# Induced spontaneous symmetry breaking chain

Eduard Boos<EMAIL_ADDRESS><EMAIL_ADDRESS>Skobeltsyn Institute
of Nuclear Physics, Lomonosov Moscow State University, 119991, Moscow,Russia

###### Abstract

The article discusses a scenario based on the idea of induced spontaneous
symmetry breaking. In this type of scenario, spontaneous symmetry breaking is
assumed at some highest energy level, which leads to a chain of several
subsequent induced symmetry violations at lower levels caused by small
mixtures between the levels. We present a simple model in which the idea is
realized by small mixing between the levels using scalar portals. In this
approach, the large difference between energy scales, for example the Planck
scale and the electroweak scale, occurs due to the product of several small
factors proportional to the mixing coefficients. Dark matter fields can be
formed in this scenario from matter fields on one or possibly several
intermediate levels between the highest one and lowest one.

Spontaneous symmetry breaking, Plank scale, Dark matter, Scalar portal

##  I Introduction

The Standard Model, quantum gauge field theory with spontaneous breaking of
electroweak symmetry, reached its logical conclusion with the discovery of the
Higgs boson at the LHC [1, 2]. The successes of the SM in describing the
processes of particle production and decay in terrestrial and space
experiments are well known. However, the SM is unable to explain a number of
observed phenomena in nature, such as baryon and lepton asymmetries in the
Universe, dark energy and dark matter, small cosmological constant, mass and
oscillations of neutrinos, etc.

There are some internal problems of the SM, such as the problem of the
negative mass parameter $\mu^{2}=-|\mu^{2}|$, artificially introduced into the
theory, the hierarchy problem associated with this parameter and the problem
of electroweak scale stabilization on a value of about 100 GeV. The negative
parameter $\mu^{2}$ is necessary for the realization of the mechanism of
spontaneous electroweak symmetry breaking (mechanism BEH [3, 4, 5]), but its
origin as well as the magnitude of the corresponding EW scale of the order of
100 GeV remain unclear.

Before the launch of the LHC, there was the so-called No-Lose-Theorem [6, 7,
8, 9], which, based on an analysis of the behavior of the scattering
amplitudes of longitudinal modes of massive electroweak SM bosons and the
requirement of perturbative unitarity, asserted that only two situations were
possible: either the existence of a sufficiently light Higgs boson with a mass
smaller $\sim$700 GeV, or a change of regime and the appearance of new physics
on a scale of the order of 1 TeV. The Higgs boson with a mass of about 125 GeV
has been discovered, and, accordingly, the argument about the magnitude of the
new scale is gone. Today we do not know, either the closest to the SM scale of
new physics, or, of course, what new physics is. The results of the search for
manifestations of new physics at the LHC are still negative, although these
negative results themselves are extremely important, narrowing the range of
parameters or closing some possible variants for new physics.

We are well aware that there is at least one scale that is significantly
larger than the electroweak scale. This is the Planck scale $\sim 10^{19}$
GeV. Note that many extensions of the Standard Model predict the presence of
other intermediate scales. Grand Unification models, their supersymmetric or
string motivated generalizations, predict the scale$\sim 10^{15}\div
10^{16}$GeV, on which the running electroweak and strong coupling constants
become equal or close. In models of neutrino physics, in particular, based on
the see-saw mechanism, the scale $\sim 10^{11}\div 10^{12}$GeV appears. In
supersymmetric models or in models with extra space-time dimensions, scales of
the order of 10 TeV are quite admissible. However, the answer to the question
of exactly what and how phase transitions occur as we move from large to
smaller scales remains very unclear.

This short letter proposes a simple scenario based on the idea of induced
spontaneous symmetry breaking. In this type of scenario, a chain of phase
transitions can be realized. Only in the theory at the highest scale, it can
be the Planck scale, there is a negative mass parameter $\mu^{2}$. Possibly
the negative sign of this parameter is due to nonlinear effects of quantum
gravity. At this highest scale, a phase transition occurs, and this transition
through mixing causes subsequent transitions in a chain in which one link
(i.e. theory at a given scale) connects to an adjacent link (i.e. theory at
the nearest scale) through a portal with a rather small mixing. Small mixing
parameters between the two nearest links (theories on the nearest scales) in
the chain lead to the fact that the farther from the lowest link a certain
link in the chain is located, the weaker the hypothetical particles of the
theory at this intermediate scale interact with the particles of the theory at
the lowest level. In this scenario, the scale of the highest level, say the
Plank scale $M_{Pl}$ and the scale of the lowest level, say the electroweak
(EW) scale of the SM are related by a coefficient proportional to the product
of small mixing parameters between theories at intermediate levels (links in
the chain).

The next section presents a simple model based on scalar portals. Models with
scalar portals, being the simplest portal extensions of the SM, allow one to
describe dark matter (DM) without contradicting existing data and theoretical
bounds in certain areas of the parameter space of these models [10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36]. For a detailed overview of the Higgs portal model in
connection with DM, including various experimental restrictions, see [34].
Higgs and scalar portals, their DM and cosmological implications are
summarized in the recent review [37]. In the presented scenario with a chain
of scalar portal models with spontaneously induced symmetry breaking, there
can obviously be candidates for dark matter particles, possibly from different
levels of the chain, due to the larger number of parameters in hand that are
usually discussed.

##  II Simple model

Consider the Lagrangian of a model that describes a chain of complex scalar
fields with mixing between the nearest neighbors:

$\displaystyle
L=(D^{(1)}_{\nu}H_{1})^{\dagger}(D^{(1)\nu}H_{1})-\mu_{1}^{2}H_{1}^{\dagger}H_{1}-\lambda_{1}(H_{1}^{\dagger}H_{1})^{2}\,\,\,\,\,\,\,\,$
(1)
$\displaystyle+\,L_{Fields(1)}+\,k_{12}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})$
$\displaystyle+(D^{(2)}_{\nu}H_{2})^{\dagger}(D^{(2)\nu}H_{2})-\mu_{2}^{2}H_{2}^{\dagger}H_{2}-\lambda_{2}(H_{2}^{\dagger}H_{2})^{2}$
$\displaystyle+\,L_{Fields(2)}+\,k_{23}(H_{2}^{\dagger}H_{2})(H_{3}^{\dagger}H_{3})$
$\displaystyle\dots$
$\displaystyle+(D^{(N-1)}_{\nu}H_{N-1})^{\dagger}(D^{(N-1)\nu}H_{N-1})-\mu_{N-1}^{2}H_{N-1}^{\dagger}H_{N-1}$
$\displaystyle-\lambda_{N-1}(H_{N-1}^{\dagger}H_{N-1})^{2}$
$\displaystyle+\,L_{Fields(N-1)}+\,k_{N-1,N}(H_{N-1}^{\dagger}H_{N-1})(H_{N}^{\dagger}H_{N})$
$\displaystyle+\,(D^{(N)}_{\nu}H_{N})^{\dagger}(D^{(N)\nu}H_{N})-\mu_{N}^{2}H_{N}^{\dagger}H_{N}-\lambda_{N}(H_{N}^{\dagger}{N})^{2}$
$\displaystyle+\,L_{Fields(N)},$

where $H_{1},...,H_{N}$ are the complex scalar fields,
$D^{(l)}_{\nu}=\partial_{\nu}-ig^{(l)}V^{(l)}_{\nu}$ is the covariant
derivative, $V^{(l)}_{\nu}$ is a gauge field interacting with the scalar field
$H_{l}$, and $L_{Fields(l)}$ is the Lagrangian of some other fields at the
level or link with number $l=1,\dots,\,N$.

Suppose all coefficients $\mu_{l}^{2}$ except one are equal to zero
$\mu_{1}^{2}=\mu_{2}^{2}=...=\mu_{N-1}^{2}=0$ and only the last coefficient is
nonzero $\mu_{N}^{2}\neq 0$ and has a negative value
$\mu_{N}^{2}=-\mid\mu_{N}^{2}\mid$. Let also all mixing parameters between the
theories at the adjacent levels (links in the chain) be small, namely:

$\displaystyle\frac{k_{12}}{\lambda_{1}},\,\,\frac{k_{12}}{\lambda_{2}},\,\,\frac{k_{23}}{\lambda_{2}},\,\,\frac{k_{23}}{\lambda_{3}},\,\,...,\,\,\frac{k_{N-1N}}{\lambda_{N-1}}\,\,\frac{k_{N-1N}}{\lambda_{N}}\ll\,1.$
(2)

The exact formulas for the case of two generations as well as a proof of the
diagonalization of the mass matrix $N\times N$ are given in the Appendix.

The boson $h_{N}$ at the last step of the chain acquires the mass

$\displaystyle m_{N}^{2}\simeq 2\lambda_{N}v_{N}^{2},$ (3)

where $v_{N}^{2}=\frac{\mid\mu_{N}^{2}\mid}{\lambda_{N}}$ as a result of the
usual mechanism of spontaneous symmetry breaking BEH [3, 4, 5]. In this case,
due to the mixing, the induced parameter
$\widetilde{\mu}_{N-1}^{2}=-k_{N-1N}\,\,v_{N}^{2}/2$ appears at the previous
step of the chain. With a positive mixing coefficient $k_{N-1,N}$ at this
level spontaneous breaking occurs, the boson $h_{N-1}$ gets mass

$\displaystyle m_{N-1}^{2}\simeq
k_{N-1N}\,\,v_{N}^{2}\simeq\frac{k_{N-1N}}{2\lambda_{N}}m_{N}^{2},$ (4)

and the parameter $\widetilde{\mu}_{N-2}^{2}$ is induced being proportional to
the mixing coefficient $k_{N-2N-1}$ between levels $N-1$ and $N-2$ of the
chain. This process continues and, as a result, at the first step of the
chain, the $h_{1}$-boson mass appears

$\displaystyle
m_{1}^{2}\simeq\frac{k_{12}}{2\lambda_{2}}\,\,\frac{k_{23}}{2\lambda_{3}}\,\,...\,\,\frac{k_{N-1N}}{2\lambda_{N}}\,m_{N}^{2}$
(5)

with the induced vacuum expectation value

$\displaystyle
v_{1}^{2}\simeq\frac{k_{12}}{2\lambda_{1}}\,\,\frac{k_{23}}{2\lambda_{2}}\,\,...\,\,\frac{k_{N-1N}}{2\lambda_{N-1}}\,v_{N}^{2}$
(6)

As one can see, the vacuum expectation value $v_{1}$ and the Higgs mass
$m_{1}$ can be significantly smaller than the scale $v_{N}$ and the mass
$m_{N}$, respectively, due to the product of a number of factors proportional
to small mixing coefficients. This observation could explain the hierarchy
between a small scale of the order of $v_{1}$ and a very large scale of the
order of $v_{N}$. The scalar field potentials before and after spontaneous
symmetry breaking corresponding to the discussed scenario are illustrated in
Fig. 1.

Figure 1:  Illustrative form of potentials before and after spontaneous
symmetry breaking induced at the lower levels by symmetry breaking at the
highest level. The lowest level corresponds to the smallest scale, and is
designated conventionally as electroweak (”EW”). The uppermost level
corresponds to the largest scale, and is conventionally designated as the
Planck scale (”Plank”).

If we match the smallest vacuum expectation value $v_{1}$ in the chain with
the vacuum expectation value of the Standard Model $v_{SM}$ ($v_{1}=v_{SM}$) ,
the mass $m_{1}$ of the boson $h_{1}$ with the SM Higgs boson mass
$M_{h_{SM}}$ ($m_{1}=M_{h_{SM}}$), and the coupling $\lambda_{1}$ with the SM
quartic coupling $\lambda_{SM}$ ($\lambda_{1}=\lambda_{SM}$), the SM relation
must take place:

$\displaystyle M_{h_{SM}}^{2}\,=\,2\lambda_{SM}v_{SM}^{2}.$ (7)

It is easy to see that, if we divide the left- and right-hand sides of eq. (5)
by the left- and right-hand sides of eq. (6), correspondingly, then taking
into account eq. (7) we get

$2\lambda_{N}=\frac{m_{N}^{2}}{v_{N}^{2}},$

which exactly matches eq. (3), as it should.

Suppose that the largest scale in the chain $v_{N}$ and the corresponding mass
$m_{N}$ are of the order of Planck scale $\sim 10^{19}$ GeV. Then the product
of the factors

$\displaystyle\varepsilon_{12}\cdot\varepsilon_{23}\,\,...\,\,\cdot\varepsilon_{i\,i+1}\,\,...\,\,\cdot\varepsilon_{N-1\,N},$
(8)

where $\varepsilon_{i\,i+1}=\sqrt{\frac{k_{i\,i+1}}{2\lambda_{i}}}$, should be
about $10^{-17}\div 10^{-16}$ assuming that the scale $v_{1}$ is of the order
of the electroweak scale $\sim 10^{2}$ GeV.

The model (1) is renormalizable if the Lagrangians $L_{Fields(i)}$, i=1,…,N
involve the operators with dimension 4 or less. On may expect a serious
hierarchy problem appearing in the model. Indeed, the correction to the mass
squared of the scalar $h_{1}$ from the loop contribution of the scalar $h_{i}$
is proportional to

$m_{i}^{2}\cdot log(m_{i}^{2}/m_{1}^{2})$

leading to a very large shift. However in the model (1) after the
diagonalization the interaction vertex of the scalars $h_{1}$ and $h_{i}$
contains the coupling constant
$\varepsilon_{12}\cdot\,\,...\,\,\cdot\varepsilon_{i-1\,i}$. Therefore, the
loop contribution is proportional to

$\frac{k_{12}}{2\lambda_{2}}\,\,...\,\,\frac{k_{i-1i}}{2\lambda_{N}}\,\cdot
m_{i}^{2}\cdot log(m_{i}^{2}/m_{1}^{2})\simeq m_{1}^{2}\cdot
log(m_{i}^{2}/m_{1}^{2}),$

demonstrating that the little hierarchy problem does not show up in the model
under consideration. The same argument also holds for the case with possible
loop contributions to the scalar $h_{1}$ mass parameter related to some other
fields from $L_{Fields(i)}$ when the potentially large correction is
suppressed by the product of small mixing parameters.

##  III Conclusion

In this short letter, we discuss a scenario in which a spontaneous symmetry
breaking or spontaneous phase transition occurs at some large energy scale.
Due to a low mixing, such a phase transition leads to an induced symmetry
breaking at some smaller induced scale, which, in turn, due to the next mixing
leads to the next induced symmetry breaking at an even smaller scale, and so
on. The result is a chain of induced spontaneous symmetry breaking scales. In
such a scenario, the large difference between the largest and smallest scales
is due to the product of several small factors proportional to the small
mixing coefficients, but each of these factors may not be that small.

In each level of induced symmetry breaking there might be its own gauge
theory. Obviously that the interactions of particles at some level with
particles at the lowest level will be smaller and smaller for higher and
higher levels being proportional to smaller and smaller mixing coefficients.
Once the interactions get small enough particles at the corresponding level
may play a role of Dark matter particles.

We give an example of a very simple model in which small mixing between the
levels in a chain of spontaneous gauge symmetry breaking is due to scalar
portals. As some speculation, an example is given that the difference between
the Planck scale and the electroweak scale can be due to about nine or five
induced spontaneous transitions, assuming the same order of small coefficients
$10^{-2}$ and $10^{-4}$, respectively.

Obviously, that having several levels weaker and weaker interacting with the
first level associated with the SM, Dark matter might be easily accumulated at
far enough levels with small enough couplings to the SM via such a multi-
portal scenario.

We considered an example of a chain with mixing between the nearest neighbors,
i.e. theories at the adjacent levels connected by mixing through scalar
portals. Of course, one can think about more complex mixing not only with the
nearest neighbors and not only through scalar portals. In this case, more
complex chains with branching and more complex mixing arise.

##  IV Acknowledgements

I am grateful to Sergey Keyzerov and Igor Volobuev for useful discussions. The
study was supported by RFBR and CNRS grant 20-52-15005. The research has been
done in the framework of the Interdisciplinary Scientific and Educational
School of Moscow University ”Fundamental and Applied Space Research”.

##  V Appendix

Let the chain contains two levels only as given by the eq. (9)

$\displaystyle
L=(D^{(1)}_{\nu}H_{1})^{\dagger}(D^{(1)\nu}H_{1})-\mu_{1}^{2}H_{1}^{\dagger}H_{1}-\lambda_{1}(H_{1}^{\dagger}H_{1})^{2}\,\,\,\,\,\,\,\,$
(9) $\displaystyle+\,\,k_{12}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})$
$\displaystyle+\,\,(D^{(2)}_{\nu}H_{2})^{\dagger}(D^{(2)\nu}H_{2})-\mu_{2}^{2}H_{2}^{\dagger}H_{2}-\lambda_{2}(H_{2}^{\dagger}H_{2})^{2},$

where we set $\mu_{1}$ equal to zero ($\mu_{1}^{2}=0$). In the unitary gauge
on both levels the quadratic form of the scalar fields reads as follows:

$\displaystyle
2\lambda_{1}v_{1}^{2}\widetilde{h_{1}}^{2}-2k_{12}v_{1}v_{2}\widetilde{h_{1}}\widetilde{h_{2}}+2\lambda_{2}v_{2}^{2}\widetilde{h_{2}}^{2}$
(10)

where

$\displaystyle
v_{1}=\sqrt{\frac{k_{12}}{2\lambda_{1}}}v_{2},\,\,\,\,\,\,v_{2}=\sqrt{\frac{4\lambda_{1}\mid\mu_{2}^{2}\mid}{4\lambda_{1}\lambda_{2}-k_{12}^{2}}}$
(11)

and the mixing coefficient $k_{12}$ as well as the quartic couplings
$\lambda_{1}$ and $\lambda_{2}$ are chosen to be positive. Rotation from the
unphysical fields $\widetilde{h_{1}}$ and $\widetilde{h_{2}}$ to physical
fields $h_{1}$ and $h_{2}$ with definite masses leads to the following
expressions for the masses squared of eigenstates:

$\displaystyle
m_{1,2}^{2}=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(12)
$\displaystyle\left[\lambda_{2}+\frac{k_{12}}{2}\mp\lambda_{2}\sqrt{\left(1-\frac{k_{12}}{2\lambda_{2}}\right)^{2}+\frac{k_{12}}{2\lambda_{1}}\frac{k_{12}^{2}}{\lambda_{2}^{2}}}\right]v_{2}^{2}.$

Assuming

$\frac{k_{12}}{\lambda_{1}},\,\,\frac{k_{12}}{\lambda_{2}}\,\,\ll\,1$

one gets the following values for the masses:

$\displaystyle
m_{2}^{2}=2\lambda_{2}v_{2}^{2},\,\,m_{1}^{2}=k_{12}\,\,v_{2}^{2}$ (13)

in complete agreement with the eqs. (3) and (4).

The mixing matrix corresponding to the above example with two levels looks as
follows:

$\begin{pmatrix}2\lambda_{1}v_{1}^{2}&-k_{12}v_{1}v_{2}\\\
-k_{12}v_{1}v_{2}&2\lambda_{2}v_{2}^{2}\end{pmatrix}$

This is easily generalized to the model given in (1). The mixing matrix has
the following form:

$\begin{pmatrix}x_{11}&x_{12}&0&0&0&...&0\\\ x_{21}&x_{22}&x_{23}&0&0&...&0\\\
0&x_{32}&x_{33}&x_{34}&0&...&0\\\ ...&...&...&...&...&...&...\\\
0&0&...&0&x_{N-2N-1}&x_{N-1N-1}&x_{N-1N}\\\
0&0&0&...&0&x_{NN-1}&x_{NN}\end{pmatrix}$

,

where $x_{11}=2\lambda_{1}v_{1}^{2}$, $x_{12}=x_{21}=-k_{12}v_{1}v_{2}$,
$x_{22}=\lambda_{2}v_{2}^{2}$, $x_{23}=x_{32}=-k_{23}v_{2}v_{3}$,
$x_{33}=2\lambda_{3}v_{3}^{2}$, $x_{34}=x_{43}=-k_{34}v_{3}v_{4}$,
$x_{N-1N-1}=2\lambda_{N-1}v_{N-1}^{2}$,
$x_{N-2N-1}=x_{N-1N-2}=-k_{N-2N-1}v_{N-2}v_{N-1}$,
$x_{NN}=2\lambda_{N}v_{N}^{2}$, $x_{N-1N}=x_{NN-1}=-k_{N-1N}v_{N-1}v_{N}$.

Such a N$\times$N matrix can be diagonalized using the product of rotation
matrices that sequentially lead to a diagonal form of the 2$\times$2 matrix
blocks:

$\begin{pmatrix}cos{\theta_{1}}&sin{\theta_{1}}&0&0&0&...&0\\\
-sin{\theta_{1}}&cos{\theta_{1}}&0&0&...&0\\\ 0&0&1&0&0&...&0\\\
...&...&...&...&...&...&...\\\ 0&0&...&0&0&1&0\\\
0&0&0&...&0&0&1\end{pmatrix}$ $\times$
$\begin{pmatrix}1&0&0&0&0&...&0\\\
0&cos{\theta_{2}}&sin{\theta_{2}}&0&0&...&0\\\
0&-sin{\theta_{2}}&cos{\theta_{2}}&0&0&...&0\\\ ...&...&...&...&...&...&...\\\
0&0&...&0&0&1&0\\\ 0&0&0&...&0&0&1\end{pmatrix}$ $\times$
…

$\begin{pmatrix}1&0&0&0&0&...&0\\\ 0&1&0&0&0&...&0\\\ 0&0&1&0&0&...&0\\\
...&...&...&...&...&...&...\\\ 0&0&...&0&0&cos{\theta_{N}}&sin{\theta_{N}}\\\
0&0&0&...&0&-sin{\theta_{N}}&cos{\theta_{N}}\end{pmatrix}$,

where $\theta_{i}$, $i=1...N$ are the rotation angles. Under conditions (2)
and (3) one gets expressions (4) and (5) for the mass eigenstates.

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