region at the reach
of collider experiments, thus from LEP we learned that the Higgs mass has to
be above 105 GeV, while Tevatron helped to exclude the range around the ZZ
threshold. Finally, the debate about whether nature likes or not elementary
scalars, was closed after the discovery of the Higgs boson at the CERN LHC,
with a mass $m_{h}=125$ GeV and properties in agreement with the SM.

Further conceptual developments helped to understand better the nature of
quadratic divergences. In particular, we learned that the conformal symmetry
[2] could help to understand the origin of the Higgs mass as a soft-breaking
effect, and therefore the SM itself can be considered as natural. Furthermore,
after having provided a method to identify quadratic divergences within the
dimensional regularization method [3], Veltman departed from the common lore
and claimed that there are no quadratic divergences in the SM [4]. Thus, it is
appropriate to say that the SM is natural, as the radiative contribution to
the Higgs mass coming from its heavier particle, the top quark, keeps the
Higgs mass of the order of the EW scale, suggesting the absence of the
hierarchy problem within the SM [6, 5].

The real problem appears for extensions of the SM that include heavy states,
with mass and couplings given by $M_{X}$ and $g_{x}$, respectively; after
integrating them out, they leave a potential large correction to the Higgs
mass ($m_{h}$), which is of the order $\delta m^{2}_{h}\simeq g_{x}M^{2}_{X}$,
and for $M_{X}>>m_{W}$, there appears to be a large correction to the Higgs
mass. The traditional solutions to this problem assume that the coupling
constant $g_{X}$ of the new heavy particles has a value similar to the other
couplings in the SM. In SUSY models, for instance, one has both fermions and
sfermions [7], and their contribution to the Higgs mass have opposite signs,
and thus cancel each other. In the conformal solution [8], the tree-level
contribution vanishes, and the Higgs mass itself is generated as a radiative
effect. But we also know that, apart from the naturalness problem, there are
other problems in the SM (flavor, generations, unification), while the
evidence for neutrino masses and the dark cosmos, i.e. dark matter and dark
energy, as well as the early universe inflation and the matter-antimatter
asymmetry, which can not be explained by the SM. Thus, it seems there are good
reasons to expect that some form of new physics should exist. One could go
ahead studying models that address some Beyond the SM issues, without caring
about the solution to the naturalness problem, hoping that some additional new
physics will bring a solution to the problem. Alternatively, we can take to
the heart the issue of naturalness, and consider only extended models that
incorporate a solution to the problem, as it was done in the past with
supersymmetry or other BSM scenarios.

So far we have a few experimental facts that could guide us in the search for
BSM physics, namely, the detection of a light Higgs, the absence of BSM
signals at LHC and the lack of direct evidence for dark matter WIMPS. In this
paper we shall try to incorporate these three hints into a single framework.
We shall assume that there exists heavy particles beyond the SM, which play a
role for the solution of some of its open problems, that induce a correction
to the Higgs mass of the order $\delta m^{2}_{h}\simeq g_{x}M^{2}_{X}$, but
this is natural because rather than having a coupling of O(1), the coupling
$g_{x}$ is feeble, such that $g_{x}M^{2}_{X}\simeq m^{2}_{h}$, which can be
considered as a solution to the naturalness problem. We argue that this
solution with q natural Feeble Coupled Sector (FECOS), should be consider on
the same foot, or perhaps even more solid, than the other ones, because it is
based on the current observations; furthermore, it provides a framework for
model building, like SUSY or Techinicolor. FECOS solution can occur in many
models, such as the Type-I see-saw model for neutrino masses, or in models
where a Feeble-interacting massive particle (FIMP) is the candidate for dark
matter; moreover it can also work for generic models that include heavy states
with suppressed couplings to the SM, including models with the gravity portal.

We can consider a generic FECOS model, and notice that when its heavy modes
are integrated out, it should leave an effective Lagrangians extension of the
SM, which includes a whole tower of higher dimensional operators suppressed by
a scale $\Lambda\simeq M_{X}$. Within this affective Lagrangian, there should
be no problem with quadratic divergences, and the corrections to the Higgs
mass should be at most of the order $O(v/\Lambda)$, and similarly for the
Higgs self coupling, a prediction that can be tested at a future Higgs
factory. Furthermore, one can build a specific model that include a FECOS
aimed to describe aspects of the Dark Cosmos (DC), for instance by including
extra scalars and fermions to account for dark matter and the inflaton. This
model, which we call the DC-SMEFT, includes renormalizable Lagrangians for
both SM and DC sectors, but it also contains higher-dimensional operators for
both sectors, including their mixture. This model provides some distinctive
signatures that could be probed in diverse fronts, ranging form colliders to
cosmology.

2\. Natural models with Feeble Coupled Sector. To discuss the problem of
naturalness, let us consider a theory that includes a SM-like Higgs boson
($h$) which interacts with a heavy scalar $S$, gauge field $V$ and fermion
$F$. The corresponding expression for the associated quadratic divergences is:

$m^{2}_{h}(\Lambda,\mu)=m^{2}_{h}(\Lambda)+\sum_{X=S,V,F}(-1)^{2J_{x}}(2J_{x}+1)\frac{g_{x}}{16\pi^{2}}[\Lambda^{2}-m^{2}_{h}(\Lambda)\log\frac{\Lambda^{2}}{\mu^{2}}]$
(1)

where $\mu$ denotes the renormalization scale, and $\Lambda$ denotes the
momentum cutoff. Nowadays we understand better the meaning of this equation.
Namely, when $\Lambda$ corresponds to the UV cutoff, one simply has to
renormalize this effect on the Higgs mass. Given that the SM is a
renormalizable theory, even when one takes $\Lambda\to\infty$, no observable
effect is left. However, the real problem resides when $\Lambda$ represents
the mass of a heavy particle i.e. $\Lambda=M$, and when it is integrated out,
it leaves a correction to the Higgs proportional to the scale $\Lambda=M$,
which could be very large. However, having a large but finite value for
$\Lambda$, is different from having to deal with a divergent term, even when
it is of the order of the Planck mass, i.e. $\Lambda<M_{pl}$; this was used in
the past to constrain on the Higgs boson mass [9]. In order to cancell the
correction to the Higgs mass, which has to be the order of the EW scale, one
can have some relationship among the couplings or the masses [10]. The
traditional solutions to this problem assume that the coupling constant
$g_{X}$ has a value similar to the other couplings in the SM, and imply the
existence of new heavy particles, which can be searched at collider or
cosmolgy. SUSY [7] corresponds to the first case, and conformal symmetry is of
the second type, and these are two well-known possibilities to cancel the
quadratic divergence.

Here we want to argue that there is also another possibility to bring under
control the naturalness problem. We shall assume that some generic extensions
of the SM includes heavy particles, which induce a natural correction to the
Higgs mass, of the order $O(m_{W})$, because rather than having a coupling
$g_{x}$ of O(1), we assume that this coupling is feeble, such that
$g_{x}M^{2}_{X}\simeq m^{2}_{h}$ even for $M_{X}>>m_{W}$, which solves the
naturalness problem. We argue that this resolution to the naturalness problem
with Feeble-Coupled Sectors (FECOS), should be consider on the same foot as
the other solutions, or perhaps even better, because its motivation is based
on the current observations. One example that proves this mechanism is
feasible corresponds to the SM augmented with neutrino masses obtained through
the see-saw mechanism [11]. In this case it happens that the correction to the
Higgs mass is given by $\delta m^{2}_{h}=y_{\nu}M^{2}/16\pi^{2}$, with the
neutrino Yukawa coupling being given by $y_{\nu}=Mm_{\nu}/v$, for
$m_{\nu}=0.05$ GeV, $\delta m^{2}_{h}$ is of the order of the EW scale even
for $M\simeq 10^{8}$; and the $\nu SM$ is natural even when it contains a
heavy RH neutrino. Another class of models that falls within the FECOS
category is the so called Feeble Interacting massive particle (FIMP), which is
a viable solution to the dark matter problem [12, 13]. The viability of this
scenario is based on a different calculation of the DM relic density, the
freeze-in mechanism, which relies on the feeble coupling of the FIMP
candidate.

Furthermore, the FECOS solution to the naturalness problem, can be a generic
one, it is of a bottom-up type, which accommodates a light Higgs boson, and
within specific models it can explain neutrino masses and dark matter. In the
FECOS models we can have heavy particles, but with very small couplings, even
suppressed by the Planck mass, $m^{2}/M_{pl}$, as it occurs in supergravity.
We can also consider models with a gravitational sector that includes other
FECOS fields that couple gravitationally with the SM, like the heavy graviton
that appears in extra dimensional models [14], with couplings suppressed by
the volume of the extra-dimensional space, or by the curvature of the extra-
dimension in Randall-Sundrum 5D models, where the SM resides on a EW brane,
while the graviton and other fields propagate in the bulk. It is also
interesting to discuss the effect of gravity on quadratic divergences, but as
shown in [15], these corrections do not affect the Higgs mass.

3\. Naturalness in the SMEFT and the Dark Cosmos. We would like to apply the
FECOS solution of naturalness, to discuss some aspects of the Dark Cosmos
(DC), in particular dark matter. Our complete model will contain the SM
sector, as well as a new DC sector of the FECOS type. One can look directly at
the effects of the new sector, for instance in neutrino physics or dark matter
searches. It is also possible to calculate their effects on the properties of
the SM particles, in particular the Higgs boson, which can be described with
an effective Lagrangian. In fact, we should notice that the level of
divergence in the fundamental theory could be less severe than one can expect
from the effective lagrangian [16]. The effective Lagrangian includes
renormalizable parts, for both the SM and for the new particles of the FECOS
type, but it contains as well higher-dimensional operators for both sectors.
For instance, the renormalizable part can include a mixing term of the form
$(\Phi^{\dagger}\Phi)S^{2}$, between the SM Higgs doublet $\Phi$ and an extra
singlet $S$. Then, after integrating out the singlet $S$, one gets a new
operator of the form $(\Phi^{\dagger}\Phi)^{3}$, which modifies the Higgs
properties. However this operator comes from a sector with suppressed
coupling, and will be highly suppressed. Thus, within the FECO models the
modification of the Higgs properties, such as the 3-particle self-coupling,
will only show tiny deviations from the SM, probably unobservable. However,
besides the naturalness problem, there are other open issues in the SM, that
also demand some physics beyond the SM, such as the origin of flavor, CPV,
unification, etc. As we do not have a solution to these problems, for the sake
of generality, we shall consider an effective Lagrangian that describes the
effects of those new particles and interactions with the SM particles. This
model could include extra fermions $\psi_{a}$ and some scalars $S_{i}$, within
a minimal setting of the FECOS type. So far we do not have enough information
to include gauge interactions for these particles, unlike the SM, so we can
start by including a number of fields that play a role for the resolution of
the problems of Dark matter, inflation, and possibly the matter-antimatter
asymmetry, similar to the $\nu SM$ [17]. We call this model the Dark Cosmos
extension of the SMEFT (DC-SMEFT), and its lagrangian has the general form:

${\cal{L}}_{DC-
SMEFT}={\cal{L}}_{SM}+{\cal{L}}_{DC}+\sum_{i,d}[\frac{\alpha_{i,d}}{\Lambda^{d-4}}O^{sm}_{d,i}+\frac{\beta_{i,d}}{\Lambda^{d-4}}O^{sm}_{d,i}]$
(2)

Here ${\cal{L}}_{SM}$ and ${\cal{L}}_{DC}$ represent the renormalizable
Lagrangians for the SM and Dark cosmos; for which we can include kinetic
terms, Yukawa interactions and an scalar potential in ${\cal{L}}_{DC}$. The
higher d-dimensional operators $O^{SM}_{i,d}$, involve only the SM fields,
while $O^{DC}_{i,d}$ involve the extra fields, as well as its mixing with SM
sector. For $O^{SM}_{i,6}$ we can include the whole list of dimension-six
operators [18], and as mentioned before, we are interested in the operators
that modify the Higgs properties, which is discussed for the complete set of
associated operators in [19]. One possible realization of this approach is the
specific FIMP model presented in [20], where the DC sector includes three
right-handed neutrinos ($N_{i}$) and a real scalar ($S$), two of the RH
neutrinos participate in the see-saw mechanism to generate light neutrino
masses, while the third one is the DM candidate. The vev of the scalar singlet
generates the light neutrino masses through dimension-5 operators, and several
scenarios with a stable FIMP are found to satisfy current constraints.

The associated phenomenology of these models can be studied, for instance by
looking for small deviations from the SM predictions. However, it could be
even more interesting to look for rare effects that are included in the
higher-dimensional operators. For instance, one can look at the effects of the
operators included in the model of [20], but for an unstable FIMP particle,
which can provide a distinctive signature of the model. In particular, one has
a dimension-five term of the form: $L\tilde{\Phi}N_{i}S$, and after both the
SM Higgs doublets $\Phi$ and the singlet $S$, aquire a vev, there is a mixing
between the third RH neutrino, which is assumed to be the FIMP DM, and this
will induce decays of the dark matter. The corresponding life-time should be
larger than the age of the universe, for this scenario to be a viable, but
even after such constraint is satisfied, it is possible to observe effects of
decaying DM. One could also decouple the neutrino mass problem from the Dark
Matter issue, by adding an extra fermion, different from the RH neutrinos,
which will play de role of FIMP dark matter, while the singlet scalar could
still participate in the generation of the see-saw mechanism. Detailed
analysis of the model is under study. A different model, that also shows the
viability of the scenario with a decaying FIMP is presented in [21].

4\. Conclusions. We presented a discussion of the naturalness problem, and
argued that besides the well-known solutions to the naturalness problem (such
as supersymmetry, Conformal symmetry, Compositeness, etc), there is also
another solution, namely the case when a new sector with heavy particles have
feeble-couplings with the Standard Model (SM), which induce natural
corrections to the Higgs mass. This solution with Feeble-coupled sectors
(FECOS), can be used for model building too, for instance in the see-saw
mechanism for neutrino masses, or the FIMP dark matter model, but it is a
generic one, and one can include models with heavy states that have Planck-
suppressed couplings with the SM. The FECOS solution to the naturalness
problem, can be a generic one, it is of a bottom-up type, which accommodates a
light Higgs boson, and within specific models it can explain neutrino masses
and dark matter, while being in agreement with the absence of signals in
collider and direct search for DM.

Further aspects of naturalness can be discussed within the SMEFT framework,
with higher-dimensional operators that induce small corrections to both the
Higgs mass and its self-coupling, and measuring it provides a strong
motivation for a future Higgs factory. We present a natural extension of the
SMEFT that includes a FIMP dark matter candidate, which can provide new
signals that test this whole new framework. In particular, decay of the DM
particle could be a distinctive signature of this model.

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