#  A road to an elementary particle physics model
with no Higgs - I

G.C. Rossi  Dipartimento di Fisica, Università di Roma “Tor Vergata”
INFN, Sezione di Roma “Tor Vergata”
Via della Ricerca Scientifica - 00133 Roma, Italy
Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche E. Fermi,
Via Panisperna 89a, 00184 Roma, Italy

###### Abstract

This is the first of two companion papers in which we prove that the recently
discovered non-perturbative mechanism capable of giving mass to elementary
fermions can also generate a mass for the electro-weak bosons, when weak
interactions are switched on, and can thus be used as a viable alternative to
the Higgs scenario. It can be demonstrated that the non-perturbatively
generated fermion and $W$ masses have the expression $m_{f}\sim
C_{f}(\alpha)\Lambda_{RGI}$ and $M_{W}\sim g_{w}c_{w}(\alpha)\Lambda_{RGI}$,
respectively, where $C_{f}(\alpha)$ and $c_{w}(\alpha)$ are functions of the
gauge couplings, $g_{w}$ is the weak coupling and $\Lambda_{RGI}$ the RGI
scale of the theory. In view of this parametric structure we are led to argue
that to get top quark and $W$ boson masses of the correct order of magnitude,
a realistic model must include a yet unobserved sector of massive fermions
(Tera-fermions) subjected, besides Standard Model interactions, to some kind
of super-strong gauge interactions (Tera-interactions) so that the full theory
(including Standard Model and Tera-particles) will have an RGI scale (we call
it $\Lambda_{T}$) much larger than $\Lambda_{QCD}$, located in the few TeV
region. The further extension of the model with the introduction of
hypercharge interactions and particles singlets under strong interactions
(leptons and Tera-leptons) is the object of the companion paper, where we also
discuss some interesting phenomenological implications of the scenario we are
advocating. As for the relation of our approach to the Standard Model, we show
that, upon integrating out the (heavy) Tera-degrees of freedom, the resulting
low energy effective Lagrangian closely resembles the Standard Model
Lagrangian. The argument rests on the key conjecture that the 125 GeV
resonance detected at LHC is a $W^{+}W^{-}/ZZ$ composite state, bound by Tera-
particle exchanges, and not an elementary particle.

Though limited in its scope (for the moment we restrict to the case of only
one family, neglecting weak isospin splitting), the present approach has a
certain number of merits with respect to the Standard Model. Lacking a
fundamental Higgs, it offers a radical solution of the Higgs mass tuning
problem. It provides a physical interpretation of the electro-weak scale, with
the latter being (a fraction of) the scale, $\Lambda_{T}$, of a new
interaction. It helps reducing the number of the Standard Model parameters, as
elementary particle masses are non-perturbatively fully determined by the
dynamics of the theory.

##  1 Introduction

The authors of ref. [1] introduced an interesting field theoretical
renormalizable model where an SU(2) fermion doublet, subjected to non-abelian
gauge interactions of the QCD type, is coupled to a complex scalar field via a
$d=4$ Yukawa term and an “irrelevant” $d>4$ Wilson-like operator [2].
Notwithstanding the fact that both terms break chiral invariance, it was shown
in [1], and numerically checked in [3] in a set of dedicated lattice
simulations, that there exists a critical value of the Yukawa coupling where
chiral symmetry is recovered, up to corrections formally vanishing as the UV
cut-off is removed.

Although an exact $\chi_{L}\times\chi_{R}$ symmetry, that rotates all the
matter fields of the Lagrangian, forbids the existence of standard fermionic
mass operators, i.e. operators of the form $m\bar{\psi}\psi$, it can be shown
that in the Nambu–Goldstone (NG) phase of the critical theory where the
$\chi_{L}\times\chi_{R}$ symmetry is spontaneously broken, peculiar non-
perturbative (NP) mass terms emerge of the kind $C\Lambda_{\rm
RGI}(\bar{\psi}_{L}U\psi_{R}+\bar{\psi}_{R}U^{\dagger}\psi_{L})$, where
$U=\exp{i\vec{\tau}\vec{\zeta}/C\Lambda_{\rm RGI}}$ is the exponential of the
Goldstone bosons associated with the spontaneously broken
$\chi_{L}\times\chi_{R}$ symmetry, $\Lambda_{\rm RGI}$ is the RGI scale of the
theory and $C$ is a computable coefficient, function of the gauge couplings.
Upon expanding $U$ around the identity we see that NP O($\Lambda_{\rm RGI}$)
masses for the elementary fermions get dynamically generated (see below eq.
(1.1)) plus a wealth of non-linear NG bosons interactions.

###  1.1 NP mass generation mechanism

NP masses emerge as a consequence of a sort of “interference” between residual
UV chiral breaking effects left behind at the critical value of the Yukawa
coupling in the NG phase of the theory, and IR features triggered by the
phenomenon of spontaneous breaking of (the recovered) chiral symmetry which
standardly takes place in a strongly interacting theory.

A detailed analysis of this subtle field theoretical interplay shows that the
non perturbatively (NP-ly) generated elementary fermion masses have the
parametric expression [1, 4]

$m_{f}\sim C_{f}(\alpha)\Lambda_{\rm RGI}\,,$ (1.1)

where the coefficient $C_{f}(\alpha)$ is a function of the gauge coupling. If
we take the “irrelevant” chiral breaking Wilson-like term to be a $d=6$
operator (as we do for illustrative purposes in eqs. (2.3) and (3.19) below),
one finds at the lowest loop order $C_{f}(\alpha)={\mbox{O}}(\alpha^{2})$.

A far reaching phenomenological implication of the formula (1.1), when
referred to the top quark, is that the relation $m_{\rm
top}\sim{\mbox{O}}(\Lambda_{\rm RGI})$ requires $\Lambda_{\rm
RGI}\gg\Lambda_{QCD}$ and of the order of a few TeVs if eq. (1.1) has to
reproduce the correct (order of magnitude of the) top mass experimental value.
We are thus led to conjecture that a new sector of super-strongly interacting
particles 111To avoid any misunderstanding we explicitly remark that here
”super” has nothing to do with super-symmetry., gauge-invariantly coupled to
standard matter, needs to exist so that the complete theory, encompassing the
new and the Standard Model (SM) particles, will have an RGI scale in the TeV
region [5, 6, 7, 8, 9].

We want to immediately remark that the need for assuming the existence of a
super-strongly interacting sector here is totally different from the reason
why Technicolor was introduced in refs. [10, 11]. Technicolor was invoked as a
way to provide mass to the electroweak (EW) bosons in the first place, while
in the present approach super-strong interactions need to be introduced to get
the right order of magnitude of the dynamically generated top quark mass and,
as we will show in sect. 3.4, also of the $W$ mass.

As we have said above, a key feature of our model is that (irrespective of the
value of the Yukawa coupling) the Lagrangian enjoys an exact symmetry,
$\chi_{L}\times\chi_{R}$, protecting elementary particle masses against power
divergent quantum corrections, unlike what happens in Wilson lattice QCD
(WLQCD) where the fermion mass is affected by a linearly divergent power
correction [2]. At variance with Technicolor, the same exact symmetry makes
all fermion–anti-fermion vev’s (condensates) vanishing, thus precluding the
possibility of using them to generate masses.

In any case, as suggested by Glashow [12], in the following to avoid
confusion, we will refer to these new super-strong interactions as Tera-
interactions and to the new set of particles as Tera-particles.

In the present paper we show that the model proposed in [1] can be naturally
extended to incorporate weak interactions and Tera-particles (see refs. [5, 6]
for an early formulation of the extended model). The $W$ bosons as well as the
Tera-fermions will acquire a NP mass O($\Lambda_{\rm RGI}$) via the same
mechanism that leads to eq. (1.1). For the $W$ mass one gets (see sect. 3)

$M_{W}\sim g_{w}c_{w}(\alpha)\Lambda_{\rm RGI}\,,$ (1.2)

where $g_{w}$ is the weak coupling and $\alpha$ is a short-hand for the set of
gauge couplings $[\alpha_{w},\alpha_{s},\alpha_{T}]$ with $\alpha_{w}$,
$\alpha_{s}$ and $\alpha_{T}$ referring to weak, strong and Tera-strong gauge
interactions, respectively. In the following we will denote by $\Lambda_{T}$
the RGI scale of the whole theory where the subscript $T$ is to remind us that
we are including Tera-particles in the theory. The lowest loop order
parametric expression of $c_{w}(\alpha)$ and that of the NP Tera-fermion mass
are derived in sect. 3.5.

In this first paper we limit ourselves to discuss how the model of ref. [1]
can be extended to incorporate weak interactions and Tera-particles. In the
companion paper [13] (hereafter referred to as (II)), besides taking up a
number of technical issues left aside here, we address the fundamental issue
of adding hypercharge interactions and particles, singlets under strong
interactions (leptons and Tera-leptons), and we present some interesting
phenomenological implications of the model.

###  1.2 Comparing with the Standard Model

Eqs. (1.1)-(1.2) are very similar to the expression of the SM Higgs-like mass
of fermions and $W$’s, respectively, with however two fundamental differences.
The first is that the scale of the masses is not the vev of the Higgs field
(whose value in the SM is, for instance, determined by fitting the $W$ mass),
but a dynamical quantity associated to a new interaction. The second, related
difference is that the values of the Yukawa couplings, which in the SM are
tuned by hand to match the phenomenological values of quark masses (and, if
present, also of lepton masses, see (II)) in the present approach are not free
parameters. They are in principle calculable and are determined by the
dynamics of the theory.

From the above discussion we conclude that the general NP scenario for
elementary particle mass generation we are describing in this investigation
can be considered as a valid alternative to the Higgs mechanism, with the
extra advantage that lacking a fundamental Higgs, we will not have to worry
about the fine tuning problem of its mass. Though we may need to reassess the
question of the stability of the EW vacuum [14, 15, 16, 17]. A further
conceptual bonus is that we have a natural interpretation of the magnitude of
the electroweak scale (EW) scale, as (a fraction of the) the physical
dynamical parameter, $\Lambda_{T}$.

####  1.2.1 A few remarks

There is a number of additional interesting features and implications of the
approach we are advocating in this work that are worth mentioning.

First of all, lacking the need for the existence of a fundamental Higgs boson
for elementary particle mass generation, we are in the obligation of finding a
convincing interpretation for the 125 GeV resonance identified at LHC [18,
19]. We conjecture that this particle is not a fundamental object. We propose
to interpret it as a $W^{+}W^{-}/ZZ$ composite state, bound by exchanges of
Tera-particles which, being charged under EW interactions, can couple to the
EW bosons. If the above interpretation is correct, this scalar particle must
be incorporated in the low energy effective Lagrangian (LEEL) of the theory
that one gets by integrating out the heavy Tera-degrees of freedom (Tera-
dof’s) since its experimental mass is much smaller than the $\Lambda_{T}$
scale. Not unexpectedly, at (momenta)${}^{2}\ll\Lambda_{T}^{2}$ the resulting
$d=4$ part of the LEEL of the model is seen to resemble very much the SM
Lagrangian. We shall discuss this issue in detail in sect. 4.

Secondly, as it was shown in ref. [20], with a reasonable choice of the
elementary particle content, a realistic theory extending the SM with the
inclusion of the new Tera-sector leads to gauge coupling unification at a
$\sim 10^{18}$ GeV scale. This somewhat large unification scale is likely to
yield a proton life-time comfortably larger than the present bound $\tau_{\rm
prot}>1.7\times 10^{34}$ years [21].

We end by observing that, as discussed in sect. 5, the problem of
understanding the multiplicity of families (and possibly cope with weak
isospin splitting) is related to the questions of the universality of the NP
mass generation mechanism we are describing in this work, and the predicting
power of the present approach. Both issues are under active investigation.

###  1.3 Outline of the paper

The outline of this first paper is as follows. In sect. 2 for completeness and
to set our notations we review the key features of the model introduced in
ref. [1]. In sect. 3 we discuss how to extend it with the introduction of weak
and Tera-interactions. In sect. 4 we prove that the LEEL one obtains by
integrating out the heavy Tera-dof’s closely resembles to the SM Lagrangian.
In sect. 5 we touch the issue of universality and predictive power. A few
conclusions and an outlook of our future lines of investigation can be found
in sect. 6. In Appendix A we recall the procedure that leads to the equations
determining the values of the Lagrangian parameters, which specify the
critical theory endowed with an enlarged symmetry of chiral nature. In
Appendix B we justify the form that the quantum effective Lagrangian (QEL)
functional of the theory 222By QEL we mean the generating functional of the
1PI vertices from which one can directly extract the full quantum information
of the model. To be precise in this paper we make a distinction between the
QEL functional that represents the Legendre transform of the partition
function and the LEEL of the theory, valid at small momenta, that is obtained
upon integrating out all the dof’s heavier than the chosen momentum scale.
takes in the NG phase at the critical point. In Appendix C we discuss the form
of the effective scalar propagator in the critical limit. The way in which the
transversality property of the $W$ polarization amplitude is implemented in
the present NP approach to mass generation is explained in Appendix D.

##  2 A toy-model

For the reader’s convenience we summarize in this section the results of ref.
[1]. The simplest situation in which the NP mass generation mechanism outlined
in the Introduction takes place is realized in a field theory where an SU(2)
fermion doublet, subjected to non-abelian gauge interactions of the QCD type,
is coupled to a complex scalar field via a $d=4$ Yukawa term and an
“irrelevant” $d=6$ Wilson-like operator [2], which both break chiral
invariance. The Lagrangian of this “toy-model” reads

$\displaystyle{\cal L}_{\rm toy}(q,A;\Phi)={\cal L}_{K}(q,A;\Phi)+{\cal
V}(\Phi)+{\cal L}_{Yuk}(q;\Phi)+{\cal L}_{Wil}(q,A;\Phi)$ (2.3)
$\displaystyle\quad\bullet\,{\cal
L}_{K}(q,A;\Phi)=\frac{1}{4}F^{A}\\!\cdot\\!F^{A}+\big{(}\bar{q}_{L}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{L}+\bar{q}_{R}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{R}\big{)}+\frac{1}{2}{\,\hbox{tr}\,}\big{[}\partial_{\mu}\Phi^{\dagger}\partial_{\mu}\Phi\big{]}$
(2.4) $\displaystyle\quad\bullet\,{\cal
V}(\Phi)=\frac{\mu_{0}^{2}}{2}{\,\hbox{tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}+\frac{\lambda_{0}}{4}\big{(}{\,\hbox{tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}\big{)}^{2}$
(2.5) $\displaystyle\quad\bullet\,{\cal
L}_{Yuk}(q;\Phi)=\eta\,\big{(}\bar{q}_{L}\Phi
q_{R}+\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}$ (2.6)
$\displaystyle\quad\bullet\,{\cal
L}_{Wil}(q,A;\Phi)=\frac{b^{2}}{2}\rho\,\big{(}\bar{q}_{L}\overleftarrow{\cal
D}^{A}_{\mu}\Phi{\cal D}^{A}_{\mu}q_{R}+\bar{q}_{R}\overleftarrow{\cal
D}^{A}_{\mu}\Phi^{\dagger}{\cal D}^{A}_{\mu}q_{L}\big{)}\,,$ (2.7)

where $q_{L}=(u_{L},d_{L})^{T}$ and $q_{R}=(u_{R},d_{R})^{T}$ are fermion iso-
doublets and
$\Phi=\varphi_{0}1\\!\\!1+i\varphi_{j}\tau^{j}=[-i\tau_{2}\varphi^{\star}|\,\varphi]$
is a $2\times 2$ matrix with
$\varphi=(\varphi_{2}-i\varphi_{1},\varphi_{0}-i\varphi_{3})^{T}$ a complex
scalar doublet, singlet under color SU($N_{c}$) gauge transformations. The
length scale $b$ is the inverse of the UV cutoff, $b^{-1}\sim\Lambda_{UV}$,
$\eta$ is the Yukawa coupling, $\rho$ is introduced to keep track of ${\cal
L}_{Wil}$ and ${\cal D}_{\mu}^{A}$ is the gauge-covariant derivative.

A few remarks are in order here. First of all, we want to stress that the
distinctive feature of the Lagrangian (2.3) is the presence of the
“irrelevant” $d>4$ chiral breaking Wilson-like term. Secondly, because of its
presence we have immediately introduced a Yukawa term in the fundamental
Lagrangian as it will be anyway generated by loop corrections. Finally, we
observe that, despite the presence of the $d=6$ ${\cal L}_{Wil}$ term, ${\cal
L}_{\rm toy}$ is power counting renormalizable, because the $d=6$ Wilson-like
operator appears in the Lagrangian multiplied by two inverse powers of the UV
cutoff. It is worth recalling that a similar situation occurs in WLQCD where
the $d=5$ Wilson term in the lattice action is multiplied by one power of the
lattice spacing, thus making the lattice action renormalizable and amenable to
Monte Carlo simulations [2, 22].

###  2.1 Symmetries

Among other obvious symmetries, ${\cal L}_{\rm toy}$ is invariant under the
(global) transformations $\chi_{L}\times\chi_{R}$ involving fermions and
scalars given by ($\Omega_{L/R}\in{\mbox{SU}}(2)$)

$\displaystyle\chi_{L}\times\chi_{R}=[\tilde{\chi}_{L}\times(\Phi\to\Omega_{L}\Phi)]\times[\tilde{\chi}_{R}\times(\Phi\to\Phi\Omega_{R}^{\dagger})]$
(2.8)
$\displaystyle\qquad\tilde{\chi}_{L}:q_{L}\rightarrow\Omega_{L}q_{L}\,,\qquad\qquad\,\bar{q}_{L}\rightarrow\bar{q}_{L}\Omega_{L}^{\dagger}$
(2.9)
$\displaystyle\qquad\tilde{\chi}_{R}:q_{R}\rightarrow\Omega_{R}q_{R}\,,\qquad\qquad\bar{q}_{R}\rightarrow\bar{q}_{R}\Omega_{R}^{\dagger}$
(2.10)

The exact $\chi_{L}\times\chi_{R}$ symmetry can be realized either á la Wigner
or á la NG depending on the shape of the scalar potential, ${\cal V}(\Phi)$.
In any case, as we mentioned in the Introduction, no linearly divergent
fermion mass can be generated by quantum corrections because the mass operator
$\bar{q}_{L}q_{R}+\bar{q}_{R}q_{L}$ is not invariant under
$\chi_{L}\times\chi_{R}$. Obviously, this also means that no fermion–anti-
fermion vev’s of the kind $\langle\bar{q}_{L}q_{R}+\bar{q}_{R}q_{L}\rangle$
(condensates) can ever be non-vanishing.

###  2.2 Chiral invariance

For generic values of $\eta$ (at fixed $\rho$), the Lagrangian ${\cal L}_{\rm
toy}$ is not invariant under the fermionic chiral transformations
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ of eqs. (2.9)-(2.10), because of the
presence of the (chiral breaking) operators ${\cal L}_{Wil}$ and ${\cal
L}_{Yuk}$. Nevertheless, invariance under
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ can be recovered (up to O($b^{2}$)
terms) by enforcing the conservation of the corresponding currents. This can
be seen to occur [1, 4] at a “critical” value of the Yukawa coupling,
$\eta_{cr}$, where the $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ rotations of
the Yukawa and Wilson-like operators, that separately break current
conservation, “compensate” each other.

The situation here is very similar to what happens in WLQCD where fermionic
chiral symmetry is recovered (up to O($a$) cutoff effects) by tuning the bare
quark mass to a critical value, $m_{cr}$, at which the chiral rotations of the
Wilson and mass operators “compensate” each other, yielding the conservation
of the chiral currents [2, 23]. As remarked above, a key difference between
the model (2.3) and WLQCD is, however, that in WLQCD no symmetry exists that
can prevent the appearance of a linearly divergent quantum correction to the
fermion mass, unlike what happens in the case of the model (2.3).

Naturally, as was done in ref. [3], the determination of the critical value of
the Yukawa coupling needs to be performed in a non-perturbative way, similarly
to what is currently done in WLQCD in the calculation of the critical mass.
Nevertheless, in order to get a feeling of what this criticality condition
implies, it is interesting to see what happens at the lowest loop order.

$\bullet$ In the Wigner phase (where $\langle|\Phi|^{2}\rangle=0$) the
condition enforcing the conservation of the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ currents (derived in [1] along the
lines of the strategy worked out in [23] and summarized in Appendix A) gives
at 1-loop

$\eta_{cr}^{(\rm 1-loop)}(\alpha_{s},\rho)=\rho\,\alpha_{s}N_{c}\eta_{1}\,,$
(2.11)

where $\eta_{1}$ is a computable coefficient. The lowest loop order mechanism
underlying this cancellation is sketched in fig. 1. At the NP level the
critical condition implies the vanishing of the effective Yukawa coupling in
the QEL of the theory.

Figure 1: The lowest loop order mechanism behind the cancellation of the
Yukawa interaction in the Wigner phase occurring at $\eta=\eta_{cr}$. Grey
disc and box stand for the insertion of the Yukawa and Wilson-like Lagrangian
vertices, respectively. The solid line represents the propagation of a
fermion, the curly line of a gluon and the dotted line of a scalar. R and L
under the fermion line denote chirality. There is a similar condition with $R$
and $L$ exchanged.

$\bullet$ In the NG phase (where $\langle|\Phi|^{2}\rangle\neq 0$ and at tree-
level equal to $v^{2}=|\mu_{0}^{2}|/\lambda_{0}$), the $\eta$ criticality
condition leads at the lowest loop order to the vanishing of the sum of
diagrams shown in fig. 2. The figure is obtained from fig. 1 by setting the
scalar field equal to its vev. We thus see that, remarkably, enforcing
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ invariance implies that in the QEL of
the critical theory precisely the Higgs-like mass of the fermion gets
cancelled out.

Figure 2: The lowest loop order mechanism behind the cancellation of the
Higgs-like mass of the quark in the NG phase, occurring at $\eta=\eta_{cr}$.
There is a similar condition with $R$ and $L$ exchanged.

We end this section with a few important remarks.

1) The possibility of recovering $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$
invariance rests on the crucial fact that the quadratic divergency of the loop
integral in fig. 1, as well as in fig. 2, is exactly compensated by the
$b^{2}$ factor coming from the insertion of the Wilson-like box, leaving
behind a finite result. This balance of UV and IR effects (which has its
ultimate basis on dimensional grounds) should not come as a surprise when
irrelevant operators are present in the Lagrangian. Indeed, compensations of
this kind also occur in WLQCD and are, for instance, at the origin of the fact
that the (partially) conserved axial and vector flavour currents suffer a
finite renormalization 333Actually in WLQCD in the vector case one can
construct a current that, being exactly conserved at any value of the lattice
spacing, does not need any renormalization.. In the following we shall
encounter other instances of this sort of “UV vs. IR compensation”.

2) We see thus that (owing to the cancellation pictorially illustrated at the
lowest loop order in fig. 2) in the present approach $\Phi$ is not going to
play the role the Higgs field does in the SM, rather its presence should be
viewed as an effective, simple way to model a $\chi_{L}\times\chi_{R}$
invariant UV completion of ${\cal L}_{\rm toy}$.

3) According to the ’t Hooft criterion [24], tuning $\eta$ at its critical
value is a costless and natural step as it leads to an enlargement of the set
of symmetries of the Lagrangian.

###  2.3 The QEL of the critical theory in the Wigner phase

In this section we provide the form that the QEL takes in the Wigner phase at
the critical value of the Yukawa coupling. As we said, enforcing invariance
under $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations means that
$\tilde{\chi}$-violating operators should be absent from the expression of the
QEL of the critical theory. The $d\leq 4$ piece of the QEL of the critical
model in the Wigner phase will then have the simple form

$\displaystyle{\Gamma}_{4\,cr}^{Wig}=\frac{1}{4}F^{A}\\!\cdot\\!F^{A}+\big{(}\bar{q}_{L}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{L}+\bar{q}_{R}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{R}\big{)}+\frac{1}{2}{\,\hbox{tr}\,}\big{[}\partial_{\mu}\Phi^{\dagger}\partial_{\mu}\Phi\big{]}+{\cal
V}(\Phi)\,,$ (2.12)

from which we see that scalars are completely decoupled from fermions. We
should observe that in the Wigner phase the critical theory is kind of
“unstable”. In fact, strictly speaking, at the critical point no seed exists
that can trigger the phenomenon of the spontaneous breaking of the (restored)
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ chiral symmetry. However, any chiral
breaking disturbance, no matter how small, would make spontaneous breaking to
take place [25] with the vacuum choosing its orientation.

###  2.4 NP mass generation in the NG phase

The situation one encounters in the NG phase of the critical theory is totally
different. We shall see that, despite the fact that in the NG phase the Higgs-
like mass of the fermion gets cancelled (see fig. 2), a non-vanishing NP
fermion mass-like term emerges.

Because of the subtle UV vs. IR interplay that we have seen is at work in the
model, in order to get some understanding of how a non-vanishing NP fermion
mass can emerge, it is crucial to start the discussion by determining the
structure of the possible operators of NP origin, formally of O($b^{2}$),
arising in the regularized theory. The analysis can be properly done by making
use of the Symanzik expansion technique [26, 27].

A study of this kind was carried out in ref. [1] where it was shown that, as a
consequence of the occurrence of the spontaneous breaking of the (restored)
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ symmetry (which is in turn triggered
by residual O($b^{2}v$) chiral breaking terms surviving at $\eta_{cr}$ in the
NG phase), NP-ly generated Symanzik operators of O($b^{2}\Lambda_{s}$) emerge.
These NP effects can be described and included in the theory if, following
refs. [26, 27], we allow ourself to work with the “augmented” Lagrangian

$\displaystyle{\cal L}_{\rm toy}\to{\cal L}_{\rm toy}+\Delta{\cal
L}_{NP}\,,\qquad\Delta{\cal
L}_{NP}=\gamma_{\bar{q}q}(\alpha_{s})O_{6,\bar{q}q}+\gamma_{AA}(\alpha_{s})O_{6,AA}\,,$
(2.13) $\displaystyle
O_{6,\bar{q}q}=b^{2}\Lambda_{s}|\Phi|\,\big{(}\bar{q}_{L}\,\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{L}\\!+\\!\bar{q}_{R}\,\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{R}\big{)}\,,\qquad
O_{6,AA}=b^{2}\Lambda_{s}|\Phi|\,F^{A}\\!\cdot\\!F^{A}\,,$ (2.14)

where the coefficients $\gamma_{\bar{q}q}(\alpha_{s})$ and
$\gamma_{FF}(\alpha_{s})$ are functions of the gauge coupling 444As usual, we
define $\alpha_{s}=g^{2}_{s}/4\pi$.. More in detail, extending the arguments
given in ref. [1], one can show that at lowest order we find that both
$\gamma_{\bar{q}q}(\alpha_{s})$ and $\gamma_{AA}(\alpha_{s})$ are
${\mbox{O}}(\alpha_{s})$, if a $d=6$ Wilson-like term, as in eq. (2.7),
appears in the fundamental Lagrangian (2.3). Complementary arguments
supporting the emergence of such NP operators are developed in Appendix B of
(II).

The structure of the operators $O_{6,\bar{q}q}$ and $O_{6,AA}$ is completely
dictated by symmetries (in particular $\chi_{L}\times\chi_{R}$) and
dimensional considerations. The presence of the $\Lambda_{s}$ factor signals
their NP origin. According to the rules of the Symanzik analysis [26, 27],
making reference to ${\cal L}_{\rm toy}+\Delta{\cal L}_{NP}$ should be seen as
a bookkeeping of the NP operator insertions occurring in the correlators of
the fundamental Lagrangian. Extending in this way the allowed set of diagrams
is necessary to fully describe all the NP features of the theory in the NG
phase. In fact, though formally of O($b^{2}$), the insertions of $\Delta{\cal
L}_{NP}$ cannot be ignored, because, as we have repeatedly said, explicit
$b^{2}$ factors can be compensated by UV power divergencies in loop integrals,
eventually leading to finite NP contributions to correlators.

The remarkable fact about the “augmented” Lagrangian (2.13) is that it
generates among others, new diagrams of NP origin (surviving in the $b\to 0$
limit) that contribute to the quark self-energy. Two instances of such
diagrams are drawn in fig. 3. An explicit calculation of the amputated zero
momentum diagram shown in the right panel of fig. 3 gives to the effective
fermion mass the finite contribution 555The vertical dotted line means
amputation of the external leg. We have nevertheless left explicit the ingoing
and outgoing lines to help the reader recognizing the particles one is
referring to.

Figure 3: Two typical lowest loop order NP quark self-energy diagrams. The
blobs represent insertions of the operators $O_{6,FF}$ and $O_{6,\bar{q}q}$,
respectively, and the dotted line the propagation of the singlet scalar,
$\zeta_{0}$, defined in eq. (2.16) below. The vertical lines mean amputation.
The rest of the notations is as in fig. 1.

$\displaystyle
m_{q}\propto\alpha_{s}^{2}\,\,\hbox{tr}\,\\!\\!\int^{1/b}\\!\frac{d^{4}k}{k^{2}}\frac{\gamma_{\mu}k_{\mu}}{k^{2}}\\!\int^{1/b}\\!\\!\\!\\!\frac{d^{4}\ell}{\ell^{2}+m^{2}_{\zeta_{0}}}\frac{\gamma_{\nu}(k+\ell)_{\nu}}{(k+\ell)^{2}}\,b^{2}\gamma_{\rho}(k+\ell)_{\rho}\,b^{2}{\Lambda_{s}}\gamma_{\lambda}(2k+\ell)_{\lambda}\\!\sim{\alpha_{s}^{2}\Lambda_{s}}\,,$
(2.15)

where one factor of $\alpha_{s}$ comes from the gluon connecting a standard
QCD-like vertex with the Wilson-like box and another from the insertion of the
$O_{6,\bar{q}q}$ operator in eq. (2.14) with the coefficient
$\gamma_{\bar{q}q}$ taken at its lowest order. In the integrand of eq. (2.15)
$m^{2}_{\zeta_{0}}=2|\mu^{2}_{0}|$ is the (square) mass of the singlet
$\zeta_{0}$ which is standardly defined by the polar decomposition

$\displaystyle\Phi=RU\,,\quad R=v+\zeta_{0}\,,$ (2.16) $\displaystyle
U=\exp[i\vec{\tau}\,\vec{\zeta}/c\Lambda_{s}]\,,$ (2.17)

with the (arbitrary) scale in the exponential conveniently chosen with an eye
to the QEL in eq. (2.18) below, so as to have the effective NG fields
$\zeta^{i},i=1,2,3$ canonically normalized. Obviously $U$ can only be defined
in the NG phase of the theory where $\langle|\Phi|^{2}\rangle\\!\neq\\!0$.

We end the section with a few remarks.

1) First of all, since fermion masses are running quantities, we must specify
which scale the mass computed in eq. (2.15) refers to. We note that $m_{q}$
emerges from diagrams that are completely dominated by the UV behaviour of the
integrated loop momenta. Thus we conclude that the self-energy diagrams of
fig. 3 provide the fermion mass at asymptotically large scale, or better, if
the theory unifies, at the GUT scale. We will make use of this important
observation in the phenomenological considerations we will present in sect. 5
of ref. (II).

2) Secondly, we need to explain what we mean by “$\zeta_{0}$-propagator”,
given the fact that the critical theory is defined precisely at the value of
$k_{b}$ where the kinetic term of $\Phi$ is canceled against the similar
operator with which the Wilson-like terms mix. This point is mentioned in
sect. 3.4.2 and fully clarified in Appendix C.

3) For phenomenological considerations (see sect. 5 in (II)) it is important
to decide whether we should read the coupling constant dependence in the last
line of eq. (2.15) as $\alpha_{s}^{2}$ (as indicated), or as
$\alpha_{s}\,g_{s}^{2}$, or rather as $(g_{s}^{2})^{2}$, because these choices
differ by non-irrelevant numerical factors. This question will be taken up in
sect. 3.5 and discussed in detail in Appendix B of ref. (II) where a criterion
to determine the form of the “optimal” gauge coupling dependence of the
$\gamma$-coefficients (i.e. the choice that correctly absorbs the $1/4\pi$
factors arising in loops) is proposed/provided.

4) To simplify the line of reasoning leading to the key eq. (2.15) we have
provisionally ignored the fact that in the operators (2.14) it is $|\Phi|$
(i.e. the modulus of $\Phi$) that appears and not $\Phi$ itself. This implies
that in the diagrams of fig. 3 after the $\zeta_{0}$ contraction a “dandling”
$U$ factor is left out exiting the Wilson-like box (recall eq. (2.16)). We did
not show this field dependence in fig. 3, because for the calculation of the
mass one can set the external field $U$ equal to the identity. However, the
complete $U$ dependence of these diagrams needs to be reinserted in order to
be able to generate a fully $\chi_{L}\times\chi_{R}$ invariant fermion mass
operator, like the first term in the second line of eq. (2.18) below.

###  2.5 The QEL of the critical theory in the NG phase

The existence of a NP fermion mass term, the occurrence of which in the NG
phase of the model (2.3) was also successfully checked in ref. [3] by explicit
lattice simulations, can be incorporated in the QEL that describes the physics
of the theory (denoted by $\Gamma^{NG}_{cr}$ in the following) with the help
of the non-analytic field $U$ introduced in eq. (2.16). Owing to the fact that
$U$ has the same transformation properties under
$\chi_{L}\\!\times\\!\chi_{R}$ as $\Phi$ does, new operators (with respect to
those appearing in the critical QEL of the Wigner phase, eq. (2.12)) invariant
under $\chi_{L}\\!\times\\!\chi_{R}$ can be constructed, leading for the
$d\leq 4$ piece of $\Gamma^{NG}_{cr}$ to the expression

$\displaystyle\Gamma_{4\,cr}^{NG}\\!=\\!\frac{1}{4}F^{A}\\!\cdot\\!F^{A}\\!+\\!\big{(}\bar{q}_{L}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{L}\\!+\\!\bar{q}_{R}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A}q_{R}\big{)}\\!+$
$\displaystyle\qquad+c_{q}\Lambda_{s}\big{(}\bar{q}_{L}Uq_{R}\\!+\\!\bar{q}_{R}U^{\dagger}q_{L}\big{)}\\!+\\!\frac{c^{2}\Lambda_{s}^{2}}{2}\,\hbox{tr}\,[\partial_{\mu}U^{\dagger}\partial_{\mu}U]\\!+\ldots$
(2.18)

with $c_{q}$ and $c$ functions of the gauge coupling. The form of
$\Gamma_{4\,cr}^{NG}$ is purely geometrical. It is constrained by dimensional
and symmetry arguments and the observation that at $\eta=\eta_{cr}$ the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ breaking Yukawa term should not be
present. The expression (2.18) is obtained by including all the operators of
dimension $d\leq 4$, invariant under $\chi_{L}\times\chi_{R}$ that can be
constructed in terms of $q$, $\bar{q}$, $A_{\mu}$ and $U$.

Dots in (2.18) are there to recall us that actually there exist other
operators invariant under $\chi_{L}\\!\times\\!\chi_{R}$ that we have not
indicated. At $d=4$ we have the scalar kinetic term
$\frac{1}{2}{\,\hbox{tr}\,}\big{[}\partial_{\mu}\Phi^{\dagger}\partial_{\mu}\Phi\big{]}$
as well as the operator
$\Lambda_{s}R\,\,\hbox{tr}\,[\partial_{\mu}U^{\dagger}\partial_{\mu}U]$ and
the scalar potential ${\cal V}(\Phi)$. However, as we shall argue in the next
section (see Appendix B), in the presence of weak interactions, the
restoration of $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ makes the first two
operators to disappear from the QEL. At the same time, at the critical point
the effective singlet field $R=v+\zeta_{0}$ (eq. (2.16)) becomes infinitely
massive and decouples.

Naturally, the third term in the r.h.s. of eq. (2.18) (which is not invariant
under $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations) is of special
interest because, upon expanding
$U=\\!1\\!\\!1\\!+i\vec{\tau}\,\vec{\zeta}/c\Lambda_{s}+\ldots$ (see eq.
(2.17)), it gives rise to a mass for the fermion plus a wealth of NG boson
non-linear interactions.

We observe that, despite the fact that the tuning of the Yukawa coupling
enforces the conservation of the chiral
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ currents, we have discovered that
peculiar mass-like terms are dynamically generated which represent NP
obstructions to the exact realization of chiral symmetry 666Actually also in
QCD mass terms break chiral symmetry. The difference is that, while in QCD
they can be put to zero by hand, here masses are NP effects that cannot be
eliminated, as they arise as soon as $\rho\neq 0$ no matter how small $|\rho|$
might be.. A suggestive way to describe the nature of this dynamically
generated mass term is to say that the latter appears as a sort of “NP
anomaly” preventing the full recovery of the chiral
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ symmetry.

##  3 Introducing weak and Tera-interactions

To proceed to the construction of a realistic model for elementary particles
physics the next step is to introduce weak interactions. At the same time, as
mentioned in the Introduction, it is also necessary to extend the model by
incorporating a super-strongly interacting sector, so that the whole theory
will have an RGI scale, $\Lambda_{T}$, much larger than $\Lambda_{QCD}$ and,
to match phenomenology, of the order of a few TeVs. Only in this way eqs.
(1.1) and (1.2) can possibly yield the correct order of magnitude of the top
quark and $W$ boson mass, respectively.

The desired extension of the model is obtained by doubling the structure of
fermions to encompass Tera-particles ($Q\\!=$ Tera-quarks and $G\\!=$ Tera-
gluons) and introducing weak interactions by gauging the exact $\chi_{L}$
symmetry. The resulting Lagrangian will have the expression

$\displaystyle{\cal L}(q,Q;A,G,W;\Phi)\\!=$ $\displaystyle\qquad={\cal
L}_{K}(q,Q;A,G,W;\Phi)\\!+\\!{\cal V}(\Phi)\\!+\\!{\cal
L}_{Yuk}(q,Q;\Phi)\\!+\\!{\cal L}_{Wil}(q,Q;A,G,W;\Phi)$ (3.19)
$\displaystyle\,\,\bullet\,\,{\cal
L}_{K}(q,Q;A,G,W;\Phi)=\frac{1}{4}\Big{(}F^{A}\cdot F^{A}+F^{G}\cdot
F^{G}+F^{W}\cdot F^{W}\Big{)}+$ (3.20)
$\displaystyle\,\,\quad+\big{(}\bar{q}_{L}\,/\\!\\!\\!\\!{\cal
D}^{AW}q_{L}\\!+\\!\bar{q}_{R}\,/\\!\\!\\!\\!{\cal
D}^{A}q_{R}\big{)}\\!+\\!\big{(}\bar{Q}_{L}\,/\\!\\!\\!\\!{\cal
D}^{AGW}Q_{L}\\!+\\!\bar{Q}_{R}\,/\\!\\!\\!\\!{\cal
D}^{AG}Q_{R}\big{)}\\!+\\!\frac{k_{b}}{2}{\,\hbox{tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}\Phi)^{\dagger}{\cal D}^{W}_{\mu}\Phi\big{]}$
$\displaystyle\,\,\bullet\,\,{\cal
V}(\Phi)=\frac{\mu_{0}^{2}}{2}k_{b}{\,\hbox{tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}+\frac{\lambda_{0}}{4}\big{(}k_{b}{\,\hbox{tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}\big{)}^{2}$
(3.21) $\displaystyle\,\,\bullet\,\,{\cal
L}_{Yuk}(q,Q;\Phi)=\eta_{q}\,\big{(}\bar{q}_{L}\Phi\,q_{R}+\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}+\eta_{Q}\,\big{(}\bar{Q}_{L}\Phi\,Q_{R}+\bar{Q}_{R}\Phi^{\dagger}Q_{L}\big{)}$
(3.22) $\displaystyle\,\,\bullet\,\,{\cal
L}_{Wil}(q,Q;A,G,W;\Phi)=\frac{b^{2}}{2}{\rho_{q}}\,\big{(}\bar{q}_{L}{\overleftarrow{\cal
D}}\,^{AW}_{\mu}\Phi{\cal D}^{A}_{\mu}q_{R}+\bar{q}_{R}\overleftarrow{\cal
D}\,^{A}_{\mu}\Phi^{\dagger}{\cal D}^{AW}_{\mu}q_{L}\big{)}+$
$\displaystyle\,\,\quad+\frac{b^{2}}{2}{\rho_{Q}}\,\big{(}\bar{Q}_{L}{\overleftarrow{\cal
D}}\,^{AGW}_{\mu}\Phi{\cal D}^{AG}_{\mu}Q_{R}+\bar{Q}_{R}\overleftarrow{\cal
D}\,^{AG}_{\mu}\Phi^{\dagger}{\cal D}^{AGW}_{\mu}Q_{L}\big{)}$ (3.23)

with obvious notations for the covariant derivatives. As for the symmetries of
the Lagrangian (3.19), the previous form of the $\chi_{L}\times\chi_{R}$ (eq.
(2.8)) and $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ (eqs. (2.9)-(2.10))
transformations, besides the obvious extension necessary to let them act also
on Tera-fermions, need to be modified in the presence of $W$ bosons to insure
invariance of the fermionic kinetic terms under
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$. The analog of the set of eqs. (2.8),
(2.9) and (2.10) then reads ($\Omega_{L/R}\\!\in$SU(2))

$\displaystyle\chi_{L}\times\chi_{R}=[\tilde{\chi}_{L}\times(\Phi\to\Omega_{L}\Phi)]\times[\tilde{\chi}_{R}\times(\Phi\to\Phi\,\Omega_{R}^{\dagger})]\,,$
(3.24)
$\displaystyle\tilde{\chi}_{L}:\left\\{\begin{array}[]{l}q_{L}\rightarrow\Omega_{L}q_{L}\qquad\qquad\quad\bar{q}_{L}\rightarrow\bar{q}_{L}\Omega_{L}^{\dagger}\\\
Q_{L}\rightarrow\Omega_{L}Q_{L}\qquad\qquad\,\bar{Q}_{L}\rightarrow\bar{Q}_{L}\Omega_{L}^{\dagger}\\\
W_{\mu}\rightarrow\Omega_{L}W_{\mu}\Omega_{L}^{\dagger}\\\ \end{array}\right.$
(3.28)
$\displaystyle\tilde{\chi}_{R}:\left\\{\begin{array}[]{l}q_{R}\rightarrow\Omega_{R}q_{R}\qquad\qquad\quad\bar{q}_{R}\rightarrow\bar{q}_{R}\Omega_{R}^{\dagger}\\\
Q_{R}\rightarrow\Omega_{R}Q_{R}\qquad\qquad\,\bar{Q}_{R}\rightarrow\bar{Q}_{R}\Omega_{R}^{\dagger}\end{array}\right.$
(3.31)

###  3.1 The critical theory

Besides the Yukawa (eq. (3.22)) and the Wilson-like (eq. (3.23)) operators,
now also the kinetic term of the scalar field (last term in eq. (3.20)) breaks
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ and mixes with ${\cal L}_{Yuk}$ and
${\cal L}_{Wil}$. Thus to get the critical theory (invariant under
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$), on top of the Yukawa couplings
$\eta_{q}$ and $\eta_{Q}$, a further parameter, $k_{b}$, has been introduced
which needs to be appropriately tuned. For convenience the coefficient $k_{b}$
is let to appear also in the expression of the scalar potential (3.21),
because with this choice the bare (vev)2 (in the NG phase) and the quartic
coupling of the canonically normalized scalar field will keep their standard
definitions, i.e. $v^{2}=|\mu^{2}_{0}|/\lambda_{0}$ and $\lambda_{0}$.

The tuning conditions determining $\eta_{q\,cr}$, $\eta_{Q\,cr}$ and
$k_{b\,cr}$ come from enforcing in the Wigner phase the conservation of the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ currents [23]. In Appendix A for
completeness we sketch the procedure that, extending the strategy proposed in
[1], leads to the tuning equations for $\eta_{q}$, $\eta_{Q}$ and $k_{b}$. The
conditions determining the critical theory physically correspond to have no
scalar kinetic term in the QEL of the theory and, similarly to what we saw
happening in sect. 2 in the case of the model (2.3), vanishing effective
Yukawa interactions.

Naturally, as was done in ref. [3] for the Yukawa coupling, the computation of
the critical values of $\eta_{q}$, $\eta_{Q}$ and $k_{b}$ needs to be
performed in a non-perturbative way, like it is currently done in WLQCD in the
determination of the critical mass. Nevertheless, as we did in sect. 2.2, in
order to get a feeling of what these criticality conditions mean, it is
interesting to examine what happens at the lowest loop order.

###  3.2 The critical conditions in the Wigner phase

Solving, with steps similar to those leading to eq. (2.11), the tuning eqs.
(A.61)-(A.65) one finds for $\eta_{q\,cr}$, $\eta_{Q\,cr}$ and $k_{b\,cr}$ at
1-loop in the Wigner phase the parametric expressions

$\displaystyle\eta_{q\,cr}^{(\rm
1-loop)}=\rho_{q}\,\alpha_{s}N_{c}\eta_{q}^{(1)}\,,\qquad\qquad\qquad\eta_{Q\,cr}^{(\rm
1-loop)}=\rho_{Q}\,\alpha_{T}N_{T}\eta_{Q}^{(1)}\,,$ (3.32)
$\displaystyle\hskip 56.9055ptk_{b\,cr}^{(\rm
1-loop)}=\rho_{q}^{2}N_{c}k_{b\,q}^{(1)}+\rho_{Q}^{2}N_{c}N_{T}k_{b\,Q}^{(1)}\,,$
(3.33)

with $\eta_{q}^{(1)}$, $\eta_{Q}^{(1)}$, $k_{b\,q}^{(1)}$ and $k_{b\,Q}^{(1)}$
computable coefficients and SU($N_{T}$) the Tera-strong gauge group.

In figs. 4 and 5 we display at the lowest loop order the mechanism leading to
the cancellation of the effective quark and Tera-quark Yukawa terms in the QEL
of the critical theory. The figures show the mixing between the quark (Tera-
quark) Yukawa operator (grey disks) and the quark (Tera-quark) Wilson-like
operator (grey box) at 1-loop.

In fig. 6 we report the leading 1-loop diagrams yielding the cancellation
between the scalar kinetic term (grey disk) and the contribution from the sum
of the quark and Tera-quark Wilson-like operators (grey boxes). The empty
boxes represent the insertion of Wilson-like vertices from the Lagrangian
necessary to close the fermion loops. As before, the loop diagrams in the
figs. 4, 5 and 6 yield finite results because the loop UV power divergencies
are exactly compensated by the $b^{2}$ factors coming from the insertion of
Wilson-like vertices.

Figure 4: The cancellation of the quark Yukawa vertex implied by the tuning
conditions determining $\eta_{q\,cr}$ at the lowest loop order in the Wigner
phase. The grey box, labelled by $\rho_{q}$, represents the quark Wilson-like
vertex. The rest of the notations is as in fig. 1.

Figure 5: The cancellation of the Tera-quark Yukawa vertex implied by the
tuning conditions determining $\eta_{Q\,cr}$ at the lowest loop order in the
Wigner phase. Double lines represent Tera-particles. The grey box, labelled by
$\rho_{Q}$, represents the Tera-quark Wilson-like vertex. The rest of the
notations is as in fig. 4.

Figure 6: The cancellation of the scalar kinetic term implied by the tuning
condition determining $k_{b\,cr}$ at the lowest loop order in the Wigner
phase. The integers $\nu_{q}$ and $\nu_{Q}$ are the multiplicities of quarks
and Tera-quarks running in the loops with $N_{c}$ and $N_{T}$ the number of
colours and Tera-colours, respectively. The grey disc, labelled by $k_{b}$,
represents the insertion of the scalar kinetic term. The empty boxes represent
the insertion of Wilson-like vertices from the Lagrangian. The rest of the
notations is as in figs. 4 and 5.

###  3.3 The critical conditions in the Nambu–Goldstone phase

In the NG phase the criticality conditions for $\eta_{q}$, $\eta_{Q}$ and
$k_{b}$ imply the cancellations shown in the figs. 7, 8 and 9, respectively.
Figs. 7 and 8, as in sect. 2, are directly obtained from the figs. 4 and 5 by
setting the scalar field equal to its vev. Fig. 9 follows from fig. 6 using
SU(2)L gauge invariance and setting the scalar at its vev.

Like we observed in sect. 2.2, also here the key feature is that the NP
enforcement of the $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ invariance implies
that in the QEL of the critical theory precisely the Higgs-like masses of
quarks, Tera-quarks and $W$’s are canceled out.

Figure 7: The mechanism underlying the cancellation of the Higgs-like mass
term of quarks at the lowest loop order occurring in the NG phase of the
critical theory. Notations are as in fig. 4.

Figure 8: The mechanism underlying the cancellation of the Higgs-like mass
term of Tera-quarks at the lowest loop order occurring in the NG phase of the
critical theory. Notations are as in fig. 5.

Figure 9: The mechanism underlying the cancellation of the Higgs-like $W$ mass
term at the lowest loop order occurring in the NG phase of the critical
theory. Wiggly lines are $W$’s. The rest of the notations is as in figs. 4, 5
and 6.

###  3.4 The NP emergence of elementary particles masses

By extending to the present case the analysis that in sect. 2.4 has led us to
identify the operators (2.14), one finds at O($b^{2}$) the following set of NP
$\chi_{L}\times\chi_{R}$ invariant operators

$\displaystyle\qquad
O_{6,\bar{Q}Q}^{T}=b^{2}\Lambda_{T}|\Phi|\,\big{(}\bar{Q}_{L}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{AGW}Q_{L}+\bar{Q}_{R}\,{\mbox{$\not\\!\\!\,$}{\cal D}}^{AG}Q_{R}\big{)}$
(3.34) $\displaystyle\qquad
O_{6,GG}=b^{2}\Lambda_{T}|\Phi|\,F^{G}\\!\cdot\\!F^{G}$ (3.35)
$\displaystyle\qquad O_{6,AA}=b^{2}\Lambda_{T}|\Phi|\,F^{A}\\!\cdot\\!F^{A}$
(3.36)

Although formally of O($b^{2}$), the effect of insertion of these operators in
correlators cannot be ignored because of the kind of IR-UV compensation we
have seen occurring when “irrelevant” operators are present in the fundamental
Lagrangian. Taking into account the operators (3.34)–(3.36) is necessary to
fully describe the NP features of the theory related to the spontaneous
breaking of the (restored) chiral symmetry.

Similarly to the situation we discussed in sect. 2.4, the occurrence of the
Symanzik NP operators (3.34)–(3.36) can be properly taken into account by
adding them to the fundamental Lagrangian, yielding the “augmented” Lagrangian
[26, 27]

$\displaystyle{\cal L}\to{\cal L}+\Delta{\cal L}_{NP}\,,\quad\Delta{\cal
L}_{NP}=\gamma_{\bar{Q}Q}^{T}O_{6,\bar{Q}Q}^{T}+\gamma_{GG}O_{6,GG}+\gamma_{AA}O_{6,AA}\,,$
(3.37)

where the coefficients $\gamma^{T}_{\bar{Q}Q}$, $\gamma_{GG}$ and
$\gamma_{AA}$ are functions of the gauge couplings. In Appendix B of (II) we
provide arguments supporting the existence of the NP operators (3.34)–(3.36)
and derive the lowest loop order behaviour of the $\gamma$-coefficients in eq.
(3.37) upon the gauge couplings 777Actually there exist other NP operators. An
example is given by
$\gamma_{WW}b^{2}\Lambda_{T}|\Phi|\,F^{W}\\!\cdot\\!F^{W}$, with
$\gamma_{WW}={\mbox{O}}(g_{w}^{2})$. The list above is limited to the
operators that give rise to the leading diagrams contributing to the NP masses
of fig. 10. . As noted in sect. 2.4, where we discussed the similar case of
eq. (2.13), eq. (3.37) is purely formal, in the sense that, according to the
rules of the Symanzik expansion [26, 27], introducing ${\cal L}+\Delta{\cal
L}_{NP}$ should be seen as a bookkeeping of the NP operator insertions
occurring in the correlators of the fundamental Lagrangian, ${\cal L}$.

At his point (see also our comment at the point 3) in the list at the end of
sect. 2.4) it is crucial to remark that, although unimportant for highlighting
the parametric dependence of $\Delta{\cal L}_{NP}$ in eq. (3.37) upon the
gauge couplings, a key question for phenomenology is to decide whether we
should use $g^{2}$ or $\alpha$ in order to best (“optimally”) parametrize the
coefficients $\gamma$ with which the NP Symanzik operators (3.34), (3.35) and
(3.36) enter in ${\cal L}+\Delta{\cal L}_{NP}$. By “optimal parametrization”
we mean choosing either $g^{2}$ or $\alpha=g^{2}/4\pi$ to absorb the factors
$1/4\pi$ arising in loops. In Appendix B of ref. (II) we show that, in the
case of $d=6$ Wilson-like terms as those in eq. (3.23), at lowest order, the
optimal choice corresponds to the following behaviour

$\gamma^{T}_{\bar{Q}Q}={\mbox{O}}(\alpha_{T})\,,\qquad\gamma_{GG}={\mbox{O}}(g_{T}^{2})\,,\qquad\gamma_{AA}={\mbox{O}}(g^{2}_{s})\,.$
(3.38)

####  3.4.1 NP mass generation

The presence of new vertices leads among other contributions, to self-energy
diagrams capable of generating NP masses not only for quarks, but also for
Tera-quarks and weak bosons. At the lowest loop order one finds the typical
amputated self-energy diagrams displayed in the four panels of fig. 10, where
on a case by case basis the operators (3.34), (3.35) or (3.36) are inserted
together with the Wilson-like vertices necessary to close the loops. All these
diagrams are finite owing to the by now familiar UV-IR compensation, and all
of O($\Lambda_{T}$) times gauge coupling dependent coefficients.

Figure 10: Examples of the lowest loop order self-energy diagrams giving rise
(from top to bottom) to NP effective masses for quarks, Tera-quarks and $W$.
Blobs represent the insertion of the appropriate NP operators among those
displayed in equations from (3.34) to (3.36), and boxes the insertion of the
Wilson-like vertices necessary to close the loops. The vertical dotted line
means amputation of the external leg.

In sect. 2.4 we have sketched the calculation of the quark mass. The
calculation of the Tera-quark mass exactly parallels what we did there. As for
the $W$ mass, from the kind of diagrams displayed in the bottom panel of fig.
10 we find

$\displaystyle
M_{W}^{2}\,\propto\,g_{w}^{2}\alpha_{T}^{2}\Lambda_{T}\,(b^{2})^{4}\\!\int^{\pi/b}\\!\frac{d^{4}k}{(2\pi)^{4}}\int^{\pi/b}\\!\frac{d^{4}\ell}{(2\pi)^{4}}\int^{\pi/b}\\!\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{k^{2}\\!+\\!m^{2}_{\zeta_{0}}}$
(3.39) $\displaystyle{\rm
tr}\Big{[}\gamma\\!\cdot\\!(q\\!-\\!k)\frac{1}{\gamma\\!\cdot\\!(q\\!-\\!k)}q_{\mu}\frac{1}{\gamma\\!\cdot\\!q}\gamma\\!\cdot\\!(q\\!-\\!\ell)\frac{1}{\gamma\\!\cdot\\!(q\\!-\\!\ell)}(q\\!-\\!\ell)_{\mu}\frac{1}{\gamma\\!\cdot\\!q}\Big{]}\frac{1}{\ell^{2}\\!+\\!m^{2}_{\zeta_{0}}}\sim
g_{w}^{2}\alpha_{T}^{2}\Lambda_{T}^{2}\,.$

After a few obvious simplifications, one sees that a finite result emerges,
like in the case of $m_{q}$, because the $b^{-8}$ three-loop UV-power
divergency is exactly compensated by the factor $b^{8}$ coming from the
insertions of the Wilson-like vertices and the NP operators (3.35), at the end
yielding an UV-finite square $W$ mass of
O($g_{w}^{2}\alpha_{T}^{2}\Lambda_{T}^{2}$).

From the observation that the NP building blocks appearing in the $W$ mass
diagrams in the fourth panel of fig. 10 are among those occurring in the
second and third panel, we conclude that the dynamical $W$ boson mass is
generated by the same NP mechanism that gives mass to fermions and Tera-
fermions.

As we remarked at the end of sect. 2.4, to be complete we should have drawn in
each diagram of fig. 10 also the field $U$, exiting from each Wilson-like
term. We did not do so because the presence of $U$ is not relevant for what
concerns the calculation of the NP effective masses (it is enough to set $U$
equal to the identity). Reinserting the whole dependence upon $U$ is, however,
necessary to give rise to the fully $\chi_{L}\times\chi_{R}$ invariant mass
operators of fermions and Tera-fermions as well as the gauge invariant
structure related to the $W$ mass of eq. (3.43) below.

####  3.4.2 The $\zeta_{0}$ propagator

We need to conclude the analysis of these NP mass estimates answering a
question that can naturally arise concerning the precise expression of the
$\zeta_{0}$ propagator in the diagrams of fig. 10. In fact, one might wonder
what do we really mean by “$\zeta_{0}$ propagator”, given the fact that the
critical value of $k_{b}$ in eq. (3.20) was fixed to precisely cancel the
scalar kinetic term against the similar operator with which the Wilson-like
terms mix. The cancellation is pictorially represented in fig. 6 at the lowest
loop order. We clarify this important issue in Appendix C by disentangling the
delicate interplay between the critical and the $b\to 0$ limit in determining
the effective expression of the critical $\zeta_{0}$ propagator.

###  3.5 The NP masses of elementary particles

From the analysis of the building blocks entering the diagrams in fig. 10 one
can determine the lowest loop order parametric expression of the effective NP
quark, Tera-quark and $W$ mass, finding

$\displaystyle
m_{q}=C_{q}\,\Lambda_{T}\,,\qquad\qquad\,\,\,C_{q}={\mbox{O}}(g_{s}^{2}\,\alpha_{s})$
(3.40) $\displaystyle
m_{Q}=C_{Q}\,\Lambda_{T}\,,\qquad\qquad\,\,C_{Q}={\mbox{O}}(g_{T}^{2}\,\alpha_{T},\alpha_{T}^{2},g_{s}^{2}\,\alpha_{s},g_{s}^{2}\,\alpha_{T})$
(3.41) $\displaystyle
M_{W}=C_{w}\,\Lambda_{T}\,,\qquad\qquad\,\,C_{w}=g_{w}\,c_{w}\,,\qquad
c_{w}={\mbox{O}}(\alpha_{T})\,,$ (3.42)

where the specific choice between $g^{2}$ and $\alpha$ we made in the
coefficients $C_{q},C_{Q}$ comes from taking the product of the factors
indicated within the gray blob times either the $\alpha_{s}$ or the
$\alpha_{T}$ accordng to whether the particle running in the external loop is
a SM or a Tera-dof (see Appendix B in (II)).

As for $C_{w}^{2}$ (it is more convenient to refer to $M_{W}^{2}$), the
$\alpha_{T}^{2}$ factor coming from the gray blobs, get simply multiplied by a
$g_{w}^{2}$ factor, because there is no weak interaction loop in the last
panel of fig. 10, in other words, because the weak couplings are external
factors and do not pick up the $1/4\pi$ loop-factor.

Considerations of this kind will be important in the sect. (3.2) of (II)
where, after extending the model to encompass hypercharge interactions
introducing leptons and Tera-leptons, we address some interesting
phenomenological questions providing a crude estimate of $\Lambda_{T}$ as well
as of some elementary particle mass ratios. Already at the level of the
analysis presented above, we can conclude that eqs. (3.40)-(3.42) allow
identifying the EW scale as (a fraction of) the physical parameter
$\Lambda_{T}$, associated to a new interaction.

###  3.6 The critical QEL in the NG phase

We now discuss the form of the QEL which is expected to describe the physics
of the renormalized critical model (3.19) in the NG phase at momenta well
below the UV cutoff scale but above $\Lambda_{T}$, i.e. in the range
$b^{-2}\gg p^{2}>\Lambda_{T}^{2}$. Following the same line of arguments we
developed in sect. 2.5 to derive eq. (2.18), we get for the $d\leq 4$ piece of
the QEL the expression

$\displaystyle{\Gamma}_{4\,cr}^{NG}(q,Q;A,G,W;\Phi)=\frac{1}{4}\Big{(}F^{A}\\!\cdot\\!F^{A}+F^{G}\\!\cdot\\!F^{G}+F^{W}\\!\cdot\\!F^{W}\Big{)}+$
$\displaystyle+\big{(}\bar{q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{WA}q_{L}+\bar{q}_{R}\,\,/\\!\\!\\!\\!{\cal
D}^{A}q_{R}\big{)}+C_{q}\Lambda_{T}\,\big{(}\bar{q}_{L}Uq_{R}+\bar{q}_{R}U^{\dagger}q_{L}\big{)}+$
$\displaystyle+\big{(}\bar{Q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{WAG}Q_{L}\\!+\\!\bar{Q}_{R}\,\,/\\!\\!\\!\\!{\cal
D}^{AG}Q_{R}\big{)}+C_{Q}\Lambda_{T}\,\big{(}\bar{Q}_{L}UQ_{R}\\!+\\!\bar{Q}_{R}U^{\dagger}Q_{L}\big{)}\\!+$
$\displaystyle+\frac{1}{2}c_{w}^{2}\Lambda_{T}^{2}{\,\hbox{tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}U)^{\dagger}{\cal D}^{W}_{\mu}U\big{]}\,,$ (3.43)

where (compare with the expression (2.17))

$U=\exp[i\vec{\tau}\,\vec{\zeta}/c_{w}\Lambda_{T}]\,.$ (3.44)

In eq. (3.43) we have included $d\leq 4$ operators with up to two derivatives.
Actually there exists a further $d=4$ operator invariant under
$\chi_{L}\times\chi_{R}$, namely ${\,\hbox{tr}\,}\big{[}({\cal
D}^{W}_{\mu}U)^{\dagger}{\cal D}^{W}_{\mu}U({\cal
D}^{W}_{\nu}U)^{\dagger}{\cal D}^{W}_{\nu}U\big{]}$ which has, however, four
derivatives. This term will not impact in an essential way on our
considerations in sect. 4 where we compare the SM Lagrangian with the
expression of the LEEL resulting upon integrating out the (heavy) Tera-dof’s.
It will contribute higher order weak corrections to the LEEL coefficients
and/or higher dimensional $d>4$ operators scaled by powers of $\Lambda_{T}$.

In Appendix B we justify the form of the expression (3.43) showing that,
despite the fact that also the term $\Lambda_{T}R\,\,\hbox{tr}\,[({\cal
D}_{\mu}^{W}U)^{\dagger}{\cal D}^{W}_{\mu}U]$ ($R$ is the singlet radial
scalar defined in eq. (2.16)), is invariant under $\chi_{L}\times\chi_{R}$, it
cannot appear in ${\Gamma}_{4\,cr}^{NG}$. The proof of this statement is based
on the decoupling theorem [28] which holds in the critical limit, based to the
fact that in the QEL the effective field $\zeta_{0}$ can be shown to get an
infinite mass and decouple.

Naturally at $d>4$ there will obviously be infinitely many other operators
contributing to ${\Gamma}_{cr}^{NG}$, scaled by larger and larger inverse
powers of $\Lambda_{T}$. A few examples are

$\displaystyle\frac{1}{\Lambda_{T}}\bar{Q}_{L}{\overleftarrow{\cal
D}}\,_{\mu}^{AGW}\\!U{\cal
D}_{\mu}^{AG}Q_{R}\\!+\\!{\mbox{hc}}\,,\,\,\frac{1}{\Lambda_{T}}\bar{q}_{L}{\overleftarrow{\cal
D}}\,_{\mu}^{AW}\\!U{\cal
D}_{\mu}^{A}q_{R}\\!+\\!{\mbox{hc}}\,,\,\,\frac{1}{\Lambda_{T}^{2}}(\bar{Q}_{L}UQ_{R})(\bar{Q}_{L}UQ_{R})\\!+\\!{\mbox{hc}}$
$\displaystyle\frac{1}{\Lambda_{T}^{2}}(\bar{q}_{L}Uq_{R})(\bar{q}_{L}Uq_{R})\\!+\\!{\mbox{hc}}\,,\quad\frac{1}{\Lambda_{T}^{4}}(\bar{Q}_{L}{\overleftarrow{\cal
D}}\,_{\mu}^{AGW}UQ_{R})(\bar{q}_{L}{\overleftarrow{\cal
D}}\,_{\mu}^{AW}\\!Uq_{R})\\!+\\!{\mbox{hc}}\,,\quad\ldots\,.$ (3.45)

###  3.7 Transversality of the $W$ polarization amplitude

As in the SM the Goldstone bosons $\zeta_{i},i=1,2,3$ (see eq. (3.44)) are
eaten up to give the longitudinal dof’s of the massive weak bosons. In
Appendix D we illustrate how the transversality property of the $W$
polarization amplitude (which in the end is a direct consequence of the SU(2)L
gauge symmetry) is realized in this model. The argument is not totally trivial
because in the present situation the $W$ mass is not generated at tree level
by the standard Higgs mechanism, but rather arises from NP spontaneous chiral
symmetry breaking effects and virtual particle exchanges like we see in the
lowest panel of fig. 10.

##  4 Integrating out Tera-degrees of freedom and the SM

We show in this section that, upon integrating out the (heavy) Tera-dof’s, in
the NG phase the resulting LEEL of the critical model (3.19) closely resembles
the SM Lagrangian. The argument rests on the key conjecture that the 125 GeV
resonance detected at LHC [18, 19] is a $W^{+}W^{-}/ZZ$ composite state, bound
by Tera-particle exchanges, and not an elementary particle. A non-relativistic
argument as well as results based on a simplified Bethe–Salpeter approach,
supporting this hypothesis, are presented in (II).

This state, which we shall denote by $h$, is (assumed to be) a singlet under
all the symmetries of the theory. Since its mass is (experimentally found to
be) $\ll\Lambda_{T}$, it must be included in the LEEL of the theory, valid for
(momenta)${}^{2}\ll\Lambda_{T}^{2}$. In these kinematical conditions the most
general LEEL invariant under the symmetries of the critical model (3.19), in
particular under the $\chi_{L}\times\chi_{R}$ transformations, and including
$h$, takes the form [29, 30]

$\displaystyle{\cal
L}_{LE}^{NG}(q;A,W;U,h)=\frac{1}{4}\Big{(}F^{A}\\!\cdot\\!F^{A}+F^{W}\\!\cdot\\!F^{W}\Big{)}+\big{(}\bar{q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{AW}q_{L}+\bar{q}_{R}\,\,/\\!\\!\\!\\!{\cal D}^{A}q_{R}\big{)}+$
$\displaystyle\quad+\,(y_{q}h+k_{q}k_{v})\,\big{(}\bar{q}_{L}Uq_{R}+\bar{q}_{R}U^{\dagger}q_{L}\big{)}+$
(4.46)
$\displaystyle\quad+\frac{1}{2}\partial_{\mu}h\partial_{\mu}h+\frac{1}{2}(k_{v}^{2}+2k_{v}k_{1}h+k_{2}h^{2}){\,\hbox{tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}U)^{\dagger}{\cal D}^{W}_{\mu}U\big{]}+\widetilde{\cal
V}(h)+\ldots\,,$

where dots represent $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ violating
operators of dimension $d>4$, like for instance

$\displaystyle\bar{q}_{L}{\overleftarrow{\cal D}}\,_{\mu}^{AW}U{\cal
D}_{\mu}^{A}q_{R}+{\mbox{hc}}\,,\qquad d=5$ (4.47)
$\displaystyle(\bar{q}_{L}Uq_{R})\,(\bar{q}_{R}U^{\dagger}q_{L})\,,\qquad\quad
d=6\,.$ (4.48)

They appears in the LEEL multiplied by inverse powers of the fundamental scale
$\Lambda_{T}$ ($\Lambda_{T}^{-1}$ and $\Lambda_{T}^{-2}$, respectively). We
note that the $1/\Lambda_{T}$ expansion entailed by the integration over Tera-
dofs provides a physical interpretation and a theoretical basis for the
occurrence of the higher dimensional operators which are standardly introduced
to describe possible new physics beyond the SM.

The scalar potential $\widetilde{\cal V}(h)$ comprises the cubic and quartic
self-interactions of the $h$ field, as well as the $h$ mass term,
$m_{h}^{2}h^{2}/2$. The $k_{v},k_{1},k_{2},y_{q},k_{q}$ coefficients and the
$\widetilde{\cal V}$-couplings are parameters that need to be fixed by
matching onto the underlying (renormalizable and unitary) fundamental critical
theory (3.19).

Tree-level matching of ${\cal L}_{4,LE}^{NG}$ in eq. (4.46) with the QEL
$\Gamma^{NG}_{4\,cr}$ in eq. (3.43) requires for the masses of quarks and $W$
(see eqs. (3.40) and (3.42)) the identifications

$\displaystyle m_{q}=C_{q}\Lambda_{T}=k_{q}k_{v}\,,\qquad
M_{W}=g_{w}\,c_{w}\Lambda_{T}=g_{w}k_{v}\,,$ (4.49)

while the unitarity of the mother theory (3.19) implies the constraints [31,
32]

$y_{q}=k_{q}\,,\qquad k_{1}=k_{2}=1\,.$ (4.50)

The above relations are expected to hold up to small loop effects controlled
by the couplings $g_{w}$ and $y_{q}$. Neglecting these corrections, one
recognizes that, with the exception of the scalar potential $\widetilde{\cal
V}(h)$, precisely the combination

$\Phi\equiv(k_{v}+h)U$ (4.51)

enters the $d\leq 4$ part of ${\cal L}_{LE}^{NG}$ (see eq. (4.46)). The latter
can thus be rewritten in the suggestive form

$\displaystyle{\cal
L}_{4,LE}^{NG}(q;A,W;U,h)=\frac{1}{4}\Big{(}F^{A}\\!\cdot\\!F^{A}+F^{W}\\!\cdot\\!F^{W}\Big{)}+\big{(}\bar{q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{AW}q_{L}+\bar{q}_{R}\,\,/\\!\\!\\!\\!{\cal D}^{A}q_{R}\big{)}+$
$\displaystyle\quad+y_{q}\,\big{(}\bar{q}_{L}\Phi
q_{R}+\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}+\frac{1}{2}{\,\hbox{tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}\Phi)^{\dagger}{\cal D}^{W}_{\mu}\Phi\big{]}+\widetilde{\cal
V}(h)\,.$ (4.52)

From eq. (4.52) we see that (up to corrections suppressed as O($\alpha_{w}$)
or O($y_{q}^{2}/4\pi)$ which we have not included) ${\cal L}_{4,LE}^{NG}$
looks very much like the SM Lagrangian (more precisely, like the Lagrangian of
the oversimplified version of the SM where the existence of families,
hypercharge interactions, leptons and weak isospin splitting is ignored). In
particular, we see that just like it happens in the case of the Higgs
mechanism in the SM, the effective Yukawa coupling of $h$ to fermions is given
by (see eqs. (4.49) and (4.50))

$y_{q}=k_{q}=\frac{m_{q}}{k_{v}}=\frac{m_{q}}{c_{w}\Lambda_{T}}\,,$ (4.53)

where $k_{v}=c_{w}\Lambda_{T}$ is what in the SM is the Higgs field vev. This
simple analysis shows that like in the SM, also here the Yukawa coupling is
proportional to the fermion mass.

There are, however, two important differences between the LEEL of our model
and the SM Lagrangian that we must point out. First of all, the
proportionality factor between the Yukawa coupling (4.53) and the fermion mass
is not in our hands, rather it is completely fixed by the NP dynamics.
Secondly, since the scalar potential $\widetilde{\cal V}(h)$ is supposed to
describe, besides the mass, the self-interactions of the (composite) $h$
field, there is no reason why it should have the same form as the SM Higgs
potential. This implies that, even if it yields $m_{h}\\!\simeq\\!125$ GeV,
differences with respect to the case of the SM may well appear in the
trilinear and quadrilinear $h$ self-couplings.

##  5 Universality

A key point that needs to be thoroughly discussed, if we want to put the
present approach on a firm conceptual basis and make it useful for
phenomenology, is to what extent the NP mass formulae we have derived above
are “universal”, or in other words to what extent their expression depends on
the specific form of the “irrelevant” $d>4$ Wilson-like chiral breaking terms
that one decides to introduce in the fundamental Lagrangian.

It turns out that the leading (lowest) power of the gauge coupling dependence
of the $C$ coefficients in eqs. (3.40), (3.41) and (3.42) depends on the
operator dimension of the Wilson-like terms entering the diagrams of fig. 10.
One can prove that generically the higher is the dimension of the Wilson-like
operator the larger will be the leading (lowest) power of the gauge coupling
controlling the behaviour of the $C$ coefficients.

Let us then start by analyzing the case in which associated to each fermion
species there is a single Wilson-like operator of dimension $d=6$. We prove in
Appendix C of (II) that in the case of a theory with only one family of quarks
and weak interactions (but neither Tera-quarks nor leptons), physics is $\rho$
independent. If other fermions are present, be them quarks belonging to other
families, Tera-particles or leptons, observables will only depend on the
ratios of the various $\rho$ parameters.

Notice that in the case numerically studied in ref. [3], where weak
interactions are absent, the quark mass is instead proportional to $\rho^{2}$.
The reason is that the scalar kinetic term is invariant under
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations. Hence there is no
condition constraining the factor in front of it which is then taken to be
equal to unit. On the contrary, in the presence of weak interactions one also
needs to introduce and tune an extra coefficient, $k_{b}$, in front of the
scalar kinetic term, because the latter is not invariant under
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations. The $k_{b}$ critical
value turns out to be proportional to $\rho^{2}$, as one can see from eq.
(3.33). This yields a compensation of $\rho$ factors in the self-energy
diagrams of fig. 10, making the quark and the $W$ mass $\rho$ independent if
there is only one fermion species or dependent on $\rho$ ratios if we have
more than one fermion species.

If the chiral breaking Wilson-like term associated to a fermion is a linear
combination of operators of different dimensions, the situation is a bit more
complicated to analyze because of mixing. However, one can say the following.
It is always the dimension of the Wilson-like operator of lowest dimension
that determines the leading (smallest) value of the power of the gauge
coupling in the coefficients of the NP mass formulae (3.40), (3.41) and
(3.42). Higher and higher dimensional Wilson-like operators affect terms of
higher and higher order in the gauge coupling power expansion.

###  5.1 Rescuing universality?

At first sight the dependence of the value of the NP masses on the precise
form of the chiral breaking Wilson-like terms that we have outlined above
might appear as a blunt violation of universality. As we have said above, in
fact, the $C$ coefficients in the eqs. (3.40), (3.41) and (3.42) depend not
only on the value of the (ratios of the) $\rho$ parameters, but they also have
a gauge coupling power dependence correlated with the operator dimension of
the various Wilson-like terms in play.

Let us examine the two issues, $\rho$ dependence and operator dimension,
separately. As for the $\rho$ dependence, the problem could be mitigated or
even eliminated by conjecturing that some symmetry exists which, putting
constraints on the $\rho$’s, restricts the variability range of their ratios.
For instance, in the case where all Wilson-like terms are $d=6$ operators, an
extreme but appealing situation would the one in which all the $\rho$’s are
equal because of some GUT symmetry. In this case the whole $\rho$ dependence
would completely drop out from physical observables.

Concerning the dependence on the dimensions of the Wilson-like operators,
which, as we said, directly impacts on the parametric gauge coupling power
dependence of the NP-ly generated masses, the situation may not be as bad as
it looks. On the contrary, in our opinion the liberty in the choice of the
dimension of the chiral breaking Wilson-like terms might give us an unexpected
handle to understand flavour. The idea is to interpret the family mass
hierarchy (from heavy to light) as related to Wilson-like terms of increasing
dimensions. In this way we may hope to get flavour as an emerging quantum
number that identifies different families with a mechanism somehow related to
the “geometry” of the UV completion of the theory. Possibly also weak isospin
breaking could be understood along these lines. Interesting examples where
ideas of this kind are exploited for phenomenology are presented in sect. 5 of
(II).

##  6 Conclusions and Outlook

In this paper we have shown that, as a viable alternative to the Higgs
scenario, a recently discovered [1, 3] NP field theoretical mechanism capable
of generating masses for elementary fermions can also provide a mass to the
$W$ when the model is extended to include weak interactions. Such a NP
phenomenon takes place in strongly interacting theories where the fermion
chiral symmetry (or an appropriate extension of it in the presence of weak
interactions), broken at the UV cutoff level, is recovered at low energy owing
to the tuning of certain Lagrangian parameters.

As a matter of fact the existence of this NP mass generation mechanism was
already noticed in WLQCD, like the compilation of $m_{cr}$ data as function of
the lattice spacing reported in fig. 1 of [1] shows. However, no use of this
NP feature was ever made for the purpose of giving mass to quarks in QCD,
because the breaking of chirality induced by the lattice regularization (due
to the presence of the Wilson term) has the well-known side effect of
generating a linearly divergent term in the mass of the fermion that
completely hide any finite underlying NP mass contribution. The standard
lattice procedure is thus to subtract the whole $m_{cr}\bar{q}q$ operator,
while at the same time adding by hand an explicit mass term for the quarks.
The peculiar property of the kind of models discussed in the present paper is
that the exact $\chi_{L}\times\chi_{R}$ symmetry precisely forbids the
existence of linearly divergent quantum corrections to fermion masses, thus
opening the possibility of exploiting possible NP finite effects related to
the spontaneous breaking of chiral symmetry to provide mass terms to
elementary particles.

The approach we are advocating in the present work, which extends ideas
already proposed in ref. [1], has also the virtue of offering interesting
hints towards the solution of some of the conceptual problems left open by the
current formulation of the SM.

  1. 1.

First of all, owing to the fact that in our model there is no elementary
Higgs, there isn’t anymore a “Higgs mass fine tuning problem”, hence no need
to explain why the Higgs mass is so much smaller than the Planck scale [10,
24]. Naturally it remains to explain why the weak force is so much stronger
than the gravitational one. In this framework, lacking a fundamental scalar,
the whole issue of the stability of the EW vacuum [14, 15, 16, 17] may need to
be revised.

  2. 2.

Secondly, unlike the SM, masses are not free parameters, rather they are NP-ly
determined by the dynamics of the theory 888For completeness, we remark that,
as we shall show in (II) (see eqs. (2.7) and (2.8)), in the present approach
neutrinos are exactly massless because with the standard hypercharge
assignments we are assuming for SM fermions (see Table 1 in ref. [20] or in
Appendix A of (II)) the right-handed neutrino Weyl component is completely
sterile..

  3. 3.

Finally, since the exact $\chi_{L}\times\chi_{R}$ symmetry protects elementary
particle masses from quantum power divergencies, the former are “naturally
small”, i.e. ${\mbox{O}}(\Lambda_{\rm RGI})$, and indeed the massless critical
theory enjoys an enhanced symmetry of chiral nature. From this point of view
the present approach to mass generation complies with the ’t Hooft criterion
of naturalness [24].

We have also seen that to cope with the magnitude of the top and $W$ mass, a
super-strongly interacting sector, gauge invariantly coupled to SM matter,
must be conjectured to exist, in order for the full theory, including SM
matter and Tera-particles, to have an RGI scale $\Lambda_{\rm
RGI}\\!\sim\\!\Lambda_{T}\\!\gg\\!\Lambda_{QCD}$ of the order of a few TeVs.
We have proved in this paper that the simplest model, introduced in [1] and
endowed with the above NP mass generation mechanism, can indeed be extended to
incorporate weak interactions and the conjectured Tera-dof’s. In a similar way
one could include leptons and the hypercharge interaction. This further step
is discussed in (II).

As for the relation of the present scheme to the SM, we have observed in sect.
4 that upon integrating out the (heavy) Tera-dof’s, under the assumption that
a “light” (compared to the TeV scale, $\Lambda_{T}$) $W^{+}W^{-}/ZZ$ composite
scalar singlet state bound by Tera-particle exchanges gets formed (that we
conjecture should be identified with the 125 GeV resonance discovered at LHC
[18, 19]), one ends up with a LEEL valid for
(momenta)${}^{2}\ll\Lambda_{T}^{2}$, which, ignoring small perturbative
corrections, resembles very much the SM Lagrangian, except possibly for the
effective trilinear and quadrilinear self-couplings of the composite scalar.

##  A Symmetries, currents and tuning

In this Appendix we provide a derivation of the criticality conditions which
allow enforcing the invariance of the Lagrangian (3.19) under the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations (3.28)-(3.31).

The exactly conserved currents associated with the transformations
$\chi_{L}\times\chi_{R}$ of eq. (3.24) have the expression ($i=1,2,3$)

$\displaystyle
J_{\mu}^{L\,i}=K_{\mu}^{i}+\bar{q}_{L}\gamma_{\mu}\frac{\tau^{i}}{2}q_{L}+\bar{Q}_{L}\gamma_{\mu}\frac{\tau^{i}}{2}Q_{L}-\frac{k_{b}}{4}{\,\hbox{tr}\,}\big{[}\Phi^{\dagger}\frac{\tau^{i}}{2}{\cal
D}^{W}_{\mu}\Phi-(\Phi\overleftarrow{\cal
D}\,^{W}_{\mu})^{\dagger}\frac{\tau^{i}}{2}\Phi\big{]}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{q}\,\big{(}\bar{q}_{L}\frac{\tau^{i}}{2}\Phi{\cal
D}^{A}_{\mu}q_{R}-\bar{q}_{R}\overleftarrow{\cal
D}\,^{A}_{\mu}\Phi^{\dagger}\frac{\tau^{i}}{2}q_{L}\big{)}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{Q}\,\big{(}\bar{Q}_{L}\frac{\tau^{i}}{2}\Phi{\cal
D}^{AG}_{\mu}Q_{R}-\bar{Q}_{R}\overleftarrow{\cal
D}\,^{AG}_{\mu}\Phi^{\dagger}\frac{\tau^{i}}{2}Q_{L}\big{)}\,,$ (A.54)
$\displaystyle
J_{\mu}^{R\,i}\\!=\\!\bar{q}_{R}\gamma_{\mu}\frac{\tau^{i}}{2}q_{R}+\bar{Q}_{R}\gamma_{\mu}\frac{\tau^{i}}{2}Q_{R}-\frac{k_{b}}{4}{\,\hbox{tr}\,}\big{[}(\Phi\overleftarrow{\cal
D}\,^{W}_{\mu})^{\dagger}\Phi\frac{\tau^{i}}{2}-\frac{\tau^{i}}{2}\Phi^{\dagger}({\cal
D}^{W}_{\mu}\Phi)\big{]}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{q}\,\big{(}\bar{q}_{R}\frac{\tau^{i}}{2}\Phi^{\dagger}{\cal
D}\,^{AW}_{\mu}q_{L}\\!-\\!\bar{q}_{L}\overleftarrow{\cal
D}\,^{AW}_{\mu}\Phi\frac{\tau^{i}}{2}q_{R}\big{)}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{Q}\,\big{(}\bar{Q}_{R}\frac{\tau^{i}}{2}\Phi^{\dagger}{\cal
D}\,^{AGW}_{\mu}Q_{L}\\!-\\!\bar{Q}_{L}\overleftarrow{\cal
D}\,^{AGW}_{\mu}\Phi\frac{\tau^{i}}{2}Q_{R}\big{)}\,,$ (A.55)

where

$K_{\mu}^{i}=g_{w}\,\hbox{tr}\,\Big{(}[W_{\nu},F^{W}_{\mu\nu}]\frac{\tau^{i}}{2}\Big{)}\,.$
(A.56)

We stress that the conserved currents, $J^{L\,i}$, to which weak bosons are
coupled, do not coincide with the conserved left-handed currents of the SM
because of the presence of Tera-particle contributions and of the terms
originating from the variation of the $d=6$ Wilson-like operators.

The key observation here is that, unlike the case where weak interactions are
absent, in the Lagrangian (3.19), neither the Yukawa and Wilson-like
operators, nor the $\Phi$ kinetic term are invariant under the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations as the latter do not
act on the scalar field (see eqs. (3.28)-(3.31)). The (non-conserved) currents
associated to the (chiral) $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$
transformations read

$\displaystyle\tilde{J}_{\mu}^{L\,i}=K_{\mu}^{i}+\bar{q}_{L}\gamma_{\mu}\frac{\tau^{i}}{2}q_{L}+\bar{Q}_{L}\gamma_{\mu}\frac{\tau^{i}}{2}Q_{L}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{q}\big{(}\bar{q}_{L}\frac{\tau^{i}}{2}\Phi{\cal
D}^{A}_{\mu}q_{R}-\bar{q}_{R}\overleftarrow{\cal
D}\,^{A}_{\mu}\Phi^{\dagger}\frac{\tau^{i}}{2}q_{L}\big{)}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{Q}\,\big{(}\bar{Q}_{L}\frac{\tau^{i}}{2}\Phi{\cal
D}^{AG}_{\mu}Q_{R}-\bar{Q}_{R}\overleftarrow{\cal
D}\,^{AG}_{\mu}\Phi^{\dagger}\frac{\tau^{i}}{2}Q_{L}\big{)}\,,$ (A.57)
$\displaystyle\tilde{J}_{\mu}^{R\,i}=\bar{q}_{R}\gamma_{\mu}\frac{\tau^{i}}{2}q_{R}+\bar{Q}_{R}\gamma_{\mu}\frac{\tau^{i}}{2}Q_{R}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{q}\big{(}\bar{q}_{R}\frac{\tau^{i}}{2}\Phi^{\dagger}{\cal
D}^{AW}_{\mu}q_{L}\\!-\\!\bar{q}_{L}\overleftarrow{\cal
D}\,^{AW}_{\mu}\Phi\frac{\tau^{i}}{2}q_{R}\big{)}+$
$\displaystyle\quad-\frac{b^{2}}{2}\rho_{Q}\,\big{(}\bar{Q}_{R}\frac{\tau^{i}}{2}\Phi^{\dagger}{\cal
D}^{AGW}_{\mu}Q_{L}\\!-\\!\bar{Q}_{L}\overleftarrow{\cal
D}\,^{AGW}_{\mu}\Phi\frac{\tau^{i}}{2}Q_{R}\big{)}\,.$ (A.58)

They differ from the conserved currents, $J^{L\,i}_{\mu}$ and
$J^{R\,i}_{\mu}$, only because the terms bilinear in the scalar field coming
from variation of the $\Phi$-kinetic term (proportional to $k_{b}$) are absent
in (A.57) and (A.58).

###  A.1 Enforcing invariance under the chiral
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations

Generalizing the strategy proposed in [1], restoration of chirality (up to
O($b^{2}$) cutoff effects) is obtained by appropriately tuning the Yukawa
couplings $\eta_{q}$ and $\eta_{Q}$ and the coefficient $k_{b}$ to critical
values determined by enforcing the conservation of the currents associated
with the $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations.

In order to identify under which conditions the (chiral)
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations (3.28) and (3.31) can
become an invariance of the Lagrangian (3.19) we start by writing down the
Schwinger–Dyson equations (SDEs) stemming from the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ transformations 999We refrain from
calling Ward–Takahashi identities (WTIs) eqs. (A.59) and (A.60) below, because
they refer to transformations that are not symmetries of the Lagrangian. For
generic values of $\eta_{q}$, $\eta_{Q}$ and $k_{b}$ we prefer to talk of
Schwinger–Dyson equations. Only at the critIcal point where the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ currents are conserved one is
entitled to talk of WTIs.. Following the steps outlined in [1], based on the
strategy devised in [23, 33], we arrive at the (renormalized) SDEs

$\displaystyle\partial_{\mu}\langle
Z_{\tilde{J}}\widetilde{J}^{L\,i}_{\mu}(x)\,\hat{O}(0)\rangle=\langle\widetilde{\Delta}_{L}^{i}\hat{O}(0)\rangle\delta(x)+$
(A.59)
$\displaystyle\quad-(\eta_{q}-\bar{\eta}^{L}_{q})\,\langle\Big{(}\bar{q}_{L}\frac{\tau^{i}}{2}\Phi
q_{R}-\bar{q}_{R}\Phi^{\dagger}\frac{\tau^{i}}{2}q_{L}\Big{)}(x)\,\hat{O}(0)\rangle+$
$\displaystyle\quad-(\eta_{Q}-\bar{\eta}^{L}_{Q})\,\langle\Big{(}\bar{Q}_{L}\frac{\tau^{i}}{2}\Phi
Q_{R}-\bar{Q}_{R}\Phi^{\dagger}\frac{\tau^{i}}{2}Q_{L}\Big{)}(x)\,\hat{O}(0)\rangle+$
$\displaystyle\quad+\frac{i}{2}g_{w}(k_{b}\\!-\\!\bar{k}_{b}^{L})\langle\,\hbox{tr}\,\\!\Big{(}\Phi^{\dagger}[\frac{\tau^{i}}{2},\\!W_{\mu}]{\cal
D}_{\mu}^{W}\Phi\\!+\\!\Phi^{\dagger}\overleftarrow{\cal
D}\,^{W}_{\mu}[W_{\mu},\\!\frac{\tau^{i}}{2}]\Phi\Big{)}(x)\hat{O}(0)\rangle\\!+\\!{\mbox{O}}(b^{2})\\!+\\!\ldots\,,$
$\displaystyle\partial_{\mu}\langle
Z_{\widetilde{J}}\tilde{J}^{R\,i}_{\mu}(x)\,\hat{O}(0)\rangle=\langle\widetilde{\Delta}_{R}^{i}\hat{O}(0)\rangle\delta(x)+$
(A.60)
$\displaystyle\quad-(\eta_{q}-\bar{\eta}^{R}_{q})\,\langle\Big{(}\bar{q}_{R}\frac{\tau^{i}}{2}\Phi^{\dagger}q_{L}-\bar{q}_{L}\Phi\frac{\tau^{i}}{2}q_{R}\Big{)}(x)\,\hat{O}(0)\rangle+$
$\displaystyle\quad-(\eta_{Q}-\bar{\eta}^{R}_{Q})\,\langle\Big{(}\bar{Q}_{R}\frac{\tau^{i}}{2}\Phi^{\dagger}Q_{L}-\bar{Q}_{L}\Phi\frac{\tau^{i}}{2}Q_{R}\Big{)}(x)\,\hat{O}(0)\rangle+{\mbox{O}}(b^{2})\\!+\\!\ldots\,,$

where $\hat{O}(0)$ is a generic local operator and the quantities
$\bar{\eta}^{L,R}_{q}$, $\bar{\eta}^{L,R}_{Q}$ and $\bar{k}_{b}^{L}$ are the
mixing coefficients of the $\tilde{\chi}_{L,R}$ variations of the Wilson-like
and Tera-Wilson-like terms with the variations of the quark and Tera-quark
Yukawa operators and the scalar kinetic term, respectively.

The tuning conditions yielding the conservation of the
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ currents that determine the values of
the parameters $\eta_{q}$, $\eta_{Q}$ and $k_{b}$ of the critical theory then
take the form

$\displaystyle\eta_{q}-\bar{\eta}^{L}_{q}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.61)
$\displaystyle\eta_{q}-\bar{\eta}^{R}_{q}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.62)
$\displaystyle\eta_{Q}-\bar{\eta}^{L}_{Q}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.63)
$\displaystyle\eta_{Q}-\bar{\eta}^{R}_{Q}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.64) $\displaystyle
k_{b}\,-\,\bar{k}_{b}^{L}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.65)

where for short we have set $\\{g\\}=(g_{s},g_{T},g_{w})$. Notice that, unlike
what happens in the absence of weak interactions where parity is unbroken, we
have for both quarks and Tera-quarks a “$Left$” and a “$Right$” equation
determining the Yukawa couplings. Thus one may wonder whether in the present
situation, where parity is not an exact symmetry of the basic Lagrangian, the
sets of $Left$ (eqs. (A.61) and (A.63) and $Right$ (eqs. (A.62) and (A.64))
constraints are compatible with each other. In the next subsection we prove
that $Left-Right$ compatibility is a consequence of the exact
$\chi_{L}\times\chi_{R}$ symmetry of the theory.

####  A.1.1 Compatibility between $Right$ and $Left$ tuning conditions

In this section we discuss the compatibility of the constraints from (A.61) to
(A.65) in the situation where, because of the presence of weak interactions,
parity is not an exact symmetry of the basic Lagrangian. We will show that,
owing to the exact $\chi_{L}\times\chi_{R}$ invariance (plus standard
symmetries and dimensionality arguments) the set of $\eta_{f}^{L}$,
$\eta_{f}^{L}$ ($f=q,Q$) and $k_{b}$ parameters that satisfies the subset of
conditions following from the requirement of, say, $\tilde{\chi}_{L}$ symmetry
restoration alone, automatically satisfies also the conditions corresponding
to $\tilde{\chi}_{R}$ symmetry restoration.

To reduce the argument to its essentials, for simplicity of notations we
imagine working in a model where only quarks and $W$’s are present. In this
situation the tuning conditions reduce to the following set of equations

$\displaystyle\eta_{q}-\bar{\eta}_{q}^{L}(\\{g\\};\eta_{q},\rho;k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.66)
$\displaystyle\eta_{q}-\bar{\eta}_{q}^{R}(\\{g\\};\eta_{q},\rho;k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.67) $\displaystyle
k_{b}\,-\,\bar{k}^{L}_{b}(\\{g\\};\eta_{q},\rho;k_{b};\mu_{0},\lambda_{0})=0\,,$
(A.68)

where $\\{g\\}=(g_{s},g_{w})$. Let us start by provisionally identifying as
$\eta_{q\,cr}$ and $k_{b\,cr}$ the values obtained by solving, say, eqs.
(A.66) and (A.68) which enforce $\tilde{\chi}_{L}$ invariance. We notice that
a bit away from the critical limit, owing to dimensional considerations,
hermiticity and exact $\chi_{L}\times\chi_{R}$ invariance, the $d=4$ QEL of
the theory in the Wigner phase must have the form

$\displaystyle{\Gamma}_{4}^{Wig}=\frac{1}{4}\Big{(}F^{A}\\!\cdot\\!F^{A}+F^{W}\\!\cdot\\!F^{W}\Big{)}+\Big{[}\bar{q}_{L}\,{\mbox{$\not\\!\\!\,$}{\cal
D}}^{A,W}q_{L}+\bar{q}_{R}\,{\mbox{$\not\\!\\!\,$}{\cal D}}^{A}q_{R}\Big{]}+$
$\displaystyle\quad+\eta_{q}^{red}\,\big{(}\bar{q}_{L}\Phi\,q_{R}+\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}+\frac{k_{b}^{red}}{2}{\,\hbox{tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}\Phi)^{\dagger}{\cal D}^{W}_{\mu}\Phi\big{]}\,+{\cal
V}(\Phi)\,,$ (A.69)

where $\eta_{q}^{red}$ and $k_{b}^{red}$ are the “reduced” couplings

$\displaystyle\eta_{q}^{red}=\eta_{q}-\bar{\eta}_{q}^{L}(\\{g\\};\eta_{q},\rho;k_{b};\mu_{0},\lambda_{0})\,,$
(A.70) $\displaystyle
k_{b}^{red}=k_{b}-\bar{k}^{L}_{b}(\\{g\\};\eta_{q},\rho;k_{b};\mu_{0},\lambda_{0})\,.$
(A.71)

Setting $\eta_{q}$ and $k_{b}$ equal to the solutions of the eqs. (A.66) and
(A.68), the restoration of the $\tilde{\chi}_{L}$ symmetry entails the
vanishing of the reduced coefficients

$\displaystyle\eta_{q}^{red}\,\stackrel{{\scriptstyle\eta_{q}\to\eta_{q\,cr},\,k_{b}\to
k_{b\,cr}}}{{\longrightarrow}}\,0\,,$ (A.72) $\displaystyle
k_{b}^{red}\,\stackrel{{\scriptstyle\eta_{q}\to\eta_{q\,cr},\,k_{b}\to
k_{b\,cr}}}{{\longrightarrow}}\,0^{+}\,.$ (A.73)

Looking at the form of eq. (A.69) at $y_{q}^{red}=0$ and $k_{b}^{red}\to
0^{+}$, it is apparent that the $\tilde{\chi}_{R}$ invariance is also
automatically restored and the current $\tilde{J}_{\mu}^{R,\,i}$ is
consequently conserved because the vanishing of $\eta_{q}^{red}$ and
$k_{b}^{red}$ makes all chiral breaking terms disappear from the QEL (A.69).
This fact in turn implies that the values of $\eta_{q\,cr}$ and $k_{b\,cr}$
determined from the eqs. (A.66) and (A.68) will also fulfil eq. (A.67). The
extension of this argument to the case of the model considered in the main
text, which also includes Tera-fermions and Tera-strong interactions, is
straightforward.

##  B The QEL of the critical model in the NG phase

In this Appendix we want to justify the form (3.43) that the QEL of the
renormalizable model (3.19) takes in the NG phase at the critical point. In
particular we want to prove that, despite the fact that the operator
$\Lambda_{T}R\,\,\hbox{tr}\,[({\cal D}_{\mu}^{W}U)^{\dagger}{\cal
D}^{W}_{\mu}U]$ is invariant under $\chi_{L}\times\chi_{R}$ transformations,
it cannot appear in $\Gamma^{NG}_{cr}$ (see eq. (3.43)). To this purpose it is
convenient to start by examining the structure of the QEL, $\Gamma^{NG}$,
slightly away from the critical point. Using the definition introduced in eq.
(2.16), we can write

$\displaystyle\Gamma^{NG}\\!=\\!\Gamma^{NG}_{4\,cr}\\!+\\!\frac{\mu_{\Phi}^{2}}{2}R^{2}\\!+\\!\frac{\lambda}{4}R^{4}\\!+\\!\frac{1}{2}k_{b}^{red}\Big{(}\partial_{\mu}R\,\partial_{\mu}R\\!+\\!R^{2}\,\hbox{tr}\,[({\cal
D}_{\mu}^{W}U^{\dagger}){\cal D}^{W}_{\mu}U]\Big{)}\\!+\\!$
$\displaystyle+\widetilde{c}_{\Phi}\,\Lambda_{T}R\,\,\hbox{tr}\,[({\cal
D}_{\mu}^{W}U)^{\dagger}{\cal
D}^{W}_{\mu}U]+\eta_{q}^{red}R\Big{(}\bar{q}_{L}Uq_{R}+\bar{q}_{R}U^{\dagger}q_{L}\Big{)}+$
$\displaystyle+\eta_{Q}^{red}R\Big{(}\bar{Q}_{L}UQ_{R}+\bar{Q}_{R}U^{\dagger}Q_{L}\Big{)}+\ldots\,,$
(B.74)

where ${\Gamma}_{4\,cr}^{NG}$ is the $d=4$ QEL of the model in the critical
limit, given in eq. (3.43). It includes all the NP
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ breaking terms of O($\Lambda_{T}$)
and O($\Lambda_{T}^{2}$). Dots stand for $d>4$ operators describing further NP
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ breaking operators that are
suppressed by inverse powers of $\Lambda_{T}$ (like for instance those in eq.
(3.45)). The “reduced” coefficients $\eta_{q}^{red}$, $\eta_{Q}^{red}$ and
$k_{b}^{red}$ are defined by the relations (which generalize eqs. (A.70) and
(A.71))

$\displaystyle\eta_{q}^{red}=\eta_{q}-\bar{\eta}_{q}^{L}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})\,,$
(B.75)
$\displaystyle\eta_{Q}^{red}=\eta_{Q}-\bar{\eta}_{Q}^{L}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})\,,$
(B.76) $\displaystyle
k_{b}^{red}=k_{b}-\bar{k}^{L}_{b}(\\{g\\};\eta_{q},\eta_{Q},\rho_{q},\rho_{Q};k_{b};\mu_{0},\lambda_{0})\,,$
(B.77)

respectively, with the critical limit determined by the conditions

$\eta_{q}^{red}=\eta_{Q}^{red}=0\,,\qquad k_{b}^{red}\to 0^{+}\,.$ (B.78)

The purpose of this Appendix is to show that also the coupling
$\widetilde{c}_{\Phi}$ vanishes in the critical limit, implying that the
undesired contribution of order $g_{w}^{2}v\Lambda_{T}$ to $M_{W}^{2}$, that
would arise from this term, cannot be present, making $M_{W}$ independent of
the vev of the scalar.

Since the QEL is a functional from which one is supposed to directly extract
the full quantum information of the model, it is mandatory to normalize
canonically the effective scalar field $R$ appearing in eq. (B.74). Upon
introducing

$\Phi_{c}=\Phi\sqrt{k_{b}^{red}}\,,\qquad R_{c}=R\sqrt{k_{b}^{red}}\,,$ (B.79)

with the subscript “$c$” standing for “canonical”, $\Gamma^{NG}$ takes the
form

$\displaystyle\Gamma^{NG}\\!=\\!\Gamma^{NG}_{4\,cr}+\frac{\mu_{\Phi}^{2}}{2k_{b}^{red}}R_{c}^{2}+\frac{\lambda}{4(k_{b}^{red})^{2}}R_{c}^{4}+\frac{1}{2}\Big{(}(\partial_{\mu}R_{c})^{2}+R_{c}^{2}\,\hbox{tr}\,[({\cal
D}_{\mu}^{W}U)^{\dagger}{\cal D}^{W}_{\mu}U]\Big{)}\\!+$
$\displaystyle+\frac{\widetilde{c}_{\Phi}}{\sqrt{k_{b}^{red}}}\,\Lambda_{T}R_{c}\,\hbox{tr}\,[({\cal
D}_{\mu}^{W}U)^{\dagger}{\cal
D}^{W}_{\mu}U]+\frac{\eta_{q}^{red}}{\sqrt{k_{b}^{red}}}R_{c}\Big{(}\bar{q}_{L}Uq_{R}+\bar{q}_{R}U^{\dagger}q_{L}\Big{)}+$
$\displaystyle+\frac{\eta_{Q}^{red}}{\sqrt{k_{b}^{red}}}R_{c}\Big{(}\bar{Q}_{L}UQ_{R}+\bar{Q}_{R}U^{\dagger}Q_{L}\Big{)}+\ldots\,,$
(B.80)

where, recalling $v^{2}=|\mu_{\Phi}^{2}|/\lambda$, we have

$R_{c}=v_{c}+\zeta_{0c}\,,\qquad\zeta_{0c}=\zeta_{0}\sqrt{k_{b}^{red}}\,,\qquad
v_{c}^{2}=\frac{|\mu_{\Phi}^{2}|}{\lambda}k_{b}^{red}\,.$ (B.81)

From the definition of $R_{c}$, the expression (B.80) of $\Gamma^{NG}$ and eq.
(B.81), it is apparent that a peculiar non-linear sigma-model is realized in
the scalar sector as $k_{b}^{red}\to 0^{+}$ because

$v_{c}\sim\sqrt{k_{b}^{red}}\to 0^{+}\,,\quad
m^{2}_{\zeta_{0c}}=\frac{2|\mu_{\Phi}^{2}|}{k_{b}^{red}}\to+\infty\,.$ (B.82)

We see that in the critical limit the squared mass of the effective
$\zeta_{0c}$ mode is a (real positive) divergent quantity while the vev of
$R_{c}$ vanishes because its effective quartic coupling,
$\lambda/4(k_{b}^{red})^{2}$, diverges faster than $m^{2}_{\zeta_{0c}}$. As
for the canonical reduced Yukawa couplings of quarks and Tera-quarks
${\eta_{q}^{red}}/{\sqrt{k_{b}^{red}}}$ and
${\eta_{Q}^{red}}\\!/\\!{\sqrt{k_{b}^{red}}}$, they can be safely set to zero
before taking the limit $k_{b}^{red}\\!\to\\!0^{+}$.

We are now ready to show that in the critical limit (B.78) the coupling
$\widetilde{c}_{\Phi}/\sqrt{k_{b}^{red}}$ appearing in the QEL (B.80) is
finite, or equivalently that

$\widetilde{c}_{\Phi}\sim\sqrt{k_{b}^{red}}\to 0\,,$ (B.83)

which validates the expression of $\Gamma^{NG}_{4\,cr}$ given in (3.43) where
the term proportional to $\widetilde{c}_{\Phi}$ was omitted.

The key remark on which the proof is based is that, owing to the decoupling
theorem [28], the QEL $\Gamma^{NG}$ (see eq. (B.80)) must be such as to yield
in the critical limit amplitudes with the virtual exchange of one $\zeta_{0c}$
particle that vanish at least like $1/m^{2}_{\zeta_{0c}}$. To see the
implications of this constraint let us for instance consider the $WW\to WW$
scattering amplitude, which receives a tree-level contribution from the
exchange of a $\zeta_{0c}$ particle. Taking the $WW\zeta_{0c}$-vertex from the
term $\propto\widetilde{c}_{\Phi}\Lambda_{T}$ in the second line of (B.80),
one gets

$A(WW\to WW)\propto
g_{w}^{2}\frac{\widetilde{c}_{\Phi}\Lambda_{T}}{\sqrt{k_{b}^{red}}}\,\frac{1}{s+m^{2}_{\zeta_{0c}}}\,\frac{\widetilde{c}_{\Phi}\Lambda_{T}}{\sqrt{k_{b}^{red}}}\,g_{w}^{2}\,.$
(B.84)

Since for large $m^{2}_{\zeta_{0c}}$ and at fixed $s$, the decoupling theorem
requires

$A(WW\to
WW)\propto(g_{w}^{2}\Lambda_{T})^{2}\frac{(\widetilde{c}_{\Phi})^{2}}{k_{b}^{red}}\frac{1}{s+m^{2}_{\zeta_{0c}}}\stackrel{{\scriptstyle
k_{b}^{red}\to
0^{+}}}{{\longrightarrow}}{\mbox{O}}\Big{(}{\frac{1}{m^{2}_{\zeta_{0c}}}}\Big{)}\,,$
(B.85)

or faster, it follows that indeed eq. (B.83) must hold, otherwise as
$k^{red}_{b}\to 0$ this contribution to $A(WW\to WW)$, instead of vanishing
like $1/m^{2}_{\zeta_{0c}}$, would diverge. As we said, the vanishing of
$\widetilde{c}_{\Phi}$, that was already accounted for in the expression of
${\Gamma}_{4\,cr}^{NG}$ we gave in eq. (2.18), tells us that $M_{W}$ does not
depend on the scalar vev.

A physical consequence of paramount importance of the analysis carried out
above is that the (canonically normalized) singlet scalar mode, $\zeta_{0c}$,
becomes an infinitely massive field with vanishing vev, decoupled from
fermions and gauge bosons, with no dynamics on any physical scale. This in
turn implies that there isn’t any dependence of physical observables on the
scalar quartic coupling, $\lambda_{0}$ and that, as announced, the $d=4$ QEL
of the critical theory in the NG phase is given by the functional
${\Gamma}_{4\,cr}^{NG}$ reported in eq. (3.43).

##  C The $\zeta_{0}$ critical propagator

An apparently tricky question is what is the expression of the $\zeta_{0}$
propagator in the critical limit, in view of the fact that the value of
$k_{b}$ in eq. (3.20) is fixed to precisely cancel the scalar kinetic term
against the similar operators with which the Wilson-like terms mix. At the
lowest loop order this cancellation is pictorially represented in fig. 6.

To answer this question we need to start from the form of the quadratic part
of the scalar field Lagrangian before tuning, which reads

${\cal L}_{\Phi}^{(2)}=\frac{k_{b}}{2\delta}{\,\hbox{tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}\Phi)^{\dagger}{\cal
D}^{W}_{\mu}\Phi\big{]}+\frac{m_{\zeta_{0}}^{2}}{2}{\,\hbox{tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}\,,$
(C.86)

where $\delta$ is a parameter that has been momentarily introduced to
canonically normalize $\zeta_{0}$ (recall $\Phi=(v+\zeta_{0})U$).

Looking at fig. 6, we can write for the amputated $\zeta_{0}$-propagator

$\Pi(p^{2})\Big{|}^{1-loop}_{amp}=\frac{1}{\delta}(k_{b}p^{2}+m_{\zeta_{0}}^{2})-K(p^{2})\,,$
(C.87)

where for the 1-loop correction we get

$K(p^{2})=k_{0}b^{-2}+k_{1}p^{2}+k_{2}b^{2}p^{4}+\ldots$ (C.88)

Inserting this expansion in eq. (C.87), we obtain

$\displaystyle\Pi(p^{2})\Big{|}^{1-loop}_{amp}=\frac{1}{\delta}(k_{b}p^{2}+m_{\zeta_{0}}^{2})-k_{0}b^{-2}-k_{1}p^{2}-k_{2}b^{2}p^{4}+\ldots=$
$\displaystyle=\Big{(}\frac{k_{b}}{\delta}-k_{1}\Big{)}p^{2}+\Big{(}\frac{m_{\zeta_{0}}^{2}}{\delta}-k_{0}b^{-2}\Big{)}-k_{2}b^{2}p^{4}+\ldots\,.$
(C.89)

These equations tell us that the tuning condition is

$k_{b\,cr}=\delta k_{1}\,,$ (C.90)

from which we have

$\displaystyle\Pi(p^{2})\Big{|}^{1-loop}_{amp}=\Big{(}\frac{k_{b}}{\delta}-k_{1}\Big{)}p^{2}+\Big{(}\frac{m_{\zeta_{0}}^{2}}{\delta}-k_{0}b^{-2}\Big{)}-k_{2}b^{2}p^{4}+\ldots=$
$\displaystyle=\frac{k_{b}-k_{b\,cr}}{\delta}p^{2}+\Big{(}\frac{m_{\zeta_{0}}^{2}}{\delta}-k_{0}b^{-2}\Big{)}-k_{2}b^{2}p^{4}+\ldots\,.$
(C.91)

To canonically normalize the scalar kinetic term we need to take

$\delta=k_{b}-k_{b\,cr}\,.$ (C.92)

The canonically normalized, amputated $\zeta_{0}$-propagator then reads

$\Pi(p^{2})\Big{|}^{1-loop}_{amp}=p^{2}+\Big{(}\frac{m_{\zeta_{0}}^{2}}{k_{b}-k_{b\,cr}}-k_{0}b^{-2}\Big{)}-k_{2}b^{2}p^{4}+\ldots\,.$
(C.93)

We now note that the self-energy diagrams (like those shown in fig. 10) have a
finite value which is independent of the $\zeta_{0}$ mass, if the limit $b\to
0$ is taken before setting $k_{b}-k_{b\,cr}\to 0$.

The result of this analysis is that the “effective” mass of $\zeta_{0}$
diverges in the critical limit, without affecting the value of the self-energy
diagrams, provided, we insist, the limit $b\to 0$ is taken before letting
$k_{b}-k_{b\,cr}\to 0$, as we should, and we have actually implicitly done in
the analysis presented in Appendix B which is based on the continuum-like form
of the QEL.

##  D Transversality of $W$ polarization amplitude

In this Appendix we want to illustrate how the expected transversality
property of the $W$ polarization amplitude is realized in the critical limit
of the model (3.23) and discuss how the Goldstone fields $\zeta_{i},i=1,2,3$
(see eq. (3.44)) get eaten up to become the longitudinal $W$ modes [34]. In
particular, we will show that the sum of the amputated diagrams displayed in
the top panel of fig. 11 has the expected transverse structure, namely

$\displaystyle\langle\widehat{W}^{i}_{\mu}(p)\widehat{W}^{j}_{\nu}(-p)\rangle\Big{|}^{\rm
amp}_{p\to
0}\\!\\!=\\!g_{w}^{2}\langle\widehat{J}^{L\,i}_{\mu}(p)\widehat{J}^{L\,j}_{\nu}(-p)\rangle\Big{|}_{p\to
0}\\!\\!\\!\to\\!c^{2}_{w}\Lambda_{T}^{2}g_{w}^{2}\Big{[}\delta_{\mu\nu}\\!-\\!\frac{p_{\mu}p_{\nu}}{p^{2}}\Big{]}\delta_{ij}\,.$
(D.94)

In eq. (D.94) we have dropped an irrelevant $\delta(0)$ factor and indicated
by $\widehat{W}^{i}_{\mu}(p)$ ($\widehat{J}^{L\,i}_{\mu}(p)$) the Fourier
transform of $W^{i}_{\mu}(x)$ ($J^{L\,i}_{\mu}(x)$). The leftmost diagram of
the top panel of fig. 11 was already computed (see eq. (3.42)) and contributes
the first term in eq. (D.94). To compute the rightmost diagram we need to
evaluate in the critical limit the (amputated) $W$-NG boson two-point function
as well as the NG-boson propagator (the latter is depicted in the bottom panel
of fig. 11).

These are pretty straightforward calculations which do not differ much from
the analogous ones we would do in the SM 101010For instance, as in the SM,
from eq. (D.94) we can read off the value of the squared $W$ mass (see eq.
(20.11) of [34]).. The only difference is that here we need to remember that
these two-point functions come completely from NP effects, as the perturbative
contribution (recall eq. (3.33))

$\Gamma^{PT}_{\rm
kin}(\zeta)=\frac{v^{2}}{F^{2}}\Big{[}k_{b}-(\rho_{q}^{2}N_{c}k_{bq}^{(1)}+\rho_{Q}^{2}N_{c}N_{T}k_{bQ}^{(1)})+\ldots\Big{]}\,\partial_{\mu}\vec{\zeta}\,\partial_{\mu}\vec{\zeta}\,\stackrel{{\scriptstyle
k_{b}\to k_{b\,cr}}}{{\longrightarrow}}0\,,$ (D.95)

vanishes in the critical limit. In eq. (D.95) we have used the standard polar
decomposition

$\displaystyle\Phi=R\,U\equiv(v+\zeta_{0})U\,,\qquad
U=\exp[i\vec{\tau}\vec{\zeta}/F]\,,$ (D.96)

where for a while $F$ is left as a not yet specified scale. The vanishing of
the quantity in the square parenthesis in eq. (D.95) is precisely the
condition that determine the critical value of $k_{b}$ in eq. (3.33). We
recall from sect. 3.1 that the first contribution in the parenthesis comes
from the NG-boson kinetic term present in the fundamental Lagrangian (3.23),
the second from the 1-loop fermion correction of fig. 9 (see eq. (3.33)) and
the dots from the rest of the loop expansion.

From the NP expression of the QEL of the critical theory given in eq. (3.43),
we get

$\langle\widehat{W}^{i}_{\mu}(p)\widehat{\zeta}^{j}(-p)\rangle\Big{|}^{\rm
amp}_{p\to 0}\to c^{2}_{w}\Lambda_{T}^{2}g_{w}\frac{p_{\mu}}{F}\delta_{ij}$
(D.97)

for the $W$-NG boson two-point function and

$\langle\widehat{\zeta}^{i}(p)\widehat{\zeta}^{j}(-p)\rangle\Big{|}^{\rm
amp}_{p\to
0}=\frac{c^{2}_{w}\Lambda_{T}^{2}}{F^{2}}\frac{1}{p^{2}}\delta_{ij}$ (D.98)

for the (tree-level) NG-boson propagator.

Figure 11: Top panel: the diagrams making transverse the $W$ polarization
amplitude. The doubly dotted lines represent the propagation of a NG boson.
Bottom panel: the NG-boson propagator. The rest of the notation is as in the
figures in sects. 3.3 and 3.4.

Putting everything together, one obtains for the sum of the two diagrams in
the top panel of fig. 11 the desired transverse expression

$\displaystyle
g_{w}^{2}\langle\widehat{J}^{L\,i}_{\mu}(p)\widehat{J}^{L\,j}_{\nu}(-p)\rangle\Big{|}_{p\to
0}\\!\\!\\!\to\\!\Big{[}c^{2}_{w}\Lambda_{T}^{2}g_{w}^{2}\delta_{\mu\nu}\\!-\\!\Big{(}c^{2}_{w}\Lambda_{T}^{2}g_{w}\frac{p_{\mu}}{F}\Big{)}\frac{F^{2}}{c^{2}_{w}\Lambda_{T}^{2}}\frac{1}{p^{2}}\Big{(}\frac{p_{\nu}}{F}c^{2}_{w}\Lambda_{T}^{2}g_{w}\Big{)}\Big{]}\delta_{ij}\\!=$
$\displaystyle\qquad=c^{2}_{w}\Lambda_{T}^{2}g_{w}^{2}\Big{[}\delta_{\mu\nu}\\!-\\!\frac{p_{\mu}p_{\nu}}{p^{2}}\Big{]}\delta_{ij}\,.$
(D.99)

It should be stressed that, as expected, the arbitrary scale $F$ introduced in
the parametrization (D.96) completely disappears from the physical formula
(D.99). Any way, as we have already remarked, in view of the form (3.43) of
the QEL it is convenient to set $F=c_{w}\Lambda_{T}$ in eq. (D.96) so as to
have canonically normalized Goldstone fields.

Acknowledgments - We are indebted to R. Frezzotti for his interest in this
work and for infinitely many comments and suggestions on the issues presented
in this paper. We wish to thank R. Barbieri, M. Bochicchio, G. Martinelli, C.
T. Sachrajda, N. Tantalo, M. Testa and especially G. Veneziano for many useful
discussions. We thank M. Garofalo for a careful reading of the manuscript and
for illuminating correspondence.

## References

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