#  DESY 21-073, HU-EP-21/12 May 2021

The Standard Model of particle physics as a conspiracy theory and the possible
role of the Higgs boson in the evolution of the early universe ††thanks: To
the memory of our distinguished colleague and guide Prof. Martinus Veltman
with his remarkable record in particle physics

Fred Jegerlehner  Humboldt-Universität zu Berlin, Institut für Physik,
Newtonstrasse 15,
D-12489 Berlin, Germany
Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6,
D-15738 Zeuthen, Germany

###### Abstract

I am considering Veltman’s “The Infrared - Ultraviolet Connection” addressing
the issue of quadratic divergences and the related huge radiative correction
predicted by the electroweak Standard Model (SM) in the relationship between
the bare and the renormalized theory, commonly called “the hierarchy problem”
which usually is claimed that this has to be cured. After the discovery of the
Higgs particle at CERN, which essentially completed the SM, an amazing
interrelation of the leading interaction strengths of the gauge bosons, the
top-quark, and the Higgs boson showed up amounting that the SM allows for a
perturbative extrapolation of the running couplings up to the Planck scale.
The central question concerns the stability of the electroweak vacuum, which
requires that the running Higgs self-coupling stays positive. Although several
evaluations seem to favor the meta-stability within the experimental and
theoretical parameter-uncertainties, one should not exclude the possibility
that other experiments and improved matching conditions will be able to
establish the absolute stability of the SM vacuum in the future. I will
discuss the stable vacuum scenario and its impact on early cosmology,
revealing the Higgs boson as the inflaton. It turns out that the Standard
Model’s presumed “hierarchy problem” and similarly the “cosmological constant
problem” resolve themselves when we understand the SM as a low energy
effective tail that is emergent from a cutoff-medium at the Planck scale. “The
Infrared - Ultraviolet Connection” conveyed by the Higgs boson mass
renormalization appears in a new light when the energy dependence of the SM
couplings is taken into account. The bare Higgs boson mass square then changes
sign below the Planck scale where it is activating the Higgs mechanism. At the
same time, it reveals that the SM towards the Planck scale is in the symmetric
phase, where the Higgs potential provides a high dark energy density
triggering inflation, and four heavy Higgs bosons which decay and thereby are
reheating the inflated early universe.

11.15.-q,11.10.Hi,12.15.Lk,98.80.Cq

##  1 Introduction and Overview

I review some points advanced by Martinus Veltman [1, 2] which allow us to
understand better what is behind the electroweak (EW) Standard Model’s
peculiar structure and how this fits perfectly into attempts to consider the
Standard Model (SM) as a low energy effective theory of a cutoff fine-grained
“ether” at the Planck scale $\Lambda_{\rm Pl}$. The discovery of the Higgs
boson [3, 4] by the ATLAS and CMS Collaborations at the LHC at CERN [5, 6]
opened a new book in particle physics. We now see the SM in a new light, not
so much because the Higgs boson has been found finally, but much more because
of the very special mass, the Higgs boson proved to have. In fact the heaviest
SM states, the weak gauge bosons, the top-quark, and the Higgs boson, interact
with each other in such a way that the leading SM couplings fall into a very
narrow window which allows for a stable Higgs vacuum up to the Planck scale
and, equally surprising, the SM remains accessible to perturbation theory up
to the Planck scale.

The reason seems to be a kind of self-organized conspiracy between SM
couplings with an amazing balance between bosonic and fermionic contributions
in the renormalization group evolution of parameters as a function of the
energy. With the other couplings given, the Higgs boson self-coupling
$\lambda$ appears to be self-tuned to let us understand the SM as emergent
from the Planck medium. If the Higgs vacuum remains stable up to the Planck
scale, the SM would shape the dynamics of the early universe, with inflation,
reheating, and what else triggered by the Higgs system. The Higgs potential
inevitably would provide a huge strongly time-dependent cosmological constant
that shapes inflation and consequently also modifies the time-dependence of
the Hubble constant (see e.g. [7]). While effective SM parameters change
little after the time of Big-Bang nucleosynthesis, early cosmology may be
shaped in a previously unexpected manner. Indeed, the Higgs boson is the only
SM particle that directly couples to gravity, as a leading contribution to the
energy-momentum tensor in Einstein’s field equations. The relevant object is
the Higgs potential

$V(\phi)=\frac{m^{2}(\mu^{2})}{2}\>\phi^{2}+\frac{\lambda(\mu^{2})}{24}\>\phi^{4}\,,$

depending on the renormalization scale $\mu$, and, provided $\lambda$ stays
positive, the SM vacuum remains stable. The Higgs boson mass renormalization
relation

$m^{2}_{H0}-m^{2}_{H}=\delta m_{H}^{2}=\frac{\Lambda_{\rm
Pl}^{2}}{(16\pi^{2})}\,C(\mu)\,,$

relating the bare mass $m_{H0}$ to the renormalized $m_{H}$, which Veltman
addressed in [8] as “Infrared - Ultraviolet Connection”, obtains a new deeper
meaning if we are taking into account that SM parameters are scale-dependent.
It happens that the renormalization counterterm, depending sensitively on the
precise actual SM parameters, shows a zero $C(\mu_{0})=0$ at about
$\mu_{0}\sim 10^{16}~{}\mbox{GeV}$ at which bare and renormalized Higgs masses
are coinciding. Below the zero, where the bare mass square is negative, the SM
falls into the broken phase where the Higgs boson, like all other massive SM
particles, acquires a mass set by the Higgs boson vacuum expectation value
(VEV) $v\sim 246~{}\mbox{GeV}$ as the relevant scale. Above the zero, in the
symmetric phase, the effective mass is
$m_{0}^{2}=m^{2}_{H0}/2=(m_{H}^{2}+\delta m_{H}^{2})/2$ effectively determined
by the dominating counterterm:

$m_{0}^{2}\sim\delta m^{2}\simeq\frac{\Lambda_{\rm
Pl}^{2}}{32\pi^{2}}\,C(\mu)\,.$

Together with the vacuum expectation value of the potential the Higgs system
provides a sufficient amount of dark energy to trigger inflation, while the
four heavy unstable Higgs states, which are existing in the symmetric phase,
via their decays are reheating the universe. SM reheating is efficient again
by peculiarities of SM physics. The existence of the third family with a
strong coupling to the Higgs field let the Higgs bosons decay predominantly
into top/anti-top, top/anti-bottom, etc. quark pairs and their charge
conjugates; this heavy quark radiation later materializes during the EW phase
transition by cascading into the much lighter quark species which are forming
ordinary matter. While dark energy is an SM product the origin of the dark
matter still remains a mystery in this scenario.

##  2 Veltman’s “derivation” of the Standard Model in the light of the
Standard Model as a low energy effective field theory

One of Martinus Veltman’s dedications has been his drive to find a
renormalizable theory for the weak interactions and to scrutinize the
electroweak theory which after all culminated in the SM of particle physics.
The younger particle physicists learning the SM from textbooks may have little
knowledge about the obstacles one had to surmount before the established
structure had been unveiled as the true theory of electroweak phenomena. The
maybe strange-looking structure of the SM as a gauge theory based on the local
symmetry group $\mathcal{G}_{\rm SM}=SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}$ (gauge couplings $g_{3}$, $g_{2}$ and $g_{1}$) broken down to
$SU(3)_{c}\otimes U(1)_{\rm e.m.}$ by the Higgs mechanism exhibits a pattern
that really cries for explanations. Veltman’s curiosity has been focused more
on the question of what makes the SM so unique, rather than to question its
validity and to declare its structure unnatural. His cardinal point in
defending the SM and questioning most of the proposed SM extensions has been
that, due to the minimal Higgs structure, the SM unavoidably predicts a
massless photon [9], which in most other models have to be imposed as an extra
condition, which then is causing a fine-tuning issue111 The simplest
supersymmetric model accidentally escapes this problem and predicts a zero
photon mass. [1]. In a remarkable paper [2]222For related considerations see
[10]. he has given a partial but very enlightening answer by listing the
general conditions which essentially allow us to derive the SM. Veltman’s list
of assumptions is the following:

1) local field theory
2) interactions follow from a local gauge principle
3) renormalizability
4) masses derive from the minimal Higgs system (one physical scalar only)
5) the right handed singlet neutrino $\nu_{R}$, which we know must exist,
does not carry hypercharge.

The last assumption looks somewhat ad hoc, but we accept it. The consequences
of the assumptions stated above are remarkable333The interested reader really
should take the time and have a look at Veltman’s surprisingly simple
derivation. (see also [11, 12]):

  1. [1]

breaking $\mathit{SU}(2)_{L}$ by a minimal Higgs sector automatically leads to
a global $U(1)_{Y}$, which can be gauged,

  2. [2]

maximal parity violation of the weak interaction $\mathit{SU}(2)_{L}$,

  3. [3]

with $\Theta_{W}$ the electroweak mixing angle, and $M_{W},\>M_{Z}$ the
masses444I denote physical masses by capital the related
$\overline{\mathrm{MS}}$ masses by lower case letters. of the $W$ and $Z$
bosons, the tree-level relation
$\rho=M_{W}^{2}/(M_{Z}^{2}\cos^{2}\Theta_{W})=1$, is a parameter independent
number and hence not subject to renormalization. It derives from the
accidental global “custodial” $\mathit{SU}(2)_{\rm cust}$ symmetry of the
Higgs system.

  4. [4]

The existence of the strictly massless photon (one zero-eigenvalue in the
spin-1 boson mass-matrix),

  5. [5]

parity conservation of QED,

  6. [6]

the validity of the Gell-Mann-Nishijima relation $Q=T_{3}+\frac{Y}{2}$,

  7. [7]

fermion family structure (lepton-quark conspiracy –
$U(1)_{Y}$ anomaly cancellation),

  8. [8]

charge quantization (if $Y_{\nu R}=0$ then $Q_{i}=T_{3i}+Y_{i}/2$ fixes
$Q_{i}$).

Here we have to add that the requirement of renormalizability also enforces
the existence of a physical neutral spin 0 particle, the Higgs boson555 If not
generated by the Higgs mechanism, the particle masses affect renormalizability
in the $\mathit{SU}(2)_{L}$ weak-interaction sector, which means that it is
the $\mathit{SU}(2)_{L}$ which has to be spontaneously broken by the minimal
Higgs system.. We do not know why right-handed neutrinos are sterile i.e. do
not couple to gauge bosons; they only couple to the Higgs field. Nevertheless,
this property fits with a minimality principle “not more than necessary”. In
the SM of electroweak interactions, neutrinos originally were assumed to be
massless although it has not been required by the SM gauge symmetry structure
but had to be imposed by an extra global $U(1)_{e}\otimes U(1)_{\mu}\otimes
U(1)_{\tau}$, implying that right-handed neutrinos did not exist (or did not
couple to anything) such that neutrinos had to be massless. This is definitely
ruled out by the observation of neutrino oscillations, meaning that each of
the subgroups $U(1)_{\ell}$ is only partially conserved. Indeed, lepton-flavor
number conservation is not an emergent property but is resulting from the
smallness of the neutrino masses. In other words, $e$, $\mu$, and $\tau$ only
differ in the way they couple to the Higgs boson, while they have identical
couplings to the gauge bosons.

Another striking property of the SM is the GIM-mechanism [13], the tree-level
absence of flavor changing neutral currents, which in the SM by its
renormalizability is manifestly implemented, but which requires fine-tuning in
most extensions of the SM. The discrete $Z_{2}$ symmetry (called R-parity)
often imposed on the extended two-doublet Higgs system is not an emergent
property (i.e. renormalizability does not require it).

One of the most important insights into how we perceive nature’s underlying
structure has been discovered by Ken Wilson (Nobel Price 1982 for his theory
for critical phenomena in connection with phase transitions) in the early
1970s when he developed the systematic approach to understanding
quantitatively the emergence of long-distance physics (macroscopic properties)
from the short distance properties of the underlying microscopic systems [14]
(see e.g. [15, 16]). Quite generally, the microscopic systems are exhibiting
an intrinsic short distance scale i.e. they represent systems exhibiting a
cutoff $\Lambda$ and allow for a low-energy expansion in $x=E/\Lambda$.
Dropping the suppressed terms (i.e. the positive powers in x), the expansion
yields a low energy effective tail which appears as an emergent low energy
structure. In case of statistical mechanics type systems the low-energy
effective structure turns out to be given by a renormalizable Euclidean
Quantum Field Theory (QFT) which is universal for a wide class of systems of
different short-distance properties. A groundbreaking discovery. Thereby, a
non-trivial long-range tail emerges as a $D=4$ world because in $D>4$ only
non-interacting free field tails are left666In condensed matter physics this
is known as the Landau criterion [17], and is the basis for the
$\varepsilon$-expansion about $D=4$ in critical phenomena [18].. The $D-4$
extra dimensions are hidden, unobservable when watched from far away. No need
for curled-up (compactified) extra-dimensions. Renormalizable $D=4$ Euclidean
QFTs are equivalent to Minkowski QFTs (Osterwalder-Schrader theorem), the
basic structure on which our understanding of particle physics is relying.
Note that this mathematical equivalence is a property of the long-distance
tail and thus a Euclidean $D\geq 4$ dimensional short distance system may
naturally show its appearance to us as a $(3,1)$-dimensional Minkowski space-
time, i.e. also Lorentz-invariance (pseudo-rotations) is an emergent symmetry,
means, also here we got rid of the “ether”. This could mean that also time is
an emergent property that only exists in the low energy tail perceived.
Thereby renormalizability and analyticity are the key ingredients. Based on
the Euclidean-Minkowskian equivalence Wilson himself developed the tool for
solving non-perturbative hadron physics based on Euclidean lattice Quantum
Chromodynamics (QCD)777In contrast to the Planck cutoff regularized SM case
(discussed below), here the cutoff provided by the inverse lattice spacing
can’t be chosen very large, which causes serious problems in controlling
lattice artifacts in the continuum limit. Therefore it is mandatory to keep
the $\mathit{SU}(3)_{c}$ non-Abelian local gauge symmetry exact on the
lattice, because symmetry violating effects could not be controlled by an
expansion in the cutoff in an efficient manner. In practice, the expansion
parameter $E/\Lambda$ would by far not be small enough such that the symmetry
breaking terms would be suppressed as much that the symmetry would be restored
accurately..

Originally, the SM [19, 9, 13] is a minimal renormalizable completion of
Fermi’s weak interaction + QED, supplemented by QCD [20] and two more fermion
families. But now “what is not capable of surviving at long distances does not
exist there” (Darwin revisited). The SM appears as a natural minimal emergent
structure in a low energy expansion from a cutoff system sitting at the Planck
scale! But this is not the only consequence: in fact, in this kind of
scenario, the SM which we supplemented by the Planck cutoff, leads to
unexpected consequences concerning its high-energy behavior. The latter is
governed by the bare (unrenormalized) theory, which must have been the
relevant theory in the very hot early universe.

Important points are:

  * -

the relationship between the renormalized and bare theory is calculable,

  * -

taking into account the energy dependence of parameters is mandatory now, this
also concerns possible power corrections (relevant operators: Higgs mass term
and vacuum energy given by the Higgs potential and its VEV $\langle
V(\phi)\rangle$,

  * -

leading irrelevant operators of dim 5 (may affect neutrino masses) and dim 6
(baryon number violating, required in baryogenesis) [21] naturally can play
significant roles during the advent of the EW phase transition expected at
about $10^{16}~{}\mbox{GeV}$. Especially the dim 6 operators are completely
suppressed at experimentally accessible energies, but in case the EW phase
transition happened at very high energies not too far below the Planck mass
$M_{\rm Pl}$, they are still significant enough to play their role in
producing the baryon asymmetry,

  * -

a surprising outcome of the conspiracies between SM couplings is that within
the SM charge screening is an exceptional property, anti-screening the rule.
Apart from the non-perturbative QCD confinement regime at energies below the
hadron mass scales, the SM and in particular, its high energy phase (early
cosmology) are under control by perturbation theory.

I see plenty of evidence that the SM is emergent in the Wilsonian sense [14]
as a renormalizable low energy tail of a cutoff system, the “ether”, sitting
at the Planck scale $\Lambda_{\rm Pl}$. The equivalent Planck mass $M_{\rm
Pl}=(c\hbar/G_{N})^{\frac{1}{2}}$ $\simeq 1.22\times 10^{19}~{}\mbox{GeV}$,
dimensionally determined by Newtons gravitational constant $G_{N}$, is one of
the fundamental parameters in physics and the only one representing a
fundamental cutoff equivalent to a minimal length. In an expansion in
$E/\Lambda_{\rm Pl}$ most short distance details get lost and we only see what
is not suppressed by positive powers of $E/\Lambda_{\rm Pl}$. Note that the
cutoff in our case is not just a tool to regularize ultra-violet (UV)
singularities, but represents a true physical reality behind what we see from
far away. Such a UV-completed system allows us to perform a low energy
expansion in the cutoff. Thereby, in the low energy tail the symmetries emerge
which usually are absent within the “ether”. Since in the low energy expansion
all details concerning the UV behavior are lost, it is not a problem that
details about the UV completion remains unknown. Here the universality of the
long-range tail is a key phenomenon well known from condensed matter systems.
There is a big variety of systems having an identical tail but differ in the
infinite tower of irrelevant dimension $d_{\mathcal{O}}>4$ operators. When we
attempt to extrapolate the SM to very high scales we expect the next leading
$d_{\mathcal{O}}=6$ operators to show up when the expansion parameter
$x=E/\Lambda_{\rm Pl}$ is about 0.1 producing a 1% effect, i.e. SM physics can
be valid up to not far below the Planck scale. When going too close to the
Planck scale the expansion of course ceases to make sense. This does not mean
that we lose control of the physics which derives from the fine-grained Planck
system (think for example of a lattice-type SM as a prototype; that the
symmetries of the continuum SM cannot be fully kept on a lattice [22] does not
harm the emergent SM as a long-distance tail, as we will argue below). The SM
together with its cutoff UV-completion we call Low Energy Effective Standard
Model (LEESM)888While the SM as a renormalizable QFT by itself makes
predictions free from any cutoff effects (for renormalized and observable
quantities parametrized in terms of measured renormalized parameters), the
LEESM is an extension of the SM where the relationship between renormalized
and bare parameters has a physical meaning, which also has been broached by
Veltman’s “The Infrared - Ultraviolet Connection” [8]..

What makes Veltman’s “derivation” of the SM structure so instructive? As
mentioned before, the SM gauge and Higgs structures, and the derived
consequences just listed, at first sight, look unexpected and motivated many
possible repairs: mirror fermions, grand unification, supersymmetric
extensions, and many more. However, the peculiar features of the SM are
immediate consequences of the general assumptions 1) to 5) above, which
(besides the last point) are emergent structures when we accept that the SM is
a low energy effective theory of a far away cutoff system. Maybe the most
striking insight is that non-Abelian local gauge symmetries, mathematics-wise
a self-evident generalization of the Abelian local gauge symmetry familiar
from QED, are emergent. The question about the origin of non-Abelian local
gauge symmetries has attracted attention after ’t Hooft’s proof of the
renormalizability of gauge theories [23]. Calculations [24, 25, 26, 27, 28,
29, 30, 31] (also see [32]) have shown that non-Abelian symmetry is a
consequence of dropping the non-renormalizable terms showing up in the high
energy behavior of tree-level matrix elements of physical processes, by
imposing a tree-unitarity requirement. In fact in the LEESM scenario
“dropping” is an automatic feature that derives from the strong suppression of
the questionable terms. Obviously, the non-Abelian gauge structures require
team-play between entries in gauge group multiplets. What is more natural than
excitation modes of the hot Planck medium grouped in doublets and triplets as
the simplest choices besides possible singlets? Still, what at first sight
looks almost impossible, namely that the strange-looking SM structure may
naturally emerge as a long-distance structure turns out to be the consequence
of the fact that in the low energy expansion only a relatively small number of
effective interaction vertices are seen, a tremendous simplification as we
cast away an infinite tower of power-suppressed terms, related to higher-
dimensional operators. The renormalizable tail, which we can see, naturally
must satisfy all requirements renormalizability imposes if we, also, accept
that this is to be achieved in the simplest possible (minimal) way.
Remarkably, local non-Abelian gauge symmetry, as well as chiral symmetry, are
mandatory for renormalizability and therefore both symmetries are naturally
emergent in the low energy tail. The crux in this game is that
renormalizability predicts the existence of the Higgs boson! But also that the
weak interactions are maximally P violating while the QED sector is strictly P
conserving are automatic consequences as a simple calculation shows [2]. While
the non-Abelian gauge structure implies the characteristic gauge-cancellations
resulting from the interplay of the coupling strength of all those different
vertices which involve the gauge fields, the emergence of the important
Abelian subgroup $U(1)_{Y}$ looks much more to be a mystery. But as the
calculations show, also here the minimal Higgs scenario automatically implies
a $U(1)_{Y}$ symmetry and a massless photon after the spontaneous $SU(2)_{L}$
breaking.

There is another hidden symmetry deriving from a minimal Higgs system, the
custodial symmetry, which is at the heart of the tree-level identity
$\rho=M_{W}^{2}/(M_{Z}^{2}\cos^{2}\Theta_{W})=1$ (up to finite SM radiative
corrections $\Delta\rho$) [33, 34, 35]. The $\rho$-parameter999 Here a special
feature of the weak corrections comes into play: the non-decoupling of heavy
particles. This is in contradistinction to the Appelquist-Carazzone theorem
[36], which infers that heavy states of mass $M$ are essentially without a
trace at energies sufficiently below the heavy particle threshold. More
precisely, effects are $O(E/M)$ or smaller, in reactions at energies $E\ll M$.
This does not apply in the weak sector of the SM where masses and couplings
are strongly correlated, while it is valid in QED and QCD. This means for
example that electroweak top-quark contributions do not only show up above the
top-quark threshold. In the approximation $M_{t},M_{H}\gg M_{Z}$ the top-quark
and the Higgs boson give a contribution  $\displaystyle\rho(0)$
$\displaystyle\equiv$ $\displaystyle G_{\rm
NC}(0)/G_{\mu}(0)=1+\frac{3\sqrt{2}G_{\mu}}{16\pi^{2}}\,\biggl{\\{}M_{t}^{2}$
$\displaystyle+\left(\frac{M_{W}^{2}}{1-M_{W}^{2}/M_{H}^{2}}\ln
M_{H}^{2}/M_{W}^{2}-\frac{M_{Z}^{2}}{1-M_{Z}^{2}/M_{H}^{2}}\ln
M_{H}^{2}/M_{Z}^{2}+\cdots\right)\biggr{\\}}\,,$ to the $\rho$ parameter at
zero momentum transfer. $G_{\rm NC}(0)$ and $G_{\mu}(0)=G_{\mu}$ are the
neutral and charged current effective Fermi-constants at vanishing momentum,
respectively. is the flagpole for heavy states because quantum corrections at
next to leading order are determined by the difference of the self-energies of
the $Z$ and the $W$ bosons

$\Delta\rho=\frac{\Pi_{Z}(0)}{M_{Z}^{2}}-\frac{\Pi_{W}(0)}{M_{W}^{2}}+\mathrm{subleading\
terms}$

and for dimensional reasons corrections are expected to be proportional to the
mass square $M_{X}^{2}$ of the heavy virtual state of mass $M_{X}>M_{Z}$
contributing to the gauge boson self-energies. Most prominent is the top-quark
contribution to $\Delta\rho$ proportional to $G_{\mu}M_{t}^{2}$ [35].
Precision measurements of $\Delta\rho$ by the LEP experiments provided an
important hint for the discovery of the top-quark at the Tevatron. Large
effects only show up for heavier fermion-doublets with a large mass splitting
like the top-bottom quark doublet ($t$,$b$) where $m_{t}\gg m_{b}$. Since
$G_{\mu}M_{f}^{2}=y_{f}^{2}/(2\sqrt{2})$ where $y_{f}$ is the Yukawa coupling
of a fermion $f$, $\Delta\rho$ effectively measures the weak isospin splitting
within the ($t$,$b$)-doublet which results from the difference in the top- and
bottom-quark Yukawa couplings.

The custodial symmetry is responsible also for the absence of leading virtual
Higgs boson effects which could contribute to $\Delta\rho$. There are no
corrections proportional to $G_{\mu}M_{H}^{2}$ but only proportional to the
logarithmic term $G_{\mu}M_{W}^{2}\>\log(M_{H}^{2}/M_{W}^{2})$ when
$M_{H}^{2}\gg M_{W}^{2}$, what made it more difficult to gather reliable
information about the Higgs boson mass from electroweak precision measurements
before the Higgs particle’s discovery. Remains the CP violation, which was
discovered as a per-mille effect in neutral kaon decays and is known as a
compelling condition for the dynamical emergence of the baryon-asymmetry in
the universe during baryogenesis (Sakharov conditions [37]). In the SM this is
achieved automatically by the triple-replica family structure which has been
proposed by Kobayashi-Maskawa [38] (see also [13, 39]). The predicted 3-family
quark-flavor mixing pattern (CKM -matrix) later has been confirmed in $B$
meson decays to happen precisely as predicted.

In place of a dogma believing that the misunderstood101010See my analysis in
[12, 40]. hierarchy-problem (see e.g. [41]) is an illness of the SM which must
be cured, e.g., by super-symmetrization of the SM as one possibility, Veltman
has stressed many times the fact that within the minimal SM the zero photon
mass is a prediction, not subject to renormalization, while in most SM
extensions (see however footnote${}^{\ref{photonsusy}}$ and [42]) the
prediction is lost and has to be imposed as an extra condition, i.e., it has
to be fine-tuned [1]. Indeed, most extensions of the SM require non-minimal
Higgs sectors: Two Doublet Higgs Models (TDHM), SUSY extensions like the
Minimal Supersymmetric SM (MSSM), Grand Unified Theories (GUT), left-right
symmetric models, the Peccei-Quinn approach to the strong CP problem, inflaton
models base on an extra scalar field and many more. The masslessness of the
photon is one of the most basic facts of life (no life otherwise), so why
should we make this fundamental property to be something we have to arrange by
hand?

In contrast, a minimalist emergent structure makes the SM pretty unique:

  * •

I think we now can well understand how various excitations in the hot chaotic
Planck medium can conspire to develop a pattern like the SM as a low energy
effective structure.

  * •

Renormalizability as a consequence of the low energy expansion and the very
large gap between the EW and the Planck scales plus certain minimality (not
too little but not too much e.g. only up to symmetry triplets) determines the
SM structure without much freedom. One could expect that as a next extended SM
structure confined $\mathit{SU}(4)$ fermion quartets could provide a bound
state spectrum providing dark matter [43], analogous to how QCD is supplying
the bulk of ordinary matter (98% binding energy).

  * •

Minimality is not a new concept in physics as we know e.g. from the principle
of least action or the SM as a minimal renormalizable extension of QED plus
Fermi theory.

  * •

The 3 fermion-families are required so that CP violation emerges naturally,
and to open the possibility that baryogenesis can find an explanation within
the SM. While normal matter requires sufficiently light quarks, i.e. small
Yukawa couplings, the cooperation of couplings that eventually allows the SM
to extend up to the Planck scale is only possible when Yukawa couplings in the
ballpark of the gauge-couplings and the Higgs self-coupling exist, which is
what is achieved with the third family only. A fourth family111111 Given the
LEP and LHC 4th family mass bounds at least some of the members would have to
be heavier than the top quark and the corresponding large Yukawa couplings
would imply dramatic changes in the extrapolation of the SM parameter running.
Also, the measured bounds of the $\rho$ parameter essentially rule out having
further fermion doublets with substantial mass splittings. obviously would
spoil the feature of the SM that we can understand it as emergent from a
Planck medium.

  * •

It is interesting to note that QCD based on $\mathit{SU}(3)_{c}$ local gauge-
symmetry requires massless gluons. Massless gauge fields, in this case, are
not only required by renormalizability but are also a condition for
confinement to work. We also note that, in contrast to the electroweak sector
with its 3 fermion-family structure where CP violation is automatically
generated, CP violation in QCD is obtained only if we add an extra CP
violating term to the Lagrangian. Indeed, this extra CP violation is not
something that the low energy expansion generates necessarily unless it is
inherent as a property of the “ether” itself. Indeed, CP violation seems to be
absent in strong interactions and axions may not be required to exist. Keep in
mind that $P$ and $CP$ violations are emergent in the electroweak sector and
certainly are absent in the primordial “ether” medium, if not one would expect
to find different $P$\- and $CP$-violation patterns in the SM. Interestingly,
also symmetry breaking can be an emergent feature, like maximal $P$ violation
out of a $P$ invariant medium shows.

In what follows I review the most striking consequences of a LEESM scenario
which I worked out in some details in [44, 45] (for a summary see also [46]),
for the case that SM couplings are such that an extrapolation up to $M_{\rm
Pl}$ is possible. The difference between stability and meta-stability depends
on a difference in the calculation of the $\overline{\mathrm{MS}}$ top-quark
Yukawa coupling from the top-quark mass [47]. Because of confinement in QCD,
the top-quark as a colored object is not observable by itself but only a
color-screened state of it. Therefore the on-shell top-quark mass usually
taken as an input in the matching conditions is not what the experiment sees
directly. In the end, the $\overline{\mathrm{MS}}$ top-quark Yukawa coupling
is not observable and is not unambiguously fixed by experiment. It rather
depends on theory input, which is not fully under control because the color
screening is a non-perturbative issue.

##  3 On the running SM couplings

That for a quantum field theory (QFT) coupling constants are not constant but
are renormalization scale-dependent as prescribed by the renormalization
groups (RGs) is known since renormalization of QFTs is known. Screening (like
Abelian gauge couplings) or anti screening (like non-Abelian gauge couplings)
effects let couplings grow or diminish as a function of the energy scale,
respectively. These effects are well known and for the QED fine structure
constant $\alpha_{\rm QED}(s)$ (screening) and the QCD strong interaction
constant $\alpha_{s}(s)$ (anti-screening) experimentally well established,
where $s$ is the center of mass energy square at which a process takes place.
Of interest are the gauge couplings $g_{1}$, $g_{2}$ and $g_{3}$ together with
the top-quark Yukawa coupling $y_{t}$ and the Higgs self-coupling $\lambda$,
where the latter two are determined from the top-quark mass and the Higgs
boson mass, respectively, via the mass-coupling relations. A crucial point
here concerns the matching conditions which are required to calculate the
$\overline{\mathrm{MS}}$ parameters from experimentally determined observables
like the physical masses.

The discovery of the Higgs boson with a mass of about 125 GeV revealed an
amazing conspiracy between the gauge couplings, the top-quark Yukawa coupling
and the Higgs boson self-coupling. These leading couplings turn out to be
falling into a narrow window [48, 49, 50, 51] which allows us to extrapolate
the SM parameters up to the Planck scale as displayed in Fig. 1. Whether this
window is matched perfectly or is almost missing the bottom of the stability
valley is a matter of controversy and depends on the implementation of the
matching conditions which are required to calculate the
$\overline{\mathrm{MS}}$ parameters in terms of physical couplings and/or
masses. While most analyses are based on [52, 53] and predict a meta stable
effective Higgs potential (see also [54, 55, 56]), a slightly modified
evaluation of $\overline{\mathrm{MS}}$ parameters based on [57] revealed
vacuum stability [44, 12]. I adopt the view:
“Although other evaluations of the matching conditions seem to favor the meta-
stability of the electroweak vacuum within the experimental and theoretical
uncertainties, one should not exclude the possibility that other experiments
and improved matching conditions will be able to establish the absolute
stability of the Standard Model in the future.”

Figure 1:  Running SM parameters from the $Z$ mass scale up to $M_{\rm Pl}$.
Left: on-shell vs $\overline{\mathrm{MS}}$ parameter matching based on F.J.,
Kalmykov, Kniehl [57](JKK). Right: the same in comparison with Shaposnikov et
al. [52], Degrassi et al. [53] (Deg) matching. Using JKK input parameters the
vacuum remains stable, with Deg input vacuum stability breaks down at about
$10^{9}~{}\mbox{GeV}$ and enters in a metastable state of the effective
potential [58] (i.e. including radiative corrections). Small cause, great
impact, just by changing a little the input parameters, most sensitive on
$y_{t}$, we get a completely different behavior of the Higgs system towards
the Planck era. The shaded space illustrates the parameter-matching
controversy, which concerns $y_{t}$ at $M_{Z}$. Plots adapted from [44].

Very exciting, and for me it has been completely unexpected, all SM couplings
except the $U(1)_{Y}$ coupling $g_{1}$ behave like non-Abelian ones, i.e.,
they show anti-screening properties [59]. This means that perturbation theory
works and gets better if we extrapolate to the Planck scale. Taking the
$\overline{\mathrm{MS}}$ couplings at the $Z$ boson mass scale as [44, 12]
$g_{1}\simeq 0.350$, $g_{2}\simeq 0.653$, $g_{3}\simeq 1.220$, $y_{t}\simeq
0.935$ and $\lambda\simeq 0.807$, the following picture arises121212The RG
equations $\mu\frac{d}{d\mu}\>g_{i}(\mu)=\beta_{i}(g_{i})\;(i=1,2,3)$,
$\mu\frac{d}{d\mu}y_{t}(\mu)=\beta_{y_{t}}(y_{t},\cdots)$ and
$\mu\frac{d}{d\mu}\lambda(\mu)=\beta_{\lambda}(\lambda,\cdots)$ for the
$\overline{\mathrm{MS}}$ couplings $g_{i},y_{t}$ and $\lambda$ define the
$\overline{\mathrm{MS}}$ running couplings as functions of the
$\overline{\mathrm{MS}}$ energy scale $\mu$. The $\overline{\mathrm{MS}}$
renormalization scheme is the appropriate parametrization for the high energy
behavior $\mu\gg M_{t}$ where $M_{t}$ is the top-quark mass, the largest of
the SM masses.. While the gauge couplings behave as expected, $g_{1}$ as
infrared (IR) free, $g_{2}$ and $g_{3}$ as asymptotically (ultraviolet) free
(AF), with leading coefficients exhibiting the related coupling only, and
denoting $c=\frac{1}{16\,\pi^{2}}$, we have

$\beta_{1}=\frac{41}{6}\,g_{1}^{3}\,c\simeq
0.00185\;;\;\;\beta_{2}=-\frac{19}{6}\,g_{2}^{2}\,c\simeq-0.00558\;;\;\;\beta_{3}=-7\,g_{3}^{3}\,c\simeq-0.08049\,,$

the leading top-quark Yukawa $\beta$-function given by

$\displaystyle\beta_{y_{t}}$ $\displaystyle=$
$\displaystyle(\frac{9}{2}\,y_{t}^{3}-\frac{17}{12}\,g_{1}^{2}\,y_{t}-\frac{9}{4}\,g_{2}^{2}\,y_{t}-8\,g_{3}^{2}\,y_{t})\,c$
$\displaystyle\simeq$ $\displaystyle 0.02327-0.00103-0.00568-0.07048$
$\displaystyle\simeq$ $\displaystyle-0.05391$

not only depends on $y_{t}$ but also on mixed terms with the gauge couplings
which have a negative sign. Interestingly, the QCD correction is the leading
contribution and determines the behavior, to be opposite from a pure Yukawa
behavior. Notice the critical balance between the dominating strong- and the
top-Yukawa couplings: QCD dominance requires $g_{3}>\frac{3}{4}\,y_{t}$ in the
gaugeless limit $(g_{1},g_{2}=0)$ . Similarly, the $\beta$-function of the
Higgs self-coupling, given by

$\displaystyle\beta_{\lambda}$ $\displaystyle=$
$\displaystyle(4\,\lambda^{2}-3\,g_{1}^{2}\,\lambda-9\,\lambda\,g_{2}^{2}+12\,y_{t}^{2}\,\lambda+\frac{9}{4}\,g_{1}^{4}+\frac{9}{2}\,g_{1}^{2}\,g_{2}^{2}+\frac{27}{4}\,g_{2}^{4}-36\,y_{t}^{4})\,c$
$\displaystyle\simeq$ $\displaystyle
0.01650-0.00187-0.01961+0.05358+0.00021+0.00149+0.00777$
$\displaystyle-0.17401\simeq-0.11595$

is dominated by the top-quark Yukawa contribution and not by the $\lambda$
coupling itself, again opposite from a pure Higgs system behavior. Here the
sign of the $\beta$-function flips when
$\lambda<\frac{3}{2}\,(\sqrt{5}-1)\,y_{t}^{2}\,$ in the gaugeless
$(g_{1},g_{2}=0)$ limit. This I call true teamwork. It means that running the
couplings up to the Planck scale they have to conspire there such that at long
distances, we can see what we see. This light particle low energy physics is
only possible for specific couplings and by the self-organized symmetries like
local gauge symmetry and chiral symmetry which let emerge such light states of
masses $M_{X}\ll\\!\\!\ll\cdots M_{\rm Pl}$. We note that all calculations of
the running parameters are based on full 2-loop matching-conditions [52, 53,
57, 54, 55] for the input $\overline{\mathrm{MS}}$ couplings and 3-loop RG
$\overline{\mathrm{MS}}$ $\beta$-functions [60, 61, 62]. Here one should keep
in mind, that the physical parameters extracted from experimental data and
used as an input, in general, do not include full 2-loop electroweak
corrections, so the quoted uncertainties may well be underestimated and there
is definite room for improvements. For our set of input parameters, the
relevant running $\overline{\mathrm{MS}}$ parameters at the Planck scale are
of comparable size in the range 0.51 for $g_{2}$ being the largest here and
0.35 for $y_{t}$ being the smallest, with $\sqrt{\lambda}$ at 0.375 slightly
larger in our normalization. This couplings ”pattern” at $M_{\rm Pl}$ is
supporting the view that what was able to penetrate to low energies originates
from a unifying medium. Note that approximations like the gaugeless case
$(g_{1}=g_{2}=0)$ or assuming $\lambda\approx 0$ are not viable approximations
near $M_{\rm Pl}$ neither anywhere at lower energy scales.

##  4 Exploiting “The Infrared - Ultraviolet Connection”

Veltman’s “The Infrared - Ultraviolet Connection” concerns the only quadratic
UV divergences of the SM which in the symmetric phase concerns the mass square
term in the Higgs potential

$\displaystyle m_{0}^{2}=m^{2}+\delta m^{2}\;;\;\;\delta
m^{2}=\frac{\Lambda_{\rm Pl}^{2}}{32\pi^{2}}\,C$ (1)

which communicates the relationship between the bare $m_{0}$ (short distance
UV) and the renormalized mass $m$ (low energy IR). The one-loop coefficient
function $C_{1}$ [8] may be written as (neglecting the small light fermion
Yukawa couplings)

$\displaystyle
C_{1}={\frac{6}{v^{2}}(M_{H}^{2}+M_{Z}^{2}+2M_{W}^{2}-4M_{t}^{2})}=2\,\lambda+\frac{3}{2}\,{g^{\prime}}^{2}+\frac{9}{2}\,g^{2}-12\,y_{t}^{2}$
(2)

and is uniquely determined by dimensionless couplings. Surprisingly, taking
into account the scale dependence of the SM couplings, the coefficient of the
quadratic divergence of the Higgs mass counterterm exhibits a zero (see Fig. 2
left panel). This has been emphasized in [63, 64] (see also [65]), where the
2-loop result $C_{2}$ is also given but represents a minor correction only,
i.e. we have $C(\mu)\simeq C_{1}(\mu)\simeq C_{2}(\mu)$. The important point
is that in the LEESM setup the cutoff is physical, meaning that UV
singularities have turned into finite but eventually large numbers.

According to (1), the SM for the given parameters makes a prediction for the
bare mass parameter $m^{2}_{0}=\mbox{sign}(m^{2}_{0})\times 10^{X}$ of the
Higgs potential, where $X$ is displayed in the right panel of Fig. 2.

Figure 2: The EW phase transition in the SM. Left: the zero in $C_{1}$ and
$C_{2}$ for $M_{H}=125.10\pm 0.14\mbox{GeV}$ . Right: shown is
$X=\mbox{sign}(m^{2}_{0})\times\log_{10}(|m^{2}_{0}|)$. At $M_{\rm Pl}$ we
have $m_{0}^{2}\sim\delta m^{2}\simeq\frac{\Lambda_{\rm
Pl}^{2}}{32\pi^{2}}\,C(\mu=\Lambda_{\rm Pl})\simeq\left(0.0295\,\Lambda_{\rm
Pl}\right)^{2}\,.$ A difference between $C_{1}$ and $C_{2}$ is barely visible
on the plot. The shaded bands illustrate the Higgs mass dependence in a range
[124,126] GeV. Adapted from [12].

In the broken phase we have $m^{2}_{0}=\frac{1}{2}\,(m^{2}_{H}+\delta
m^{2}_{H})$, which is calculable! What happens is:

  * •

the coefficient $C(\mu)\simeq C_{1}(\mu)\simeq C_{2}(\mu)$ exhibits a zero,
for $M_{H}=125~{}\mbox{GeV}$ at about $\mu_{0}\sim 1.4\times
10^{16}~{}\mbox{GeV}$, clearly but not far below $\mu=M_{\rm Pl}$,

  * •

at the zero of the coefficient function the counterterm $\delta
m^{2}=m^{2}_{0}-m^{2}=0$ ($m$ the $\overline{\mathrm{MS}}$ mass) vanishes and
the bare mass changes sign,

  * •

the sign change triggers a phase transition (PT), the Higgs mechanism, that is
inducing masses simultaneously for the weak gauge-bosons and for all fermions,
note that the Higgs field exhibits as many different couplings as there are
different massive SM particles,

  * •

the large $m_{0}$ and the large $V(0)=\langle V(\phi)\rangle$ (see below) for
$\mu>\mu_{0}$ in the symmetric phase initiate cosmic inflation when
temporarily the potential dominates the kinetic term
$V(\phi)\gg\frac{1}{2}\,\dot{\phi}^{2}$, where
$\dot{\phi}=\mathrm{d}\phi/\mathrm{d}t\,.$

  * •

At the transition point $\mu_{0}$ we have $m_{0}=m(\mu_{0}^{2})$ and a bare
Higgs field VEV $v_{0}=v(\mu_{0}^{2})$, where $v(\mu^{2})$ is the
$\overline{\mathrm{MS}}$ renormalized VEV; the power cutoff-effects appear
nullified at $\mu_{0}$! Note that $v$ is characteristic for long-range order
(order parameter) and shows-up for $\mu<\mu_{0}$ only. In the high energy
phase, $\mu>\mu_{0}$ which extends up to $M_{\rm Pl}$ we have $v\equiv 0$, no
point to expect $v=O(M_{\rm Pl})$. Where is the hierarchy problem?

  * •

Furthermore there is a jump in the vacuum density, which agrees with the
renormalized one: $-\Delta\rho_{\rm
vac}=\frac{\lambda(\mu^{2}_{0})}{24}\,v^{4}(\mu_{0}^{2})\,,$ and thus is being
$O(v^{4})$ and not $O(M^{4}_{\rm Pl})\,$ as often claimed.

The second important quantity we have not taken into account so far is the
vacuum energy $V(0)=\langle V(\phi)\rangle$ related to the quartic UV
singularity [45]. Again, in the LEESM scenario, the vacuum energy is a
calculable quantity. Considering the Higgs boson doublet-field $\Phi(x)$, in
the symmetric phase $\mathit{SU}(2)$ symmetry implies that while
$\langle{\Phi(x)}\rangle\equiv 0$ the composite field $\Phi^{+}\Phi(x)$ is a
singlet such that the invariant vacuum energy is represented just by simple
Higgs-field loops, the self-contractions of the Higgs fields in the potential.
With

$\displaystyle\langle{0}|\Phi^{+}\Phi|{0}\rangle=\frac{1}{2}\langle{0}|H^{2}|{0}\rangle\equiv\frac{1}{2}\,\Xi\;;\;\;\Xi=\frac{\Lambda_{\rm
Pl}^{2}}{16\pi^{2}}\,,$ (3)

we obtain a cosmological constant (CC) given by

$\displaystyle
V(0)=\langle{V(\phi)}\rangle=\frac{m^{2}}{2}\,\Xi+\frac{\lambda}{8}\,\Xi^{2}=\frac{\Lambda_{\rm
Pl}^{4}}{(16\pi^{2})^{2}}\,\frac{1}{8}(2C+\lambda)\,.$ (4)

A Wick ordering type of rearrangement of the Lagrangian also leads to a shift
of the effective mass

$\displaystyle{m^{\prime}}^{2}=m^{2}+\frac{\lambda}{2}\,\Xi=m^{2}+\frac{\Lambda_{\rm
Pl}^{2}}{32\pi^{2}}\,(C+\lambda)\,\,.$ (5)

For our values of the $\overline{\mathrm{MS}}$ input parameters the zero in
the Higgs mass counter term and hence the phase transition point gets shifted
downwards as follows

$\displaystyle{\mu_{0}\approx 1.4\times
10^{16}~{}\mbox{GeV}}\to{\mu^{\prime}_{0}\approx 7.7\times
10^{14}~{}\mbox{GeV}\,.}$ (6)

The shift is shown in the right panel of Fig. 3 together with finite
temperature effects displayed separately in the left panel.

Figure 3: $X$ as displayed in the right panel of Fig. 2 including leading
finite temperature correction to the potential
$V(\phi,T)=\frac{1}{2}\,(g_{T}\,T^{2}+m_{0}^{2})\,\phi^{2}+\frac{\lambda}{24}\,\phi^{4}+\cdots$
with
$g_{T}=\frac{1}{16}\left[3g_{2}^{2}+g_{1}^{2}+4y_{t}^{2}+\frac{2}{3}\,\lambda\right]$
from [66] affecting the phase transition point. Left: for the bare case
[$m^{2},C_{1}$ ]. Right: with adjusted effective mass from vacuum
rearrangement [$m^{2}_{\rm eff}={m^{\prime}}^{2},C^{\prime}_{1}=C_{1}+\lambda$
]. In the case $\mu_{0}$ sufficiently below $M_{\rm Pl}$ , the case displayed
here, finite temperature effects affect the position of the phase transition
little, while the change of the effective mass by the vacuum rearrangement is
more efficient. The finite temperature effect with our parameters is barely
visible. Updated from [12] (where $X$ has been displayed on a log axis,
falsely labled as log10).

We notice that the SM predicts a huge CC at $M_{\rm Pl}$ :

$\displaystyle\rho_{\phi}\simeq V(\phi)\sim(1.29\,\Lambda_{\rm Pl})^{4}\sim
6.13\times 10^{76}~{}\mbox{GeV}^{4}\,,$ (7)

which is exhibiting a very weak scale dependence (running couplings) such that
we are confronted with the question how to get rid of this huge quasi-
constant? Remember that $\rho_{\phi}$ has no direct dependence of $a(t)$ which
is the decreasing radius of the Friedman universe. An intriguing structure
again solves the puzzle. The effective CC counterterm has a zero, which again
is a point where renormalized and bare quantities are in agreement:

$\displaystyle\rho_{\Lambda
0}=\rho_{\Lambda}+\delta\rho_{\Lambda}\;;\;\;\delta\rho_{\Lambda}=\frac{\Lambda_{\rm
Pl}^{4}}{(16\pi^{2})^{2}}\,X(\mu)$ (8)

with $X(\mu)\simeq\frac{1}{8}\,(2C(\mu)+\lambda(\mu))$ which has a zero close
to the zero of $C(\mu)$ when $2\,C(\mu)=-\lambda(\mu)$. Note that
$C(\mu)=-\lambda(\mu)$ is the shifted Higgs transition point.

Again we find a matching point $\rho_{\Lambda 0}=\rho_{\Lambda}$ between the
low-energy and the high-energy world. At this point, the memory of the quartic
Planck scale enhancement gets lost, as it should be since we know that the low
energy phase does not provide access to cutoff effects.

Crucial point is that

$\displaystyle{X(\mu)=2C+\lambda=5\,\lambda+3\,g_{1}^{2}+9\,g_{2}^{2}-24\,y_{t}^{2}}$
(9)

acquires positive bosonic contribution and negative fermionic ones, with
different scale-dependence131313Unbroken SUSY would require a perfect
cancellation to happen at all scales. Broken SUSY would largely eliminate the
quadratic and quartic enhancements which are the driving effects for our
scenery. Within the SM the cancellation between bosonic and fermionic entries
only happens at some point which seems to require fine-tuning. No problem, the
energy scan, the evolution of the universe provides, accomplishes it for us..
$X$ can vary dramatically (pass a zero), while individual couplings are weakly
scale-dependent with $y_{t}(M_{Z})/y_{t}(M_{\rm Pl})\sim 2.7$ the biggest and
$g_{1}(M_{Z})/g_{1}(M_{\rm Pl})\sim 0.76$ the smallest change. Obviously, the
energy dependence of any of the individual couplings would by far not be able
to sufficiently diminish the originally huge cosmological constant. Only the
existence of a zero in the coefficient function $X(\mu)$ is able to provide
the dramatic reduction of the effective CC, by nullifying the huge cutoff-
sensitive prefactor. Since inflation is tuning the total energy density into
the critical density, the one of a flat geometry, the renormalized dark energy
density can only be a fraction of the critical density, what we know it is.
Fig. 4 illustrates the behavior of the power enhanced quantities $m^{2}_{\rm
eff}(\mu)=m^{2}_{0}(\mu)$ and $\rho_{\Lambda}(\mu)$ as a function of “time” in
units $1/\log_{10}\mu\,,$ where $\mu$ is the energy scale, decreasing as the
universe is cooling down.

Figure 4: The Higgs potential effective $m^{2}_{\rm eff}$ [left] and the dark
energy density $\rho_{\Lambda}$ [right], in units of $\Lambda_{\rm Pl}$ , as
functions of “time”, represented here by $1/\log_{10}\mu\>,$ where $\mu$
represents the energy at that time. Below the matching point $\mu_{\rm CC}$,
where $\rho_{\Lambda}\simeq 1.6\times 10^{-47}$ in Planck mass units, we show
a scaled up to $\rho_{\Lambda}\times 10^{13}$ value of the present dark energy
density $\mu_{0\Lambda}^{4}$ with $\mu_{0\Lambda}\simeq
0.00171~{}\mathrm{eV}.$ Note: $\rho_{\Lambda}(t)$ includes besides the large
positive $V(0)$ also negative contributions from vacuum condensates, like
$\Delta\rho_{\rm EW}$ from the Higgs mechanism and $\Delta\rho_{\rm QCD}$ from
the chiral phase transition. Reproduced from [12].

##  5 Remark on the impact on inflation

Everybody knows the SM hierarchy problem. True, the SM predicts a large bare
Higgs potential mass in the symmetric phase i.e. for $\mu>\mu_{0}$. Is this a
problem? Is this unnatural? In any case, it is a prediction of the SM!
Formally, for $\mu<\mu_{0}$, in the low energy phase, (1) predicts a large
negative $m_{0}^{2}$ which triggers the Higgs mechanism, but it has no
physical meaning because the physical Higgs boson mass now is determined by
the curvature of the Higgs potential at its rearranged minimum at $\phi=v$.

At low energy, we see what we see (what is to be seen): the renormalizable,
renormalized SM as it describes close to all we know up to LHC
energies141414Apart from such never clarified 2 $\sigma$ deviations like e.g.
the $\sin^{2}\Theta_{W}$ measurement by SLD and LEP, the new measurement of
the muon $g-2$ at Fermilab [67] now reveals a 4.2 $\sigma$ deviation.
Similarly, the LHCb Collaboration observes a 3.1 $\sigma$ deviation in lepton
universality [68]. and which is devoid of cutoff effects. What if we go to
very very high energies even close to the Planck scale? Since for
$\mu>\mu_{0}$ the SM predicts the bare $m^{2}_{0}$ to be positive, we are in
the symmetric phase with zero Higgs boson VEV $v=0$ where we start to see the
bare theory i.e. the SM with its bare short distance effective parameters. In
the symmetric phase, all particle masses vanish except for the now four mass-
degenerate very heavy Higgs bosons. A calculation shows that near below the
Planck scale the potential mass term is dominating and the Higgs particles can
be moving at most very slowly, i.e. the potential energy151515By
$:\\!\phi^{n}\\!:$ we denote Wick ordering, which means that $\phi$-field
self-contraction are to be omitted.

$V(\phi)=\frac{m^{2}}{2}\,\phi^{2}+\frac{\lambda}{24}\,\phi^{4}=V(0)+\frac{{m^{\prime}}^{2}}{2}:\\!\phi^{2}\\!:+\frac{\lambda}{24}:\\!\phi^{4}\\!:$

is large while the kinetic energy $\frac{1}{2}\dot{\phi}^{2}$ is small. The
Higgs boson contributes to the energy-momentum tensor in Einstein’s gravity
equations, providing a pressure $p=\frac{1}{2}\,\dot{\phi}^{2}-V(\phi)$ and an
energy density $\rho=\frac{1}{2}\,\dot{\phi}^{2}+V(\phi)$. A tick after the
Big Bang at Planck time $t_{\rm Pl}$ the slow–roll condition
$\frac{1}{2}\,\dot{\phi}^{2}\ll V(\phi)$ is satisfied during some time, where
$p\approx-V(\phi)\,,\>\rho\approx+V(\phi)$ so that $p=-\rho$. This means
$\rho=\rho_{\Lambda}$ represents dark energy! i.e., the system exhibits an
unusual equation of state, which however could be known from ferromagnetic
systems [69]. So we learn that the Higgs boson in the early universe provides
huge dark energy (DE)! Note that the latter includes two different parts
$V(\phi)=V(0)+\Delta V(\phi)\,,$ a static Higgs potential VEV plus a part that
depends on the dynamics of the Higgs field.

The huge DE provides anti-gravity which is inflating the universe! Indeed the
Friedman equation $\frac{\mathrm{d}a}{a}=H(t)\,\mathrm{d}t$ predicts an
exponential growth of the radius $a(t)$ of the universe as $a(t)=\exp Ht$ ,
$H(t)$ the Hubble constant $H\propto\sqrt{V(\phi)}$ . Inflation stops quite
quickly as the field decays exponentially according to the field equation
$\ddot{\phi}+3H\dot{\phi}\simeq-V^{\prime}(\phi)$ . At some point, the Higgs
potential mass term will be the dominant term
$V(\phi)\approx\frac{m^{2}}{2}\,\phi^{2}$ and we have a harmonic oscillator
with friction which predicts Gaussian inflation, as it is observed in Cosmic
Microwave Background (CMB) sky maps [70]. LEESM inflation automatically passes
through the Gaussian phase before inflation halts as it follows from the Higgs
field dynamics. All this tells us that the Higgs boson is the inflaton, always
under the assumption the SM couplings at the end will turn out to allow for an
extrapolation up to $M_{\rm Pl}$.

If inflation happens it tunes the total energy density to be that of a flat
space, which has the particular value $\rho_{\rm crit}=\mu^{4}_{\rm crit}$
with $\mu_{\rm crit}=0.00247\,\mathrm{eV}.$ From the CMB data, we know that
today $\rho_{\Lambda}=\mu^{4}_{0,\Lambda}$ with
$\mu_{0,\Lambda}=0.00171\,\mathrm{eV}.$ With the ongoing expansion (cooling)
the dark energy density will be approaching
$\mu_{\infty,\Lambda}=0.00247\,\mathrm{eV}.$ In any case, the large early
cosmological constant gets tamed by inflation to be part of the critical flat
space density. No cosmological constant problem either? In any case, inflation
is proven to have happened by observation and inflation requires the existence
of a scalar field [71, 72]. The Higgs field is precisely such a field we need
and within the SM it has the properties which promote it to be the inflaton.

All other inflaton models must assume a least one additional scalar sectors to
exist where neither the shape of the potential nor the parameters are known a
priori. In this case, all “predictions” have to be tailored to reproduce what
you attempt to predict like e.g. the spectral indices deriving from CMB data.
Higgs inflation is very different. We know the Higgs boson properties and the
mechanism providing the dark energy. Although this looks pretty
straightforward it is delicate, because of the high sensitivity to the SM
parameters and their uncertainties and missing higher-order corrections. Higgs
inflation is an SM prediction, with all parameters calculable, except the
value of the Higgs field in the Higgs potential. To get the required amount of
inflation, about $N_{e}\approx 60$ e-folds ($N_{e}=\ln[{a(t_{\rm
end})}/{a(t_{\rm initial})}]$ with $t_{\rm initial/end}$ the time inflation
starts/ends) are required to cover the CMB causal cone, one needs
$\phi_{0}=4.5\,M_{\rm Pl}$ at Planck time. This is not unreasonable because
the medium is extremely hot with a Hubble constant of about $H_{i}\simeq
17\,M_{\rm Pl}$, given by the SM spectrum and the Stefan-Boltzmann law. The
huge field strength at $M_{\rm Pl}$ decays exponentially in a very short time.

As we know, for arbitrary scales $\mu$, the compensation between positive
bosonic terms and negative fermionic terms within the SM is incomplete. Exact
supersymmetry would amount to a perfect cancellation in both cases, for
$m_{0}^{2}$ and $\langle V(0)\rangle$, by the supersymmetric partners of the
SM particles. However, the running of the SM parameters actually also happens
to completely cancel bosonic and fermionic contributions at some point. This
depends on the proper conspiracy of the SM couplings. That the cancellation
only happens at a particular scale thereby is not important. In fact, the
existence of a matching point solves both the hierarchy problem as well as the
cosmological constant problem. The nullification of the power-enhanced terms
happens dynamically, without imposing supersymmetry or alternative mechanisms
designed to avoid quadratic divergences. Such matching points, if they exist
below the Planck scale, are always met at some point since the expansion of
the universe implies the necessary energy scan.

##  6 Conclusion

I have elaborated how Tini Veltman’s understanding of the SM structure
provides a convincing picture of the SM as a low energy effective theory
deriving from a Planck medium. Minimality seems to be the guiding principle
and favors the SM as a viable low energy emergent structure. I do not know any
SM extension which is not suffering from additional fine-tuning issues, while
pretended SM naturalness problems are used to motivate such extensions.
Admittedly, also for the SM parameters, the majority of them particle-Higgs-
boson couplings, we have no idea about what determines their values, which are
covering 14 orders of magnitude between the neutrino masses of order
$10^{-3}~{}{\rm eV}$ and the top-quark mass at $1.72\times 10^{11}~{}{\rm
eV}\,.$ However, the Higgs boson discovery has told us that there is a
”teamwork” of the leading SM couplings, which can reveal a particular link of
the SM to the Planck world. This may be seen as a hint that a kind of self-
organized system may be at work within the primordial “ether”161616Let me
quote here Wikipedia [73]: Self-organization, also called spontaneous order,
is a process where some form of overall order arises from local interactions
between parts of an initially disordered system. The process can be
spontaneous when sufficient energy is available, not needing control by any
external agent. It is often triggered by seemingly random fluctuations,
amplified by positive feedback. The resulting organization is wholly
decentralized, distributed over all the components of the system. As such, the
organization is typically robust and able to survive or self-repair
substantial perturbation. Chaos theory discusses self-organization in terms of
islands of predictability in a sea of chaotic unpredictability.. Looking at
the primordial hot chaotic medium with the eyes of a distant observer we see
the chaos through the filter which only lets through the long-distance
physics. The corresponding cooperation of the different interactions,
balancing bosonic versus fermionic contributions in running couplings, seems
to be even more striking when we are considering the consequences for the
“Infrared - Ultraviolet Connection” first broached by Veltman. The possible
impact of these conspiracies is the SM prediction of Higgs inflation and
reheating after the Big Bang. I have presented only a gross picture that
derives from the leading power-enhanced LEESM effects. For consideration of
finite temperature effects within the LEESM Higgs-inflation scenario I refer
to Fig. 3 and [44, 45], which also include a discussion of reheating and
possible impacts for explaining the baryon-asymmetry in the universe. Also,
the bare Higgs potential is subject to radiative corrections [58] which
generates an effective potential on which the meta-stable vacuum scenarios are
relying. In [12] I have shown that when power-enhanced corrections are taken
into account in the stable vacuum case these radiative corrections have little
impact on the leading pattern.

ATLAS and CMS results, the milestone discovery of the SM Higgs boson together
with the absence of any hints of beyond the SM (BSM) physics, may have
unexpectedly revolutionized particle physics, namely showing that the SM has
higher self-consistency (conspiracy) than expected and previous arguments for
the existence of new physics may turn out not to be compelling. I see the SM
as a low energy effective theory of some cutoff system at $M_{\rm Pl}$
consolidated. The crucial point is the enormous gap between $M_{\rm Pl}$ and
from what we can see, which includes all SM particles. Dark energy and
inflation are unavoidable consequences of the SM provided new physics does not
disturb the SM prediction substantially.

My main theses are:

  * 1)

The SM as a renormalized QFT has no hierarchy problem, rather its LEESM
extension provides a hierarchy solution for inflation in the early universe.
Over the fairly weak logarithmic scale dependence of the running parameters,
only the leading relevant operators like the Higgs potential and the Higgs
boson mass term are power enhanced and hence able to “talk” with the Planck
regime conspicuously.

  * 2)

A super-symmetric or any other extension of the SM cannot be motivated by the
(non-existing) hierarchy problem.

  * 3)

In the early symmetric phase the quadratically enhanced bare mass term in the
Higgs potential together with the quartically enhanced Higgs potential VEV
trigger inflation shortly after the Big Bang. The relevant time-dependent
power corrections are providing a strongly time-dependent CC, which is shown
in Fig. 4. This also implies a strongly time-dependent Hubble constant in the
early universe which possibly could solve the existing Hubble constant puzzle
[7].

  * 4)

As the hot universe cools down after the Big Bang, the running of the SM
couplings let the bare Higgs potential mass $m_{0}^{2}$ flip sign from a large
positive to a large negative value. This is triggering the Higgs mechanism at
about $\mu_{0}\sim 10^{16}~{}\mbox{GeV}$. In the broken phase171717The large
negative $m_{0}^{2}$ ceases to act as a Higgs boson mass and loses its
observability. After the memory on the cutoff has been lost at that point,
within the broken phase the cutoff has no further meaning than as a UV
regulator and there is no reason to choose it to be the Planck cutoff. the
Higgs boson is naturally as light as other SM particles which are generated by
the Higgs mechanism, i.e. including the Higgs boson mass itself all masses are
determined by the mass-coupling relations
$M_{W}^{2}=\frac{1}{4}\,g^{2}\,v^{2},\,M_{Z}^{2}=\frac{1}{4}\,(g^{2}+g^{\prime
2})\,v^{2},\,M_{f}^{2}=\frac{1}{2}\,y^{2}_{f}\,v^{2},\,{\boldsymbol{M}}_{H}^{2}=\frac{1}{3}\,{\boldsymbol{\lambda}}\,{\boldsymbol{v}}^{2}\,.$
Like all other particles, the Higgs mass appears generated by the non-
vanishing VEV $v\neq 0$ . Note that $v$ is not Planck scale enhanced as at the
Higgs phase transition point the renormalized and the bare mass coincide (the
$\Lambda_{\rm Pl}$ enhanced correction is zero at the transition point) and in
the high energy phase when $\mu>\mu_{0}$ up to $M_{\rm Pl}$ we have $v\equiv
0$. So, provided new physics does not disturb the SM pattern substantially,
dark energy and inflation are unavoidable consequences of the SM Higgs system
in our bottom-up approach.

  * 5)

The Higgs mechanism terminates inflation and triggers the electroweak phase
transition; reheating likely proceeds via the four heavy decaying Higgs
particles into fermion pairs (predominantly top/anti-top quark-pairs) just
before the system jumps into the broken phase. Again, no non-SM particles
needed to provide reheating.

All that necessitates reconsidering the early pre-Higgs phase epoch of
cosmology. The SM then is in the symmetric phase, which takes a very different
form from known physics in the broken phase. In place of the massless photon
which emerges at the electroweak phase transition, in the symmetric phase we
have all the SM gauge bosons as massless radiation fields together with the
massless Weyl fermions. At these times there are four very heavy Higgs
particles that drive inflation and reheating. A more detailed understanding of
the EW phase transition is required and could unveil surprises.

Epilogue: The sharp dependence of the Higgs vacuum stability on the SM input
parameters and possible SM extensions and the vastly different scenarios which
can result as a consequence of minor shifts in parameter space makes the
stable vacuum case a particularly interesting one and it could reveal the
Higgs particle as “the master of the universe”. After all, it is commonly
accepted that dark energy is the stuff shaping the universe both at very early
as well as at the late times.

A lot of details have to be worked out to scrutinize the whole picture. On the
SM side, the major issue for the future is the very delicate conspiracy
between SM couplings, which means that precision determinations of the
parameters now turn to be more important than ever. Precision measurements of
top-quark and Higgs boson properties should be a prime challenge for the LHC
and the ILC/FCC-ee projects. Precision values for $\lambda$ , $y_{t}$ and
$\alpha_{s}$ from the high energy frontier should go together with a program
aiming to provide more precise low energy hadronic cross section measurements
at low energy hadron facilities together with lattice QCD calculations of
hadronic vacuum polarization effects which would allow one to reduce hadronic
uncertainties in $\alpha(M_{Z})$ and $\alpha_{2}(M_{Z})$ .

The mayor open issues: where is dark matter? can we explain baryon asymmetry?
what triggers the see-saw mechanism explaining the smallness of neutrino
masses? what is tuning the strong CP problem?

While these questions could not be answered under the presumptions of the
standard paradigm based on the belief that going to higher energies reveals
more symmetry and a structurally simpler world exhibiting supersymmetry,
higher symmetry groups, super-gravity, and strings, I think one should
reconsider them under the aspect of the paradigm of emergence, where nature
uncovers more complexity at very high energy and symmetries are emergent at
low energies. It is time to reflect the prejudices which are guiding
speculations on BSM physics on the path towards higher energies and higher
precision. Searching for emergent structures beyond the SM seems to be a more
promising strategy to actually find what is missing in the SM and could be
proven one day. While I argued that the LEESM emergence scenario conflicts
with hierarchy-problem motivated SM extensions, the LEESM does not exclude BSM
physics which we know we have to incorporate in any case, like dark matter
(e.g. in the form of $\mathit{SU}(4)$ confined bound states as studied in
[43]), Majorana neutrinos181818The 3 singlet neutrinos $\nu_{R}$ needed to
allow the neutrinos to have masses generated by the Higgs mechanism are
included in the SM and must exhibit corresponding Yukawa couplings. The
neutral Weyl-fermions could be Majorana particles and if so they would also
have a singlet Majorana mass term which is allowed by renormalizability but is
not protected by any symmetry. Then like the Higgs boson mass in the symmetric
phase these Majorana particles would have Planck scale induced very heavy
masses which would induce a see-saw mechanism and explaining the lightweights
of the observed neutrinos in the broken phase [74]., axions, etc. Last but not
least the scenario is easy to falsify: find a 4th family fermion, a SUSY
particle, or whatever modifies SM quadratic effects or the SM parameter
running pattern substantially.

Since the SM parameters, as they have been fixed by experimental data within
their uncertainties and to the extent that the conversion to
$\overline{\mathrm{MS}}$ parameters theory-wise is under control, in their
cooperation are very very close (at 1.3 $\sigma$ as inferred in [54]) if not
on the spot to match the window of Higgs vacuum stability and SM extend-
ability up to the gravity ruled Planck scale, the Higgs-inflation option
remains an exciting scenario. Why the relevant parameters, the gauge
couplings, the top-quark Yukawa coupling, and the Higgs potential self-
coupling determined via the Higgs boson mass, should have values barely
missing the option that the scalar field which is required by inflation may be
identified as the SM Higgs boson field remains to be clarified. So stay tuned,
beware of an unstable vacuum. After all, it is the vacuum on which our cosmos
rests!

Acknowledgments:
I am indebted to Wolfgang Kluge and Oliver Bär for their careful reading of
the manuscript and helpful comments. I also thank Mikhail Kalmykov for many
inspiring discussions and long-time collaboration in electroweak two-loop
calculations and in particular on working out the relationship between on-
shell and $\overline{\mathrm{MS}}$ parameters in the SM, which plays a key
role in the present article.

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