# Coherent electroproduction of vector mesons on spinless targets

S.I. Manaenkov<EMAIL_ADDRESS>B.P. Konstantinov Petersburg
Nuclear Physics Institute of Research Center ”Kurchatov Institute”,
Gatchina 188300, Leningrad region, Russia

###### Abstract

The amplitude ratios of vector-meson production by heavy photons on spinless
targets are shown to can be explicitly expressed in terms of the spin-density-
matrix elements (SDMEs) only if the lepton beam is longitudinally polarized.
Making use of the amplitude ratios as free fit parameters instead of the SDMEs
reduces the number of the real parameters in data description from 23 to 8.
The exact formula for $R=\frac{d\sigma_{L}}{dt}/\frac{d\sigma_{T}}{dt}$ in
terms of the SDMEs is obtained for spinless targets and the new approximate
formula for $R$ is proposed for nucleon targets.

###### pacs:

13.60.Le, 13.88.+e, 14.40.Cs

##  I Introduction

Investigation of exclusive vector-meson production in lepton-nucleon deep-
inelastic scattering provides information both on production mechanism and on
the nucleon structure Sak ; Critt ; Bauer ; INS . This assumes that incoherent
scattering is studied if targets are nuclei considered as a system of
noninteracting nucleons. Coherent scattering from a nucleus when it is intact
can give information both on production mechanism on the nucleus and on its
structure. For instance, the leading amplitudes of the vector-meson production
on the nucleon at high energies and small Bjorken $x_{B}$ can be used to
obtain the gluon distribution in the nucleon Rysk93 ; Rysk99 , while the
amplitude of the coherent vector-meson production on the nucleus can provide
the information on the gluon distribution in the nucleus. The relation between
the gluon distribution in the nucleon and that in the nucleus is probably not
trivial. Indeed, the unexpected relation between the quark distributions in
nuclei and the nucleon was called the EMC effect. It was observed for the
first time in 1983 EMC  but up to now, the EMC effect is not ultimately
understood (see, for instance, reviews EMC-rev1 ; EMC-rev2 ).

In the present paper, the simplest case of nuclei with the spin $J=0$ will be
considered. All formulas will be obtained in the one-photon-exchange
approximation. The most popular method of phenomenological description of
exclusive vector-meson production is the spin-density-matrix-element (SDME)
method of the SDME extraction from the fit of the angular distribution of
final particles to the experimental data. This method was proposed in the
pioneering work by Schilling and Wolf SW  for unpolarized targets and
generalized to polarized targets by Fraas Fraas  and Diehl Diehl . This method
which will be called the SDME method considers the SDMEs as free independent
parameters.

The SDMEs were expressed in terms of the amplitudes of the process

$\displaystyle\gamma^{*}(\lambda_{\gamma})+N(\lambda_{N})\to
V(\lambda_{V})+N(\mu_{N})$ (1)

in SW ; Fraas ; Diehl , where $\gamma^{*}$ denotes the virtual photon created
by the beam lepton (or anti-lepton), $N$ is the nucleon, and $V$ denotes the
vector meson. The helicities of the particles in reaction in Eq.
$(\ref{gnvn})$ are given in parentheses. The relation between the SDMEs and
the helicity amplitudes is not used in the phenomenological analysis of the
data with the SDME method. In the amplitude method, the ratios of the helicity
amplitudes are extracted directly from the experimental distribution of the
final particles. This method was for the first time used independently by
collaborations H1 H1-Aar  and HERMES DC-84 .

For the spinless nucleus $S$, the helicity amplitudes
$F_{\lambda_{V}\lambda_{\gamma}}$ of the reaction

$\displaystyle\gamma^{*}(\lambda_{\gamma})+S\to V(\lambda_{V})+S$ (2)

obey the symmetry relations SW ; Diehl

$\displaystyle
F_{-\lambda_{V}-\lambda_{\gamma}}=(-1)^{\lambda_{V}-\lambda_{\gamma}}F_{\lambda_{V}\lambda_{\gamma}}$
(3)

due to parity conservation in strong and electromagnetic interactions. The
direct consequence of relations $(\ref{par-cons})$ is the following: the
number of independent helicity amplitudes is 5. We choose the amplitudes
$F_{11}=F_{-1-1}$, $F_{10}=-F_{-10}$, $F_{1-1}=F_{-11}$, $F_{01}=-F_{0-1}$,
and $F_{00}$. For high energy and large photon virtuality, the dominant
amplitude is $F_{00}$ and convenient choice for the amplitude ratios is
$F_{\lambda_{V}\lambda_{\gamma}}/F_{00}$. When the photon virtuality, $Q^{2}$
tends to zero $F_{00}\to 0$, therefore this choice becomes meaningless and the
amplitude ratios $F_{\lambda_{V}\lambda_{\gamma}}/F_{11}$ are considered.

##  II Basic equations for the helicity amplitudes

Let us first define the kinematic variables and reference systems. By
definition, the laboratory system (LS) is that where the nuclear target is at
rest. The lepton beam energy in LS is $E$, while the energy of scattered
lepton is $E^{\prime}$. The four-vector of the initial lepton momentum is
$k=(E,{\bf k})$, while the four-momentum of the final lepton is
$k^{\prime}=(E^{\prime},{\bf k}^{\prime})$, hence the photon four-momentum is
$q=k-k^{\prime}$ and the photon virtuality is $Q^{2}=-q^{2}\geq 0$. The photon
four-momentum in the laboratory system looks like $q=(\nu,0,0,q_{z})$ with the
photon 3-momentum aligned along the $Z$-axis in LS. The ratio of the virtual
photon energy to the lepton beam energy in LS will be denoted $y=\nu/E$. If
the target nucleus mass is $M_{A}$ with $A$ being the mass number of the
nucleus then the Mandelstam variable for the reaction in Eq. $(\ref{gsvs})$ is
$s=M_{A}^{2}-Q^{2}+2M_{A}\nu\equiv W^{2}$, where $W$ denotes the center-of-
mass (CM) energy of the photon-nucleus system. The variable $x$ for the target
nucleus can be defined in LS as $x=Q^{2}/(2M_{A}\nu)$, while the Bjorken
variable $x_{B}$ corresponding to scattering from the nucleon with its mass
$M_{N}$ is $x_{B}=Q^{2}/(2M_{N}\nu)$.

The unnormalized spin-density matrix
$\Upsilon_{\lambda_{V}\tilde{\lambda}_{V}}$ of the produced vector meson is
given by the von-Neumann formula

$\displaystyle\Upsilon_{\lambda_{V}\tilde{\lambda}_{V}}=\sum_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}F_{\lambda_{V}\lambda_{\gamma}}F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}\varrho_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}},$
(4)

where $F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}$ denotes the
complex-conjugate amplitude and
$\varrho_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$ is the spin-density
matrix of the virtual photon. It contains two terms SW

$\displaystyle\varrho_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}=\varrho^{U}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}+P_{b}\varrho^{L}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}},$
(5)

where the matrix $\varrho^{U}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$
describes the unpolarized beam case while
$\varrho^{L}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$ corresponds to the
additional term due to the longitudinal beam polarization, $P_{b}$. The matrix
$\varrho^{U}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$ corresponds to the
unit flux of the transversely polarized virtual photons, while the ratio of
fluxes of the longitudinally to transversely polarized virtual photons is

$\epsilon=\frac{1-y-Q^{2}/(4E^{2})}{1-y+y^{2}/2+Q^{2}/(4E^{2})}.$ (6)

Both matrix $\varrho^{U}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$ and
$\varrho^{L}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$can be found in SW ;
DC-24 . They depend on $\epsilon$ and $\Phi$ that denotes the angle between
the vector-meson-production plane and the lepton-scattering plane (see
definition in DC-24 ; Joos ).

To be more specific, we consider $\rho^{0}$-meson production and the angular
distribution of pions from the decay $\rho^{0}\to\pi^{+}+\pi^{-}$, though all
our results are valid for $\phi$-meson production followed by the decay
$\phi\to K^{+}K^{-}$. The angular distribution of the pions in the rest system
of the $\rho^{0}$-meson normalized to unity, $\mathcal{W}$ is DC-24

$\displaystyle\mathcal{N}\mathcal{W}=\sum_{\lambda,\tilde{\lambda}}Y_{1\lambda}(\vartheta,\varphi)Y^{*}_{1\tilde{\lambda}}(\vartheta,\varphi)\Upsilon_{\lambda\tilde{\lambda}},$
(7)

where $\vartheta$ and $\varphi$ are the polar and azimuthal angles of the
$\pi^{+}$-meson 3-momentum in the vector-meson-rest (VMR) system. Note, that
all the systems under consideration in the present paper are right-handed. The
spherical harmonics are chosen in a standard way Diehl

$\displaystyle Y_{1\pm
1}(\vartheta,\varphi)=\mp\sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\varphi},$
(8) $\displaystyle
Y_{10}(\vartheta,\varphi)=\sqrt{\frac{3}{4\pi}}\cos\vartheta.$ (9)

Since the normalization condition for the angular distribution looks like

$\displaystyle\int_{0}^{2\pi}\frac{d\Phi}{2\pi}\int_{0}^{\pi}\sin\vartheta
d\vartheta\int_{0}^{2\pi}d\varphi\mathcal{W}(\Phi,\vartheta,\varphi)=1,$ (10)

the normalization factor $\mathcal{N}$ is defined by

$\displaystyle\mathcal{N}=\mathcal{N}_{T}+\epsilon\mathcal{N}_{L},$ (11)
$\displaystyle\mathcal{N}_{T}=|F_{11}|^{2}+|F_{01}|^{2}+|F_{-11}|^{2}=|F_{11}|^{2}+|F_{01}|^{2}+|F_{1-1}|^{2},$
(12)
$\displaystyle\mathcal{N}_{L}=|F_{10}|^{2}+|F_{00}|^{2}+|F_{-10}|^{2}=2|F_{10}|^{2}+|F_{00}|^{2},$
(13)

where in $(\ref{def-nt})$, $(\ref{def-nl})$ the symmetry property of the
helicity amplitudes from $(\ref{par-cons})$ is used.

After substituting into formula $(\ref{nw-ups})$ Eq. $(\ref{upsil})$, using
formula $(\ref{gmul})$ and expressions for the matrix elements of the virtual-
photon spin-density matrices
$\varrho^{U}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$ and
$\varrho^{L}_{\lambda_{\gamma}\tilde{\lambda}_{\gamma}}$ from Refs. SW ; DC-24
and spherical harmonics from $(\ref{spharm1})$, $(\ref{spharm0})$ one gets
finally the standard formula for the total angular distribution $\mathcal{W}$
where

$\mathcal{W}=\mathcal{W}^{U}+P_{b}\mathcal{W}^{L}.$ (14)

The formulas for $\mathcal{W}^{U}$ and $\mathcal{W}^{L}$ with the Schilling-
Wolf SDMEs $r^{\alpha}_{\lambda_{V}\tilde{\lambda}_{V}}$ SW  presented in
DC-24  are the same for scattering both from the nucleons and from the
spinless nuclei. But the expressions for the Schilling-Wolf SDMEs in terms of
the helicity amplitudes are different. In order to obtain the formulas for the
spinless targets one should first of all put in formulas (77-99) of Ref. DC-24
all the unnatural-parity-exchange (UPE) amplitudes equal to zero. In all the
sums with the natural-parity-exchange (NPE) amplitudes, the substitution has
to be done

$\displaystyle\widetilde{\sum}\\{T_{\lambda_{V}\lambda_{\gamma}}T^{*}_{\lambda^{\prime}_{V}\lambda^{\prime}_{\gamma}}\\}\to
F_{\lambda_{V}\lambda_{\gamma}}F^{*}_{\lambda^{\prime}_{V}\lambda^{\prime}_{\gamma}}.$
(15)

The abbreviated notation applied in DC-24  means

$\displaystyle\widetilde{\sum}\\{T_{\lambda_{V}\lambda_{\gamma}}T^{*}_{\lambda^{\prime}_{V}\lambda^{\prime}_{\gamma}}\\}\equiv\frac{1}{2}\sum_{\lambda_{N},\mu_{N}}\\{T_{\lambda_{V}\mu_{N}\lambda_{\gamma}\lambda_{N}}T^{*}_{\lambda^{\prime}_{V}\mu_{N}\lambda^{\prime}_{\gamma}\lambda_{N}}\\},$
(16)
$\displaystyle\widetilde{\sum}\\{U_{\lambda_{V}\lambda_{\gamma}}U^{*}_{\lambda^{\prime}_{V}\lambda^{\prime}_{\gamma}}\\}\equiv\frac{1}{2}\sum_{\lambda_{N},\mu_{N}}\\{U_{\lambda_{V}\mu_{N}\lambda_{\gamma}\lambda_{N}}U^{*}_{\lambda^{\prime}_{V}\mu_{N}\lambda^{\prime}_{\gamma}\lambda_{N}}\\},$
(17)

where $\lambda_{N}$ denotes the helicity of the initial nucleon while
$\mu_{N}$ is the final nucleon helicity in the NPE amplitudes
$T_{\lambda_{V}\mu_{N}\lambda_{\gamma}\lambda_{N}}$ and the UPE amplitudes
$U_{\lambda_{V}\mu_{N}\lambda_{\gamma}\lambda_{N}}$. The normalization factors
for the spinless targets in (77-99) of DC-24  are given by $(\ref{norm-nl-
nt}-\ref{def-nl})$.

For the unpolarized SDMEs that determine the angular distribution
$\mathcal{W}^{U}(\Phi,\vartheta,\varphi)$, we have basic equations from which
the ratios of the helicity amplitudes will be found

$\displaystyle\mathcal{N}(1-r^{04}_{00})=|F_{11}|^{2}+|F_{1-1}|^{2}+2\epsilon|F_{10}|^{2},\;\;$
(18)
$\displaystyle\sqrt{2}\mathcal{N}r^{5}_{11}=\rm{Re}\\{(F_{11}-F_{1-1})F_{10}^{*}\\},\;\;$
(19) $\displaystyle\mathcal{N}r^{1}_{11}=\rm{Re}\\{F_{1-1}F_{11}^{*}\\},\;\;$
(20)
$\displaystyle\mathcal{N}r^{04}_{1-1}=\rm{Re}\\{F_{1-1}F_{11}^{*}\\}-\epsilon|F_{10}|^{2},\;\;$
(21)
$\displaystyle\frac{\mathcal{N}}{\sqrt{2}}\\{\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1}\\}=\rm{Re}\\{F_{10}F_{11}^{*}\\},\;\;$
(22)
$\displaystyle\frac{\mathcal{N}}{\sqrt{2}}\\{\rm{Im}(r^{6}_{1-1})+r^{5}_{1-1}\\}=\rm{Re}\\{F_{1-1}F_{10}^{*}\\},\;\;$
(23)
$\displaystyle\mathcal{N}\\{r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\\}=|F_{11}|^{2},\;\;$
(24)
$\displaystyle\mathcal{N}\\{r^{1}_{1-1}+\rm{Im}(r^{2}_{1-1})\\}=|F_{1-1}|^{2},\;\;$
(25)
$\displaystyle\mathcal{N}r^{04}_{00}=|F_{01}|^{2}+\epsilon|F_{00}|^{2},\;\;$
(26)
$\displaystyle\frac{\mathcal{N}}{\sqrt{2}}r^{5}_{00}=\rm{Re}\\{F_{00}F_{01}^{*}\\},\;\;$
(27) $\displaystyle\mathcal{N}r^{1}_{00}=-|F_{01}|^{2},\;\;$ (28)
$\displaystyle
2\mathcal{N}r^{04}_{10}=\rm{Re}\\{(F_{11}-F_{1-1})F_{01}^{*}+2\epsilon
F_{10}F_{00}^{*}\\},\;\;$ (29)
$\displaystyle\sqrt{2}\mathcal{N}\\{\rm{Re}(r^{5}_{10})-\rm{Im}(r^{6}_{10})\\}=\rm{Re}\\{F_{11}F_{00}^{*}+F_{10}F_{01}^{*}\\},\;\;$
(30)
$\displaystyle\sqrt{2}\mathcal{N}\\{\rm{Re}(r^{5}_{10})+\rm{Im}(r^{6}_{10})\\}=\rm{Re}\\{F_{10}F_{01}^{*}-F_{1-1}F_{00}^{*}\\},\;\;$
(31)
$\displaystyle\mathcal{N}\\{\rm{Im}(r^{2}_{10})-\rm{Re}(r^{1}_{10})\\}=\rm{Re}\\{F_{11}F_{01}^{*}\\},\;\;$
(32)
$\displaystyle\mathcal{N}\\{\rm{Im}(r^{2}_{10})+\rm{Re}(r^{1}_{10})\\}=\rm{Re}\\{F_{1-1}F_{01}^{*}\\}.\;\;$
(33)

For the polarized SDMEs that determine the angular distribution
$\mathcal{W}^{L}(\Phi,\vartheta,\varphi)$, we have 8 additional basic
equations

$\displaystyle\sqrt{2}\mathcal{N}r^{8}_{11}=\rm{Im}\\{(F_{11}-F_{1-1})F_{10}^{*}\\},$
(34)
$\displaystyle\mathcal{N}\rm{Im}(r^{3}_{1-1})=-\rm{Im}\\{F_{1-1}F_{11}^{*}\\},$
(35)
$\displaystyle\frac{\mathcal{N}}{\sqrt{2}}\\{\rm{Im}(r^{7}_{1-1})+r^{8}_{1-1}\\}=-\rm{Im}\\{F_{11}F_{10}^{*}\\},$
(36)
$\displaystyle\frac{\mathcal{N}}{\sqrt{2}}\\{r^{8}_{1-1}-\rm{Im}(r^{7}_{1-1})\\}=\rm{Im}\\{F_{1-1}F_{10}^{*}\\},$
(37)
$\displaystyle\frac{\mathcal{N}}{\sqrt{2}}r^{8}_{00}=\rm{Im}\\{F_{01}F_{00}^{*}\\},$
(38) $\displaystyle
2\mathcal{N}\rm{Im}(r^{3}_{10})=\rm{Im}\\{(F_{11}+F_{1-1})F_{01}^{*}\\},$ (39)
$\displaystyle\sqrt{2}\mathcal{N}\\{\rm{Im}(r^{7}_{10})+\rm{Re}(r^{8}_{10})\\}=\rm{Im}\\{F_{11}F_{00}^{*}-F_{10}F_{01}^{*}\\},$
(40)
$\displaystyle\sqrt{2}\mathcal{N}\\{\rm{Im}(r^{7}_{10})-\rm{Re}(r^{8}_{10})\\}=\rm{Im}\\{F_{10}F_{01}^{*}+F_{1-1}F_{00}^{*}\\}.$
(41)

Equations $(\ref{d1}-\ref{d24})$ for the helicity amplitudes have infinite
number of solutions. Indeed, if any set of the amplitudes
$F_{\lambda_{V}\lambda_{\gamma}}$ is a solution then

$\displaystyle\mathcal{F}_{\lambda_{V}\lambda_{\gamma}}=\Lambda
e^{i\chi}F_{\lambda_{V}\lambda_{\gamma}}$ (42)

for arbitrary real parameters $\Lambda\neq 0$ and $\chi$ is also a solution.
If we add data on the differential cross section $d\sigma/dt$ of the vector-
meson production by the virtual photon to the SDME data this fixes $\Lambda$
equal to 1, since SW

$\displaystyle\frac{d\sigma}{dt}=\frac{d\sigma_{T}}{dt}+\epsilon\frac{d\sigma_{L}}{dt}=J_{ph}\mathcal{N}=J_{ph}(\mathcal{N}_{T}+\epsilon\mathcal{N}_{L}).$
(43)

Here, $d\sigma_{T}/dt$ is the differential cross section of the vector-meson
production by the transverse photon, while $d\sigma_{L}/dt$ corresponds to the
longitudinal (scalar) photon polarization. The flux factor for the transverse
photons in $(\ref{diff-cr-sec})$ is given by SW

$\displaystyle J_{ph}=\frac{1}{32\pi^{2}(Q^{2}+\nu^{2})}.$ (44)

Measuring $d\sigma_{T}/dt$ we experimentally define the value of the
normalization factor $\mathcal{N}$ using $(\ref{diff-cr-sec})$ and
$(\ref{j-phot})$. The common phase $\chi$ of all the helicity amplitudes is
not fixed by Eqs. $(\ref{d1}-\ref{d24})$. In the future, we will say that
there is a unique solution $F_{\lambda_{V}\lambda_{\gamma}}$ if there is no
solutions $\mathcal{F}_{\lambda_{V}\lambda_{\gamma}}$ except those obeying Eq.
$(\ref{scale-phas})$ with $\Lambda=1$. Since ratios of the helicity amplitudes
are invariant under transformation in Eq. $(\ref{scale-phas})$ the meaning of
the solution uniqueness for the amplitude ratios is usual.

As is seen from Eqs. $(38-39)$ of DC-24  all SDMEs are multiplied by the
functions of the angles $\Phi$, $\vartheta$, $\varphi$ all of which are
different from each other. Hence, all
$r^{\alpha}_{\lambda_{V}\tilde{\lambda}_{V}}$ can be extracted independently.
First of all, we a going to solve the system of Eqs. $(\ref{d1}-\ref{d16})$
trying to express the amplitudes through $\mathcal{N}$ and
$r^{\alpha}_{\lambda_{V}\tilde{\lambda}_{V}}$. Note that
$r^{\alpha}_{\lambda_{V}\tilde{\lambda}_{V}}$ are dimensionless quantities and
may depend on dimensionless amplitude ratios rather than on the helicity
amplitudes themselves. The number of the independent complex amplitude ratios
is four that corresponds to 8 real independent functions of $W$, $Q^{2}$, $t$
for any fixed vector-meson mass. The total number of the Schilling-Wolf SDMEs
is 23. Hence there is 23-8 equations of constraint between the SDMEs. The
linear constraint relations follow immediately from Eqs. $(77-99)$ of Ref.
DC-24  adopted to the spinless targets, namely

$\displaystyle r^{5}_{11}=-r^{5}_{1-1},$ (45) $\displaystyle
r^{8}_{11}=-r^{8}_{1-1},$ (46) $\displaystyle
1-r^{04}_{00}=2(r^{1}_{1-1}+r^{1}_{11}-r^{04}_{1-1}).$ (47)

The nonlinear constraint relations will be presented later. The relations
between $r^{\alpha}_{\lambda_{V}\tilde{\lambda}_{V}}$ are not useless since
they permit to check the quality of the detector property description by
Monte-Carlo codes. Indeed, any real detector distorts the momentum
distribution of the final particles in any process. Therefore the
distributions $\mathcal{W}^{U}$ and $\mathcal{W}^{L}$ are obtained when the
detector efficiency is taken into account to restore these distributions from
the experimentally measured distributions of particles in the final state of
reaction under study. The detector property is usually described for modern
detectors by Monte-Carlo codes adopted for the applied detector. If the used
Monte Carlo describes the detector property not perfectly the extracted SDMEs
have systematic errors which lead to the violation of the constraint relations
between the SDMEs $r^{\alpha}_{\lambda_{V}\tilde{\lambda}_{V}}$. If the MC
codes are good enough the SDMEs directly extracted from the experimental data
with the SDME method should coincide within experimental uncertainties with
those calculated with the directly extracted helicity-amplitude ratios
(obtained by the amplitude method). A comparison of the SDMEs extracted
directly from experimental data and calculated ones with the amplitude ratios
directly extracted from the same data is also the test of a quality of the
detector property description with the MC codes.

##  III Vector-meson Production with Unpolarized Beam

For the unpolarized beam, when $P_{b}=0$ we have $\mathcal{W}=\mathcal{W}^{U}$
from Eq. $(\ref{w-total})$. The main question which must be answered in this
paragraph is the following: is it possible to extract the unique set of the
amplitude moduli and phase differences from the data on the differential cross
section (that fixes $\mathcal{N}$) and the angular distribution
$\mathcal{W}^{U}$ ? We shall show that the system of Eqs.
$(\ref{d1}-\ref{d16})$ has no unique solution, while Eqs.
$(\ref{d1}-\ref{d16})$ and $(\ref{d17}-\ref{d24})$ have the unique solution.
This means that all moduli and all the phase differences between the
amplitudes are single-valued.

It is easy to get formulas for the amplitude moduli. Formulas for
$|F_{11}|^{2}$, $|F_{1-1}|^{2}$, and $|F_{01}|^{2}$ follows immediately from
$(\ref{d7})$, $(\ref{d8})$, and $(\ref{d11})$, respectively

$\displaystyle|F_{11}|^{2}=\mathcal{N}\\{r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\\},$
(48)
$\displaystyle|F_{1-1}|^{2}=\mathcal{N}\\{r^{1}_{1-1}+\rm{Im}(r^{2}_{1-1})\\}.$
(49) $\displaystyle|F_{01}|^{2}=-\mathcal{N}r^{1}_{00}.$ (50)

Considering the sum of Eqs. $(\ref{d9})$ and $(\ref{d11})$ we get

$\displaystyle|F_{00}|^{2}=\mathcal{N}\\{r^{04}_{00}+r^{1}_{00}\\}/\epsilon,$
(51)

while the difference of Eqs. $(\ref{d3})$ and $(\ref{d4})$ gives the last
formula of interest

$\displaystyle|F_{10}|^{2}=\mathcal{N}\\{r^{1}_{11}-r^{04}_{1-1}\\}/\epsilon.$
(52)

In Eqs. $(\ref{amp-d7}-\ref{amp-d3d4})$ $\mathcal{N}$ is considered as known
from the measurement of the differential cross section and using Eqs.
$(\ref{diff-cr-sec})$ and $(\ref{j-phot})$. If $d\sigma/dt$ is not measured
the ratios of the amplitude moduli can be obtained only. For this case,
dividing any of Eqs. $(\ref{amp-d7}-\ref{amp-d3d4})$ by another formula for
the amplitude ratio is obtained. For instance, dividing Eq. $(\ref{amp-d7})$
by Eq. $(\ref{amp-d9d11})$ one gets

$\displaystyle\frac{|F_{11}|^{2}}{|F_{00}|^{2}}=\frac{\epsilon\\{r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\\}}{r^{04}_{00}+r^{1}_{00}}$
(53)

and so on.

The phase differences between the amplitudes cannot be obtained single-valued
since Eqs. $(\ref{d1}-\ref{d16})$ contain only real parts of the amplitude
products of the type
$F_{\lambda_{V}\lambda_{\gamma}}F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}$
which contain only the cosines of the phase differences. Indeed, the real part
of the product of any two complex numbers $F_{\lambda_{V}\lambda_{\gamma}}$
and $F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}$ is

$\rm{Re}\\{F_{\lambda_{V}\lambda_{\gamma}}F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}\\}=|F_{\lambda_{V}\lambda_{\gamma}}||F_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}|\cos(\delta_{\lambda_{V}\lambda_{\gamma}}-\delta_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}),$
(54)

where $\delta_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}$ denotes the phase
of the complex number $F_{\lambda_{V}\lambda_{\gamma}}$, while
$|F_{\lambda_{V}\lambda_{\gamma}}|$ is the modulus of
$F_{\lambda_{V}\lambda_{\gamma}}$. If a set of the phases
$\delta_{\lambda_{V}\lambda_{\gamma}}$ of all the amplitudes
$F_{\lambda_{V}\lambda_{\gamma}}$ is a solution of Eqs.
$(\ref{d1}-\ref{d16})$, then the set with
$-\delta_{\lambda_{V}\lambda_{\gamma}}$ is also the solution of the same
equations. Hence at least two solutions of basic equations
$(\ref{d1}-\ref{d16})$ exist. Considering Eqs. $(\ref{d17}-\ref{d24})$ which
contain imaginary parts of the bilinear amplitude products
$F_{\lambda_{V}\lambda_{\gamma}}F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}$
we can get sines of the phase differences since

$\rm{Im}\\{F_{\lambda_{V}\lambda_{\gamma}}F^{*}_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}\\}=|F_{\lambda_{V}\lambda_{\gamma}}||F_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}|\sin(\delta_{\lambda_{V}\lambda_{\gamma}}-\delta_{\tilde{\lambda}_{V}\tilde{\lambda}_{\gamma}}).$
(55)

Adding Eqs. $(\ref{d17}-\ref{d24})$ to Eqs. $(\ref{d1}-\ref{d16})$ makes the
phase differences single-valued.

Let us try to find any solution of Eqs. $(\ref{d1}-\ref{d16})$ for cosines of
phase differences using equation $(\ref{real-z1-z2})$. Substituting Eqs.
$(\ref{d15})$ for $\rm{Re}\\{F_{11}F^{*}_{01}\\}$ and $(\ref{amp-d7})$,
$(\ref{amp-d11})$ respectively for moduli of $F_{11}$ and $F_{01}$ into
$(\ref{real-z1-z2})$ one gets

$\cos(\delta_{11}-\delta_{01})=\frac{\rm{Im}(r^{2}_{10})-\rm{Re}(r^{1}_{10})}{\sqrt{\\{\rm{Im}(r^{2}_{1-1})-r^{1}_{1-1}\\}r^{1}_{00}}}.\;\;$
(56)

In an analogous way, substituting Eqs. $(\ref{d16})$ for
$\rm{Re}\\{F_{1-1}F^{*}_{01}\\}$ and $(\ref{amp-d8})$, $(\ref{amp-d11})$
respectively for moduli of $F_{1-1}$ and $F_{01}$ into $(\ref{real-z1-z2})$
one gets

$\cos(\delta_{1-1}-\delta_{01})=\frac{\rm{Im}(r^{2}_{10})+\rm{Re}(r^{1}_{10})}{\sqrt{-\\{\rm{Im}(r^{2}_{1-1})+r^{1}_{1-1}\\}r^{1}_{00}}},\;\;$
(57)

while substituting Eqs. $(\ref{d3})$ for $\rm{Re}\\{F_{1-1}F^{*}_{11}\\}$ and
$(\ref{amp-d7})$, $(\ref{amp-d8})$ respectively for moduli of $F_{11}$ and
$F_{1-1}$ into $(\ref{real-z1-z2})$ one obtains

$\cos(\delta_{11}-\delta_{1-1})=\frac{r^{1}_{11}}{\sqrt{\\{r^{1}_{1-1}\\}^{2}-\\{\rm{Im}(r^{2}_{1-1})\\}^{2}}}.\;\;$
(58)

Since
$(\delta_{11}-\delta_{1-1})=(\delta_{11}-\delta_{01})-(\delta_{1-1}-\delta_{01})$
than according to the basic trigonometric formula, we have the equation

$\cos(\delta_{11}-\delta_{1-1})=\cos(\delta_{11}-\delta_{01})\cos(\delta_{1-1}-\delta_{01})+\sin(\delta_{11}-\delta_{01})\sin(\delta_{1-1}-\delta_{01}).$
(59)

It is obvious from $(\ref{cos-del-11-1m1=sum})$ that

$\displaystyle\\{\cos(\delta_{11}-\delta_{1-1})-\cos(\delta_{11}-\delta_{01})\cos(\delta_{1-1}-\delta_{01})\\}^{2}=$
$\displaystyle=\sin^{2}(\delta_{11}-\delta_{01})\sin^{2}(\delta_{1-1}-\delta_{01})=\\{1-\cos^{2}(\delta_{11}-\delta_{01})\\}\\{1-\cos^{2}(\delta_{1-1}-\delta_{01})\\}.$
(60)

Expressing all cosines in $(\ref{dif1-fin})$ with the help of $(\ref{cos-
del-11-01})$, $(\ref{cos-del-1m1-01})$, and $(\ref{cos-del-11-1m1})$ we obtain
the first non-linear constraint equation

$\displaystyle\Bigl{\\{}r^{1}_{11}|r^{1}_{00}|+\Bigl{(}\rm{Re}(r^{1}_{10})\Bigr{)}^{2}-\Bigl{(}\rm{Im}(r^{2}_{10})\Bigr{)}^{2}\Bigr{\\}}^{2}=$
$\displaystyle=\Bigl{\\{}\Bigl{(}r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\Bigr{)}r^{1}_{00}+\Bigl{(}\rm{Im}(r^{2}_{10})-\rm{Re}(r^{1}_{10})\Bigr{)}^{2}\Bigr{\\}}\times$
$\displaystyle\times\Bigl{\\{}\Bigl{(}r^{1}_{1-1}+\rm{Im}(r^{2}_{1-1})\Bigr{)}r^{1}_{00}+\Bigl{(}\rm{Im}(r^{2}_{10})+\rm{Re}(r^{1}_{10})\Bigr{)}^{2}\Bigr{\\}}.$
(61)

Let us find $\cos(\delta_{11}-\delta_{10})$ and
$\cos(\delta_{1-1}-\delta_{10})$ respectively from Eqs. $(\ref{d5})$,
$(\ref{amp-d7})$, $(\ref{amp-d3d4})$ and Eqs. $(\ref{d6})$, $(\ref{amp-d8})$,
$(\ref{amp-d3d4})$, while $\cos(\delta_{11}-\delta_{1-1})$ from $(\ref{cos-
del-11-1m1})$. Since
$(\delta_{11}-\delta_{1-1})=(\delta_{11}-\delta_{10})-(\delta_{1-1}-\delta_{10})$
we can obtain the second non-linear constraint relation

$\displaystyle\Bigl{\\{}r^{1}_{11}\Bigl{(}r^{1}_{11}-r^{04}_{1-1}\Bigr{)}-\frac{\epsilon}{2}\Bigl{[}\Bigl{(}\rm{Im}(r^{6}_{1-1})\Bigr{)}^{2}-\Bigl{(}r^{5}_{1-1})\Bigr{)}^{2}\Bigr{]}\Bigr{\\}}^{2}=$
$\displaystyle=\Bigl{\\{}\Bigl{(}r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\Bigr{)}\Bigl{(}r^{1}_{11}-r^{04}_{1-1}\Bigr{)}-\frac{\epsilon}{2}\Bigl{(}\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1})\Bigr{)}^{2}\Bigr{\\}}\times$
$\displaystyle\times\Bigl{\\{}\Bigl{(}r^{1}_{1-1}+\rm{Im}(r^{2}_{1-1})\Bigr{)}\Bigl{(}r^{1}_{11}-r^{04}_{1-1}\Bigr{)}-\frac{\epsilon}{2}\Bigl{(}\rm{Im}(r^{6}_{1-1})+r^{5}_{1-1})\Bigr{)}^{2}\Bigr{\\}}$
(62)

in an analogous way as Eq. $(\ref{first-non-lin})$ was obtained and so on.

##  IV Vector-meson Production with Polarized Beam

For polarized beam, the angular distribution $\mathcal{W}^{L}$ is added to
$\mathcal{W}^{U}$ in Eq. $(\ref{w-total})$. From Eqs. $(\ref{d1}-\ref{d16})$
and $(\ref{d17}-\ref{d24})$, the relations for complex numbers follow, namely,
relation

$\displaystyle
F_{1-1}F_{11}^{*}=\mathcal{N}\Bigl{\\{}r^{1}_{11}-i\rm{Im}(r^{3}_{1-1})\Bigr{\\}}$
(63)

can be obtained from Eqs. $(\ref{d3})$ and $(\ref{d18})$, while formula

$\displaystyle
F_{01}F_{00}^{*}=\frac{\mathcal{N}}{\sqrt{2}}\Bigl{\\{}r^{5}_{00}+ir^{8}_{00}\Bigr{\\}}$
(64)

follows from Eqs. $(\ref{d10})$ and $(\ref{d21})$. If we combine Eq.
$(\ref{d2})$ with Eq. $(\ref{d5})$ to get a real part of amplitude products
and Eq. $(\ref{d17})$ with $(\ref{d20})$ to get the imaginary part of the
amplitude products we obtain easily

$\displaystyle(F_{11}+F_{1-1})F_{10}^{*}=\mathcal{N}\sqrt{2}\Bigl{\\{}\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1}-r^{5}_{11}+i\Bigl{(}r^{8}_{11}+r^{8}_{1-1}-\rm{Im}(r^{7}_{1-1})\Bigr{)}\Bigr{\\}}.$
(65)

If we take into account in Eq. $(\ref{f11+f1m1f10--1})$ relations $(\ref{u2})$
and $(\ref{u3})$ we simplify it and obtain

$\displaystyle(F_{11}+F_{1-1})F_{10}^{*}=\mathcal{N}\sqrt{2}\Bigl{\\{}\rm{Im}(r^{6}_{1-1})-i\rm{Im}(r^{7}_{1-1})\Bigr{\\}},$
(66)

while relation

$\displaystyle(F_{11}+F_{1-1})F_{01}^{*}=2\mathcal{N}\Bigl{\\{}\rm{Im}(r^{2}_{10})+i\rm{Im}(r^{3}_{10})\Bigr{\\}}$
(67)

follows from the sum of Eqs. $(\ref{d15})$, $(\ref{d16})$ and also
$(\ref{d22})$. Calculating the difference of Eqs. $(\ref{d13})$, $(\ref{d14})$
and the sum of Eqs. $(\ref{d23})$, $(\ref{d24})$ the formula

$\displaystyle(F_{11}+F_{1-1})F_{00}^{*}=\mathcal{N}\sqrt{8}\Bigl{\\{}-\rm{Im}(r^{6}_{10})+i\rm{Im}(r^{7}_{10})\Bigr{\\}}$
(68)

can be obtained. Considering the difference of Eqs. $(\ref{d5})$ and
$(\ref{d2})$ one obtains the real part while Eq. $(\ref{d20})$ gives the
imaginary part of the product $F_{1-1}F_{10}^{*}$, therefore

$\displaystyle
F_{1-1}F_{10}^{*}=\frac{\mathcal{N}}{\sqrt{2}}\Bigl{\\{}\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1}-2r^{5}_{11}+i\Bigl{(}r^{8}_{1-1}-\rm{Im}(r^{7}_{1-1})\Bigr{)}\Bigr{\\}}.$
(69)

The equation

$\displaystyle
F_{11}F_{10}^{*}=\frac{\mathcal{N}}{\sqrt{2}}\Bigl{\\{}\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1}+i\Bigl{(}2r^{8}_{11}+r^{8}_{1-1}-\rm{Im}(r^{7}_{1-1})\Bigr{)}\Bigr{\\}}$
(70)

follows immediately from Eqs. $(\ref{d5})$ and the sum of $(\ref{d17})$,
$(\ref{d20})$. Substitution of Eq. $(\ref{u2})$ into $(\ref{f1m1f10-01})$ and
Eq. $(\ref{u3})$ into $(\ref{f11f10-01})$ leads respectively to the more
simple formulas

$\displaystyle
F_{1-1}F_{10}^{*}=\frac{\mathcal{N}}{\sqrt{2}}\Bigl{\\{}\rm{Im}(r^{6}_{1-1})-r^{5}_{11}+i\Bigl{(}r^{8}_{1-1}-\rm{Im}(r^{7}_{1-1})\Bigr{)}\Bigr{\\}},$
(71) $\displaystyle
F_{11}F_{10}^{*}=\frac{\mathcal{N}}{\sqrt{2}}\Bigl{\\{}\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1}+i\Bigl{(}r^{8}_{11}-\rm{Im}(r^{7}_{1-1})\Bigr{)}\Bigr{\\}}.$
(72)

Dividing Eq. $(\ref{f01f00})$ by Eq. $(\ref{amp-d9d11})$ we get the first
amplitude ratio

$\displaystyle\frac{F_{01}}{F_{00}}\equiv\frac{F_{01}F_{00}^{*}}{F_{00}F_{00}^{*}}\equiv\frac{F_{01}F_{00}^{*}}{|F_{00}|^{2}}=\frac{\epsilon\bigl{\\{}r^{5}_{00}+ir^{8}_{00}\bigr{\\}}}{\sqrt{2}\bigl{\\{}r^{04}_{00}+r^{1}_{00}\bigr{\\}}}.$
(73)

Dividing Eq. $(\ref{f11+f1m1f10})$ by Eq. $(\ref{f11+f1m1f00})$ and
considering the complex conjugate of the SDME-linear-combination ratio we
obtain the second amplitude ratio

$\displaystyle\frac{F_{10}}{F_{00}}=-\frac{\rm{Im}(r^{6}_{1-1})+i\rm{Im}(r^{7}_{1-1})}{2\bigl{\\{}\rm{Im}(r^{6}_{10})+i\rm{Im}(r^{7}_{10})\bigr{\\}}}.$
(74)

Dividing Eq. (63) by (48) we get the ratio $F_{1-1}/F_{11}$

$\displaystyle\frac{F_{1-1}}{F_{11}}\equiv\frac{F_{1-1}F_{11}^{*}}{|F_{11}|^{2}}=\frac{r^{1}_{11}-i\rm{Im}(r^{3}_{1-1})}{\bigl{\\{}r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\bigr{\\}}}.$
(75)

Dividing Eq. (68) by (51) one gets a chain of equalities

$\displaystyle\frac{(F_{11}+F_{1-1})F_{00}^{*}}{|F_{00}|^{2}}\equiv\Bigl{\\{}1+\frac{F_{1-1}}{F_{11}}\Bigr{\\}}\frac{F_{11}F_{00}^{*}}{|F_{00}|^{2}}\equiv$
$\displaystyle\equiv\Bigl{\\{}1+\frac{F_{1-1}}{F_{11}}\Bigr{\\}}\frac{F_{11}}{F_{00}}=\frac{\epsilon\sqrt{8}\bigl{\\{}-\rm{Im}(r^{6}_{10})+i\rm{Im}(r^{7}_{10})\bigr{\\}}}{r^{04}_{00}+r^{1}_{00}}.$
(76)

Substituting the ratio $F_{1-1}/F_{11}$ given by auxiliary Eq. (75) in Eq.
(76) we rewrite Eq. (76) as

$\displaystyle\Bigl{\\{}1+\frac{r^{1}_{11}-i\rm{Im}(r^{3}_{1-1})}{r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})}\Bigr{\\}}\frac{F_{11}}{F_{00}}=\frac{\epsilon\sqrt{8}\bigl{\\{}-\rm{Im}(r^{6}_{10})+i\rm{Im}(r^{7}_{10})\bigr{\\}}}{r^{04}_{00}+r^{1}_{00}}.$
(77)

We obtain finally from Eq. $(\ref{ex1-f11-f00})$ that

$\displaystyle\frac{F_{11}}{F_{00}}=\frac{\epsilon\sqrt{8}\bigl{\\{}-\rm{Im}(r^{6}_{10})+i\rm{Im}(r^{7}_{10})\bigr{\\}}\bigl{\\{}r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\bigr{\\}}}{\bigl{\\{}r^{04}_{00}+r^{1}_{00}\bigr{\\}}\bigl{\\{}r^{1}_{11}+r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})-i\rm{Im}(r^{3}_{1-1})\bigr{\\}}}.$
(78)

Multiplying (75) and (78) we get the last linearly independent ratio of the
helicity amplitudes to $F_{00}$

$\displaystyle\frac{F_{1-1}}{F_{00}}=\frac{\epsilon\sqrt{8}\bigl{\\{}-\rm{Im}(r^{6}_{10})+i\rm{Im}(r^{7}_{10})\bigr{\\}}\bigl{\\{}r^{1}_{11}-i\rm{Im}(r^{3}_{1-1})\bigr{\\}}}{\bigl{\\{}r^{04}_{00}+r^{1}_{00}\bigr{\\}}\bigl{\\{}r^{1}_{11}+r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})-i\rm{Im}(r^{3}_{1-1})\bigr{\\}}}.$
(79)

Formulas $(\ref{f01-f00})$, $(\ref{f10-f00})$, $(\ref{f11-f00})$,
$(\ref{f1m1-f00})$ provide ratios of the amplitudes to $F_{00}$, which are
convenient for deep-inelastic scattering since $F_{00}$ dominates at
$Q^{2}\to\infty$ and is non-zero at $v_{T}=0$. When $Q^{2}\to 0$ these ratios
become meaningless since $F_{00}\to 0$. The $S$-Channel Helicity Conservation
(SCHC) approximation assumes that nonzero amplitudes at $v_{T}=0$ are only
amplitudes with $\lambda_{V}=\lambda_{\gamma}$. It is convenient to divide all
helicity amplitudes by another SCHC amplitude which is nonzero at $Q^{2}=0$.
It is the amplitude $F_{11}$. The first such ratio is given by Eq.
$(\ref{f1m1-f11})$, while the ratio $F_{00}/F_{11}$ gives the formula inverse
to Eq. $(\ref{f11-f00})$. Dividing the relation complex-conjugate to Eq.
$(\ref{f11f10})$ by Eq. $(\ref{amp-d7})$, we get the final result for
$F_{10}/F_{11}$

$\displaystyle\frac{F_{10}}{F_{11}}=\frac{F_{10}F_{11}^{*}}{|F_{11}|^{2}}=\frac{\rm{Im}(r^{6}_{1-1})-r^{5}_{1-1}-i\bigl{(}r^{8}_{11}-\rm{Im}(r^{7}_{1-1})\bigr{)}}{\sqrt{2}\bigl{\\{}r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})\bigr{\\}}}.$
(80)

Analogously, dividing the relation complex-conjugate to Eq.
$(\ref{f11+f1m1f01})$ by Eq. $(\ref{amp-d7})$, we get

$\displaystyle\frac{F_{01}(F_{11}+F_{1-1})^{*}}{|F_{11}|^{2}}=\frac{F_{01}}{F_{11}}\Bigl{[}1+\Bigl{(}\frac{F_{1-1}}{F_{11}}\Bigr{)}^{*}\Bigr{]}=\frac{2\bigl{\\{}\rm{Im}(r^{2}_{10})-i\rm{Im}(r^{3}_{10})\bigr{\\}}}{r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})}.$
(81)

Substituting for $F_{1-1}/F_{11}$ in the square brackets in Eq.
$(\ref{ex-f01-f11})$ formula $(\ref{f1m1-f11})$ we obtain for $F_{01}/F_{11}$
the final result

$\displaystyle\frac{F_{01}}{F_{11}}=\frac{2\bigl{\\{}\rm{Im}(r^{2}_{10})-i\rm{Im}(r^{3}_{10})\bigr{\\}}}{r^{1}_{11}+r^{1}_{1-1}-\rm{Im}(r^{2}_{1-1})+i\rm{Im}(r^{3}_{1-1})}.$
(82)

##  V Virtual-photon Longitudinal-to-transverse Cross-section Ratio

The ratio of the differential cross sections of vector-meson production by
longitudinally-polarized ($\frac{d\sigma_{L}}{dt}$) to transversely-polarized
($\frac{d\sigma_{T}}{dt}$) virtual photons is equal to SW

$\displaystyle
R\equiv\frac{d\sigma_{L}}{dt}\Big{/}\frac{d\sigma_{T}}{dt}=\frac{\mathcal{N}_{L}}{\mathcal{N}_{T}}=\frac{2|F_{10}|^{2}+|F_{00}|^{2}}{|F_{11}|^{2}+|F_{01}|^{2}+|F_{1-1}|^{2}}.$
(83)

Here, Eqs. $(\ref{def-nt})$, $(\ref{def-nl})$, and $(\ref{diff-cr-sec})$ are
used. Substituting formulas for the moduli of the helicity amplitudes
$(\ref{amp-d7}-\ref{amp-d3d4})$ into Eq. $(\ref{def-sl-st})$ we get

$\displaystyle
R=\frac{r^{04}_{00}+r^{1}_{00}+2(r^{1}_{11}-r^{04}_{1-1})}{\epsilon(2r^{1}_{1-1}-r^{1}_{00})}.$
(84)

We conclude from Eq. $(\ref{r-less})$ that the formula for the ratio of the
cross sections, $R$ in terms of the SDMEs for spinless targets is exact,
therefore we do not need the Rosenbluth decomposition Ros  to extract $R$ from
the experimental data. In order to obtain the cross sections
$\frac{d\sigma_{T}}{dt}$ and $\frac{d\sigma_{L}}{dt}$ from the measured value
of $\frac{d\sigma}{dt}=\frac{d\sigma_{T}}{dt}+\epsilon\frac{d\sigma_{L}}{dt}$
data for the same $\nu$, $Q^{2}$, $t$ but for at least two beam energies (and
hence two values of $\epsilon$ according to Eq. $(\ref{def-eps})$) are needed.
We remind that the only way to extract the exact value for $R$ from the
electroproduction data on the nucleon is to use the Rosenbluth decomposition,
while any formula for $R$ in terms of the SDMEs, obtained at one beam energy,
is approximate.

Using constraint relation $(\ref{u1})$ it is possible to rewrite Eq.
$(\ref{r-less})$ in another form, namely

$\displaystyle
R=\frac{1}{\epsilon}\Bigl{\\{}\frac{1}{(2r^{1}_{1-1}-r^{1}_{00})}-1\Bigr{\\}}.$
(85)

Formulas $(\ref{r-less})$ and $(\ref{r-less-upe})$ are to give the same values
for $R$ withing experimental uncertainties. Since there are only two SDMEs in
the expression for $R$ in Eq. $(\ref{r-less-upe})$ the cross-section ratio has
usually the better accuracy than $R$ defined by Eq. $(\ref{r-less})$ which
contains five SDMEs. We conclude that for mutually dependent SDMEs there is no
unique formula for $R$. Various formulas for $R$ may express it in terms of
SDME sets different from each other and these $R$ can have different
experimental uncertainties.

Originally, the SDMEs in the Schilling-Wolf representation were used for the
vector-meson production on the nucleons. The simplest idea is to use Eq.
$(\ref{r-less})$ or Eq. $(\ref{r-less-upe})$ for an improvement of the
description of the virtual-photon longitudinal-to-transverse cross-section
ratio on the nucleon. In order to distinguish observables for the nucleon from
the same quantities for spinless targets we will mark the former by tilde.
Using formulas for the SDMEs
$\widetilde{r}_{\lambda_{V}\tilde{\lambda}_{V}}^{\alpha}$ (see SW ; DC-24 ) in
terms of the helicity amplitudes
$T_{\lambda_{V}\mu_{N}\lambda_{\gamma}\lambda_{N}}$ and
$U_{\lambda_{V}\mu_{N}\lambda_{\gamma}\lambda_{N}}$ we obtain the formula for
the virtual-photon longitudinal-to-transverse cross-section ratio on the
nucleon from Eq. $(\ref{r-less})$

$\displaystyle\widetilde{R}_{U}=\frac{\widetilde{r}^{04}_{00}+\widetilde{r}^{1}_{00}+2(\widetilde{r}^{1}_{11}-\widetilde{r}^{04}_{1-1})}{\epsilon(2\widetilde{r}^{1}_{1-1}-\widetilde{r}^{1}_{00})}=$
$\displaystyle=\frac{\widetilde{\sum}\bigl{\\{}\epsilon\bigl{(}|T_{00}|^{2}+2|T_{10}|^{2}-2|U_{10}|^{2}\bigr{)}+2|U_{01}|^{2}+4\rm{Re}\bigl{(}U_{1-1}U_{11}^{*}\bigr{)}\bigr{\\}}}{\epsilon\widetilde{\sum}\bigl{\\{}|T_{11}|^{2}+|T_{01}|^{2}+|T_{-11}|^{2}-|U_{11}|^{2}-|U_{01}|^{2}-|U_{-11}|^{2}\bigr{\\}}}.$
(86)

The formulas for the true ratio of the cross sections on the nucleon are SW ;
DC-24

$\displaystyle\widetilde{R}=\widetilde{\mathcal{N}}_{L}/\widetilde{\mathcal{N}}_{T},$
(87)
$\displaystyle\widetilde{\mathcal{N}}_{L}=\widetilde{\sum}\bigl{(}|T_{00}|^{2}+2|T_{10}|^{2}+2|U_{10}|^{2}\bigr{)},\;\;$
(88)
$\displaystyle\widetilde{\mathcal{N}}_{T}=\widetilde{\sum}\bigl{(}|T_{11}|^{2}+|T_{01}|^{2}+|T_{-11}|^{2}+|U_{11}|^{2}+|U_{01}|^{2}+|U_{-11}|^{2}\bigr{)},\;\;\;$
(89)
$\displaystyle\widetilde{\mathcal{N}}=\widetilde{\mathcal{N}}_{T}+\epsilon\widetilde{\mathcal{N}}_{L}.\;\;\;$
(90)

A comparison of $\widetilde{R}_{U}$ given by Eq. $(\ref{rs-amp})$ and the true
ratio $\widetilde{R}$ defined by Eqs. $(\ref{r-nl-nt-nucl}-\ref{nt-1/2})$
shows that the contributions of the UPE-amplitudes are taken into account in
Eq. $(\ref{rs-amp})$ incorrect. For the collider experiments at very high
energies, the contribution of the UPE-amplitudes is strongly suppressed (as
usually expected in Regge phenomenology KS ; IW ) then
$\widetilde{R}_{U}=\widetilde{R}$ with a good accuracy. But for fixed-target
experiments, a neglect of the UPE-amplitudes especially the greatest among
them, $U_{1\frac{1}{2}1\frac{1}{2}}$ can give an appreciable discrepancy. For
instance, the contribution of $U_{1\frac{1}{2}1\frac{1}{2}}$ is a measurable
effect at $W\approx 6$ GeV for $\rho^{0}$-meson production at HERMES DC-24 ;
DC-84  and is crucially important for $\omega$-production DC-95 ; DC-95-err .

Now, we are going to improve Eq. $(\ref{rs-amp})$ for $\widetilde{R}_{U}$. Let
us consider the numerator of Eq. $(\ref{rs-amp})$ and express it in terms of
the helicity amplitudes using standard equations for
$\widetilde{r}_{\lambda_{V}\tilde{\lambda}_{V}}^{\alpha}$ from SW ; DC-24

$\displaystyle
D_{S}\equiv\widetilde{r}^{04}_{00}+\widetilde{r}^{1}_{00}+2(\widetilde{r}^{1}_{11}-\widetilde{r}^{04}_{1-1})=$
$\displaystyle=\\{\epsilon\widetilde{\mathcal{N}}_{L}+\widetilde{\sum}[-4\epsilon|U_{10}|^{2}+2|U_{01}|^{2}+4\rm{Re}\bigl{(}U_{1-1}U_{11}^{*}\bigr{)}]\\}/\widetilde{\mathcal{N}}=$
$\displaystyle=\frac{\epsilon\widetilde{\mathcal{N}}_{L}}{\widetilde{\mathcal{N}}}+\beta_{D}=\frac{1}{1+1/(\epsilon\widetilde{R})}+\beta_{D},$
(91)

where $\widetilde{\mathcal{N}}_{L}$, $\widetilde{\mathcal{N}}_{T}$, and
$\widetilde{\mathcal{N}}$ are defined by Eqs. $(\ref{nl-1/2}-\ref{n=nt-
exnl})$, while $\widetilde{R}$ is given by Eq. $(\ref{r-nl-nt-nucl})$. The
quantity $\beta_{D}$ according to Eq. $(\ref{rs-numera})$ is

$\displaystyle\beta_{D}=\bigl{\\{}\widetilde{\sum}[-4\epsilon|U_{10}|^{2}+2|U_{01}|^{2}+4\rm{Re}\bigl{(}U_{1-1}U_{11}^{*}\bigr{)}]\bigr{\\}}/\widetilde{\mathcal{N}}.$
(92)

The exact formula for $\widetilde{R}$ follows from the right-hand side of Eq.
$(\ref{rs-numera})$

$\displaystyle\widetilde{R}=\frac{1}{\epsilon}\frac{D_{S}-\beta_{D}}{1-D_{S}+\beta_{D}}.$
(93)

The parameter $\beta_{D}$ at high energy is small since it is the ratio of
bilinear products of the UPE-amplitudes to $\widetilde{\mathcal{N}}$ which
contains the sum of squared moduli of all NPE- and UPE-amplitudes. We would
like to stress that the numerator in formula $(\ref{def-beta-u})$ for
$\beta_{D}$ does not contain the modulus square of the largest UPE-amplitude
$U_{1\frac{1}{2}1\frac{1}{2}}$. Neglecting the value of $\beta_{D}$ in Eq.
$(\ref{ds-r-beta-prec})$ one obtains the approximate formula of interest

$\displaystyle\widetilde{R}\approx\widetilde{R}_{D}=\frac{1}{\epsilon}\frac{D_{S}}{1-D_{S}}=\frac{\widetilde{r}^{04}_{00}+\widetilde{r}^{1}_{00}+2(\widetilde{r}^{1}_{11}-\widetilde{r}^{04}_{1-1})}{\epsilon\\{1-\widetilde{r}^{04}_{00}-\widetilde{r}^{1}_{00}-2(\widetilde{r}^{1}_{11}-\widetilde{r}^{04}_{1-1})\\}}.$
(94)

Equation $(\ref{ds-r-beta-prec})$ is exact and can be used to calculate
correction to $\widetilde{R}_{D}$ due to the contribution of $\beta_{D}$. The
ratio $\widetilde{R}_{D}$ approximates $\widetilde{R}$ better than
$\widetilde{R}_{U}$ since the numerator in formula $(\ref{def-beta-u})$ for
the small parameter $\beta_{D}$ does not contain the contribution of
$|U_{1\frac{1}{2}1\frac{1}{2}}|^{2}$, while the correction to
$\widetilde{R}_{U}$ contains the contribution of
$|U_{1\frac{1}{2}1\frac{1}{2}}|^{2}$.

In the SCHC approximation, when all amplitudes with
$\lambda_{V}\neq\lambda_{\gamma}$ are assumed to be negligibly small at small
$v_{T}$ one has to put $\widetilde{r}^{1}_{00}$, $\widetilde{r}^{1}_{11}$,
$\widetilde{r}^{04}_{1-1}$ equal to zero in Eq. $(\ref{r-ds-appr})$ according
to the formulas for $\widetilde{r}_{\lambda_{V}\tilde{\lambda}_{V}}^{\alpha}$
presented in SW ; DC-24 . This leads to the relation
$\widetilde{R}\approx\widetilde{R}_{04}$ less precise at high energies than
$\widetilde{R}\approx\widetilde{R}_{D}$, where

$\displaystyle\widetilde{R}_{04}=\frac{\widetilde{r}^{04}_{00}}{\epsilon(1-\widetilde{r}^{04}_{00})}.$
(95)

This statement will be proved below. The quantity $\widetilde{R}_{04}$ is
commonly used (see, for instance, Critt ; INS ; SW ; H1-Aar ; DC-24 ; DC-95 )
to estimate the ratio $\widetilde{R}$ from the experimental data. We note,
that the improvement which gives formula $(\ref{r-ds-appr})$ in comparison to
$(\ref{r04-appr})$ will be important for more precise data than the modern
ones. Even at intermediate energies the difference
$\widetilde{R}_{04}-\widetilde{R}_{D}$ is of the order of the total
experimental uncertainty. Let us consider $\widetilde{R}_{04}$ for three bins
in $Q^{2}$ with mean values $<Q^{2}>=1.19$ GeV2, $1.66$ GeV2, and $3.06$ GeV2
for the proton target from Ref. DC-24 . The values of $<\epsilon>$ are
$0.831$, $0.825$, and $0.790$, while the ratios $\widetilde{R}_{04}$ are
$0.701\pm 0.063$, $0.798\pm 0.080$, and $1.053\pm 0.074$. The values of
$\widetilde{R}_{D}$ $0.617$, $0.795$, and $1.004$ are systematically smaller
than the values of $\widetilde{R}_{04}$ by $0.084$, $0.003$, and $0.049$ but
the differences are comparable to the total experimental uncertainties of
$\widetilde{R}_{04}$ within less than 1.3 of the standard deviation.

The difference between the exact ratio $\widetilde{R}$ and the approximate
quantity $\widetilde{R}_{a}$ for $a=D$ or $a=04$ can be expressed with the
exact relation

$\displaystyle\widetilde{R}-\widetilde{R}_{a}=-\frac{\beta_{a}}{\epsilon}\frac{(1+\epsilon\widetilde{R}_{a})^{2}}{1+\beta_{a}(1+\epsilon\widetilde{R}_{a})},$
(96)

where $\beta_{a}$ and $\widetilde{R}_{a}$ for $a=D$ are respectively defined
by Eqs. $(\ref{def-beta-u})$ and $(\ref{r-ds-appr})$, while for $a=04$
$\beta_{04}$ is

$\displaystyle\beta_{04}=\widetilde{\sum}\bigl{\\{}-2\epsilon|T_{10}|^{2}-2\epsilon|U_{10}|^{2}+|T_{01}|^{2}+|U_{01}|^{2}\bigr{\\}}/\widetilde{\mathcal{N}}$
(97)

and $\widetilde{R}_{04}$ is given by Eq. $(\ref{r04-appr})$. A comparison of
Eq. $(\ref{def-beta-u})$ with Eq. $(\ref{def-beta-04})$ shows that $\beta_{D}$
contains only the UPE-amplitude contribution (without the contribution of
$|U_{1\frac{1}{2}1\frac{1}{2}}|^{2}$) in the numerator of Eq. $(\ref{def-
beta-u})$ which goes to zero at $W\to\infty$, while $\beta_{04}$ contains the
NPE-amplitude contribution ($|T_{01}|^{2}$ and $|T_{10}|^{2}$) in addition to
the UPE-amplitude contribution in the numerator of Eq. $(\ref{def-beta-04})$,
therefore $\beta_{04}$ remains, generally speaking, nonzero when $W\to\infty$.
This means that $|\beta_{D}|\ll|\beta_{04}|$ at high energies of the virtual
photon.

##  VI Conclusion

The exact formulas for the helicity-amplitude ratios of the vector-meson
production by a polarized lepton beam on spinless targets in terms of the
vector-meson spin-density-matrix elements are obtained. It is shown also that
data for an unpolarized beam are insufficient to find single-valued solutions
for the amplitude ratios. The fit of the angular distribution normalized to
unity in the amplitude method needs 8 free real parameters (4 real and 4
imaginary parts of the complex amplitude ratios), while the number of free
parameters $r_{\lambda_{V}\tilde{\lambda}_{V}}^{\alpha}$ in the SDME method is
23. The SDMEs are not independent, hence 23-8 constraint relations between
SDMEs exist. Tests of the constraint relation validity permit to estimate the
precision with which the Monte-Carlo codes describe the detector properties.
It is reasonable to use both the SDME and the amplitude method since the
differences between SDMEs extracted directly from data and calculated from the
amplitude ratios permit also to estimate the quality of detector property
description with the Monte-Carlo codes. The exact formulas $(\ref{r-less})$
and $(\ref{r-less-upe})$ for the virtual-photon longitudinal-to-transverse
differential-cross-section ratio, $R$ in terms of the SDMEs are found for
spinless targets. The new approximate formula $(\ref{r-ds-appr})$ more precise
for high energies than the standard Eq. $(\ref{r04-appr})$ for the ratio
$\widetilde{R}$ in term of the SDME $\widetilde{r}^{04}_{00}$ applicable to
scattering off the nucleon targets is proposed.

## Acknowledgements

I would like to thank Prof. O.L. Fedin, Prof. V.T. Kim, Dr. M.B. Zhalov, Dr.
E.N. Komarov, Dr. I.B. Smirnov and all other participants of the High Energy
Physics Division seminar in Petersburg Nuclear Physics Institute for
constructive discussion. I am also much obliged to Prof. V.R. Shaginyan for
advises that improved the manuscript.

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Hendrik Schröter1, Alberto N. Escalante-B.2, Tobias Rosenkranz2, Andreas
Maier1

# DeepFilterNet: Perceptually Motivated Real-Time Speech Enhancement

###### Abstract

Multi-frame algorithms for single-channel speech enhancement are able to take
advantage from short-time correlations within the speech signal. Deep
Filtering (DF) was proposed to directly estimate a complex filter in frequency
domain to take advantage of these correlations.

In this work, we present a real-time speech enhancement demo using
DeepFilterNet. DeepFilterNet's efficiency is enabled by exploiting domain
knowledge of speech production and psychoacoustic perception. Our model is
able to match state-of-the-art speech enhancement benchmarks while achieving a
real-time-factor of 0.19 on a single threaded notebook CPU. The framework as
well as pretrained weights have been published under an open source license.

Index Terms: speech enhancement, multi-frame filtering, deep filtering

##  1 Introduction

Recently, various speech enhancement models take advantage of properties of
psychoacoustic perception. That is, a speech model consisting of a periodic
(voiced) and a noisy component is assumed [1, 2, 3]. Further, it can be
assumed that the exact phase of the noisy speech component is not relevant.
Rather reconstructing the speech envelope at a course frequency resolution is
sufficient so that the resulting component sounds like the original speech.
For the voiced speech however, reconstructing the phase, or, improving the
periodicity is important in low signal-to-noise ratio (SNR) conditions. Multi-
frame (MF) filtering in frequency domain was proposed to take advantage of
short-time speech correlations, since speech can be decomposed into a
correlated and an interfering component [4]. Deep filtering (DF) [5, 6] has
been recently used to directly estimate the complex frequency domain filter
that is able to enhance the periodic component. It has been shown, that deep
filtering outperforms the mostly used complex ratio mask (CRM) [3] and
achieves state-of-the-art results [7].

In this work, we summarize our proposed DeepFilterNet framework and show some
improved results over previous work [7]. Due to its efficiency, we can use the
model for real-time noise reduction e.g. in video-calls.

##  2 Deep Filtering

###  2.1 Signal Model

Let $x(k)$ be a mixture signal

$x(k)=s(k)+z(k)\text{,}\vspace{-.2em}$ (1)

where $s(k)$ is a clean speech signal and $z(k)$ an interfering background
noise. Typically, noise reduction operates in time/frequency domain:

$X(t,f)=S(t,f)+Z(t,f)\text{,}\vspace{-.2em}$ (2)

where $X(t,f)$ is the STFT representation of the time domain signal $x(k)$ and
$t$, $f$ are the time and frequency bins. To simplify the following, we omit
the frequency index $f$ since all frequency bins are processed equivalently.
Further, with filter length $N$, we define the noisy multi-frame vector as
$\bm{\bar{x}}_{t}\in\mathbb{C}^{N}$:

$\bm{\bar{x}}(t)=[X(t+l),X(t-1+l),\dots,X(t-N+1+l)]^{\text{T}}\text{\
,}\vspace{-.2em}$ (3)

where $l$ is an optional look-ahead parameter. Due to the look-ahead, the
filter will include non-causal taps which introduces additional latency. And
with the complex filter $\bm{\bar{w}}(t)\in\mathbb{C}^{N}$

$\bm{\bar{w}}_{\text{DF}}(t)=[W_{0}(t),W_{1}(t),\dots,W_{N-1}(t)]^{\text{T}}\vspace{-.2em}$
(4)

we define the deep filtering as:

$\boxed{Y(t)=\bm{\bar{w}}_{\text{DF}}(t)^{\text{H}}\bm{\bar{x}}(t)}\text{\
,}\vspace{-.2em}$ (5)

where $\circ^{\text{H}}$ denotes the conjugate transpose operator. As
mentioned above, deep filtering directly estimates the complex filter
$\bm{\bar{w}}_{\text{DF}}(t)$.

##  3 DeepFilterNet Framework

Figure 1: DeepFilterNet Framework.

Figure 1 shows a schematic depiction of the DeepFilterNet noise reduction
framework. The model operates in two stage: 1. Enhancing the speech envelope
at a coarse frequency resolution, and 2. enhancing the speech periodicity
using DF.

DeepFilterNet operates in $48\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ sampling
rate on $20\text{\,}\mathrm{ms}$ windows with a hop size of
$10\text{\,}\mathrm{ms}$. Additionally, a look-ahead of 2 frames is used
resulting in an overall latency of $40\text{\,}\mathrm{ms}$. The first stage
operates in real-valued ERB (equivalent rectangular bandwidth) domain and
predicts 32 ERB scaled gains that are pointwise multiplied with the noisy
spectrum, which allows to reconstruct the overall speech envelope. The second
stage predicts a complex filter $N=5$ tap filter in frequency domain. This
filter is only applied for the lowest 96 frequency bins, i.e. up to
$4.8\text{\,}\mathrm{kHz}$. The final enhanced output spectrum is then
constructed from DF output for the lower frequencies and the ERB gain output
for the higher frequencies.

We take advantage of several properties of speech production model and
psychoacoustics:

  * •

The speech consists of a noisy and a periodic (short-time correlated)
component [8].

  * •

Loudness perception is logarithmic. This is used in the ERB feature
calculation and the loss.

  * •

Frequency perception is logarithmic. ERB feature logarithmically compress the
input spectrum with 481 frequency bins to 32 ERB bands which reduces
dimensions of the neural network input and output [9].

  * •

Most of voiced (periodic) speech energy is below $5\text{\,}\mathrm{kHz}$.

  * •

Moreover, the human ear is most sensitive around
$2\text{\,}\mathrm{kHz}5\text{\,}\mathrm{kHz}$ [10]. Thus, the second stage
enhancing the speech periodicity via DF is only applied up to approx.
$5\text{\,}\mathrm{kHz}$. This further reduces input and output dimension of
the neural network.

We further predict the local SNR $\xi\in[-15,35]$ $\mathrm{dB}$ within the
encoder network on frame level. This allows to completely disable the ERB or
the DF decoder depending on the current noise conditions. That is, we define
the following criteria:

  1. $\xi<$-10\text{\,}\mathrm{dB}$$:

Disable both decoders, return silent spectrum.

  2. $\xi>$20\text{\,}\mathrm{dB}$$:

Disable DF decoder. Only low noise condition, enhancing the periodicity is not
necessary.

  3. else:

Run all stages for best noise reduction.

We implemented a real-time loop in Rust and use
tract111https://github.com/sonos/tract as DNN inference framework.
DeepFilterNet achieves a single-threaded real-time factor of 0.19 on a
notebook i5-8250U CPU and is published under a permissive
license222https://github.com/Rikorose/DeepFilterNet.

##  4 Experiments and Results

We train the slightly modified DeepFilterNet model on the full multi-lingual
DNS4 dataset [11], while oversampling the high-quality PTDB and VCTK datasets
by a factor of 10 and evaluate on the unseen VCTK/DEMAND test set (Table 1).

Table 1: Objective results on Voicebank+Demand test set

Model | PESQ | CSIG | CBAK | COVL | STOI
---|---|---|---|---|---
DeepFilterNet [3] | $2.81$ | $4.14$ | $3.31$ | $3.46$ | $0.942$
DeepFilterNet2 [7] | $3.08$ | $4.30$ | $3.40$ | $3.699$ | $0.9429$
DeepFilterNet3 | $3.17$ | $4.34$ | $3.61$ | $3.77$ | $0.944$

Figure 2 shows the proposed demo for live background noise suppression. Due to
its efficiency, we can listen to the audio in real-time. Further, we can
dynamically configure the noise attenuation and modify thresholds where the
first and second denoising stage are disabled. Moreover, the DeepFilterNet
real-time implementation can also be used for live noise reduction by adding a
virtual microphone on Linux systems with pipewire via an LADSPA plugin. This
allows for real-time noise reduction e.g. during video calls.

Figure 2: DeepFilterNet Live Demo.

## References

  * [1] J.-M. Valin, ``A hybrid DSP/deep learning approach to real-time full-band speech enhancement,'' in _2018 IEEE 20th International Workshop on Multimedia Signal Processing (MMSP)_. IEEE, 2018, pp. 1–5.
  * [2] J.-M. Valin, U. Isik, N. Phansalkar, R. Giri, K. Helwani, and A. Krishnaswamy, ``A Perceptually-Motivated Approach for Low-Complexity, Real-Time Enhancement of Fullband Speech,'' in _INTERSPEECH 2020_ , 2020.
  * [3] H. Schröter, A. N. Escalante-B, T. Rosenkranz, and A. Maier, ``Deepfilternet: A low complexity speech enhancement framework for full-band audio based on deep filtering,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 7407–7411.
  * [4] Y. A. Huang and J. Benesty, ``A multi-frame approach to the frequency-domain single-channel noise reduction problem,'' _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 20, no. 4, pp. 1256–1269, 2011.
  * [5] H. Schröter, T. Rosenkranz, A. Escalante Banuelos, M. Aubreville, and A. Maier, ``CLCNet: Deep learning-based noise reduction for hearing aids using complex linear coding,'' in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2020. [Online]. Available: https://rikorose.github.io/CLCNet-audio-samples.github.io/
  * [6] W. Mack and E. A. Habets, ``Deep Filtering: Signal Extraction and Reconstruction Using Complex Time-Frequency Filters,'' _IEEE Signal Processing Letters_ , vol. 27, pp. 61–65, 2020.
  * [7] H. Schröter, A. Escalante-B, T. Rosenkranz, and A. Maier, ``Deepfilternet2: Towards real-time speech enhancement on embedded devices for full-band audio,'' in _2022 International Workshop on Acoustic Signal Enhancement (IWAENC)_. IEEE, 2022, pp. 1–5.
  * [8] T. F. Quatieri, _Discrete-time speech signal processing: principles and practice_. Pearson Education India, 2002\.
  * [9] B. C. Moore and B. R. Glasberg, ``A comparison of four methods of implementing automatic gain control ( agc) in hearing aids,'' _British Journal of Audiology_ , vol. 22, no. 2, pp. 93–104, 1988.
  * [10] Y. Suzuki and H. Takeshima, ``Equal-loudness-level contours for pure tones,'' _The Journal of the Acoustical Society of America_ , vol. 116, no. 2, pp. 918–933, 2004.
  * [11] H. Dubey, V. Gopal, R. Cutler, A. Aazami, S. Matusevych, S. Braun, S. E. Eskimez, M. Thakker, T. Yoshioka, H. Gamper _et al._ , ``Icassp 2022 deep noise suppression challenge,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 9271–9275.


IIT-BHU

Discrete origins of matter

Gauhar Abbasa111email<EMAIL_ADDRESS>
Neelam Singha222email<EMAIL_ADDRESS>
a Department of Physics, Indian Institute of Technology (BHU), Varanasi
221005, India

Abstract

> We discuss models of the flavour problem and dark matter based on the
> discrete $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}\times\mathcal{Z}_{\rm
> P}$ flavour symmetry. A new class of dark-matter emerges out of these
> models, which is defined as the flavonic dark matter. An ultra-violet
> completion of these models based on the dark-technicolour paradigm is also
> presented.

> ##  1 Introduction

The incompleteness of the standard model (SM) manifests itself in many ways.
For instance, the absence of any mechanism for the origin of the fermionic
mass spectrum and the mixing patterns, including that of leptons. This is
known as the flavour problem. For a review of the flavour problem, see
reference [1]. Moreover, discovery of dark matter extends the span of the
incompleteness of the SM to a point where the need to go beyond the SM becomes
inevitable.

In this work, we discuss some new developments to solve the problem of flavour
and dark matter together through a generic $\mathcal{Z}_{\rm
N}\times\mathcal{Z}_{\rm M}\times\mathcal{Z}_{\rm P}$ discrete flavour
symmetry [2, 3, 4]. The ultra-violet origin of this symmetry lies in a dark-
technicolour scenario where multi-fermion chiral condensates provide a
solution of the flavour problem [2, 4, 5, 6]. A subset of this symmetry,
$\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$, is capable of providing a dark
matter candidate referred to as the `` flavonic dark matter" together with a
solution to the flavour problem [7].

We shall present our discussion along the following track: In section 2, we
present a phenomenological investigation of two proto-type $\mathcal{Z}_{\rm
N}\times\mathcal{Z}_{\rm M}$ flavour symmetries by deriving the bounds of the
flavon field of the Froggatt-Nielsen (FN) mechanism [8] using the flavour
physics data. In section 3, we discuss a joint solution of the flavour problem
and dark matter, which gives rise to the flavonic dark matter. We discuss an
atypical solution to the flavour problem based on the VEV hierarchy in section
4, which is based on the $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ flavour symmetry. A dark-technicolour paradigm
providing a UV completion of the $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ flavour symmetry and the VEV hierarchy is
discussed in section 5.

##  2  $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ flavour symmetry

The $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ flavour symmetry
effectively determines flavour structure of the SM, including that of
neutrinos [3]. The key idea is to extend SM symmetry by introducing a complex
singlet scalar flavon field $\chi$, which transforms under the
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ symmetry of the SM as,

$\displaystyle\chi\mathrel{\mathop{\mathchar 58\relax}}(1,1,0).$ (1)

The Lagrangian that provides masses to the charged fermions of the SM reads
as,

$\displaystyle-{\mathcal{L}}_{\rm Yukawa}$ $\displaystyle=$
$\displaystyle\left[\dfrac{\chi(\chi^{\dagger})}{\Lambda}\right]^{n_{ij}^{u}}y_{ij}^{u}\bar{\psi}_{L_{i}}^{q}\tilde{\varphi}\psi_{R_{j}}^{u}+\left[\dfrac{\chi(\chi^{\dagger})}{\Lambda}\right]^{n_{ij}^{d}}y_{ij}^{d}\bar{\psi}_{L_{i}}^{q}\varphi\psi_{R_{j}}^{d}$
$\displaystyle+$
$\displaystyle\left[\dfrac{\chi(\chi^{\dagger})}{\Lambda}\right]^{n_{ij}^{\ell}}y_{ij}^{\ell}\bar{\psi}_{L_{i}}^{\ell}\varphi\psi_{R_{j}}^{\ell}+{\rm
H.c.},$ $\displaystyle=$ $\displaystyle
Y^{u}_{ij}\bar{\psi}_{L_{i}}^{q}\tilde{\varphi}\psi_{R_{j}}^{u}+Y^{d}_{ij}\bar{\psi}_{L_{i}}^{q}\varphi\psi_{R_{j}}^{d}+Y^{\ell}_{ij}\bar{\psi}_{L_{i}}^{\ell}\varphi\psi_{R_{j}}^{\ell}+\text{H.c.}.$

Employing the principle of minimum suppression [9], it turns out that
$\mathcal{Z}_{2}\times\mathcal{Z}_{5}$ is the simplest realization of
$\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ symmetry. We term it as the
minimal $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ model. Under this
symmetry, the charges to various fields are assigned in the way, as depicted
in Table 1.

Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{5}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{5}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{9}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{9}$
---|---|---|---|---|---|---|---|---|---|---|---
$u_{R},c_{R},t_{R}$ | + | $\omega^{2}$ |  $\psi_{L_{2}}^{q}$, $\psi_{L_{2}}^{\ell}$ | + | $\omega^{4}$ | $u_{R},t_{R}$ | + | $1$ |  $\psi_{L_{1}}^{q}$, $\psi_{L_{1}}^{\ell}$ | + | $\omega$
$d_{R},s_{R},b_{R}$ | - | $\omega$ |  $\psi_{L_{3}}^{q}$, $\psi_{L_{3}}^{\ell}$ | + | $\omega^{2}$ | $c_{R}$ | + | $\omega^{4}$ |  $\psi_{L_{2}}^{q}$, $\psi_{L_{2}}^{\ell}$ | + | $\omega^{8}$
$e_{R},\mu_{R},\tau_{R}$ | - | $\omega$ | $\chi$ | - | $\omega$ | $d_{R},s_{R},b_{R}$ | - | $\omega^{3}$ |  $\psi_{L_{3}}^{q}$, $\varphi$ | + | $1$
$\nu_{e_{R}},\nu_{\mu_{R}},\nu_{\tau_{R}}$ | - | $\omega^{3}$ | $\varphi$ | + | 1 | $e_{R},\mu_{R},\tau_{R}$ | - | $\omega^{3}$ | $\psi_{L_{3}}^{\ell}$ | + | $\omega^{6}$
$\psi_{L_{1}}^{q}$, $\psi_{L_{1}}^{\ell}$ | + | $\omega$ |  |  |  | $\nu_{e_{R}},\nu_{\mu_{R}},\nu_{\tau_{R}}$ | - | $\omega^{7}$ | $\chi$ | - | $\omega$
Table 1: Assigned charges to SM and flavon fields under $\mathcal{Z}_{2}$,
$\mathcal{Z}_{5}$, and $\mathcal{Z}_{9}$ symmetries, where $\omega$ is the
fifth or ninth root of unity.

It is observed that certain Yukawa couplings in the minimal model based on
$\mathcal{Z}_{2}\times\mathcal{Z}_{5}$ flavour symmetry are not order-one,
which is conventionally favoured in literature. We adopt a non-minimal form of
$\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ paradigm,
$\mathcal{Z}_{2}\times\mathcal{Z}_{9}$ flavour symmetry, where the Yukawa
couplings are indeed order-one, and assign charges to the SM and flavon
fields, as shown in Table 1.

###  2.1 Masses and mixing patterns

The minimal model based on $\mathcal{Z}_{2}\times\mathcal{Z}_{5}$ flavour
symmetry allows following mass matrices for up and down-type quarks, and
charged leptons,

${\cal
M}_{u}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{u}\epsilon^{4}&y_{12}^{u}\epsilon^{4}&y_{13}^{u}\epsilon^{4}\\\
y_{21}^{u}\epsilon^{2}&y_{22}^{u}\epsilon^{2}&y_{23}^{u}\epsilon^{2}\\\
y_{31}^{u}&y_{32}^{u}&y_{33}^{u}\end{pmatrix},{\cal
M}_{d}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{d}\epsilon^{5}&y_{12}^{d}\epsilon^{5}&y_{13}^{d}\epsilon^{5}\\\
y_{21}^{d}\epsilon^{3}&y_{22}^{d}\epsilon^{3}&y_{23}^{d}\epsilon^{3}\\\
y_{31}^{d}\epsilon&y_{32}^{d}\epsilon&y_{33}^{d}\epsilon\end{pmatrix},{\cal
M}_{\ell}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\ell}\epsilon^{5}&y_{12}^{\ell}\epsilon^{5}&y_{13}^{\ell}\epsilon^{5}\\\
y_{21}^{\ell}\epsilon^{3}&y_{22}^{\ell}\epsilon^{3}&y_{23}^{\ell}\epsilon^{3}\\\
y_{31}^{\ell}\epsilon&y_{32}^{\ell}\epsilon&y_{33}^{\ell}\epsilon\end{pmatrix}.$
(3)

And for the non-minimal $\mathcal{Z}_{2}\times\mathcal{Z}_{9}$ flavour
symmetry, the corresponding mass matrices are

$M_{u}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{u}\epsilon^{8}&y_{12}^{u}\epsilon^{6}&y_{13}^{u}\epsilon^{8}\\\
y_{21}^{u}\epsilon^{8}&y_{22}^{u}\epsilon^{4}&y_{23}^{u}\epsilon^{8}\\\
y_{31}^{u}&y_{32}^{u}\epsilon^{4}&y_{33}^{u}\end{pmatrix},M_{d}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{d}\epsilon^{7}&y_{12}^{d}\epsilon^{7}&y_{13}^{d}\epsilon^{7}\\\
y_{21}^{d}\epsilon^{5}&y_{22}^{d}\epsilon^{5}&y_{23}^{d}\epsilon^{5}\\\
y_{31}^{d}\epsilon^{3}&y_{32}^{d}\epsilon^{3}&y_{33}^{d}\epsilon^{3}\end{pmatrix},M_{\ell}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\ell}\epsilon^{7}&y_{12}^{\ell}\epsilon^{7}&y_{13}^{\ell}\epsilon^{7}\\\
y_{21}^{\ell}\epsilon^{5}&y_{22}^{\ell}\epsilon^{5}&y_{23}^{\ell}\epsilon^{5}\\\
y_{31}^{\ell}\epsilon^{3}&y_{32}^{\ell}\epsilon^{3}&y_{33}^{\ell}\epsilon^{3}\end{pmatrix}.$
(4)

The masses of the quarks and charged leptons are obtained in terms of the
expansion parameter $\epsilon$. Their expressions at the leading-order take up
the form as shown in table 2 [10] [9].

Masses | $\mathcal{Z}_{2}\times\mathcal{Z}_{5}$ | $\mathcal{Z}_{2}\times\mathcal{Z}_{9}$
---|---|---
$\\{m_{t},m_{c},m_{u}\\}$ | $\simeq\\{|y_{33}^{u}|,~{}|y_{22}^{u}|\epsilon^{2},~{}|y_{11}^{u}|\epsilon^{4}\\}v/\sqrt{2}$ | $\simeq\\{|y_{33}^{u}|,~{}|y_{22}^{u}|\epsilon^{4},~{}|y_{11}^{u}|\epsilon^{8}\\}v/\sqrt{2}$
$\\{m_{b},m_{s},m_{d}\\}$ | $\simeq\\{|y_{33}^{d}|\epsilon,~{}|y_{22}^{d}|\epsilon^{3},~{}|y_{11}^{d}|\epsilon^{5}\\}v/\sqrt{2}$ | $\simeq\\{|y_{33}^{d}|\epsilon^{3},~{}|y_{22}^{d}|\epsilon^{5},~{}|y_{11}^{d}|\epsilon^{7}\\}v/\sqrt{2}$
$\\{m_{\tau},m_{\mu},m_{e}\\}$ | $\simeq\\{|y_{33}^{l}|\epsilon,~{}|y_{22}^{l}|\epsilon^{3},~{}|y_{11}^{l}|\epsilon^{5}\\}v/\sqrt{2}$ | $\simeq\\{|y_{33}^{l}|\epsilon^{3},~{}|y_{22}^{l}|\epsilon^{5},~{}|y_{11}^{l}|\epsilon^{7}\\}v/\sqrt{2}$
Table 2: Leading order mass patterns of quarks and leptons in terms of
parameter $\epsilon$ for minimal and non-minimal models.

The quark mixing angles up to the leading order are approximately as given in
table 3 [10].

Quark mixing angles | $\mathcal{Z}_{2}\times\mathcal{Z}_{5}$ | $\mathcal{Z}_{2}\times\mathcal{Z}_{9}$
---|---|---
$\sin\theta_{12}\simeq|V_{us}|$ | $\simeq\left|{y_{12}^{d}\over y_{22}^{d}}-{y_{12}^{u}\over y_{22}^{u}}\right|\epsilon^{2}$ | $\simeq\left|{y_{12}^{d}\over y_{22}^{d}}-{y_{12}^{u}\over y_{22}^{u}}\right|\epsilon^{2}$
$\sin\theta_{23}\simeq|V_{cb}|$ | $\simeq|{y_{23}^{d}\over y_{33}^{d}}-{y_{23}^{u}\over y_{33}^{u}}|\epsilon^{2}$ | $\simeq\left|{y_{23}^{d}\over y_{33}^{d}}\right|\epsilon^{2}$
$\sin\theta_{13}\simeq|V_{ub}|$ | $\simeq\left|{y_{13}^{d}\over y_{33}^{d}}-{y_{12}^{u}y_{23}^{d}\over y_{22}^{u}y_{33}^{d}}-{y_{13}^{u}\over y_{33}^{u}}\right|\epsilon^{4}$ | $\simeq\left|{y_{13}^{d}\over y_{33}^{d}}-{y_{12}^{u}y_{23}^{d}\over y_{22}^{u}y_{33}^{d}}\right|\epsilon^{4}$
Table 3: Leading order expressions for quark mixing angles in terms of the
parameter $\epsilon$ for minimal and non-minimal models.

For our numerical analysis, the values of the expansion parameter $\epsilon$
turn out to be 0.1 and 0.23 for the minimal and non-minimal models,
respectively [9].

###  2.2 The scalar potential

The scalar potential of the model can be written as,

$\displaystyle-V$
$\displaystyle=-\mu^{2}\varphi^{\dagger}\varphi+\lambda(\varphi^{\dagger}\varphi)^{2}-\mu_{\chi}^{2}\,\chi^{*}\chi+\lambda_{\chi}\,(\chi^{*}\chi)^{2}+(\rho\
\chi^{2}+\mathrm{H.c.})+\lambda_{\varphi\chi}(\chi^{*}\chi)(\varphi^{\dagger}\varphi).$
(5)

Under the assumption that there is no Higgs-flavon mixing,
$\lambda_{\varphi\chi}=0$, flavon field can be parametrized by excitations
around its VEV in the following way,

$\displaystyle\chi(x)=\frac{f+s(x)+i\,a(x)}{\sqrt{2}}.$ (6)

where $f$ represents the VEV of the flavon field, while $s$ and $a$ denote its
scalar and pseudoscalar components, respectively.

The minimization conditions for the scalar potential yield the following
expressions for the masses of the scalar and pseudoscalar components of the
flavon,

$\displaystyle m_{s}$
$\displaystyle=\sqrt{\mu_{\chi}-2\rho}=\sqrt{\lambda_{\chi}}f,$ $\displaystyle
m_{a}$ $\displaystyle=\sqrt{-2\rho}.$ (7)

It is evident from equation 7 that the mass of the pseudoscalar is contingent
on the soft-symmetry breaking parameter $\rho$, establishing it as a free
parameter of the model, alongside the VEV $f$.

###  2.3  Bounds on the flavour scale of the $\mathcal{Z}_{\rm
2}\times\mathcal{Z}_{\rm M}$ flavour symmetry

Now we discuss the flavour bounds on the parameter space of the
$\mathcal{Z}_{\rm 2}\times\mathcal{Z}_{\rm M}$ flavour symmetries [9]. We show
the summary of the most stringent bounds on the parameter space of both the
minimal and non-minimal $\mathcal{Z}_{\rm 2}\times\mathcal{Z}_{\rm M}$ models,
where $M=5,9$, in figures 1.

(a)

(b)

Figure 1: Summarized representation of constrained $m_{a}-f$ parameter space
for both the minimal $\mathcal{Z}_{2}\times\mathcal{Z}_{5}$ and non-minimal
$\mathcal{Z}_{2}\times\mathcal{Z}_{9}$ models, depicted in the left and right
panels, respectively.

Currently, the quark flavour observables from $K^{0}-\bar{K}^{0}$ mixing and
$D^{0}-\bar{D}^{0}$ mixing impose stringent constraints on the parameter space
for both the minimal and non-minimal models. The observable $R_{\mu\mu}$,
which is ratio of $BR(B_{d}\rightarrow\mu^{+}\mu^{-})$ and
$BR(B_{s}\rightarrow\mu^{+}\mu^{-})$ turns out to be crucial in the phase-ii@
of the LHCb as it eliminates a large parameter space, as shown in figure 1.

##  3  Flavonic dark matter

The $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ flavour symmetry framework
unveils a remarkable scenario by offering a novel configuration to address
both the flavour problem and the existence of dark matter within a unified and
comprehensive framework. This is achieved by adding the following term to the
scalar potential [7],

$V=-\lambda{\chi^{\tilde{N}}\over\Lambda^{\tilde{N}-4}}+\text{H.c.},$ (8)

where $\lambda=|\lambda|e^{i\alpha}$, and $\tilde{N}$ denotes the least common
multiple of $\rm N$ and $\rm M$ in the $\mathcal{Z}_{\rm
N}\times\mathcal{Z}_{\rm M}$ framework. For a value of $\tilde{N}$ being
larger than a particular threshold, the axial flavon field becomes light
enough to be a DM candidate. We refer to it as the flavonic dark matter (FDM)
[7].
The flavon attains VEV as $\langle\chi\rangle=f/{\sqrt{2}}$, leading to
$\mathcal{Z}_{\rm N}$ symmetry breaking. The axial flavon mass is expressed as
[7],

$m_{a}^{2}={1\over 8}|\lambda|\tilde{N}^{2}\epsilon^{\tilde{N}-4}f^{2}.$ (9)

To achieve the cold dark matter density correctly, the mass of the flavonic
dark matter is determined as,

$m_{a}=3.4\times 10^{-3}{\rm eV}\left(10^{12}{\rm
GeV}\over\varphi_{0}\right)^{4},$ (10)

where $\varphi_{0}$ represents the initial misalignment of the axial flavon
field from its true vacuum during the inflationary phase. For details, see
[7]. Taking $\varphi_{0}=a_{0}f/\tilde{N}$, the required axial flavon mass
$m_{a}$, and VEV $f$ turn out to be,

$m_{a}=0.88\times
10^{16}\left(\epsilon^{\tilde{N}-4}\tilde{N}^{4}\frac{|\lambda|}{a_{0}^{2}}\right)^{2/5}{\rm
eV},$ (11) $f=2.5\times
10^{7}\left(\frac{\tilde{N}^{6}}{a_{0}^{8}|\lambda|\epsilon^{\tilde{N}-4}}\right)^{1/10}{\rm
GeV}.$ (12)

We utilize a framework based on $\mathcal{Z}_{\rm 8}\times\mathcal{Z}_{\rm
22}$ flavour symmetry that furnishes a set-up to impart dark-matter candidate
with simultaneously addressing the flavour problem. Unlike the two prototypes
discussed in the section 2 , the $\mathcal{Z}_{\rm 8}\times\mathcal{Z}_{\rm
22}$ flavour symmetry prohibits the generation of mass of the top quark
through tree-level SM Yukawa operator rather, they are generated by the non-
renormalizable dimension-5 operator.

Fields | $\mathcal{Z}_{8}$ | $\mathcal{Z}_{22}$ | Fields | $\mathcal{Z}_{8}$ | $\mathcal{Z}_{22}$ | Fields | $\mathcal{Z}_{8}$ | $\mathcal{Z}_{22}$ | Fields | $\mathcal{Z}_{8}$ | $\mathcal{Z}_{22}$ | Fields | $\mathcal{Z}_{8}$ | $\mathcal{Z}_{22}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$u_{R}$ | $\omega^{2}$ | $\omega^{2}$ | $c_{R}$ | $\omega^{5}$ | $\omega^{5}$ | $t_{R}$ | $\omega^{6}$ | $\omega^{6}$ | $d_{R}$ | $\omega^{3}$ | $\omega^{3}$ | $s_{R}$ | $\omega^{4}$ | $\omega^{4}$
$b_{R}$ | $\omega^{4}$ | $\omega^{4}$ | $\psi_{L,1}^{q}$ | $\omega^{2}$ | $\omega^{10}$ | $\psi_{L,2}^{q}$ | $\omega$ | $\omega^{9}$ | $\psi_{L,3}^{q}$ | $\omega^{7}$ | $\omega^{7}$ | $\psi_{L,1}^{\ell}$ | $\omega^{3}$ | $\omega^{3}$
$\psi_{L,2}^{\ell}$ | $\omega^{2}$ | $\omega^{2}$ | $\psi_{L,3}^{\ell}$ | $\omega^{2}$ | $\omega^{2}$ | $e_{R}$ | $\omega^{4}$ | $\omega^{12}$ | $\mu_{R}$ | $\omega^{7}$ | $\omega^{7}$ | $\tau_{R}$ | $\omega^{7}$ | $\omega^{21}$
$\nu_{e_{R}}$ | $\omega^{2}$ | $1$ | $\nu_{\mu_{R}}$ | $\omega^{5}$ | $\omega^{3}$ | $\nu_{\tau_{R}}$ | $\omega^{6}$ | $\omega^{4}$ | $\chi$ | $\omega$ | $\omega$ | $H$ | 1 | 1
Table 4: Assignment of charges to the SM and the flavon fields under the
$\mathcal{Z}_{8}$ and $\mathcal{Z}_{22}$ symmetries, where $\omega$ is the 8th
or 22nd root of unity.

Following the charge assignments to the SM and flavon fields under these
symmetries, as given in table 4, the mass matrices for the charged fermions
can be written as,

$M_{u}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{u}\epsilon^{8}&y_{12}^{u}\epsilon^{5}&y_{13}^{u}\epsilon^{4}\\\
y_{21}^{u}\epsilon^{7}&y_{22}^{u}\epsilon^{4}&y_{23}^{u}\epsilon^{3}\\\
y_{31}^{u}\epsilon^{5}&y_{32}^{u}\epsilon^{2}&y_{33}^{u}\epsilon\end{pmatrix},M_{d}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{d}\epsilon^{7}&y_{12}^{d}\epsilon^{6}&y_{13}^{d}\epsilon^{6}\\\
y_{21}^{d}\epsilon^{6}&y_{22}^{d}\epsilon^{5}&y_{23}^{d}\epsilon^{5}\\\
y_{31}^{d}\epsilon^{4}&y_{32}^{d}\epsilon^{3}&y_{33}^{d}\epsilon^{3}\end{pmatrix},M_{\ell}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\ell}\epsilon^{9}&y_{12}^{\ell}\epsilon^{4}&y_{13}^{\ell}\epsilon^{4}\\\
y_{21}^{\ell}\epsilon^{10}&y_{22}^{\ell}\epsilon^{5}&y_{23}^{\ell}\epsilon^{3}\\\
y_{31}^{\ell}\epsilon^{8}&y_{32}^{\ell}\epsilon^{5}&y_{33}^{\ell}\epsilon^{3}\end{pmatrix}.$
(13)

The approximate expressions for masses of the quarks and leptons at the
leading-order are [7],

$\displaystyle\\{m_{t},m_{c},m_{u}\\}$
$\displaystyle\simeq\\{|y_{33}^{u}|\epsilon,~{}\left|y_{22}^{u}\right|\epsilon^{4},~{}\left|y_{11}^{u}\right|\epsilon^{8}\\}v/\sqrt{2},$
$\displaystyle\\{m_{b},m_{s},m_{d}\\}$
$\displaystyle\simeq\\{|y_{33}^{d}|\epsilon^{3},~{}\left|y_{22}^{d}\right|\epsilon^{5},\left|y_{11}^{d}\right|\epsilon^{7}\\}v/\sqrt{2},$
(14) $\displaystyle\\{m_{\tau},m_{\mu},m_{e}\\}$
$\displaystyle\simeq\\{|y_{33}^{l}|\epsilon^{3},~{}\left|y_{22}^{l}\right|\epsilon^{5},~{}\left|y_{11}^{l}\right|\epsilon^{9}\\}v/\sqrt{2}.$

The quark mixing angles are determined as [7],

$\displaystyle\sin\theta_{12}\simeq$
$\displaystyle|V_{us}|\simeq\left|{y_{12}^{d}\over
y_{22}^{d}}-{y_{12}^{u}\over y_{22}^{u}}\right|\epsilon,\sin\theta_{23}\simeq$
$\displaystyle|V_{cb}|\simeq\left|{y_{23}^{d}\over
y_{33}^{d}}-{y_{23}^{u}\over y_{33}^{u}}\right|\epsilon^{2},$
$\displaystyle\sin\theta_{13}\simeq$
$\displaystyle|V_{ub}|\simeq\left|{y_{13}^{d}\over
y_{33}^{d}}-{y_{12}^{u}y_{23}^{d}\over y_{22}^{u}y_{33}^{d}}-{y_{13}^{u}\over
y_{33}^{u}}\right|\epsilon^{3}.$ (15)

For $\mathcal{Z}_{\rm 8}\times\mathcal{Z}_{\rm 22}$ flavour symmetry, the
value of $\tilde{N}$ is determined to be 88, which using in equations 11 and
12 result in,

$\displaystyle f\approx 1.0\times
10^{14}\,\rm{GeV},~{}~{}\text{and}~{}~{}m_{a}\approx 1.9\times
10^{-3}\,\rm{eV},$ (16)

assuming $\epsilon=0.225$ with $|\lambda|=1$ and $a_{0}=1$. To ensure the
prolonged stability of flavonic dark matter, its decay into electrons and
positrons pair must be prohibited, implying $m_{a}<2m_{e}$. This condition
sets a rigorous constraint on the parameter $\tilde{N}$ and VEV $f$ for the
existence of flavonic dark matter, specifically requiring that [7],

$\tilde{N}>53,~{}~{}\mbox{and}~{}~{}f>4\times 10^{11}\mbox{GeV}.$ (17)

The decays of FDM to neutrino is highly suppressed due to coupling of the
neutrinos to DM being of the order of $\epsilon^{20}$. The axial coupling of
photon to DM turns out to be $g_{\varphi\gamma\gamma}=\frac{\alpha}{2\pi
v_{F}}\frac{5}{3}$, which results in the corresponding decay width
$\Gamma(a\rightarrow\gamma\gamma)=1.69\times 10^{-68}\rm GeV$ [7]. These
results suggest that the lifetimes of these processes exceed the age of the
universe, indicating the stability of flavonic dark matter.

Figure 2:  Predicted mass ranges for detection of flavonic dark matter
(depicted by the thick green line) and axion-like particles. [14].

The predicted photon coupling versus the flavonic dark matter (FDM) mass is
presented in figure 2. It reveals that FDM masses exceeding approximately 1
keV, corresponding to $\tilde{N}<67$ and $v_{F}<4\times 10^{12}$ GeV, are
excluded. This exclusion aligns with recent constraints from INTEGRAL/SPI data
[11]. Notably, these predictions also coincide with that of the GUT-scale QCD
axion mass at around $10^{-9}$ eV, leading to a prospect for investigations in
future [12]. The mass range above the keV level can be explored further
through the upcoming THESEUS experiment [13].

##  4 The framework of the VEV hierarchy

In this section, we turn up to an approach that stands in stark contrast to
the FN mechanism relying on $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$
flavour symmetry but shares the use of discrete symmetries. The Hierarchical
VEVs model (HVM) firstly proposed in [4] redefines the flavour problem by
incorporating three Abelian discrete symmetries $\mathcal{Z}_{2}$,
$\mathcal{Z}_{2}^{\prime}$, $\mathcal{Z}_{2}^{\prime\prime}$ and six gauge
singlet scalar fields $\chi_{i},(i=1-6)$, whose VEVs account for both the
fermionic mass spectrum and flavour mixing. The masses of the SM fermions are
produced by the following dimension-5 operators,

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Lambda}\Bigl{[}y_{ij}^{u}\bar{\psi}_{L_{i}}^{q}\tilde{\varphi}\psi_{R_{i}}^{u}\chi_{i}+y_{ij}^{d}\bar{\psi}_{L_{i}}^{q}\varphi\psi_{R_{i}}^{d}\chi_{i+3}+y_{ij}^{\ell}\bar{\psi}_{L_{i}}^{\ell}\varphi\psi_{R_{i}}^{\ell}\chi_{i+3}\Bigr{]}+{\rm
H.c.}.$ (18)

The hierarchy in the masses and mixing patterns is explained by VEVs hierarchy
in a way that $\langle\chi_{4}\rangle\sim\langle\chi_{1}\rangle$,
$\langle\chi_{2}\rangle>>\langle\chi_{5}\rangle$,
$\langle\chi_{3}\rangle>>\langle\chi_{6}\rangle$,
$\langle\chi_{3}\rangle>>\langle\chi_{2}\rangle>>\langle\chi_{1}\rangle$, and
$\langle\chi_{6}\rangle>>\langle\chi_{5}\rangle>>\langle\chi_{4}\rangle$.

To account for the neutrino masses, an additional gauge singlet scalar field
$\chi_{7}$ is added, and neutrino mass patterns are recovered through the
type-i@ seesaw mechanism. The HVM framework, however, does not offer any
explanation to the leptonic mixing angles. This problem is addressed by a
standard HVM (SHVM)[5], which provides remarkably precise predictions of
leptonic mixing angles by linking them to the Cabibbo angle in the form of
inequalities.

The realization of SHVM involves enlarging the HVM framework by extending the
SM through the generic $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ symmetry. To produce the flavour structure of the
SM assuming normal hierarchy for neutrino mass patterns, one must ensure that
$\rm N=2$, $\rm M\geq 3$, and $\rm P\geq 14$ [5]. We take the
$\mathcal{Z}_{2}\times\mathcal{Z}_{4}\times\mathcal{Z}_{14}$ symmetry, and the
charges to different SM fermionic and scalar fields under this symmetry are
assigned, as shown in table 5.

Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$u_{R}$ | - | $1$ | $\omega^{11}$ |  $d_{R}$, $s_{R}$, $b_{R}$ | + | $1$ | $\omega^{12}$ | $\psi_{L_{3}}^{q}$ | + | $1$ | $\omega^{2}$ | $\tau_{R}$ | + | $\omega^{3}$ | $\omega$
$c_{R}$ | + | $1$ | $\omega^{6}$ | $\chi_{4}$ | + | $1$ | $\omega^{13}$ | $\psi_{L_{1}}^{\ell}$ | + | $\omega^{3}$ | $\omega^{12}$ | $\nu_{e_{R}}$ | + | $\omega$ | $\omega^{8}$
$t_{R}$ | + | $1$ | $\omega^{4}$ | $\chi_{5}$ | + | $1$ | $\omega^{11}$ | $\psi_{L_{2}}^{\ell}$ | + | $\omega^{3}$ | $\omega^{10}$ | $\nu_{\mu_{R}}$ | - | $\omega$ | $\omega^{3}$
$\chi_{1}$ | - | $1$ | $\omega^{2}$ | $\chi_{6}$ | + | $1$ | $\omega^{10}$ | $\psi_{L_{3}}^{\ell}$ | + | $\omega^{3}$ | $\omega^{6}$ | $\nu_{\tau_{R}}$ | - | $\omega$ | $\omega^{3}$
$\chi_{2}$ | + | $1$ | $\omega^{5}$ | $\psi_{L_{1}}^{q}$ | + | $1$ | $\omega^{13}$ | $e_{R}$ | - | $\omega^{3}$ | $\omega^{10}$ | $\chi_{7}$ | - | $\omega^{2}$ | $\omega^{8}$
$\chi_{3}$ | + | $1$ | $\omega^{2}$ | $\psi_{L_{2}}^{q}$ | + | $1$ | $\omega$ | $\mu_{R}$ | + | $\omega^{3}$ | $\omega^{13}$ | $\varphi$ | + | 1 | 1
Table 5: Charges assigned to SM fermions and scalar fields under
$\mathcal{Z}_{2}$, $\mathcal{Z}_{4}$ and $\mathcal{Z}_{14}$ symmetries,
considering normal mass ordering for neutrinos.

The mass patterns of charged fermions are obtained through dimension-5
operators in the Lagrangian given in equation 18, considering the VEVs pattern
of $\chi_{i}$ fields as discussed above. The corresponding mass matrices are,

$\displaystyle{\cal M}_{\cal U}$
$\displaystyle=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{u}\epsilon_{1}&0&y_{13}^{u}\epsilon_{2}\\\
0&y_{22}^{u}\epsilon_{2}&y_{23}^{u}\epsilon_{5}\\\
0&y_{32}^{u}\epsilon_{6}&y_{33}^{u}\epsilon_{3}\end{pmatrix},{\cal M}_{\cal
D}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{d}\epsilon_{4}&y_{12}^{d}\epsilon_{4}&y_{13}^{d}\epsilon_{4}\\\
y_{21}^{d}\epsilon_{5}&y_{22}^{d}\epsilon_{5}&y_{23}^{d}\epsilon_{5}\\\
y_{31}^{d}\epsilon_{6}&y_{32}^{d}\epsilon_{6}&y_{33}^{d}\epsilon_{6}\\\
\end{pmatrix},{\cal
M}_{\ell}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\ell}\epsilon_{1}&y_{12}^{\ell}\epsilon_{4}&y_{13}^{\ell}\epsilon_{5}\\\
0&y_{22}^{\ell}\epsilon_{5}&y_{23}^{\ell}\epsilon_{2}\\\
0&0&y_{33}^{\ell}\epsilon_{2}\\\ \end{pmatrix},$ (19)

The masses of the charged fermions are retained as [5],

$\displaystyle m_{t}$ $\displaystyle=$ $\displaystyle\
\left|y^{u}_{33}\right|\epsilon_{3}v/\sqrt{2},~{}m_{c}=\
\left|y^{u}_{22}\epsilon_{2}-\dfrac{y_{23}^{u}y_{32}^{u}\epsilon_{5}\epsilon_{6}}{y_{33}^{u}\epsilon_{3}}\right|v/\sqrt{2},~{}m_{u}=|y_{11}^{u}|\,\epsilon_{1}v/\sqrt{2},$
$\displaystyle m_{b}$ $\displaystyle\approx$ $\displaystyle\
|y^{d}_{33}|\epsilon_{6}v/\sqrt{2},m_{s}\approx\
\left|y^{d}_{22}-\dfrac{y_{23}^{d}y_{32}^{d}}{y_{33}^{d}}\right|\epsilon_{5}v/\sqrt{2},$
$\displaystyle m_{d}$ $\displaystyle\approx$
$\displaystyle\left|y_{11}^{d}-{y_{12}^{d}y_{21}^{d}\over
y^{d}_{22}-\dfrac{y_{23}^{d}y_{32}^{d}}{y_{33}^{d}}}-{{y_{13}^{d}(y_{31}^{d}y_{22}^{d}-y_{21}^{d}y_{32}^{d})-y_{31}^{d}y_{12}^{d}y_{23}^{d}}\over{(y^{d}_{22}-\dfrac{y_{23}^{d}y_{32}^{d}}{y_{33}^{d}})y^{d}_{33}}}\right|\,\epsilon_{4}v/\sqrt{2},$
$\displaystyle m_{\tau}$ $\displaystyle\approx$ $\displaystyle\
|y^{\ell}_{33}|\epsilon_{2}v/\sqrt{2},~{}m_{\mu}\approx\
|y^{\ell}_{22}|\epsilon_{5}v/\sqrt{2},~{}m_{e}=|y_{11}^{\ell}|\,\epsilon_{1}v/\sqrt{2}.$
(20)

The quark mixing angles are obtained as,

$\displaystyle\sin\theta_{12}$
$\displaystyle\simeq\left|\frac{y_{12}^{d}\epsilon_{4}}{y_{22}^{d}\epsilon_{5}}\right|=\frac{\epsilon_{4}}{\epsilon_{5}},\quad\sin\theta_{23}\simeq\left|\frac{y_{23}^{d}\epsilon_{5}}{y_{33}^{d}\epsilon_{6}}-\frac{y_{23}^{u}\epsilon_{5}}{y_{33}^{u}\epsilon_{3}}\right|\approx\left|\frac{y_{23}^{d}}{y_{33}^{d}}\right|\sin\theta_{12}^{2},$
(21) $\displaystyle\sin\theta_{13}$
$\displaystyle\simeq\left|\frac{y_{13}^{d}\epsilon_{4}}{y_{33}^{d}\epsilon_{6}}-\frac{y_{13}^{u}\epsilon_{2}}{y_{33}^{u}\epsilon_{3}}\right|\geq\left|\frac{y_{13}^{d}}{y_{33}^{d}}\right|\sin\theta_{12}^{3}-\left|\frac{y_{13}^{u}}{y_{33}^{u}}\right|\frac{m_{c}}{m_{t}}\approx\sin\theta_{12}^{3}-2\frac{m_{c}}{m_{t}},$
(22)

where $\epsilon_{i}=\dfrac{\langle\chi_{i}\rangle}{\Lambda}$,
$\epsilon_{i}<1$, and we made the assumptions that
$\epsilon_{4}/\epsilon_{5}=\sin\theta_{12}=0.225$,
$\epsilon_{5}/\epsilon_{6}=\sin\theta_{23}$, $|y_{13}^{d}/y_{33}^{d}|\approx
1$, and $|y_{13}^{u}/y_{33}^{u}|\approx 2$. Consequently, the SHVM predicts
the quark mixing angle $\sin\theta_{13}$ to be approximately
$\sin\theta_{12}^{3}-2(m_{c}/m_{t})$, aligning well with its experimentally
observed value.

###  4.1 Neutrino masses and leptonic mixing parameters

To recover neutrino masses, we add three right-handed neutrinos $\nu_{eR}$,
$\nu_{\mu R}$, $\nu_{\tau R}$ to the SM and write the corresponding Yukawa
Lagrangian as,

$\displaystyle-{\mathcal{L}}_{\rm Yukawa}^{\nu}$ $\displaystyle=$
$\displaystyle
y_{ij}^{\nu}\bar{\psi}_{L_{i}}^{\ell}\tilde{\varphi}\nu_{f_{R}}\left[\dfrac{\chi_{i}\chi_{j}(\text{or}~{}\chi_{i}\chi_{j}^{\dagger})}{\Lambda^{2}}\right]+{\rm
H.c.}.$ (23)

The Dirac mass matrix turn out to be,

${\cal M}_{{\cal
D}}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\nu}\epsilon_{1}\epsilon_{7}&y_{12}^{\nu}\epsilon_{4}\epsilon_{7}&y_{13}^{\nu}\epsilon_{4}\epsilon_{7}\\\
0&y_{22}^{\nu}\epsilon_{4}\epsilon_{7}&y_{23}^{\nu}\epsilon_{4}\epsilon_{7}\\\
0&y_{32}^{\nu}\epsilon_{5}\epsilon_{7}&y_{33}^{\nu}\epsilon_{5}\epsilon_{7}\end{pmatrix},$
(24)

which results in following mass expressions for neutrinos,

$\displaystyle m_{3}$ $\displaystyle\approx$
$\displaystyle|y^{\nu}_{33}|\epsilon_{5}\epsilon_{7}v/\sqrt{2},m_{2}\approx\left|y^{\nu}_{22}-\dfrac{y_{23}^{\nu}y_{32}^{\nu}}{y_{33}^{\nu}}\right|\epsilon_{4}\epsilon_{7}v/\sqrt{2},m_{1}\approx|y_{11}^{\nu}|\,\epsilon_{1}\epsilon_{7}v/\sqrt{2}.$
(25)

For the benchmark values of $y_{ij}^{\nu}$ obtained through our numerical
analysis [5], the masses of neutrinos turn out to have the values
$\\{m_{3},m_{2},m_{1}\\}=\\{5.05\times 10^{-2},8.67\times 10^{-3},2.67\times
10^{-4}\\}\,\text{eV}$. The predictions for leptonic mixing are pivotal in
SHVM. Assuming all the couplings to be of order one, the leptonic mixing
angles can be expressed in terms of the Cabibbo angle in the following way
[5],

$\displaystyle\sin\theta_{12}^{\ell}$ $\displaystyle\simeq$
$\displaystyle\left|-{y_{12}^{\nu}\over
y_{22}^{\nu}}+{y_{12}^{\ell}\epsilon_{4}\over
y_{22}^{\ell}\epsilon_{5}}+{y_{23}^{\ell*}y_{13}^{\nu}\epsilon_{4}\over
y_{33}^{\ell}y_{33}^{\nu}\epsilon_{5}}\right|\geq\left|-{y_{12}^{\nu}\over
y_{22}^{\nu}}\right|-\left|{y_{12}^{\ell}\over
y_{22}^{\ell}}+{y_{23}^{\ell*}y_{13}^{\nu}\over
y_{33}^{\ell}y_{33}^{\nu}}\right|{\epsilon_{4}\over\epsilon_{5}}\geq
1-2\sin\theta_{12},$ (26) $\displaystyle\sin\theta_{23}^{\ell}$
$\displaystyle\simeq$ $\displaystyle\left|{y_{23}^{\ell}\over
y_{33}^{\ell}}-{y_{23}^{\nu}\epsilon_{4}\over
y_{33}^{\nu}\epsilon_{5}}\right|\geq\left|{y_{23}^{\ell}\over
y_{33}^{\ell}}\right|-\left|{y_{23}^{\nu}\over
y_{33}^{\nu}}\right|{\epsilon_{4}\over\epsilon_{5}}\geq 1-\sin\theta_{12},$
$\displaystyle\sin\theta_{13}^{\ell}$ $\displaystyle\simeq$
$\displaystyle\left|-{y_{13}^{\nu}\epsilon_{4}\over
y_{33}^{\nu}\epsilon_{5}}+{y_{13}^{\ell}\epsilon_{5}\over
y_{33}^{\ell}\epsilon_{2}}\right|\geq\left|-{y_{13}^{\nu}\over
y_{33}^{\nu}}\right|{\epsilon_{4}\over\epsilon_{5}}-\left|{y_{13}^{\ell}\over
y_{33}^{\ell}}\right|{\epsilon_{5}\over\epsilon_{2}}\geq\sin\theta_{12}-\frac{m_{s}}{m_{c}},$

where $m_{s}/m_{c}=\epsilon_{5}/\epsilon_{2}$. The latest measurement of
Cabibbo angle yields $\sin\theta_{12}=(0.225\pm 0.00067)$ [15]. It is clear
from above equations that $\sin\theta_{12}^{\ell}$ as well as
$\sin\theta_{23}^{\ell}$ shares the same precision of the Cabibbo angle's
experimental precision. Using the Cabibbo angle measurement, leptonic mixing
angles are obtained as $\sin\theta_{12}^{\ell}=0.55\pm 0.00134$ and
$\sin\theta_{23}^{\ell}=0.775\pm 0.00067$.

Using running masses of strange and charm quarks at 1 TeV
($m_{c}=0.532^{+0.074}_{-0.073}$ GeV, $m_{s}=4.7^{+1.4}_{-1.3}\times 10^{-2}$
GeV) [16], we obtain $\sin\theta_{13}^{\ell}=(0.1169-0.1509)$. However, the
experimental lower end excludes $(0.1169-0.1413)$, and the theory rules out
the upper end $(0.1509-0.1550)$. Thus, the final conclusive range for
$\sin\theta_{13}^{\ell}$ is predicted to be $0.1413-0.1509$. The upcoming
neutrino experiments like DUNE, Hyper-Kamiokande, and JUNO may test these
predictions [17].
The SHVM rules out the possibility of neutrino masses originating from
Majorana-type neutrinos, as the charge assignment of SHVM forbids both the
Majorana Lagrangian and the Weinberg operators [5]. Consequently, the SHVM
predicts neutrinos to be of the Dirac-type [5].

##  5  The dark technicolour paradigm

A UV completion of the SHVM can be obtained through the dark-technicolour
framework [6]. The dark-technicolour model consists of the symmetry
$\mathcal{S}=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times
SU(\rm{N}_{\rm{TC}})\times SU(\rm{N}_{\rm{DTC}})\times SU(\rm{N}_{\rm{F}})$
symmetry, where TC stands for technicolour, DTC is for the dark-technicolour,
and $\rm F$ is a strong dynamics of vector-like fermions [4]. The TC quarks
transform under the symmetry $\mathcal{S}$ as [4],

$\displaystyle{T_{L}}^{i}$ $\displaystyle\equiv$
$\displaystyle\begin{pmatrix}T_{i}\\\
B_{i}\end{pmatrix}_{L}\mathrel{\mathop{\mathchar
58\relax}}(1,2,0,\rm{N}_{TC},1,1),T_{i,R}\mathrel{\mathop{\mathchar
58\relax}}(1,1,1,\rm{N}_{TC},1,1),$ (27) $\displaystyle B_{i,R}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle(1,1,-1,\rm{N}_{TC},1,1),$

where electric charges $+\frac{1}{2}$ for $T$ and $-\frac{1}{2}$ for $B$.

The $\rm DTC$ dynamics has the following fermions,

$\displaystyle D$ $\displaystyle\equiv$ $\displaystyle
C_{i,L,R}\mathrel{\mathop{\mathchar 58\relax}}(1,1,1,1,\rm{N}_{\rm
DTC},1),~{}S_{i,L,R}\mathrel{\mathop{\mathchar 58\relax}}(1,1,-1,1,\rm{N}_{\rm
DTC},1),$ (28)

where electric charges of the quark $\mathcal{C}$ is $+\frac{1}{2}$, and that
of the $\mathcal{S}$ is $-\frac{1}{2}$.

The vector-like fermions of $SU(\rm N_{\rm F})$ dynamics transform as,

$\displaystyle F_{L,R}$ $\displaystyle\equiv$ $\displaystyle
U_{L,R}^{i}\equiv(3,1,\dfrac{4}{3},1,1,\rm{N}_{\rm
F}),D_{L,R}^{i}\equiv(3,1,-\dfrac{2}{3},1,1,\rm{N}_{\rm F}),$ (29)
$\displaystyle N_{L,R}^{i}$ $\displaystyle\equiv$
$\displaystyle(1,1,0,1,1,\rm{N}_{\rm
F}),E_{L,R}^{i}\equiv(1,1,-2,1,1,\rm{N}_{\rm F}).$

There are three axial $U(1)_{A}^{\rm TC,DTC,F}$ symmetries in the DTC paradigm
which are broken as $U(1)_{A}^{\rm TC,DTC,F}\rightarrow\mathcal{Z}_{2\rm
K_{\rm TC,DTC,F}}$ producing the generic $\mathcal{Z}_{\rm
N}\times\mathcal{Z}_{\rm M}\times\mathcal{Z}_{\rm P}$ flavour symmetry, where
$\rm N=2\rm K_{\rm TC}$, $\rm M=2\rm K_{\rm DTC}$ and $\rm P=2\rm K_{\rm F}$
[18].

Figure 3: Feynman diagrams corresponding to charged fermion masses in the
dark-technicolour paradigm. The diagram at the top depicts generic
interactions of SM, TC, F, and DTC fermions, while diagram at the bottom shows
the formation of fermionic condensates resulting in masses for charged SM
fermions.

The mass of a charged fermion is originated from the interactions given in
fig. 3 and turns out to be [6],

$\displaystyle m_{f}$ $\displaystyle=$ $\displaystyle
y_{f}\frac{\Lambda_{\text{TC}}^{3}}{\Lambda_{\text{ETC}}^{2}}\dfrac{1}{\Lambda_{\text{F}}}\frac{\Lambda_{\text{DTC}}^{n_{i}+1}}{\Lambda_{\text{EDTC}}^{n_{i}}}\exp(n_{i}k),~{}$
(30)

where $n_{i}=2,4,\cdots 2n$ represents the number of fermions in a multi-
fermion chiral condensate playing the role of the VEV $\langle\chi_{i}\rangle$
in the SHVM [4]. The multi-fermion condensate can be parametrized as [19],

$\langle(\bar{\psi}_{L}\psi_{R})^{n}\rangle\sim\left(\Lambda\exp(k\Delta\chi)\right)^{3n},$
(31)

where $\Delta\chi$ shows the chirality of an operator, $k$ denotes a constant,
and $\Lambda$ is the scale of the gauge theory. From this, we conclude that,

$\displaystyle\epsilon_{i}=\dfrac{1}{\Lambda_{\text{F}}}\frac{\Lambda_{\text{DTC}}^{n_{i}+1}}{\Lambda_{\text{EDTC}}^{n_{i}}}\exp(n_{i}k).$
(32)

Figure 4: Feynman diagrams corresponding to the masses of neutrinos in the
dark-technicolour paradigm. The diagram at the top shows generic interactions
among the SM, TC, DTC gauge sectors, while the diagram at the bottom
represents the formation of fermionic condensates leading to neutrino masses.

The neutrino masses are recovered by the generic interactions given in figure
4, where it is assumed that the TC and DTC sectors are accommodated in a GUT
theory where the interactions between the $F_{L}$ and $F_{R}$ fermions are
mediated by the gauge bosons of the GUT. The VEV $\langle\chi_{7}\rangle$
corresponds to the chiral condensate $\langle\bar{F}_{L}F_{R}\rangle$. The
masses of neutrinos become,

$\displaystyle m_{\nu}$ $\displaystyle=$ $\displaystyle
y_{f}\frac{\Lambda_{\text{TC}}^{3}}{\Lambda_{\text{ETC}}^{2}}\dfrac{1}{\Lambda_{\text{F}}}\frac{\Lambda_{\text{DTC}}^{n_{i}+1}}{\Lambda_{\text{EDTC}}^{n_{i}}}\exp(n_{i}k)\dfrac{1}{\Lambda_{\text{F}}}\frac{\Lambda_{\text{F}}^{3}}{\Lambda_{\text{GUT}}^{2}}\exp(2k).$
(33)

For more details, see ref. [6].

##  6  Summary

In this work, we have discussed new approaches to address the flavour problem
and dark matter together. The flavour bounds on the models based on
$\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm M}$ flavour symmetries are
extremely stringent in the phase-ii@ of the LHCb, and most of the parameter
space is ruled out.

These new frameworks provides origins of the flavour problem and dark matter
based on discrete $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ flavour symmetries. Origin of these symmetries
may lie in a technicolour paradigm. We notice that these new frameworks result
in a new class of dark matter referred to as the flavonic dark matter.

## Acknowledgement

This work is supported by the Council of Science and Technology, Govt. of
Uttar Pradesh, India through the project `` A new paradigm for flavour problem
" no. CST/D-1301, and Science and Engineering Research Board, Department of
Science and Technology, Government of India through the project ``Higgs
Physics within and beyond the Standard Model" no. CRG/2022/003237. N. S.
acknowledges the support through the INSPIRE fellowship by the Department of
Science and Technology, Government of India.

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