# A symmetry perspective of the Riemann zeros

Pushpa Kalaunia<EMAIL_ADDRESS>(correspinding author) Prasanta K.
Panigrahib<EMAIL_ADDRESS>aDepartment of Physics, Indian Institute
of Space Science and Technology, Thiruvananthapuram, Kerala, 695547, India
bDepartment of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, Mohanpur 741246, West Bengal, India

###### Abstract

We study the relationship between the zeros of the Riemann zeta function and
physical systems exhibiting supersymmetry, $PT$ symmetry and $SU(2)$ group
symmetry. Our findings demonstrate that unbroken supersymmetry is associated
with the presence of non-trivial zeros of the zeta function. However, in other
cases, supersymmetry is spontaneously broken and the ground state energy of
the system is not zero. Moreover, we have established the manifestation of PT
symmetry invariance within our supersymmetric system. In addition, our
findings provide insights into a $SU(2)$ symmetry that arises within these
systems, with the Hilbert space having a two-level structure.

##  I Introduction

Supersymmetric quantum mechanics (SUSY QM) [1, 2, 3] is a theoretical
framework that combines ideas from quantum mechanics and supersymmetry, a
symmetry that relates particles with different spin values. One of the
interesting features of SUSY QM is that its ground state energy always
vanishes, a consequence of the supersymmetry of the theory.

Recently, there has been a fascinating development in the study of SUSY QM,
which connects it to the Riemann Hypothesis [4], a famous problem in
mathematics that deals with the distribution of prime numbers [5].
Specifically, it has been shown that the non-trivial zeros of the Riemann zeta
function can be described as the ground state energy of certain SUSY QM
models.

In some recent studies, the relationship between the Riemann zeta function and
various aspects of Physics has been explored [6, 7, 8, 9, 10, 11]. In another
study, the Riemann zeta function relates to scattering amplitudes in quantum
field theory [12]. Additionally, the Riemann zeta function and its relevance
to second-order supersymmetric quantum mechanics have been discussed [13].
The Riemann zeta function [14] is defined for complex values of $s$ in the
following form as

$\displaystyle\zeta(s)=$
$\displaystyle\frac{1}{(1-2^{1-s})}\sum_{n=1}^{\infty}(-1)^{n+1}n^{-s},\,\,\,\Re(s)>0.$
(1)

The Riemann zeta function can also be defined in the complex plane by the
contour integral [14]

$\displaystyle\zeta(s)$ $\displaystyle=\frac{\Gamma(1-s)}{2\pi
i}\int_{C}\frac{t^{s-1}}{e^{-t}-1}dt,$ (2)

where the contour of integration encloses the negative $t$-axis, looping from
$t=-\infty-i0$ to $t=-\infty+i0$ enclosing the point $t=0$.

The Riemann zeta function possesses both trivial and non-trivial zeros.
Trivial zeros of the zeta function occur when $s$ is negative even integer.
These zeros can be obtained from the functional relation of the Riemann zeta
function, which is given by

$\displaystyle\zeta(s)$ $\displaystyle=2^{s}\pi^{s-1}\sin(\frac{\pi
s}{2})\Gamma(1-s)\zeta(1-s),\,s<1.$ (3)

Non-trivial zeros are associated with the Riemann hypothesis, which asserts
that all non-trivial zeros of the Riemann zeta function $\zeta(s)$ lie on the
critical line, the line in the complex plane where the real part of $s$ is
equal to $1/2$. To incorporate trivial zeros into the supersymmetric model, a
modified inner product was introduced, and the Hilbert space of the
supersymmetric system was defined by implementing appropriate boundary
conditions. This was discussed in more detail in a study by Kalauni et al.
[15].

In [15], we define a supersymmetric model in which the eigenvalue of the
system arises in the form of $\zeta(s)\zeta(1-s)$ (where $s=\sigma+i\omega$,
$\sigma$ and $\omega$ are real), which is the product of the Riemann zeta
function evaluated at complex arguments $(\sigma+i\omega)$ and
$(1-\sigma-i\omega)$. It was demonstrated that the states of the
supersymmetric system under consideration form an orthonormal set of
functions, when the Hilbert space is defined for a finite interval of real
line, specifically in $L^{2}(1,a)$ with parameters $\sigma=1/2$ and
$a=e^{2\pi/\omega}$. As a result, it was shown that the ground state energy
vanishes only when $\sigma=1/2$. However, it is natural to ask whether there
are any additional indications which suggest that the ground state energy
vanishes only when $\sigma=1/2$. The aim of this paper is to gain a more
comprehensive understanding of the supersymmetric system under consideration
by exploring different symmetries within the system. To achieve this, we
employ various methods, including analyzing the Witten index, examining $PT$
symmetry, and investigating $SU(2)$ symmetry present in the system.

In the next section, we provide a review of the SUSY QM model, which yields
the Riemann zeta function as an eigenvalue of the system. In the third
section, the Witten index is used to examine the unbroken and broken
supersymmetry of the system. It is shown that for $\sigma=1/2$, it gives real
eigenvalues of the system while for other values of $\sigma$, the energy of
the system becomes complex, indicating a possible connection with $PT$
symmetry [16, 17].

$PT$ symmetry is a concept in quantum mechanics which refers to the behavior
of physical systems that are invariant under the combined operations of parity
$(P)$ and time reversal $(T)$. One of the interesting aspects of $PT$ symmetry
is that it can lead to the existence of complex eigenvalues and non-Hermitian
operators in quantum mechanics. In the fourth section, we investigate the $PT$
symmetry in our supersymmetric system and establish that it remains unbroken
only if a specific condition holds. However, for other cases, PT symmetry is
broken. In the fifth section, we show that this Hilbert space of the system
forms a $SU(2)$ symmetry (function-dependent) which is present in the system
because of the presence of the trivial and the non-trivial zeros of the zeta
function.

These results provide new insights into the underlying algebraic structures
and symmetries of the system. This gives the motivation for comprehending the
interrelations between the Riemann zeros and diverse symmetries, namely
supersymmetry, $PT$ symmetry, and $SU(2)$ symmetry.

##  II Supersymmetric System and Riemann zeta function

To begin, we review our supersymmetric model [15], which yields the Riemann
zeta function as an eigenvalue. This supersymmetric model is defined by
introducing lowering and raising operators (denoted by $A$ and $A^{\dagger}$,
respectively), which is given as

$\displaystyle A$ $\displaystyle=x^{-i\omega}\Omega,$ $\displaystyle
A^{\dagger}$ $\displaystyle=\Omega^{\dagger}x^{i\omega},$ (4)

where

$\displaystyle\Omega=$ $\displaystyle\frac{\Gamma(x\frac{d}{dx}+1)}{2\pi
i}\intop_{C}\frac{t^{-x\frac{d}{dx}-1}}{e^{-t}-1}dt,$ (5)

and $\omega$ is real and contour $C$ is as given in Eq. (2). When the operator
$\Omega$ (defined in Eq. (5)) acts on the monomial $x^{-s}$, it produces the
eigenvalues in terms of the Riemann zeta function as [18]

$\displaystyle\Omega x^{-s}$
$\displaystyle=\left[\frac{\Gamma(x\frac{d}{dx}+1)}{2\pi
i}\intop_{C}\frac{t^{-x\frac{d}{dx}-1}}{e^{-t}-1}dt\right]x^{-s},$
$\displaystyle=\left[\frac{\Gamma(1-s)}{2\pi
i}\intop_{C}\frac{t^{s-1}}{e^{-t}-1}dt\right]x^{-s},$
$\displaystyle=\zeta(s)\,x^{-s}.$ (6)

Using these operators $\Omega$ and $\Omega^{\dagger}$, it is possible to
construct a set of eigenstates for the Hamiltonian of the system, which
correspond to different energy levels of the system. These eigenstates can be
used to compute the spectrum of the system and determine its ground state
energy.

The Hamiltonian of the supersymmetric system can be written in terms of a
$2\times 2$ matrix as

$\displaystyle H=$ $\displaystyle\left[\begin{array}[]{cc}H_{-}&0\\\
0&H_{+}\end{array}\right],$ (9)

where

$\displaystyle H_{-}$ $\displaystyle=A^{\dagger}A=\Omega^{\dagger}\Omega,$
$\displaystyle H_{+}$
$\displaystyle=AA^{\dagger}=x^{-i\omega}\Omega\Omega^{\dagger}x^{i\omega},$
(10)

are supersymmetric partner Hamiltonians.

We define the wavefunction as $x^{-s}$ with $s=\sigma+i\omega$ (where $\sigma$
and $\omega$ are real). It gives eigenvalues of the system in terms of the
Riemann zeta function as

$\displaystyle H_{-}x^{-\sigma-i\omega}$
$\displaystyle=\zeta(\sigma+i\omega)\zeta(1-\sigma-i\omega)x^{-\sigma-i\omega}.$
(11)

Eq. (11) shows that $H_{-}$ has real eigenvalues only if $\sigma=1/2$. This
means that when $\sigma\neq 1/2$, then the eigenvalues of $H_{-}$ will be
complex numbers.

In our specific model [5, 15], the ground state energy of the system is
described in terms of the Riemann zeta function. Our analysis indicates that
the unbroken supersymmetry present in the system results as a vanishing ground
state energy of the system. We elaborate on the concept of unbroken
supersymmetry and its relation to the ground state energy in the subsequent
section by utilizing the Witten index.

##  III Unbroken and Broken Supersymmetry

The Witten index is a powerful tool for studying supersymmetric theories. One
of the most important applications of the Witten index is in determining
whether or not supersymmetry is broken in a given theory.

To calculate the Witten index for a given system, we count the number of
bosonic and fermionic states with zero energy, denoted as $n_{B}^{E=0}$ and
$n_{F}^{E=0}$, respectively. These counts are done under three distinct
conditions, which Witten identified in his paper [1]:

  1. 1.

If $n_{B}^{E=0}-n_{F}^{E=0}\neq 0$, then supersymmetry is unbroken.

  2. 2.

If $n_{B}^{E=0}=n_{F}^{E=0}=0$, supersymmetry is spontaneously broken.

  3. 3.

If $n_{B}^{E=0}$ and $n_{F}^{E=0}$ are equal but non-zero, supersymmetry is
unbroken.

We start by defining the wavefunction as $x^{-s}$ with $s=\sigma+i\omega$
(where $\sigma$ and $\omega$ are real) and for applying Witten index
condition, we check that

$\displaystyle H_{-}x^{-s}$ $\displaystyle=\zeta(s)\zeta(1-s)x^{-s};$
$\displaystyle H_{+}x^{s-1-i\omega}$
$\displaystyle=\zeta(s)\zeta(1-s)x^{s-1-i\omega},$ (12)

which shows that when the parameter $s$ is fixed, there exists a bosonic state
denoted as $x^{-s}$ and a corresponding fermionic state denoted as
$x^{s-1-i\omega}$, where both states contribute the same amount of energy.
However, it is important to note that the bosonic and fermionic states are
normalized and form a Hilbert space if $\text{Re}(s)=1/2$. As we see that if
$\text{Re}(s)=1/2$, $\text{Im}(s)=\omega$, and $\zeta(1/2+i\omega)=0$, we find
a bosonic state $x^{-1/2+i\omega}$ and a corresponding fermionic state
$x^{-1/2-i\omega}$, both of which contribute zero energy to the ground state.
This fulfills the third condition of the Witten index, where both
$n_{B}^{E=0}$ and $n_{F}^{E=0}$ are equal to $1$, resulting in zero difference
and indicating unbroken symmetry. However, if $\text{Re}(s)=1/2$ and $\omega$
does not correspond to the imaginary part of the non-trivial zeros of the zeta
function, then supersymmetry remains broken due to Witten’s second condition.

For values of Re(s) other than $1/2$, no solutions exist within the Hilbert
space, resulting in both $n_{B}^{E=0}$ and $n_{F}^{E=0}$ being zero, which
indicates symmetry breaking. This implies that when Re(s)$\neq$1/2, then the
eigenvalues of $H_{-}$ are complex numbers.

This peculiarity of complex eigenvalues can be linked to the $PT$ symmetry of
the system. In the following section, we elaborate on the $PT$ symmetry of the
system and its relation to the eigenvalues of the Hamiltonian.

##  IV $PT$ Unbroken and Broken Symmetry

In [15], it has been demonstrated that the $n^{th}$ states of the system are
normalized and constitute a complete set of functions in the interval $[1,a]$
of square-integrable functions $L^{2}$. The value of $a$ is related to non-
trivial zeros of the zeta function, specifically, if
$\zeta(\frac{1}{2}+i\omega)=0$ where $\omega=2\pi/\log a$. Additionally, these
functions are also orthonormal in the interval $[-a,-1]$ of square-integrable
functions $L^{2}$.

In order to define $PT$ symmetry, it is necessary to have a Hilbert space
where the operations $x\rightarrow-x$ and $p\rightarrow p$ can be utilized. It
indicates that we can use $PT$ symmetry for this particular type of function,
which may have important implications for the properties and behavior of the
system.

To incorporate both positive and negative values of $x$, we take the direct
sum of these two Hilbert spaces denoted as $L^{2}[-a,-1]\oplus L^{2}[1,a]$.
Therefore, we define wave functions in the following form

$\displaystyle\psi_{1}(x)$
$\displaystyle=|x|^{-\sigma-i\omega}\,\,\,\,\,\,\,\,\,\,\,\,\,\psi_{2}(x)=\text{sgn}(x)|x|^{-\sigma-i\omega}.$
(13)

If the Hamiltonian $H_{-}$ is invarinat under $PT$ symmetry, it satisfies,

$\displaystyle[PT,H_{-}]\psi_{1}(x)$ $\displaystyle=0,$ (14)

It follows that

$\displaystyle\zeta(\sigma+i\omega)\zeta(1-\sigma-i\omega)$
$\displaystyle-\zeta(\sigma-i\omega)\zeta(1-\sigma+i\omega)=0.$ (15)

Similarly, we can check $PT$ symmetry for $\psi_{2}(x)$ as

$\displaystyle[PT,H_{-}]\psi_{2}(x)$ $\displaystyle=0,$ (16)

which is true if Eq. (15) is satisfied.

An important insight derived from this connection is that $PT$ symmetry in the
system remains invariant in two cases: i) $\text{Re}(s)=\nicefrac{{1}}{{2}}$
and ii) $\text{Im}(s)=0$ (where $s=\sigma+i\omega)$.

This finding implies that $PT$ symmetry, similar to supersymmetry, remains
unchanged when the $\text{Re}(s)=\nicefrac{{1}}{{2}}$. However, for values of
$s$ other than those specified, the $PT$ symmetry of the system is broken.

This system also possesses an underlying $SU(2)$ algebraic structure like
Quasi-exactly solvable system [19]. In the following section, we demonstrate
this wavefunction-dependent $SU(2)$ symmetry, which arises subject to the same
constraint outlined in Eq. (15).

##  V $SU(2)$ symmetry

$SU(2)$ is a Lie group that describes the group of special unitary
transformations. The generators of $SU(2)$ are the angular momentum operators
$J_{-}$, $J_{0}$, and $J_{+}$, which satisfy the following commutation
relations

$\displaystyle[J_{+},J_{-}]$ $\displaystyle=2J_{0},$
$\displaystyle{}[J_{0},J_{\pm}]$ $\displaystyle=\pm J_{\pm}.$ (17)

To establish the link between the Riemann zeros and $SU(2)$ symmetry, we begin
by defining $J_{-}$ and $J_{+}$ using the lowering and raising operators of
our supersymmetric system, with a normalizing factor of
$1/\sqrt{\zeta(k)\zeta(1-k)}$:

$\displaystyle J_{-}=\frac{A}{\sqrt{\zeta(k)\zeta(1-k)}}=$
$\displaystyle\frac{1}{\sqrt{\zeta(k)\zeta(1-k)}}x^{-i\omega}\Omega,$
$\displaystyle J_{+}=\frac{A^{\dagger}}{\sqrt{\zeta(k)\zeta(1-k)}}=$
$\displaystyle\frac{1}{\sqrt{\zeta(k)\zeta(1-k)}}\Omega^{\dagger}x^{i\omega},$
(18)

and we chose values of $k=1/2$ for non-trivial zeros and $k=-2N-i\omega$ for
the case of trivial zeros.
The commutation relation between $J_{+}$ and $J_{-}$ leads to,

$\displaystyle[J_{+},J_{-}]=\frac{(\Omega^{\dagger}\Omega-x^{-i\omega}\Omega\Omega^{\dagger}x^{i\omega})}{\zeta(k)\zeta(1-k)}$
$\displaystyle=2J_{0},$ (19)

where

$\displaystyle J_{0}$
$\displaystyle=\frac{(\Omega^{\dagger}\Omega-x^{-i\omega}\Omega^{\dagger}\Omega
x^{i\omega})}{2\zeta(k)\zeta(1-k)}.$ (20)

We define $J_{x}$ and $J_{y}$ in the following form,

$\displaystyle J_{x}$
$\displaystyle=\frac{\Omega^{\dagger}x^{i\omega}+x^{-i\omega}\Omega}{2\sqrt{\zeta(k)\zeta(1-k)}},$
$\displaystyle J_{y}$
$\displaystyle=\frac{\Omega^{\dagger}x^{i\omega}-x^{-i\omega}\Omega}{2i\sqrt{\zeta(k)\zeta(1-k)}}.$
(21)

The action of the generators $J_{x}$, $J_{y}$, and $J_{0}$ on a function
$x^{-s}$ yields,

$\displaystyle J_{x}x^{-s}$
$\displaystyle=\frac{\zeta(1-s+i\omega)x^{-s+i\omega}+\zeta(s)x^{-s-i\omega}}{2\sqrt{\zeta(k)\zeta(1-k)}},$
$\displaystyle J_{y}x^{-s}$
$\displaystyle=\frac{\zeta(1-s+i\omega)x^{-s+i\omega}-\zeta(s)x^{-s-i\omega}}{2i\sqrt{\zeta(k)\zeta(1-k)}},$
$\displaystyle J_{0}x^{-s}$
$\displaystyle=\left[\frac{\zeta(s)\zeta(1-s)-\zeta(s-i\omega)\zeta(1-s+i\omega)}{2\zeta(k)\zeta(1-k)}\right]x^{-s}.$
(22)

For $SU(2)$, the Casimir operator is given by,

$\displaystyle J^{2}$ $\displaystyle=J_{x}^{2}+J_{y}^{2}+J_{0}^{2},$ (23)

that commutes with each of these generators $J_{x}$, $J_{y}$ and $J_{0}$.
The Casimir operator acts on the function $x^{-s}$ as

$\displaystyle J^{2}x^{-s}=$
$\displaystyle\left[\frac{\zeta(s-i\omega)\zeta(1-s+i\omega)+\zeta(s)\zeta(1-s)}{2\zeta(k)\zeta(1-k)}\right]x^{-s}$
$\displaystyle+\left[\frac{\zeta(s-i\omega)\zeta(1-s+i\omega)-\zeta(s)\zeta(1-s)}{2\zeta(k)\zeta(1-k)}\right]^{2}x^{-s},$
(24)

where $s=\sigma+i\omega$. The structures of the commutators and the Casimir
operator on this Hilbert space are indicative of a Hilbert-space-dependent
structure [20].

We will now demonstrate how the $SU(2)$ algebra is related to the existence of
both non-trivial and trivial zeros in zeta functions, possessing a Hilbert-
space-dependent spin $1/2$ structure.

###  V.1 Non-trivial Riemann zeros

We define a $SU(2)$ eigenstate as

$\displaystyle|\sigma,m\rangle$ $\displaystyle=\frac{1}{\sqrt{\log
a}}x^{-\sigma+i(m-\frac{1}{2})\omega},$ (25)

where $a=e^{2\pi/\omega}.$ The spin of a particle with quantum number $s$ can
take $2s+1$ values for the unitary irreducible representation. In the present
case, the Hilbert space has a two-level $\sigma=1/2$ representation. These two
states can be represented as $|\sigma,m\rangle=$$|1/2,1/2\rangle$ and
$|1/2,-1/2\rangle$. When $\sigma=1/2$, these states form a discrete,
orthonormal, and complete basis in a finite interval of real line [15] and the
orthonormality condition for this basis is given by

$\displaystyle\langle 1/2,m|1/2,m^{\prime}\rangle$
$\displaystyle=\delta_{mm^{\prime}}.$ (26)

We assume that $\omega$ is the imaginary part of non-trivial zeros of the zeta
function and define

$\displaystyle\zeta(\frac{1}{2}+i\omega)=\zeta(\frac{1}{2}-i\omega)$
$\displaystyle=0.$ (27)

Using $k=1/2$ and Eq. (27), we can write Eq. (22) as

$\displaystyle J_{x}|1/2,1/2\rangle$
$\displaystyle=\frac{1}{2}|1/2,-1/2\rangle,$ $\displaystyle
J_{y}|1/2,1/2\rangle$ $\displaystyle=-\frac{1}{2i}|1/2,-1/2\rangle,$
$\displaystyle J_{0}|1/2,1/2\rangle$
$\displaystyle=\frac{1}{2}|1/2,1/2\rangle,$ (28)

which provides the eigenvalues of $J{{}^{2}}$ and $J_{0}$ as

$\displaystyle J^{2}|1/2,\pm 1/2\rangle$ $\displaystyle=\frac{3}{4}|1/2,\pm
1/2\rangle,$ $\displaystyle J_{0}|1/2,\pm 1/2\rangle$
$\displaystyle=\pm\frac{1}{2}|1/2,\pm 1/2\rangle.$ (29)

From Eqs. (18-29), we can see that the operators $J_{+}$, $J_{-}$, and $J_{0}$
satisfy the well-known $SU(2)$ algebraic commutation relations

$\displaystyle[J_{+},J_{-}]|1/2,1/2\rangle$
$\displaystyle=(2J_{0})|1/2,1/2\rangle,$
$\displaystyle{}[J_{0},J_{-}]|1/2,1/2\rangle$
$\displaystyle=-J_{-}|1/2,1/2\rangle,$
$\displaystyle{}[J_{0},J_{+}]|1/2,-1/2\rangle$
$\displaystyle=J_{+}|1/2,-1/2\rangle.$ (30)

It is noteworthy that $SU(2)$ algebra in this system is dependent on the zeta
function, and the zeta function must satisfy a condition given in Eq. (27) in
order to satisfy this algebra. This constraint implies that $\sigma$ must be
$1/2$, indicating that it applies to the non-trivial zeros of the zeta
function. We explain the significance of the non-trivial zeros of the Riemann
zeta function for demonstrating the $SU(2)$ algebra. Now, we examine how the
trivial zeros help us understand the $SU(2)$ algebra in our system.

###  V.2 Trivial Riemann zeros:

It is well known that the Riemann zeta function $\zeta(\sigma+i\omega)$
possesses trivial zeros when $\sigma=-2N$ and $\omega=0$. This can be
understood through the functional relation of the zeta function, which is
given in Eq. (3).

In this case, we define the state

$\displaystyle\psi_{\sigma,m}$ $\displaystyle=\frac{1}{\sqrt{\log
a}}x^{-\sigma+i(\frac{1}{2}-m)\omega}.$ (31)

These functions do not form the orthonormal set under Dirac inner product
unless $\sigma=1/2$, which is the case for non-trivial zeros. For other values
of $\sigma$, we have introduced the modified inner product [21, 15] then it
forms an orthonormal set of functions.

Using $\sigma=-2N$, $k=-2N-i\omega$, $m=1/2$ and using Eq. (31), one can write
Eq. (22) as,

$\displaystyle J_{x}\psi_{-2N,1/2}$
$\displaystyle=\frac{\zeta(1+2N+i\omega)}{2\sqrt{\zeta(-2N-i\omega)\zeta(1+2N+i\omega)}}\psi_{-2N,-1/2},$
$\displaystyle J_{y}\psi_{-2N,1/2}$
$\displaystyle=\frac{\zeta(1+2N+i\omega)}{2i\sqrt{\zeta(-2N-i\omega)\zeta(1+2N+i\omega)}}\psi_{-2N,-1/2},$
$\displaystyle J_{0}\psi_{-2N,1/2}$
$\displaystyle=-\frac{1}{2}\psi_{-2N,1/2}.$ (32)

It leads to

$\displaystyle J^{2}=(J_{x}^{2}+J_{y}^{2}+J_{0}^{2})$
$\displaystyle=\frac{3}{4},$ (33)

which also implies that $j=\frac{1}{2}$ and $m$ varies from $-1/2$ to $1/2$.

We see that to establish the functional-dependent $SU(2)$ algebra, it is
essential to have either non-trivial or trivial zeros of the zeta function.
Without the presence of either of these types of Riemann zeros, our model
would not exhibit the algebraic structure of $SU(2)$. This indicates that the
existence of either type of zero plays a critical role in the formation of the
$SU(2)$ algebra here, which is an appealing outcome.

##  VI Conclusion

We have demonstrated that various symmetries are linked to both the Riemann
zeros and the supersymmetric system underlying it. Supersymmetry naturally
leads to the utilization of the Witten index for the characterization of
ground state energy. Interestingly, it is shown that SUSY is unbroken for
$\text{Re}(s)=\sigma=1/2$. The appearance of PT symmetry and the function-
dependent $SU(2)$ symmetries are deeply connected to the properties of the
Hilbert space, which can be characterized by the Riemann zeros. Our results
show that SUSY remains unbroken only when $\sigma=1/2$, which corresponds to
the non-trivial zeros of the zeta function. We have also demonstrated that the
$PT$ symmetry remains unbroken in two cases: i)
$\text{Re}(s)=\nicefrac{{1}}{{2}}$ and ii) $\text{Im}(s)=0$ (where
$s=\sigma+i\omega)$ while in other cases, the $PT$ symmetry is broken. We find
an underlying $SU(2)$ algebraic structure, where the algebra is realized in
the Hilbert space with suitable constraints. Intriguingly, only the spin-half
representation is realized. The Hilbert space-dependent $SU(2)$ symmetry and
the presence of only spin-half unitary irreducible representation are
indicative of a deeper structure, needing further study.

Our findings provide novel perspectives on the Riemann zeros and their
association with various symmetries. These symmetries shed light on the
fundamental connections between the Riemann hypothesis and physics, and
highlight the potential for new insights and discoveries at the intersection
of these two fields.

###### Acknowledgements.

We acknowledge Prof. Kimball A. Milton for the useful discussions. This work
is financially supported by the DST, Govt. of India under the Women Scientist
A, Ref. No. DST/WOS-A/PM-64/2019. We are thankful to the referees for
providing their valuable feedback.

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##  Appendix A Derivation Details of Equations in Section “SU(2) Symmetery”

Using Eq. (18) and Eq. (20), we get

$\displaystyle[J_{+},J_{-}]$
$\displaystyle=\frac{1}{b(k)}(\Omega^{\dagger}\Omega-x^{-i\omega}\Omega\Omega^{\dagger}x^{i\omega}),$
$\displaystyle{}[J_{0},J_{-}]$
$\displaystyle=\frac{1}{2(b(k))^{3/2}}[\Omega^{\dagger}\Omega
x^{-i\omega}\Omega-x^{-i\omega}\Omega\Omega^{\dagger}\Omega$
$\displaystyle-x^{-i\omega}\Omega\Omega^{\dagger}\Omega+x^{-i\omega}\Omega
x^{-i\omega}\Omega\Omega^{\dagger}x^{i\omega}],$
$\displaystyle{}[J_{0},J_{+}]$
$\displaystyle=\frac{1}{2(b(k))^{3/2}}[\Omega^{\dagger}\Omega\Omega^{\dagger}x^{i\omega}-x^{-i\omega}\Omega\Omega^{\dagger}x^{i\omega}\Omega^{\dagger}x^{i\omega}$
$\displaystyle-\Omega^{\dagger}x^{i\omega}\Omega^{\dagger}\Omega+\Omega^{\dagger}\Omega\Omega^{\dagger}x^{i\omega}],$
(34)

where

$\displaystyle b(k)$ $\displaystyle=\zeta(k)\zeta(1-k).$ (35)

The operator $\Omega$ and its adjoint $\Omega^{\dagger}$, operate on monomials
$x^{-s}$ and provide

$\displaystyle\Omega x^{-s}$ $\displaystyle=\zeta(s)x^{-s},$
$\displaystyle\Omega^{\dagger}x^{-s}$ $\displaystyle=\zeta(1-s)x^{-s}.$ (36)

We use Eq. (36) and calculate the commutation relations as

$\displaystyle[J_{+},J_{-}]x^{-s}$
$\displaystyle=\frac{[b(s)-b(s-i\omega)]}{b(k)}x^{-s},$
$\displaystyle{}[J_{0},J_{-}]x^{-s}$
$\displaystyle=[b(s+i\omega)+b(s-i\omega)-2b(s)]\times$
$\displaystyle\frac{\zeta(s)}{2(b(k))^{3/2}}x^{-s-i\omega},$
$\displaystyle{}[J_{0},J_{+}]x^{-s-i\omega}$
$\displaystyle=[-b(s-i\omega)-b(s+i\omega)+2b(s)]\times$
$\displaystyle\frac{\zeta(1-s)}{2(b(k))^{3/2}}x^{-s}.$ (37)

$J_{x}^{2}$ and $J_{y}^{2}$ act on the monomial $x^{-s}$ as

$\displaystyle J_{x}^{2}x^{-s}=$ $\displaystyle\frac{\zeta(1-s+i\omega)}{4}$
$\displaystyle\times$
$\displaystyle\left[\frac{\zeta(1-s+2i\omega)x^{-s+2i\omega}+\zeta(s-i\omega)x^{-s}}{b(k)}\right]$
$\displaystyle+$
$\displaystyle\frac{\zeta(s)}{4}\left[\frac{\zeta(1-s)x^{-s}+\zeta(s+i\omega)x^{-s-2i\omega}}{b(k)}\right],$
$\displaystyle J_{y}^{2}x^{-s}=$ $\displaystyle\frac{-\zeta(1-s+i\omega)}{4}$
$\displaystyle\times$
$\displaystyle\left[\frac{\zeta(1-s+2i\omega)x^{-s+2i\omega}-\zeta(s-i\omega)x^{-s}}{b(k)}\right]$
$\displaystyle+$
$\displaystyle\frac{\zeta(s)}{4}\left[\frac{\zeta(1-s)x^{-s}-\zeta(s+i\omega)x^{-s-2i\omega})}{b(k)}\right],$
(38)

where $s=\sigma+i\omega$.


# Elementary particle non-perturbative mass generation
A step towards a beyond-the-Standard-Model model

Giancarlo Rossi  Dipartimento di Fisica, Università di Roma Tor Vergata, Via
della Ricerca Scientifica, 00133 Roma, Italy
INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma,
Italy
Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico
Fermi, Via Panisperna 89a, 00184 Roma, Italy<EMAIL_ADDRESS>
###### Abstract

We show that a recently discovered non-perturbative field-theoretical
mechanism giving mass to elementary fermions, is also capable of generating a
mass for the electro-weak bosons and can thus be used as a viable alternative
to the Higgs scenario. A detailed analysis of this remarkable feature shows
that the non-perturbatively generated fermion and $W$ masses have the
parametric form $m_{f}\sim C_{f}(\alpha)\Lambda_{RGI}$ and $M_{W}\sim
g_{w}c_{w}(\alpha)\Lambda_{RGI}$, respectively, where the coefficients
$C_{f}(\alpha)$ and $c_{w}(\alpha)$ are functions of the gauge couplings,
$g_{w}$ is the weak coupling and $\Lambda_{\rm RGI}$ is the RGI scale of the
theory. In view of these expressions, we see that to match the experimental
values of the top quark and $W$ masses, we are led to conjecture the existence
of a yet unobserved sector of massive fermions (that we denote Tera-fermions)
subjected, besides ordinary Standard Model interactions, to some kind of
super-strong gauge interactions (Tera-interactions), so that the full theory
(SM plus Tera-particles) will have an RGI scale $\Lambda_{\rm
RGI}\equiv\Lambda_{T}$ in the TeV region. This approach offers a solution of
the mass naturalness problem (there is no fundamental Higgs), an understanding
of the fermion mass hierarchy and a physical interpretation of the electro-
weak scale as a fraction of $\Lambda_{T}$.

###### keywords:

BSM models, mass naturalness, fermion mass hierarchy.

##  1 Introduction

We have introduced in [1] and numerically explored in [2] a 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. Despite the
fact that both terms break chiral invariance, it was shown in [1] that there
exists a critical value of the Yukawa coupling where chiral symmetry is
recovered, up to small effects vanishing as the UV cut-off is removed. The
interesting observation is that in the Nambu–Goldstone (NG) phase of this
critical theory non-perturbative (NP) O($\Lambda_{\rm RGI}$) fermion masses
get dynamically generated.

They are a consequence of a sort of “interference” between UV chiral breaking
terms and non-perturbative (NP) effects, coming from the spontaneous breaking
of the (recovered) chiral symmetry occurring in the NG phase of the theory,
that is standardly induced by the strong interaction dynamics. A detailed
analysis of this remarkable field-theoretical feature shows that NP-ly
generated fermion masses have the parametric form

$m_{f}\sim C_{f}(\alpha)\Lambda_{\rm RGI}\,,$ (1)

where $\Lambda_{\rm RGI}$ is the RGI scale of the theory and $C_{f}(\alpha)$
is a function of the gauge coupling constants $\\{\alpha\\}$ of the theory. If
we take the irrelevant chiral breaking Wilson-like term to be a $d=6$
operator, one finds to lowest loop order
$C_{f}(\alpha)={\mbox{O}}(\alpha_{f}^{2})$, where $\alpha_{f}$ is the coupling
constant of the strongest among the gauge interactions the particle is
subjected to.

An exact symmetry protects elementary particle masses against quantum power
divergencies, at variance with what happens in Wilson lattice QCD (WLQCD).
Masses are “small”, i.e. ${\mbox{O}}(\Lambda_{\rm RGI})$, as the massless
theory enjoys an enhanced (chiral) symmetry, in line with the ’t Hooft
naturalness notion [3].

A far reaching consequence of the formula (1), if applied to the top quark, is
that since $m_{top}={\mbox{O}}(\Lambda_{\rm RGI})$, to match its experimental
values a new sector of super-strongly interacting particles, gauge-invariantly
coupled to standard matter, needs to exist so that the complete theory,
including the new sector and Standard Model (SM) particles, will have an RGI
scale $\Lambda_{\rm RGI}\equiv\Lambda_{T}\gg\Lambda_{QCD}$ and of the order of
a few TeV’s. We immediately notice that the reason for assuming the existence
of a super-strongly interacting sector here is very different from the reason
why Technicolor was introduced [4, 5]. Technicolor was invoked to give mass to
the EW bosons in the first place, while in the present approach super-strong
interactions are introduced to give the right order of magnitude to the top
quark and $W$ masses. Anyway to avoid confusion and following Glashow [6], we
will refer to this new set of particles as Tera-particles.

The model can be naturally extended to incorporate EW interactions and
leptons. EW bosons as well as leptons will acquire NP masses O($\Lambda_{T}$)
via the same mechanism that leads to eq. (1) 111With the SM hypercharge
assignment, in our model neutrinos are massless because $\nu_{R}$ is sterile.
We will not discuss here how to give mass to neutrinos in the present
framework., but with coefficient functions that scale like powers of the EW
gauge coupling, $g_{w}$. For instance, to the leading loop order one gets for
the $W$ mass (see sect. 3 and refs. [7, 8])

$M_{W}\sim\sqrt{\alpha_{w}}c_{w}(\alpha)\Lambda_{\rm RGI}\,,\quad
c_{w}(\alpha)={\mbox{O}}(\alpha)\,,$ (2)

where $\alpha$ in eq. (2) is a short-hand for the set of gauge couplings
$[\alpha_{s},\alpha_{T}]$ with $\alpha_{s}$ and $\alpha_{T}$ the strong and
Tera-strong gauge couplings, respectively.

Eqs. (1) and (2) are somewhat similar to the expression of the Higgs-masses of
fermions and $W$’s with, however, two fundamental differences. The first is
that the scale of the mass is not the vev of the Higgs field, but a dynamical
scale related to a new interaction. The second is that the modulation of the
Yukawa couplings that in the SM is introduced by hand to accomodate the values
of fermion masses, here, as we said before, is controlled to leading order by
the magnitude of the gauge coupling of the strongest among the interactions
the particle feels.

We conclude from this analysis that the NP scenario for mass generation we are
advocating can be considered as a valid alternative to the Higgs mechanism,
with the extra advantage that we will not have to deal with power divergencies
in the Higgs mass as there is no Higgs around. A further conceptual bonus is
that, in view of the above mass formulae, we have a natural interpretation of
the magnitude of the EW scale, as (a fraction of) the dynamical physical
parameter, $\Lambda_{T}$.

There is a number of further interesting features of the approach we are
describing that are worth mentioning, but which we are not going to develop in
this contribution for lack of space.

First of all, the dependence of the NP fermion masses upon the gauge couplings
(to leading order one finds $C(\alpha_{f})\\!=\\!{\mbox{O}}(\alpha_{f}^{2})$
for a $d=6$ Wilson-like operator, see eq. (7) below) may offer a hint to
understand the fermion mass hierarchy $m_{\tau}\\!\ll\\!m_{t}\\!\ll\\!m_{T}$,
as due to the ranking among weak, strong and Tera-strong gauge couplings,
$\alpha_{Y}\\!\ll\\!\alpha_{s}\\!\ll\\!\alpha_{T}$. We expect the gauge
coupling dependence of the NP fermion mass estimate (1) to be more compelling
for the heaviest of the SM fermion families which we assume the following
analysis refers to.

Secondly, lacking the need for a Higgs boson, we propose to interpret the 125
GeV resonance, recently identified at LHC, as a $W^{+}W^{-}/ZZ$ composite
state, bound by exchanges of Tera-particles. Since this bound state is light
on the $\Lambda_{T}$ scale, it should be incorporated in the low energy
effective Lagrangian (LEEL) that one gets by integrating out the “heavy” Tera-
degrees of freedom. As a result, at (momenta)${}^{2}\ll\Lambda_{T}^{2}$ the
effective Lagrangian is seen to resemble very much to the Lagrangian of the SM
(see sect. 4 and ref. [9]).

Finally it was shown in ref. [10] that with a reasonable choice of the
elementary particle content, a theory extending the SM with the inclusion of
the new super-strong sector leads to gauge coupling unification at a $\sim
10^{18}$ GeV scale. This higher than usual unification scale, irrespective of
the underlying GUT theory generically allowing for proton decay, is likely to
yield a proton life-time comfortably larger than the present limit $\tau_{\rm
prot}>1.7\times 10^{34}$ years.

It goes without saying that the issue of how eq. (1) could be extended to deal
with lighter families (and to cope with weak isospin splitting) is outside the
scope of this talk and it is still a matter of investigation.

##  2 A simple toy-model

The simplest model enjoying the NP mass generation mechanism we outlined
before is described by a Lagrangian where an SU(2) fermion doublet, subjected
to non-abelian gauge interactions (of the QCD type), is coupled to a complex
scalar via $d=4$ Yukawa and “irrelevant” $d=6$ Wilson-like chiral breaking
terms. The Lagrangian of this toy-model reads

$\displaystyle{\cal L}_{\rm toy}(q,A;\Phi)=$ $\displaystyle\quad={\cal
L}_{kin}(q,A;\Phi)\\!+\\!{\cal V}(\Phi)\\!+\\!{\cal
L}_{Yuk}(q;\Phi)\\!+\\!{\cal L}_{Wil}(q,A;\Phi)$ (3)
$\displaystyle\bullet\,{\cal L}_{kin}(q,A;\Phi)=\frac{1}{4}(F^{A}\cdot
F^{A})+\bar{q}_{L}\,{\mbox{$\not\\!\\!\\!\cal
D$}}^{A}q_{L}+\bar{q}_{R}\,{\mbox{$\not\\!\\!\\!\cal D$}}^{A}q_{R}+$
$\displaystyle\quad+\frac{1}{2}{\hbox{Tr}\,}\big{[}\partial_{\mu}\Phi^{\dagger}\partial_{\mu}\Phi\big{]}$
(4) $\displaystyle\bullet\,{\cal
V}(\Phi)=\frac{\mu_{0}^{2}}{2}{\hbox{Tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}+\frac{\lambda_{0}}{4}\big{(}{\hbox{Tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}\big{)}^{2}$
(5) $\displaystyle\bullet\,{\cal L}_{Yuk}(q;\Phi)=\eta\,\big{(}\bar{q}_{L}\Phi
q_{R}+\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}$ (6) $\displaystyle\bullet\,{\cal
L}_{Wil}(q,A;\Phi)\\!=\\!\frac{b^{2}}{2}\rho\,\big{(}\bar{q}_{L}\overleftarrow{\cal
D}^{A}_{\mu}\Phi{\cal D}^{A}_{\mu}q_{R}\\!+\\!\bar{q}_{R}\overleftarrow{\cal
D}^{A}_{\mu}\Phi^{\dagger}{\cal D}^{A}_{\mu}q_{L}\big{)}\,,$ (7)

where
$\Phi\\!=\\!\varphi_{0}1\\!\\!1\\!+\\!i\varphi_{j}\tau^{j}\\!=\\![-i\tau_{2}\varphi^{\star}|\,\varphi]$
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 the color gauge, SU($N_{c}$),
$b^{-1}\\!\sim\\!\Lambda_{UV}$ is the UV cutoff, $\eta$ is the Yukawa
coupling, $\rho$ is to keep track of ${\cal L}_{Wil}$ and ${\cal D}_{\mu}^{A}$
the covariant derivative. The Lagrangian (3) is power counting renormalizable
(like WLQCD is, despite the presence of the $d\\!=\\!5$ Wilson term).

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

$\displaystyle\chi_{L}\\!\times\\!\chi_{R}\\!=\\![\tilde{\chi}_{L}\times(\Phi\to\Omega_{L}\Phi)]\times[\tilde{\chi}_{R}\times(\Phi\to\Phi\Omega_{R}^{\dagger})]\,,$
(8)
$\displaystyle\tilde{\chi}_{L/R}:q_{L/R}\rightarrow\Omega_{L/R}q_{L/R}\,,\quad\bar{q}_{L/R}\rightarrow\bar{q}_{L/R}\Omega_{L/R}^{\dagger}\,.$
(9)

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. In any case no
power divergent fermion mass can be generated by quantum corrections as the
mass operator $(\bar{q}_{L}q_{R}+\bar{q}_{R}q_{L})$ is not invariant under
$\chi_{L}\times\chi_{R}$.

The operators ${\cal L}_{Wil}$ and ${\cal L}_{Yuk}$ break
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ and mix. Thus for generic values of
$\eta$, ${\cal L}_{\rm toy}$ is not invariant under the fermionic chiral
transformations $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$. However, one can
show [11, 1] that chiral invariance can be recovered (up to vanishingly small
O($b^{2}$) terms) at a critical value $\eta=\eta_{cr}$, where the Wilson-like
term and the Yukawa terms “compensate”, much like chiral symmetry is recovered
(up to O($a$) cutoff effects) in WLQCD by tuning the bare quark mass to a
critical value, $m_{cr}$, where the Wilson and the mass term “compensate”. The
condition determining $\eta_{cr}$ corresponds to the vanishing of the
effective Yukawa term in the Quantum Effective Lagrangian (QEL) of the theory
222By Quantum Effective Lagrangian we mean the generating functional of the
1PI vertices of the theory..

In the Wigner phase (where $\langle|\Phi|^{2}\rangle=0$) this condition
implies the cancellation diagrammatically represented at the lowest order in
the top panel of fig. 1 with complete decoupling of fermions and scalars [1].

Figure 1: Top: the cancellation of the effective Yukawa term implied by the
condition determining $\eta_{cr}$ in the Wigner phase. The disc represents the
Yukawa vertex, the box the insertion of the Wilson-like vertex. Bottom: the
mechanism behind the cancellation of the Higgs-like mass of the quark in the
NG phase occurring at the $\eta=\eta_{cr}$.

In the NG phase (where
$\langle|\Phi|^{2}\rangle\\!=\\!v^{2}\\!=\\!\mu_{0}^{2}/\lambda_{0}$) the same
condition implies that the Higgs-like mass of the fermion is cancelled out, as
schematically shown in the bottom panel of fig 1. It is a peculiar feature of
the model (3) that the (quadratic) divergencies in the loop integrals in fig.
1 are exactly compensated by the $b^{2}$ factors coming from the insertion of
the Wilson-like vertex. In the following we will encounter other instances of
this UV vs. IR compensation.

###  2.1 NP mass generation

Now the key observation is that at $\eta=\eta_{cr}$ where in the NG phase the
fermion Higgs-mass is cancelled, the quark acquires a non-vanishing NP mass
via a UV vs. IR compensation mechanism, reminiscent of the one that makes
finite the 1-loop diagrams of fig. 1.

Because of this kind of subtle UV vs. IR compensations, it is crucial to
analyze and determine the structure of the formally O($b^{2}$) terms of the
regularized theory. This can be properly done by making recourse to the
Symanzik expansion technique [12].

This analysis was carried out in ref. [1] where it was shown that, in order to
saturate the $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ Ward–Takahashi
Identities in the presence of the spontaneous breaking of the (restored)
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ symmetry, one must include in the
Symanzik expansion also formally O($b^{2}$) NP-ly generated operators of the
form

$\displaystyle O_{6,\bar{q}q}\propto
b^{2}\Lambda_{s}\alpha_{s}|\Phi|\,\Big{[}\bar{q}\,\,{\mbox{$\not\\!\\!\\!\cal
D$}}^{A}q\Big{]}\,,$ (10) $\displaystyle O_{6,FF}\propto
b^{2}\Lambda_{s}\alpha_{s}|\Phi|\,\Big{[}F^{A}\\!\cdot\\!F^{A}\Big{]}\,.$ (11)

The expression of these operators is completely fixed by symmetries (in
particular $\chi_{L}\times\chi_{R}$) and dimensional considerations. The
presence of the $\Lambda_{s}$ factor signals their NP origin. The existence of
the operators (10) and (11) can be effectively taken into account by including
in the formalism the new diagrams generated by the augmented Lagrangian

$\displaystyle{\cal L}_{\rm toy}\to{\cal L}_{\rm toy}+\Delta{\cal
L}_{NP}^{Sym}\,,$ $\displaystyle\Delta{\cal
L}_{NP}^{Sym}=b^{2}\Lambda_{s}\alpha_{s}|\Phi|\,\Big{[}c_{FF}F^{A}\\!\cdot\\!F^{A}+c_{\bar{q}q}\bar{q}\,\,{\mbox{$\not\\!\\!\\!\cal
D$}}^{A}q\Big{]}+\ldots$ (12)

The remarkable fact is that new diagrams are generated that contribute, among
other correlators, to the quark self-energy. A couple of them are drawn in
fig. 2. A straightforward calculation gives for the amputated zero momentum
diagram in the right panel of fig. 2

Figure 2: Two lowest loop order quark self-energy diagrams. The blobs
represent insertions of $\Delta{\cal L}_{NP}^{Sym}$.

$\displaystyle{{m_{q}^{NP}}}\\!\propto\\!{{\alpha_{s}^{2}}}\\!\int^{1/b}\\!\frac{d^{4}k}{k^{2}}\frac{\gamma_{\mu}k_{\mu}}{k^{2}}\int^{1/b}\\!\\!\\!\frac{d^{4}\ell}{\ell^{2}+m^{2}_{\zeta_{0}}}\frac{\gamma_{\nu}(k+\ell)_{\nu}}{(k+\ell)^{2}}\cdot$
$\displaystyle\cdot\,b^{2}\gamma_{\rho}(k+\ell)_{\rho}\,b^{2}{{\Lambda_{s}}}\gamma_{\lambda}(2k+\ell)_{\lambda}\sim{{\alpha_{s}^{2}\Lambda_{s}}}\,.$
(13)

Here again we get a finite result, owing to an exact compensation between the
UV power divergencies of the 2-loop integrals and the IR behaviour determined
by the expression of the Wilson-like vertex (7) and the physics of the
spontaneous breaking of the chiral $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$
symmetry, in turn encoded in the form of the Symanzik operators (10) and (11).
A similar compensation occurs in the calculation of the diagram in the left
panel of fig. 2.

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

The resulting non-vanishing quark mass, the occurrence of which was checked in
the explicit lattice simulations of the model (3) in ref. [2], can be
incorporated in the QEL, $\Gamma^{NG}$, of the theory, upon introducing the
non-analytic field ${U}$ in the polar decomposition

$\Phi=(v+\zeta_{0})U\,,\quad U=\exp[i\vec{\tau}\vec{\zeta}/v]\,.$ (14)

Simply on the basis of symmetries and observing that $U$ transforms like
$\Phi$, so that new $\chi_{L}\times\chi_{R}$ invariant operators can be
constructed [1, 9], one obtains

$\displaystyle\Gamma_{d=4}^{NG}\\!=\\!\Gamma_{d=4}^{Wig}\Big{|}_{\hat{\mu}_{\Phi}^{2}<0}+\frac{1}{4}(F^{A}\\!\cdot\\!F^{A})\\!+\\!\bar{q}_{L}\,{\mbox{$\not\\!\\!\\!\cal
D$}}^{A}q_{L}\\!+\\!\bar{q}_{R}\,{\mbox{$\not\\!\\!\\!\cal D$}}^{A}q_{R}+$
$\displaystyle+{{c_{q}\Lambda_{s}[\bar{q}_{L}Uq_{R}\\!+\\!\bar{q}_{R}U^{\dagger}q_{L}]}}\\!+\\!\frac{c^{2}\Lambda_{s}^{2}}{2}\hbox{Tr}\,[\partial_{\mu}U^{\dagger}\partial_{\mu}U]\,.$
(15)

Naturally, of special interest is the fourth term in the r.h.s. of eq. (15)
because, upon expanding $U=\\!1\\!\\!1\\!+\vec{\tau}\,\vec{\zeta}/v+\ldots$,
it gives rise to a mass for the fermion plus a wealth of NG boson
interactions.

##  3 Introducing weak and Tera-interactions

To proceed to the construction of a possible, realistic beyond-the-SM-model we
clearly need to introduce EW 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, in order for the whole theory to have an
RGI scale, $\Lambda_{T}$, much larger than $\Lambda_{QCD}$ and of the order of
a few TeVs. Only in this way there is the chance that eq. (1) can yield the
correct order of magnitude of the top quark mass. As proved in [13], the
extended Lagrangian is obtained by doubling the structure of quarks in order
to encompass Tera-particles ($Q$=Tera-quarks and $G$=Tera-gluons) and gauging
the exact $\chi_{L}$ symmetry to introduce weak interactions. The whole
Lagrangian then reads

$\displaystyle{\cal L}(q,Q;\Phi;A,G,W)={\cal L}_{kin}(q,Q;\Phi;A,G,W)+{\cal
V}(\Phi)+$ $\displaystyle\quad+{\cal L}_{Yuk}(q,Q;\Phi)+{\cal
L}_{Wil}(q,Q;\Phi;A,G,W)$ (16) $\displaystyle\bullet\,{\cal
L}_{kin}=\frac{1}{4}\Big{(}F^{A}\cdot F^{A}+F^{G}\cdot F^{G}+F^{W}\cdot
F^{W}\Big{)}+$ $\displaystyle\quad+\Big{[}\bar{q}_{L}\,/\\!\\!\\!\\!{\cal
D}^{AW}q_{L}\\!+\\!\bar{q}_{R}\,/\\!\\!\\!\\!{\cal
D}^{A}q_{R}\Big{]}\\!+\\!\Big{[}\bar{Q}_{L}\,/\\!\\!\\!\\!{\cal
D}^{AGW}Q_{L}\\!+\\!\bar{Q}_{R}\,/\\!\\!\\!\\!{\cal D}^{AG}Q_{R}\Big{]}+$
$\displaystyle\quad+\frac{k_{b}}{2}{\hbox{Tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}\Phi)^{\dagger}{\cal D}^{W}_{\mu}\Phi\big{]}$ (17)
$\displaystyle\bullet\,{\cal
V}=\frac{\mu_{0}^{2}}{2}k_{b}{\hbox{Tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}+\frac{\lambda_{0}}{4}\big{(}k_{b}{\hbox{Tr}\,}\big{[}\Phi^{\dagger}\Phi\big{]}\big{)}^{2}$
(18) $\displaystyle\bullet\,{\cal
L}_{Yuk}=\eta_{q}\,\big{(}\bar{q}_{L}\Phi\,q_{R}\\!+\\!\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}\\!+\\!\eta_{Q}\,\big{(}\bar{Q}_{L}\Phi\,Q_{R}\\!+\\!\bar{Q}_{R}\Phi^{\dagger}Q_{L}\big{)}$
(19) $\displaystyle\bullet\,{\cal
L}_{Wil}\\!=\\!\frac{b^{2}}{2}{\rho_{q}}\big{(}\bar{q}_{L}{\overleftarrow{\cal
D}}\,^{AW}_{\mu}\Phi{\cal
D}^{A}_{\mu}q_{R}\\!+\\!\bar{q}_{R}\overleftarrow{\cal
D}\,^{A}_{\mu}\Phi^{\dagger}{\cal D}^{AW}_{\mu}q_{L}\big{)}\\!+\\!$
$\displaystyle\quad+\frac{b^{2}}{2}{\rho_{Q}}\,\big{(}\bar{Q}_{L}{\overleftarrow{\cal
D}}\,^{AGW}_{\mu}\Phi{\cal D}^{AG}_{\mu}Q_{R}+\bar{Q}_{R}\overleftarrow{\cal
D}\,^{AG}_{\mu}\Phi^{\dagger}{\cal D}^{AGW}_{\mu}Q_{L}\big{)}$ (20)

with obvious notations for the covariant derivatives. Besides the Yukawa (eq.
(19)) and the Wilson-like (eq. (20)) operators, now also the kinetic term of
the scalar (last term in eq. (17)) breaks
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ and mixes with ${\cal L}_{Yuk}$ and
${\cal L}_{Wil}$. Thus to get to the critical theory (invariant under
$\tilde{\chi}_{L}\times\tilde{\chi}_{R}$), on top of $\eta_{q}$ and
$\eta_{Q}$, a further parameter, $k_{b}$, needs to be introduced and
appropriately tuned. The conditions determining the critical theory correspond
to have vanishing effective Yukawa interactions and vanishing scalar kinetic
term in the QEL. Notice that the coefficient $k_{b}$ appears also in the
expression of the scalar potential (18). The reason for this is that in this
way the (bare) vev2 (in the NG phase) and the quartic coupling of the
canonically normalized scalar field will have the standard definitions,
$v^{2}=|\mu^{2}_{0}|/\lambda_{0}$ and $\lambda_{0}$.

The tuning conditions that determine $\eta_{q\,cr}$, $\eta_{Q\,cr}$ and
$k_{b\,cr}$ in the Wigner phase imply the cancellations schematically
illustrated in the fig. 3. To the lowest loop order one finds

$\displaystyle\eta_{q\,cr}^{(1-loop)}\\!=\\!\rho_{q}\,\eta_{1q}\alpha_{s}\,,\qquad\eta_{Q\,cr}^{(1-loop)}\\!=\\!\rho_{Q}\,\eta_{1Q}\alpha_{T}\,,$
(21) $\displaystyle\hskip
36.98866ptk_{b\,cr}^{(1-loop)}\\!=\\![\rho_{q}^{2}N_{c}+\rho_{Q}^{2}N_{c}N_{T}]k_{1}\,,$
(22)

with $\eta_{1q}$, $\eta_{1Q}$ and $k_{1}$ computable coefficients and
SU($N_{T}$) the Tera-gauge group. As before, UV loop divergencies are exactly
compensated by the IR behaviour of the inserted Wilson-like vertices.

Figure 3: From top to bottom, the cancellation implied by the tuning
conditions determining $\eta_{q\,cr}$, $\eta_{Q\,cr}$ and $k_{b\,cr}$ at the
lowest order in the Wigner phase. Boxes represent the insertion of the quark
and Tera-quark Wilson-like vertices, respectively, the disc the insertion of
the scalar kinetic term. Double lines represent Tera-particles. The rest of
the notations is as in fig. 1.

In the NG phase the tuning conditions entail the cancellation of the Higgs-
like mass of quarks, Tera-quarks and $W$’s, as schematically represented in
figs. 4.

Figure 4: From top to bottom, the cancellation mechanism of the Higgs-like
mass of quark, Tera-quark and $W$ occurring in the NG phase of the critical
theory.

Extending to the present case the Symanzik analysis of the formally O($b^{2}$)
operators necessary to describe the NP effects related to the spontaneous
breaking of the (recovered) $\tilde{\chi}_{L}\times\tilde{\chi}_{R}$ symmetry
in the regularized theory, one finds the following set of operators

$\displaystyle
O_{6,\bar{Q}Q}^{s}=b^{2}\alpha_{s}\,\rho_{Q}\Lambda_{T}|\Phi|\Big{[}\bar{Q}_{L}\,{\mbox{$\not\\!\\!\\!\cal
D$}}^{AGW}Q_{L}+\bar{Q}_{R}\,{\mbox{$\not\\!\\!\\!\cal D$}}^{AG}Q_{R}\Big{]}$
(23) $\displaystyle
O_{6,\bar{Q}Q}^{T}=b^{2}\alpha_{T}\,\rho_{Q}\Lambda_{T}|\Phi|\Big{[}\bar{Q}_{L}\,{\mbox{$\not\\!\\!\\!\cal
D$}}^{AGW}Q_{L}+\bar{Q}_{R}\,{\mbox{$\not\\!\\!\\!\cal D$}}^{AG}Q_{R}\Big{]}$
(24) $\displaystyle
O_{6,AA}=b^{2}\alpha_{s}\,\rho_{Q}\Lambda_{T}|\Phi|F^{A}\\!\cdot\\!F^{A}$ (25)
$\displaystyle
O_{6,GG}=b^{2}\alpha_{T}\,\rho_{Q}\Lambda_{T}|\Phi|F^{G}\\!\cdot\\!F^{G}$ (26)
$\displaystyle
O_{6,WW}=b^{2}\alpha_{w}\,\rho_{Q}\Lambda_{T}|\Phi|F^{W}\\!\cdot\\!F^{W}$ (27)

Figure 5: Examples of self-energy diagrams giving NP masses to quarks, Tera-
quarks and $W$. The blobs represent the insertion of the NP Symanzik operators
(25), (26) and (24) for quark and Tera-quark, Tera-quark and $W$,
respectively. The rest of the notations is as in figs. 3 and 4.

NP masses emerge from the kind of diagrams shown in fig. 5. They are all
finite owing to the by now well-known UV-IR compensation and all of
O($\Lambda_{T}$) times gauge coupling dependent coefficient functions (see
next section).

###  3.1 The critical QEL in the NG phase

With the same line of arguments we used before to derive eq. (15), we get for
the $d=4$ piece of the QEL of the critical theory in the NG phase the
expression

$\displaystyle{\Gamma}_{4\,cr}^{NG}(q,Q;\Phi;A,G,W)\\!=\\!\frac{1}{4}\Big{(}F^{A}\cdot
F^{A}\\!+\\!F^{G}\cdot F^{G}\\!+\\!F^{W}\cdot F^{W}\Big{)}+$ (28)
$\displaystyle\quad+\Big{[}\bar{q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{WA}q_{L}+\bar{q}_{R}\,\,/\\!\\!\\!\\!{\cal
D}^{A}q_{R}\Big{]}+C_{q}\Lambda_{T}\,\Big{(}\bar{q}_{L}Uq_{R}+\bar{q}_{R}U^{\dagger}q_{L}\Big{)}+$
$\displaystyle\quad+\Big{[}\bar{Q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{WAG}Q_{L}\\!+\\!\bar{Q}_{R}\,\,/\\!\\!\\!\\!{\cal
D}^{AG}Q_{R}\Big{]}+C_{Q}\Lambda_{T}\,\Big{(}\bar{Q}_{L}UQ_{R}\\!+\\!\bar{Q}_{R}U^{\dagger}Q_{L}\Big{)}+$
$\displaystyle\quad+\frac{1}{2}c_{w}^{2}\Lambda_{T}^{2}{\hbox{Tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}U)^{\dagger}{\cal D}^{W}_{\mu}U\big{]}\,.$

From a detailed analysis of the diagrams in fig. 5 one can read-off the
parametric dependence of the mass of quarks, Tera-quarks and $W$s to the
leading loop order. One finds

$\displaystyle
m_{q}^{NP}\\!=\\!C_{q}\,\Lambda_{T}\,,\,\,\,\,\,C_{q}\\!=\\!{\mbox{O}}(\alpha_{s}^{2})$
(29) $\displaystyle
m_{Q}^{NP}\\!=\\!C_{Q}\,\Lambda_{T}\,,\,\,\,\,C_{Q}\\!=\\!{\mbox{O}}(\alpha_{T}^{2},\ldots)$
(30) $\displaystyle
M_{W}^{NP}\\!=\\!C_{w}\,\Lambda_{T}\,,\,\,\,\,C_{w}\\!=\\!\\!\sqrt{\alpha_{w}}\,c_{w}\,,\,\,c_{w}=k_{w}{\mbox{O}}(\alpha_{T},\ldots)\,.$
(31)

##  4 Integrating out Tera-particles and the SM

We show in this section that, upon integrating out the (heavy) Tera-dof’s in
the critical model (16), the resulting LEEL closely resembles the SM
Lagrangian. The argument rests on the conjecture that the 125 GeV resonance
detected at LHC is a $W^{+}W^{-}/ZZ$ composite state bound by Tera-particle
exchanges. This state, which we shall denote by $h$, is a singlet under all
the symmetries of the theory. Since its mass is $\ll\Lambda_{T}$, it must be
included in the LEEL, valid for (momenta)${}^{2}\ll\Lambda_{T}^{2}$. In these
kinematical conditions the most general Lagrangian invariant under the
symmetries of the critical model (16), in particular under the
$\chi_{L}\times\chi_{R}$ transformations, and including $h$, takes the form

$\displaystyle{\cal
L}_{LEEL}^{NG}=\frac{1}{4}\Big{(}F^{A}\\!\cdot\\!F^{A}+F^{W}\\!\cdot\\!F^{W}\Big{)}+\big{(}\bar{q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{AW}q_{L}+\bar{q}_{R}\,\,/\\!\\!\\!\\!{\cal D}^{A}q_{R}\big{)}+$
$\displaystyle+(y_{q}h+k_{q}k_{v})\,\big{(}\bar{q}_{L}Uq_{R}+\bar{q}_{R}U^{\dagger}q_{L}\big{)}+\frac{1}{2}\partial_{\mu}h\partial_{\mu}h+$
$\displaystyle+\\!\frac{1}{2}(k_{v}^{2}\\!+\\!2k_{v}k_{1}h\\!+\\!k_{2}h^{2}){\hbox{Tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}U)^{\dagger}{\cal D}^{W}_{\mu}U\big{]}\\!+\\!\widetilde{\cal
V}(h)\\!+\\!\ldots\,,$ (32)

where dots represent $\tilde{\chi}_{L}\\!\times\\!\tilde{\chi}_{R}$ violating
operators of dimension $d\\!>\\!4$. The scalar potential $\widetilde{\cal
V}(h)$ comprises the cubic and quartic self-interactions of the $h$ field, and
includes the $h$ mass term, $m_{h}^{2}h^{2}/2$. The coefficients
$k_{v},k_{1},k_{2},y_{q},k_{q}$ and the $\widetilde{\cal V}$-couplings are
parameters that need to be fixed by matching onto the underlying
(renormalizable and unitary) fundamental critical theory (16).

Tree-level matching of ${\cal L}_{LEEL}^{NG}$ (eq. (32)) with the QEL
$\Gamma^{NG}_{4\,cr}$ (eq. (28)) requires the identifications

$\displaystyle k_{q}k_{v}=C_{q}\Lambda_{T}=m_{q}^{NP}\,\quad
g_{w}k_{v}=g_{w}\,c_{w}\Lambda_{T}=M_{W}^{NP}\,.$ (33)

Tree-level unitarity implies the constraints

$y_{q}=k_{q}\,,\qquad k_{1}=k_{2}=1\,.$ (34)

These relations are expected to hold up to small loop effects controlled by
$g_{w}$ and $y_{q}$. Neglecting these corrections, one recognizes that
precisely the combination

$\Phi\equiv(k_{v}+h)U$ (35)

enters in the $d\leq 4$ part of ${\cal L}_{LEEL}^{NG}$ with the exception of
the scalar potential $\widetilde{\cal V}(h)$. Eq. (32) can then be rewritten
in the suggestive form

$\displaystyle{\cal
L}_{4,LEEL}^{NG}\\!=\\!\frac{1}{4}\Big{(}F^{A}\\!\cdot\\!F^{A}\\!+\\!F^{W}\\!\cdot\\!F^{W}\Big{)}\\!+\\!\big{(}\bar{q}_{L}\,\,/\\!\\!\\!\\!{\cal
D}^{AW}q_{L}\\!+\\!\bar{q}_{R}\,\,/\\!\\!\\!\\!{\cal D}^{A}q_{R}\big{)}\\!+$
$\displaystyle+y_{q}\,\big{(}\bar{q}_{L}\Phi
q_{R}\\!+\\!\bar{q}_{R}\Phi^{\dagger}q_{L}\big{)}\\!+\\!\frac{1}{2}{\hbox{Tr}\,}\big{[}({\cal
D}\,^{W}_{\mu}\Phi)^{\dagger}{\cal
D}^{W}_{\mu}\Phi\big{]}\\!+\\!\widetilde{\cal V}(h)\,.$ (36)

We see that ${\cal L}_{4,LEEL}^{NG}$ looks very much like the SM Lagrangian
with, however, an important difference. 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.

##  5 Conclusions and Outlook

We have shown that, as an alternative to the Higgs mechanism, elementary
particle masses can be NP-ly generated in a strongly interacting theory where
chiral symmetry, broken at the UV cutoff level by irrelevant Wilson-like
terms, is recovered at low energy owing to the tuning of certain Lagrangian
parameters. Since an exact symmetry prevents the existence of standard mass
terms of the kind $m\bar{\psi}\psi$ and in the critical theory there is no
Higgs field around, the mechanism we are advocating allows a neat solution of
the mass naturalness problem. In fact, no power divergencies can affect
physical masses and masses are “small”, i.e. ${\mbox{O}}(\Lambda_{\rm RGI})$,
as the massless theory enjoys an enhanced (chiral) symmetry, consistently with
the ’t Hooft idea of naturalness [3].

We have seen that to cope with the magnitude of the $W$ and top mass, a super-
interacting sector, gauge invariantly coupled to standard matter, must exists
leading to a theory with an RGI scale $\Lambda_{\rm RGI}=\Lambda_{T}$ of the
order of a few TeVs. The simplest model discussed in ref. [1] enjoying the NP
mass generation mechanism can be straightforwardly extended to incorporate
weak interactions and the new Tera-degrees of freedom. In a similar way one
could introduce leptons and the hypercharge interaction, an issue that we
defer to a forthcoming investigation.

## Acknowledgements

We wish to thank R. Frezzotti for infinitely many discussions on the issues
presented in this talk. Exchanges of ideas with M. Garofalo are also
acknowledged.

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