August 27, 2024

#  Cross-correlation Power Spectra and Cosmic Birefringence of the CMB via
Photon-neutrino Interaction

Roohollah Mohammadi<EMAIL_ADDRESS>Iranian National Museum of Science and
Technology (INMOST), PO Box 11369-14611, Tehran, Iran.
School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.
O. Box 19395-5531, Tehran, Iran.  Jafar Khodagholizadeh<EMAIL_ADDRESS>Farhangian University, P.O. Box 11876-13311, Tehran, Iran.  Mahdi Sadegh
<EMAIL_ADDRESS>School of Particles and Accelerators, Institute for Research
in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran.  Ali Vahedi
<EMAIL_ADDRESS>Department of Astronomy and High Energy Physics,
Faculty of Physics, Kharazmi University, P. O. Box 15614, Tehran, Iran.  S. S.
Xue<EMAIL_ADDRESS>ICRANet Piazzale della Repubblica, 10 -65122, Pescara, Italy.
Physics Department, University of Rome La Sapienza, Rome, Italy.
INFN, Sezione di Perugia, Perugia, Italy
ICTP-AP, University of Chinese Academy of Sciences, Beijing, China.

###### Abstract

In the context of the standard model of particles, the weak interaction of
cosmic microwave background (CMB) and cosmic neutrino background (C$\nu$B),
can generate non-vanishing TB and EB power spectra in the order of one loop
forward scattering, in the presence of scalar perturbation, which is in
contrast with the standard scenario cosmology. Comparing our results with the
current experimental data may provide, significant information about the
nature of C$\nu$B, including CMB-C$\nu$B forward scattering for TB, TE, and EB
power spectra. To this end, different cases were studied, including Majorana
C$\nu$B and Dirac C$\nu$B. On the other hand, it was shown that the mean
opacity due to cosmic neutrino background could behave as an anisotropic
birefringent medium and change the linear polarization rotation angle.
Considering the contributions from neutrino and anti-neutrino forward
scattering with CMB photons (in the case of Dirac neutrino), we introduce
relative neutrino and anti-neutrino density asymmetry
($\delta_{\nu}=\frac{\Delta
n_{\nu}}{n_{\nu}}=\frac{n_{\nu}-n_{\bar{\nu}}}{n_{\nu}}$). Then, using the
cosmic birefringence angle reported by the Planck data release
$\beta=0.30^{\circ}\pm 0.11^{\circ}$ ($68\%C.L.$), some constraints can be put
on $\delta_{\nu}$. Also, the value of cosmic birefringence due to Majorana
C$\nu$B medium is estimated at about $\beta|_{\nu}\simeq 0.2$ rad. In this
respect, since Majorana neutrino and anti-neutrino are exactly the same, both
CB contributions will be added together. However, this value is at least two
orders larger than the cosmic birefringence angle reported by the Planck data
release, $\beta=0.30^{\circ}\pm 0.11^{\circ}$ ($68\%C.L.$). Finally, we
shortly discussed this big inconsistency. It is noteworthy that to calculate
the contribution of photon-neutrino forward scattering for cosmic
birefringence, we just consider the standard model of particles and the
standard scenario of cosmology.

Tensor modes fluctuations, Curved cosmology, Neutrinos.

###### pacs:

13.15.+g,34.50.Rk,13.88.+e

##  I Introduction

Cosmic neutrinos can decouple deep earlier from matter about 1 sec after the
Big Bang (BB) at the temperature of $\cong 1MeV\cong 10^{10}K$. Therefore,
along with the cosmic microwave background (CMB) coming from 380000 years
after BB, the cosmic neutrino background (C$\nu$B) contains valuable
information about the history of the universe weinberg . Today, It is very
hard to detect C$\nu$B due to its very weak interactions and low energy
($1.95K\cong 1.68\times 10^{-4}eV$).
Moreover, a relativistic neutrino beam can not help make a difference, at
least in observation, between the Dirac and Majorana neutrino in the case of
electromagnetic interaction. This fact directly relates to the left-handed
interacting neutrino in the SM context, which is called the ”Practical Dirac-
Majorana confusion theorem” kayser . However the very low energy of C$\nu$B
that makes these particles is in the non-relativistic regime. Therefore, the
Dirac or Majorana nature of neutrino has different electromagnetic effects
palbook  detected by upgrading experiments. Recently, some experiments have
been conducted to detect C$\nu$B. For example, KATRIN equipped to C$\nu$B by
the most probable interaction channel to capture the electron neutrino from
the background by Tritium
($\nu_{e}(1.95Kelvin)+{}^{3}H\rightarrow{}^{3}He+e^{-}$) Faessler:2016tjf .
Nevertheless, many theoretical attempts have been proposed in the literature
to search for these relic neutrinos Faessler:2011qj ; Lazauskas:2007da ;
Munyaneza:2006kw ; Pas:2001nd ; weiler ; Wigmans:2002rb ; Ringwald:2003qa ;
Ringwald:2004np . The detection method of C$\nu$B is divided into three
categories: (1) direct detection via momentum transfer in neutrino elastic
scattering off nuclei target, (2) neutrino capture in nuclei, or (3) indirect
method by spectral distortion generated by C$\nu$B and ultra-high energy
neutrinos or protons originating from unknown sources yanagisawa . The common
issue of all these categories is that no search has been done in any of them
for the nature of Dirac or Majorana neutrinos. Another indirect detection
method of C$\nu$B was proposed by Mohammadi et al. Mohammadi:2016bxl ;
Khodagholizadeh:2014nfa ; Mohammadi:2013dea ; Mohammadi:2013ksa . In this
method, photon-neutrino forward scattering is considered a tool for probing
C$\nu$B that distinguishes the Majorana and Dirac nature of C$\nu$B.
The CMB photons can interact electromagnetically with matter. Therefore, it is
possible to test very vast theoretical models of particle physics and
cosmology Abazajian:2013oma ; Abazajian:2016yjj ; Jeong:2019zaz ;
Kamionkowski:2015yta ; Zucca:2016iur ; Kosowsky:1994cy . The intensity and
polarization of CMB are used to extract the required information.
The most commonly used observables to investigate theoretically and
experimentally are the CMB polarization power spectra. The B-mode
($C_{\ell}^{BB}$) power spectrum can be used to study primordial gravitational
waves Seljak:1996gy . In the standard scenario of cosmology, the CMB B-mode
polarization due to the Thomson scattering can only be generated in the
presence of tensor metric perturbations Zaldarriaga:1996xe ;
Zaldarriaga:1997ch ; Hu . Gravitational lensing is also another source of
B-mod SPTpol:2013omd . Thereupon, the B-mode power spectrum can calculate one
of the most important cosmological parameters as tensor-to-scalar
(r-parameter) cardoso ; Vazquez:2013dva . E-mode polarization is generated by
both scalar and tensor metric fluctuations. Nowadays, E-mode power spectrum
($C_{\ell}^{EE}$) measurements are almost consistent with the $\Lambda$CDM
cosmological model Henning:2017nuy ; Aghanim:2019ame ; Louis:2016ahn . It is
of note that temperature-polarization power spectrum ($C_{\ell}^{TE}$)
measurements can surpass our information about CMB temperature anisotropy
($C_{\ell}^{TT}$) and put some stronger constraints on cosmological model.
This effect is attributed to the lower levels of polarization foreground
measurements Galli ; Louis:2016ahn . This point motivates us to investigate
some of the main cross-correlation power spectra such as $TE$, $TB$, and $EB$.
In the standard scenario of cosmology, cross-correlations of $EB$ and $TB$ are
due to the parity invariant vanishing Kamionkowski:1996zd ; Seljak:1996gy .
Thus, these cross-correlations can be used to check some parity-violating
interactions and go beyond the standard models kamion1999 . In liu ; Carroll ,
the authors have theoretically investigated the quintessence and Axion-like
dark matter particle coupled to the electromagnetic strength tensor and it’s
dual ($\mathcal{L}=\phi(t)F_{\mu\nu}\tilde{F}^{\mu\nu}$). Besides, they showed
that can generate non-zero $EB$ and $TB$ power spectra. In these theoretical
models, the plane of CMB linear polarization (E-mode and B-mode polarizations)
can rotate by an angle proportional to $\int^{t_{0}}_{t_{\rm
LSS}}dt\leavevmode\nobreak\ \dot{\phi}$ which is known as cosmic birefringence
(CB) Gluscevic . In another paper Geng  authors proposed a CPT-even six-
dimensional effective Lagrangian to consider the effect of neutrino density
asymmetry on CB angle. The authors in Geng  did not discuss the nature of
neutrinos in their calculation. But, in this work, we combine the result of
Geng  on the density asymmetry of neutrino and anti-neutrino and the direct
one-loop calculation of neutrino (anti-neutrino)-photon scattering amplitude
for Dirac neutrino. In addition, we calculate the same method separately for
Majorana neutrino. In conventional physics, the cosmological birefringent
medium is caused by some radiation anisotropies of the background along the
cosmological path of CMB since the last scattering surface time. These
anisotropies, for example, include finite displacement of electrons in the
ionized plasma, photon-photon scattering due to the virtual electron-positron
pairs in a vacuum, and lack of hydrogen atom potential as a simple harmonic
oscillator Montero-Camacho:2018vgs .
Moreover, in the presence of cosmic neutrino background, their interaction
with cosmic microwave background via parity violating forward scattering makes
an opaque medium: this opacity acts as a birefringent medium. Using the
phenomenological value of this opacity, we can calculate the contribution of
CMB-C$\nu$B interaction at the measured CB angle. Recently, This CB has been
measured by analyzing the 2022 cosmological foreground polarization data of
Planck ($\beta=0.3^{\circ}\pm 0.11^{\circ}$) for small $l$ at a 68% confidence
level minami . Also, some other experiments have measured CB that rely on
polarization angle calibration based on hardware calibrators and astrophysical
sources Ade:2014afa ; Bianchini:2020osu ; Namikawa:2020ffr . Contrary to
minami  their results were consistent with zero.
In Mohammadi:2016bxl , we studied the generation of CMB B-mode polarization
via forward scattering of photon-neutrino in the case of tensor metric
perturbation. The result showed that the effect of Majorana neutrinos on the
$r$-parameter was different from that of the Dirac neutrinos.
Distinguishing Dirac from Majorana neutrinos can be used in the microwave
cavity Abdi:2019hvx , $SU(6)$ grand unified theories Chacko:2020tbu , weak
interaction of aggregate matter Segarra:2020rah , neutrino
decayBalantekin:2018ukw , pure leptonic decays at the LHC Arbelaez:2017zqq ,
neutrino-electron elastic scattering in the presence of all possible Lorentz-
invariant interactions Rodejohann:2017vup , and non-cyclic geometric phase for
neutrinos Capolupo:2016idi ; Lu:2021yav ; Johns:2021ets .
In this work, we will solve the quantum Boltzmann equation for the time-
evolution of matrix density (Stokes parameters) of an ensemble of CMB photons.
The CMB-C$\nu$B scattering is involved here as a collision term. The
CMB-C$\nu$B weak interaction not only can generate non-zero EB and TB power
spectra but also can identify the Dirac and Majorana nature of C$\nu$B.
Moreover, the generated neutrino opacity via CMB-C$\nu$B interaction behaves
as a birefringent medium that can change the CMB linear polarization angle.
Therefore, the observed cross-power spectra between E-mode, B-mode, and
temperature in the background of scalar perturbations have a significant
contribution to measured CB. Hence, this scattering should be regarded among
the most effective phenomena. To give an effective sense, we show that the CB
angle $\beta$ can be roughly estimated by the value of opacity in $z=1100$ due
to Majorana C$\nu$B medium $\beta|_{\nu}\simeq 1/2\kappa(z=1100)$ where
$\kappa=\int_{\eta}^{\eta_{0}}d\eta\dfrac{\sqrt{2}}{6\pi}\alpha
G^{F}\dfrac{n_{\nu}}{2}$, $\eta$ is the conformal time, $n_{\nu}$ is neutrino
number density, $G^{F}$ and $\alpha$ are Fermi coupling constant and
electromagnetic fine structure constant. The estimated CB angle in the case
C$\nu$B as Majorana particle is $\beta|_{\nu}\simeq 0.2\leavevmode\nobreak\
rad$ (see Section III for details). However this value is at least two orders
larger than the cosmic birefringence angle reported by the Planck data
release, $\beta\simeq 0.0053$rad ($68\%C.L.$). Thus, let us see what can be
the source of this big inconsistency. First, the electromagnetic interaction
of CMB and C$\nu$B is not the only source of CB angle. To better understand
the mentioned inconsistency, all other possible sources for CB should be
considered. We have shortly discussed this big inconsistency before the
conclusion.

##  II Cross power spectrum in the presence of Thomson and CMB-C$\nu$B
scattering

To calculate the cross-power spectra of CMB radiation, we start with CMB
radiation as an ensemble of photons described by

$\displaystyle\hat{\rho}_{ij}=\dfrac{1}{tr(\hat{\rho})}\int\dfrac{d^{3}k}{(2\pi)^{3}}\rho_{ij}(k)D_{ij}(k),$
$\displaystyle\hat{\rho}_{ij}\equiv\frac{1}{2}\left(\begin{matrix}I+Q&U-iV\\\
U+iV&I-Q\\\ \end{matrix}\right)$ (1)

where $I$, $Q$ ($U$) and $V$ are the Stokes parameters indicating the total
intensities, linear polarizations intensities and the difference between
clockwise and counter-clockwise circular polarization intensities,
respectively Chandra . $D_{ij}(\vec{k})\equiv
a_{i}^{\dagger}(\vec{k})a_{j}(\vec{k})$ is the photon number operator. Then,
to obtain the time evolution of Stokes parameters, we solved the following
quantum Boltzmann equation as follows:

$\displaystyle(2\pi)^{3}2k^{0}\frac{d}{dt}\rho_{ij}(\boldmath
k)=i<[H_{I}^{0},D_{ij}(\boldmath
k)]>-\frac{1}{2}\int_{-\infty}^{+\infty}dt<[H_{I}^{0}(t),[H_{I}^{0}(t),D_{ij}(\boldmath
k)]]>,$ (2)

where $H_{I}^{0}$ denotes the leading order term of the interacting
Hamiltonian. The first term on RHS of the above equation is called forward
scattering (interacting without momentum exchange) with a linear dependence on
$H_{I}^{0}$. The second one is the high-order collision terms (representing
ordinary cross-section terms) with square dependence on $H_{I}^{0}$ (see
Kosowsky:1994cy  and Appendix for more detail). As shown in Mohammadi:2013ksa
, in the case of the C$\nu$B-CMB interaction given by Fig.(1), the
contribution of forward scattering is non-zero and proportional to
($\propto\alpha G_{F}$). Note that, the usual cross-section of photon-
neutrino scattering is proportional to ($\propto\alpha^{2}G_{F}^{2}$), which
is very smaller than the forward scattering term.

$\gamma$$W^{+}$$e^{-}$$e^{-}$$e^{-}$$\nu_{e}$$\gamma$$\nu_{e}$

$\gamma$$W^{+}$$e^{-}$$e^{-}$$e^{-}$$\nu_{e}$$\gamma$$\nu_{e}$

Figure 1: Two Feynman diagrams describe the interaction between photons and
Dirac neutrinos at the one-loop level. Along the loop, there are two QED-
vertexes and two weak interacting vertexes. Note that the neutral channel
diagram with a $Z^{0}$-boson exchange does not contribute to the interacting
Hamiltonian $H^{0}_{I}$. The ”conjugated” Feynman diagrams should be
considered in the case of neutrinos as Majorana particles. This diagrams can
be extracted by replacing all the internal lines in their conjugated lines. It
is noteworthy that there are other diagrams containing more W bosons lines.
These diagrams give contributions in the order of $\mathcal{O}(G_{F}^{2})$.
Regarding the very small contribution of these diagrams, we ignore them.

The Boltzmann equation for the linear polarization and temperature anisotropy
parameters of CMB radiation in the presence of Compton scattering and photon-
neutrino forward scattering could be written asroh2

$\displaystyle\frac{d}{d\eta}\Delta_{T}^{(S)}+iK\mu\Delta_{T}^{(S)}+4[\dot{\psi}-iK\mu\varphi]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=\dot{\tau}_{e\gamma}\Big{[}-\Delta_{T}^{(S)}+\Delta_{T}^{0(S)}+i\mu
v_{b}+{1\over 2}P_{2}(\mu)\,\Pi\Big{]},$ (3)
$\displaystyle\frac{d}{d\eta}\Delta_{P}^{\pm(S)}+iK\mu\Delta_{P}^{\pm(S)}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
=\dot{\tau}_{e\gamma}\Big{[}-\Delta_{P}^{\pm(S)}-{1\over
2}[1-P_{2}(\mu)]\,\Pi\Big{]}\mp a(\eta)\dot{\kappa}\leavevmode\nobreak\
\Delta_{P}^{\pm(S)},$ (4)

where $\Delta_{P}^{\pm(S)}=Q^{(S)}\pm iU^{(S)}$,
$\Pi\equiv\Delta_{I}^{2(S)}+\Delta_{P}^{2(S)}+\Delta_{P}^{0(S)}$, $\psi$, and
$\phi$ are scalar functions defining scalar perturbation of metric in
longitudinal gauge, $v_{b}$ is neutrino bulk velocity,
$\dot{\tau}_{e\gamma}=an_{e}x_{e}\sigma_{T}$ is differential optical depth for
Thomson scattering, where $a(\eta)$ is the expansion factor normalized to
unity; $x_{e}$ is the ionization fraction and $\sigma_{T}$ is Thomson cross-
section. $\dot{\kappa}=d\kappa(\eta)/d\eta$ is opacity due to this
interaction. In the case of a Majorana neutrino, this parameter has different
behavior compared to the Dirac neutrino case.
The CMB polarization $\Delta_{p}^{\pm(S)}(\eta_{0},k,\mu)$ is separated into
the divergence-free part (B-mode) and curl-free (E-mode) as follows:

Figure 2: EB power spectrum in the presence of scalar perturbation via
C$\nu$B-CMB forward scattering is plotted in terms of $l$ by considering
C$\nu$B as Dirac for $\delta_{\nu}=0.08$ (solid line) and Majorana (dashed
line) neutrino. Note in standard scenario of cosmology, EB power spectrum in
the presence of scalar perturbation is exactly zero.

$\displaystyle\Delta_{E}^{(S)}(\eta_{0},K,\mu)$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g(\eta)\,\Pi(\eta,K)\\{\partial_{\mu}^{2}\left[(1-\mu^{2})^{2}e^{ix\mu}\cos{\kappa(\eta)}\right]\\},$
(5) $\displaystyle\Delta_{B}^{(S)}(\eta_{0},K,\mu)$ $\displaystyle=$
$\displaystyle\frac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g(\eta)\,\Pi(\eta,K)\\{\partial_{\mu}^{2}\left[(1-\mu^{2})^{2}e^{ix\mu}\sin{\kappa(\eta)}\right]\\},$
(6)

where $g(\eta)=\dot{\tau}_{e\gamma}e^{-\tau_{e\gamma}}$,
$\partial_{\mu}=\partial/\partial\mu$, and $\mu=\hat{\bf n}.\hat{\mathbf{K}}$
is the cosine angle of the CMB photon direction $\hat{\bf n}={\bf k}/|{\bf
k}|$ and the wave vectors $\bf{K}$, $x=\bf K(\eta_{0}-\eta)$ and
$\kappa(\eta)=\int_{\eta}^{\eta_{0}}d\eta a(\eta)\dot{\kappa}(\eta)$.
Eq. (6) shows that the photon-neutrino scattering ($\kappa\not=0$) results in
the non-trivial B-mode $\Delta_{B}^{(S)}$ and modifies the E-mode
$\Delta_{E}^{(S)}$. This value suggest that the Compton scattering by itself
can not generate B-mode without taking into account the tensor type of metric
perturbations zh95 ; uros ; Hu . Therefore, there are three combined power
spectra among B-mode, E-mode and T as in

Figure 3:  $\Delta C^{(S)}_{TEl}=C^{(S)}_{TEl}|_{{}_{CNB}}-C^{(S)}_{TEl}$ via
CMB-C$\nu$B forward scattering is plotted in terms of $l$ by considering
C$\nu$B as Dirac (solid line) and Majorana (dashed line) neutrino. This plot
shows just the contribution of photon-neutrino interaction. Note that
corrections from neutrino-photon scattering to TE power spectrum are very
smaller than standard cosmology $C^{(S)}_{TEl}$ contribution. Therefore,
standard cosmology $C^{(S)}_{TEl}$ contribution is subtracted from the total
$C^{(S)}_{TEl}|_{{}_{CNB}}$ power spectrum in the presence of Thomson and
neutrino-photon scattering. We can estimate the contribution of CMB-C$\nu$B
interaction on the TE cross power spectrum by $\Delta
C^{(S)}_{TEl}/C^{(S)}_{TEl}\simeq 1-\cos(\tilde{\kappa})\simeq 0.005$ in the
case of neutrino as Dirac particles (for $\delta_{\nu}=0.08$) which may be
detected by increasing the precision of future experiments. Figure 4: TB power
spectrum via CMB-C$\nu$B forward scattering is plotted in terms of $l$ by
considering C$\nu$B as Dirac (for $\delta_{\nu}=0.08$)(solid line) and
Majorana (dashed line) neutrino. Note that the TB power spectrum in the
presence of scalar perturbation is exactly zero in the standard scenario of
cosmology.

$\displaystyle C_{EB}^{\ell(S)}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{9}{16}\frac{(\ell+2)!}{(\ell-2)!}\int
K^{2}dKP_{S}(K)\Delta^{(S)}_{E}\Delta^{(S)}_{B},$ (7) $\displaystyle
C_{TE}^{\ell(S)}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\sqrt{\frac{9}{16}\frac{(\ell+2)!}{(\ell-2)!}}\int
K^{2}dKP_{S}(K)\Delta^{(S)}_{T}\Delta^{(S)}_{E},$ (8) $\displaystyle
C_{TB}^{\ell(S)}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\sqrt{\frac{9}{16}\frac{(\ell+2)!}{(\ell-2)!}}\int
K^{2}dKP_{S}(K)\Delta^{(S)}_{T}\Delta^{(S)}_{B},$ (9)

where

$\displaystyle\Delta^{(S)}_{T}(K,\eta_{0})=\int_{0}^{\eta_{0}}d\eta
g(\eta)S_{T}^{(S)}(\eta,K)j_{\ell}(x),$
$\displaystyle\Delta^{(S)}_{E}(K,\eta_{0})=\int_{0}^{\eta_{0}}d\eta\,g(\eta)\,\Pi(\eta,K)\frac{j_{\ell}(x)}{x^{2}}\cos{\kappa(\eta)},$
(10)
$\displaystyle\Delta^{(S)}_{B}(K,\eta_{0})=\int_{0}^{\eta_{0}}d\eta\,g(\eta)\,\Pi(\eta,K)\frac{j_{\ell}(x)}{x^{2}}\sin{\kappa(\eta)},$

and

$\displaystyle
S_{T}^{{}^{(S)}}(K,\eta)=g(\Delta_{T0}+2\dot{\alpha}+\dfrac{\dot{v}_{b}}{K}+\dfrac{\Pi}{4}+\dfrac{3\ddot{\Pi}}{4K^{2}})$
$\displaystyle+e^{-\tau_{e\gamma}}(\dot{\eta}+\ddot{\alpha})+\dot{g}(\alpha+\dfrac{v_{b}}{K}+\dfrac{3\dot{\Pi}}{4K^{2}})+\dfrac{3\ddot{g}\Pi}{4K^{2}},$
(11)

where $j_{\ell}(x)$ is a spherical Bessel function of rank $\ell$,
$\dot{\alpha}=(\dot{h}+6\dot{\rho})/2K^{2}$, $h$, and $\rho$ are additional
sources of temperature anisotropy. The contribution of photon-neutrino
(C$\nu$B) scattering to the cross-power spectrum is negligible for small
$\ell$. However, it is significantly large for $50<\ell<200$ ( Figs.(2-4)). As
we already know, the non-zero $C_{TB}^{\ell(S)}$ may also appear in exotic
theories due to the presence of helical flows in the primordial plasma at the
time of recombination Pogosian . Therefore the CMB-C$\nu$B scattering is a new
source of TB correlation as well.
The $\kappa$ in Eq.(II) is the effective opacity of neutrino generated by
neutrino-photon interaction from the last scattering time (redshift
$z_{l}=1100$) to the present time ($z_{0}=0$). Using the Friedmann equation in
the matter-dominated era (in the case of Majorana),
$H^{2}/H^{2}_{0}=\Omega_{M}^{0}(1+z)^{3}+\Omega_{\Lambda}^{0},\,H_{0}\approx
74\,$km/s/Mpc, $\Omega_{M}^{0}\approx 0.3\pm 0.007,\Omega_{\Lambda}^{0}\approx
0.7$ Planck:2018vyg , and $ad\eta=-dz/H(1+z)$ and conservation of the total
neutrino number $n_{\nu}=n_{\nu}^{0}(1+z)^{3}$, we obtain

$\displaystyle\kappa(z)$ $\displaystyle=$
$\displaystyle\int^{\eta_{0}}_{\eta}a\,d\eta\dot{\kappa}=\frac{\sqrt{2}}{12\pi}\alpha\,G^{F}n^{0}_{\nu}\int^{z}_{0}dz^{\prime}\frac{(1+z^{\prime})^{2}}{H(z^{\prime})}$
(12) $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{12\pi}\alpha\,G^{F}n^{0}_{\nu}\frac{2H(z^{\prime})}{3\Omega_{M}^{0}H_{0}^{2}}\Big{|}^{z}_{0}$

where the present number-density of all flavor neutrinos and anti-neutrinos is
$n^{0}_{\nu}=\sum(n^{0}_{\nu}+n^{0}_{\bar{\nu}})\approx 340\,{\rm cm^{-3}}$.
According to Eq.(12), $\kappa$ strongly depends on the red shift. Thus, it is
not wise to ignore this dependency in the integration of Eq.(II). As a result,
we can estimate the maximum value $\kappa$ near the last scattering surface
with red-shift $z_{l}=1100$, as follows:

$\displaystyle\tilde{\kappa}=\kappa(z_{l})\simeq
0.22(\frac{n_{\nu}^{0}}{340cm^{3}})\leavevmode\nobreak\ rad.$ (13)

If we consider neutrino as Dirac, according to the discussion in Geng , to
protect CP conservation, we should use the difference between neutrino and
anti-neutrino density. Therefore, we have

$\displaystyle\kappa(z)$ $\displaystyle=$
$\displaystyle\int^{\eta_{0}}_{\eta}a\,d\eta\dot{\kappa}=\frac{\sqrt{2}}{12\pi}\alpha\,G^{F}\delta_{\nu}n^{0}_{\nu}\int^{z}_{0}dz^{\prime}\frac{(1+z^{\prime})^{2}}{H(z^{\prime})}$
(14) $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{12\pi}\alpha\,G^{F}\delta_{\nu}n^{0}_{\nu}\frac{2H(z^{\prime})}{3\Omega_{M}^{0}H_{0}^{2}}\Big{|}^{z}_{0}$

where $\delta_{\nu}$ is the relative density asymmetry of neutrino and anti-
neutrino. Now, using the new measurement of the CB angle, we can constrain
$\delta_{\nu}$ as

$\displaystyle\delta_{\nu}<0.094\frac{340cm^{3}}{n^{0}_{\nu}}.$ (15)

We can approximately write $C_{EB}^{l(S)}$, $C_{TE}^{l(S)}$ and
$C_{TB}^{l(S)}$, for large scale and small $l$ moments, as follows

$\displaystyle
C_{EB}^{l(S)}\cong\dfrac{1}{2}\sin(2\tilde{\kappa})\bar{C}_{EE}^{l(S)}$
$\displaystyle C_{TE}^{l(S)}\cong\cos(\tilde{\kappa})\bar{C}_{TE}^{l(S)}$ (16)
$\displaystyle C_{TB}^{l(S)}\cong\sin(\tilde{\kappa})\bar{C}_{TE}^{l(S)}$

where $\bar{C}_{EE}^{l(S)}$ and $\bar{C}_{TE}^{l(S)}$ are power spectra of
E-mode polarization and its correlation with temperature anisotropy
contributed from the Compton scattering alone in the case of scalar
perturbation.

Figure 5:  Variation of $\beta=1/2\kappa$ in radian in terms of redshift.

##  III Cosmic birefringence calculation

In the presence of CB medium, E-mode can be converted to B-mode and vice
versa, so:

$\displaystyle\Delta_{E}^{(S)}=\bar{\Delta}_{E}^{(S)}\cos(2\beta)-\bar{\Delta}_{B}^{(S)}\sin(2\beta),$
$\displaystyle\Delta_{B}^{(S)}=\bar{\Delta}_{B}^{(S)}\cos(2\beta)+\bar{\Delta}_{E}^{(S)}\sin(2\beta),$
(17)

where $\bar{\Delta}_{E}^{(S)}$ and $\bar{\Delta}_{B}^{(S)}$ are the value of
E-mode and B-mode, respectively, in a standard scenario of cosmology without
considering CB effects Minami:2019ruj . Since $\bar{\Delta}_{B}^{(S)}=0$ in
the standard scenario of cosmology, we can rewrite the above equations as

$\displaystyle\Delta_{E}^{(S)}=\bar{\Delta}_{E}^{(S)}\cos(2\beta),$
$\displaystyle\Delta_{B}^{(S)}=\bar{\Delta}_{E}^{(S)}\sin(2\beta),$ (18)

As shown in Eq.(II), we faced an anisotropic birefringence medium. On the
other hand, the $\kappa$ value strongly depends on $\eta$ and red-shift (
Fig.(5)). The presence of Bessel functions in Eq.(II), which also depends on
$\eta$, highlights the importance of $\kappa$ dependence on red-shift.
According to this red-shift dependency in the integration of Eq.(II),
comparing the theoretical results of $C^{l(S)}_{EB}$ with its reported
measurements to extract information is more logical. To give an effective
sense, the CB angle $\beta$ can be estimated by the value of opacity
$\kappa(\eta)$ around the last scattering redshift ($z=1100$). The explanation
is that the visibility function ($g(\eta)$) in the integrated Eq.(II) has a
strong peak at that time. By substituting $\tilde{\kappa}$ given in Eq.(12)
instead of $\kappa$ in Eq.(II) and comparing the results with Eq.(18), we have
$\beta\simeq\frac{1}{2}\tilde{\kappa}$. Finally, we have

$\displaystyle C^{l(S)}_{EE}=\bar{C}^{l(S)}_{EE}\cos^{2}\tilde{\kappa},$ (19)
$\displaystyle C^{l(S)}_{BB}=\bar{C}^{l(S)}_{EE}\sin^{2}\tilde{\kappa},$ (20)
$\displaystyle C^{l(S)}_{EB}=\frac{1}{2}\bar{C}^{l(S)}_{EE}\sin
2\tilde{\kappa}.$ (21)

##  IV  Big inconsistency between CB angle generated via Majorana C$\nu$B-CMB
forward scattering and one reported CB angle

Note that from the above equations, CB angle $\beta$ can be estimated:

$\beta|_{\nu\,M}\cong\frac{1}{2}\tilde{\kappa}=12.6^{\circ},$ (22)

where $\beta|_{\nu\,M}$ refer to the contribution of C$\nu$B-CMB forward
scattering for CB angle in the case of Majorana (M). However, this value is at
least one order larger than the cosmic birefringence angle reported by the
Planck data release, $\beta=0.30^{\circ}\pm 0.11^{\circ}$ ($68\%C.L.$).

As mentioned above, $\kappa$ depends on redshift. Therefore, this point
becomes more important, when considering the properties of the Bessel
functions $j_{\ell}(x)$ in Eq.(II). Extracting Eqs.(19-21) from Eqs.(7-9) can
be relevant just in case of ignoring the red-shift dependency of $\kappa$.
Here, $\frac{1}{2}\tilde{\kappa}$ is the highest possible value for CB angle.
As a result, to have a more relevant estimation for the CB angle, it is better
to directly compare $C^{l}_{EB}$ generated via C$\nu$B-CMB forward scattering
(Fig.2) by experiment measurements. As Fig.(6) shows, for $l>20$,
$C^{l(S)}_{EB}/C^{l(S)}_{EE}\simeq 1/2\sin[2\tilde{\kappa}]$ , which is
compatible with the CB angle given in Eq.(22).

Figure 6:  The ratio of $C^{l(S)}_{EB}/C^{l(S)}_{EE}$ is plotted in terms of
$l$.

The next point is that here we only calculate the contribution of photon-
neutrino forward scattering. There may be some other mechanisms generating the
EB power spectrum that can suppress the contribution of photon-neutrino
interaction and CB angle. among these mechanisms are Chern-Simons Axion
coupling to the electromagnetic field Gluscevic , scattering of CMB photons
off polarized electrons Khodagholizadeh:2019het ; polarized , and dipole
asymmetry of temperature anisotropy of CMB dipoleasymmetry . Large-scale
magnetic fields and other Lorentz-violating interactions can also play roles
as birefringence mediums. Compton scattering in the presence of non-
commutative background and scalar mode of perturbation, in addition to
generating circularly polarized microwaves, can lead to B-mode polarization of
the Cosmic Microwave Background NC  and EB power spectrum. Therefore, it is
necessary to check its effect on the CB angle. For example, considering the CB
angle due to both C$\nu$B-CMB forward scattering and a Chern-Simons coupling
of a time-dependent Axion field to the electromagnetic tensor and comparing
the result with the reported CB angle can provide new constraints on mass and
electromagnetic coupling of axion sadeghobata . The role of other cosmological
parameters (e.g.,neutrino density, effective flavor number of neutrinos, and
Hubble parameter) cannot be ignored for the exact value of the CB angle.
Nevertheless, these effects do not change the order of CB angle due to
Majorana C$\nu$B-CMB forward scattering.

##  V  Summary

In this study, we wrote quantum Boltzmann equations in the presence of scalar
perturbation of metric for temperature fluctuations and linear polarization of
CMB radiation by using CMB Stokes parameters (Eq.(4). To this end, we
considered CMB-C$\nu$B forward scattering in the context of the standard model
of particles as the collision term. Afterward, the explicit expressions for
TE, TB, and EB cross-power spectra were derived in the form of the line-of-
sight integral solutions. We showed that the parity-violating photon-neutrino
forward scattering generated non-zero TB and EB power spectra in the presence
of scalar perturbation. TE, TB, and EB power spectra via Compton scattering
with the effect of CMB-C$\nu$B forward scattering were plotted in terms of $l$
for C$\nu$B as Dirac and Majorana particles ( Figs.(2-4)). As a result, TB and
EB power spectra in terms of $l$ behaved like TE and EE power spectra,
respectively.
Also, it was proved that C$\nu$B-CMB forward scattering can make an opaque
medium that can play as a birefringence medium and rotates the linear
polarization plane of CMB. As shown in Eq.(II), this medium is an anisotropic
birefringence medium and $\kappa$ value depends on the red-shift Fig.(5). The
presence of Bessel functions (and also dependence on $\eta$) in Eq.(II)
highlights the importance of $\kappa$ dependence on red-shift. In such a case,
comparing $C^{l(S)}_{(EB)}$ with its reported measurements is the most
reasonable decision at least in the case of $l<20$ (Fig.(6)). Our results for
$C^{l(S)}_{(EB)}$ are shown in Fig.(2).
Despite the extreme dependence of $\kappa$ on the red shift, to give an
effective sense, we showed that the CB angle $\beta$ can be roughly estimated
by the value of opacity $\beta|_{\nu}\simeq 1/2\tilde{\kappa}$ in the case of
$l>20$ (Fig.(6)). Therefore, we obtained $\beta|_{\nu\,M}\simeq 12.6^{\circ}$,
which refers to the contribution of C$\nu$B-CMB forward scattering for CB
angle in the case of C$\nu$B as Majorana particles. It seems this result is
not in agreement with the measurement reported in minami . Also, consistent
withGeng , we showed that if the nature of C$\nu$B is Dirac, the last reported
measurement of CB angle from Planck data release ($\beta=0.3^{\circ}\pm
0.11^{\circ}$) can constrain relative density asymmetry of neutrino and anti-
neutrino by the $\delta_{\nu}<0.094$.

The next point is that we only calculated the contribution of C$\nu$B-CMB
forward scattering. In this respect, some other mechanisms can reduce the
contribution of C$\nu$B-CMB interaction and CB angle. Some of these mechanisms
are Axion-CMB interaction, polarized Compton scattering, Lorentz violating
interaction, external large-scale magnetic field, and dipole asymmetry of CMB
temperature anisotropy, that generate EB power spectrum. Furthermore, the role
of other cosmological parameters cannot be ignored on the exact value of CB
angle, e.g., neutrino density and effective flavor number of neutrinos.
Hopefully, by upgrading data of EB-cross power spectrum and, theoretical and
experimental studies in the future, we can obtain more information about the
origin of the reported CB angle, the nature of C$\nu$B and Axion mass, and its
electromagnetic coupling.

##  VI Appendix

###  VI.1 Neutrino Calculation

Here, we show more details of Eq.(2) for neutrinos. Interaction Lagrangian is

$\displaystyle\mathcal{L}_{I}=\mathcal{L}_{QED}+\mathcal{L}_{e\nu}$ (23)

where the first term is QED Lagrangian and the second term is weak interaction
containing electron-neutrino-W boson vertex. The Photon field and neutrino
field are considered as follows,

$\displaystyle
A_{\mu}(x)=\int\frac{d^{3}k}{(2\pi)^{3}2k^{0}}\left[a_{i}(k)\epsilon_{i\mu}(k)e^{-ik\cdot
x}+a_{i}^{\dagger}(k)\epsilon^{*}_{i\mu}(k)e^{ik\cdot x}\right]$ (24)
$\displaystyle\psi(x)=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{\mathbf{q}}}}\sum_{r}\left[b_{r}(p)\mathcal{U}_{r}(p)e^{-iq\cdot
x}+d_{r}^{\dagger}(q)\mathcal{V}_{r}(q)e^{iq\cdot x}\right],$ (25)

where $\epsilon_{i\mu}$ is the polarization vector of the photon,
$\mathcal{U}and\mathcal{V}$ are Dirac Spinors and $a,a^{\dagger}$,
$b,b^{\dagger}$ and $d,d^{\dagger}$ are annihilation and creation operators of
photon, neutrino, and anti-neutrino respectively with the following commutator
relations,

$\displaystyle[a_{s}(k),a^{\dagger}_{s^{\prime}}(k^{\prime})]=(2\pi)^{3}2k^{0}\delta_{ss^{\prime}}\delta^{(3)}(k-k^{\prime})\leavevmode\nobreak\
,$ (26)
$\displaystyle\\{b_{r}(q),b^{\dagger}_{r^{\prime}}(q^{\prime})\\}=\\{d_{r}(q),d^{\dagger}_{r^{\prime}}(q^{\prime})\\}=(2\pi)^{3}\delta_{rr^{\prime}}\delta^{(3)}(q-q^{\prime})$

In the SM for elementary particle physics, the Hamiltonian of photon-neutrino
forward scattering at the leading order is given by the exchange of leptons
and the $W-Z$ gauge bosons, via the one-loop diagrams, as shown in Fig.(7),

$\displaystyle H_{I}^{0}$ $\displaystyle=$ $\displaystyle\int
d\mathbf{q}d\mathbf{q}^{\prime}d\mathbf{k}d\mathbf{k}^{\prime}(2\pi)^{3}\delta^{3}(\mathbf{q}^{\prime}-\mathbf{q})(2\pi)^{3}\delta^{3}(\mathbf{k}^{\prime}-\mathbf{k})\exp\left[i\left(q^{\prime
0}+p^{\prime 0}-q^{0}-p^{0}\right)t\right]$ (27)
$\displaystyle\times\left[b^{\dagger}_{r^{\prime}}(\mathbf{q}^{\prime})a^{\dagger}_{s^{\prime}}(\mathbf{p}^{\prime})T_{fi}(ps,qr,p^{\prime}s^{\prime},q^{\prime}r^{\prime})b_{r}(\mathbf{q})a_{s}(\mathbf{p})\right]\leavevmode\nobreak\
.$

where neutrino phase spaces $\int d\mathbf{q}\equiv d^{3}q/\sqrt{2q^{0}}$ and
$\int d\mathbf{q}^{\prime}\equiv d^{3}q^{\prime}/\sqrt{2q^{\prime 0}}$. The
scattering amplitude $T_{fi}$ is given by

$\displaystyle T_{fi}$ $\displaystyle=$
$\displaystyle\frac{1}{8}e^{2}g_{w}^{2}\int\frac{d^{4}l}{(2\pi)^{4}}D_{\alpha\beta}(q-l)\bar{\mathcal{U}}_{r^{\prime}}(\mathbf{q}^{\prime})\gamma^{\alpha}(1-\gamma_{5})S_{F}(l+p-p^{\prime})$
(28) $\displaystyle\times$
$\displaystyle\left[\epsilon\\!\\!\\!/_{s^{\prime}}S_{F}(l+p)\epsilon\\!\\!\\!/_{s}+\epsilon\\!\\!\\!/_{s}S_{F}(l-p^{\prime})\epsilon\\!\\!\\!/_{s^{\prime}}\right]S_{F}(l)\gamma^{\beta}(1-\gamma_{5})\mathcal{U}_{r}(\mathbf{q})\leavevmode\nobreak\
,$

where the notations $D_{\alpha\beta}$ and $S_{F}$ represent the propagators of
$W_{\mu}^{\pm}$ gauge-bosons and charged-lepton respectively. By helping
Dimensional regularization and Feynman parameters, we go forward to obtain the
leading order terms of the right side of the above equation. Then

$\displaystyle T_{fi}$ $\displaystyle=$
$\displaystyle-\frac{1}{16}\frac{1}{4\pi^{2}}e^{2}g_{w}^{2}\,\int_{0}^{1}dy\int_{0}^{1-y}dz\,\frac{(1-y-z)}{zM^{2}_{W}}\,\bar{\mathcal{U}}_{r}(q)(1+\gamma_{5})$
(29) $\displaystyle\times$
$\displaystyle\Big{(}2zq\\!\\!\\!/\epsilon_{s^{\prime}}.\epsilon_{s}+\,2z(\epsilon\\!\\!\\!/_{s^{\prime}}\,\,\mathbf{q}.\epsilon_{s}\,+\epsilon\\!\\!\\!/_{s}\,\mathbf{q}.\epsilon_{s^{\prime}}\,)+(3y-1)k\\!\\!\\!/\,(\epsilon\\!\\!\\!/_{s}\,\epsilon\\!\\!\\!/_{s^{\prime}}\,-\epsilon\\!\\!\\!/_{s^{\prime}}\,\epsilon\\!\\!\\!/_{s}\,)\,\Big{)}\mathcal{U}_{r}(q).$

Here we use the gamma-matrix identity
$A\\!\\!\\!\\!/\,B\\!\\!\\!\\!/=2A.B-B\\!\\!\\!\\!/\,A\\!\\!\\!\\!/$, the
polarization vector properties $k.\epsilon_{i}=0$ and
$\epsilon_{i}.\epsilon_{j}=-\delta_{ij}$. The straightforward calculations of
Eqs. (27) and (28) yield the one-loop effective Hamiltonian of the photon-
neutrino forward scattering Mohammadi:2013ksa

$\displaystyle H_{I}^{0}$ $\displaystyle=$
$\displaystyle\alpha\,G^{F}\frac{\sqrt{2}}{12\pi}\,\int\frac{d^{3}k}{(2\pi)^{3}}\,\left[a^{\dagger}_{s^{\prime}}(\mathbf{k})\,a_{s}(\mathbf{k})\right]\,\int\frac{d^{3}q}{(2\pi)^{3}}\,b^{\dagger}_{r}(\mathbf{q})\,b_{r}(\mathbf{q})$
(30)
$\displaystyle\times\Big{(}2\epsilon_{s^{\prime}}(\mathbf{p})\cdot\epsilon_{s}(\mathbf{p})(q^{0}-|\mathbf{q}|)+4|{\bf
q}|(\hat{\mathbf{q}}\cdot\epsilon_{s})\,(\hat{\mathbf{q}}\cdot\epsilon_{s^{\prime}})-2i\varepsilon_{\lambda\alpha\beta\mu}k^{\alpha}\,\epsilon_{s^{\prime}}^{\beta}\,\epsilon_{s}^{\mu}\,\hat{q}^{\lambda}\Big{)}.$

Now, using the above interaction Hamiltonian, commutators (26) and below
expectation values of operators are obtained

$\displaystyle\langle\,a_{1}a_{2}...b_{1}b_{2}...\,\rangle$ $\displaystyle=$
$\displaystyle\langle\,a_{1}a_{2}...\,\rangle\langle\,b_{1}b_{2}...\,\rangle,$
(31)
$\displaystyle\langle\,a^{\dagger}_{s^{\prime}}(k^{\prime})a_{s}(k)\,\rangle$
$\displaystyle=$ $\displaystyle
2k^{0}(2\pi)^{3}\delta^{3}(\mathbf{k}-\mathbf{k^{\prime}})\rho_{ss^{\prime}}(\mathbf{x},\mathbf{k}),$
(32)
$\displaystyle\langle\,b^{\dagger}_{r^{\prime}}(q^{\prime})b_{r}(q)\,\rangle$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta^{3}(\mathbf{q}-\mathbf{q^{\prime}})\delta_{rr^{\prime}}\frac{1}{2}n_{\nu}(\mathbf{x},\mathbf{q}),$
(33)
$\displaystyle\langle\,d^{\dagger}_{r^{\prime}}(q^{\prime})d_{r}(q)\,\rangle$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta^{3}(\mathbf{q}-\mathbf{q^{\prime}})\delta_{rr^{\prime}}\frac{1}{2}n_{\bar{\nu}}(\mathbf{x},\mathbf{q}),$
(34)

where $\rho_{ss^{\prime}}(\mathbf{x},\mathbf{k})$ is the local matrix density
and $n_{\nu}(\mathbf{x},\mathbf{q})$ and
$n_{\bar{\nu}}(\mathbf{x},\mathbf{q})$ are the local spatial density of
neutrinos and anti-neutrinos in the momentum state $\mathbf{q}$, respectively.
We can calculate the forward scattering term of Eq.(2). After tedious but
straightforward calculations, we have

$\displaystyle(2\pi)^{3}2k^{0}\frac{d\rho_{ij}}{dt}=i\langle[H^{0}_{I},D^{0}_{ij}({\bf
k})]\rangle=\frac{-e^{2}g_{w}^{2}}{64\pi^{2}M_{W}^{2}}\int
d\mathbf{q}\big{[}\rho_{s^{\prime}j}(\mathbf{x},{\bf
k})\delta_{is}-\rho_{is}(\mathbf{x},{\bf
k})\delta_{js^{\prime}}\big{]}n_{\nu}(\mathbf{x},\mathbf{q})$ (35)
$\displaystyle\times$
$\displaystyle\bar{\mathcal{U}}_{r}(q)(1+\gamma_{5})\Big{[}q\\!\\!\\!/\epsilon_{s^{\prime}}\cdot\epsilon_{s}+\,(\epsilon\\!\\!\\!/_{s^{\prime}}\,\,q\cdot\epsilon_{s}\,+\epsilon\\!\\!\\!/_{s}\,q\cdot\epsilon_{s^{\prime}}\,)+\frac{1}{6}k\\!\\!\\!/\,(\epsilon\\!\\!\\!/_{s}\,\epsilon\\!\\!\\!/_{s^{\prime}}\,-\epsilon\\!\\!\\!/_{s^{\prime}}\,\epsilon\\!\\!\\!/_{s}\,)\Big{]}\,\mathcal{U}_{r}(q)+{O}(\frac{m_{e}^{2}}{M_{W}^{4}}).$

The next step in the Quantum Boltzmann equation solution is to solve Eq.(3).

$k$$q-p$$p+k$$p+k+k^{\prime}$$p$$q$$k^{\prime}$$q^{\prime}$

$k$$q-p$$p-k^{\prime}$$p+k+k^{\prime}$$p$$q$$k^{\prime}$$q^{\prime}$

Figure 7: Full detail of the most contributing photon-neutrino Feynman
diagrams in $H_{I}^{0}(t)$.

###  VI.2 Photon-Neutrino Effective Lagrangian in six dimensions D=6

To give a more convincing, we expand our calculation to anti-neutrino-photon
scattering and Majorana neutrino, we have the following gauge invariant
effective Lagrangian in 6 dimensions (D=6) Geng

$\displaystyle\mathcal{L}_{\gamma\nu}\sim
G_{F}\alpha\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}\bar{\nu}_{R}\gamma_{\alpha}\nu_{L}A_{\beta}.,$
(36)

This effective Lagrangian for Majorna fermion can be written as

$\displaystyle\mathcal{L}_{\gamma\nu}\sim
G_{F}\alpha\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}\chi^{\dagger}\bar{\sigma}_{\alpha}\chi
A_{\beta},$ (37)

where $\chi$ is the massless 2-component of Majorana neutrino. According to
the Geng  paper, the final result of the Boltzmann equation solving Dirac
neutrino must be calculated using relative density asymmetry of neutrino and
anti-neutrino ($\delta_{\nu}$) to preserve CP conserving of Eq.(36). However,
since Eq.(37) is completely same for neutrino and anti-neutrino in the case of
Majorana neutrino, we should add up neutrino density and anti-neutrino density
in the final result.
The above effective Lagrangians contain CP conservation terms. Due to the CP
conservation of the above interaction, antineutrino-photon forward scattering
in the Dirac case, can generate a CB angle in the opposite sign and equal
amplitude in comparison with neutrino-photon forward scattering,

$\displaystyle(2\pi)^{3}2k^{0}\frac{d\rho_{ij}}{dt}=i\langle[H^{0}_{I},D^{0}_{ij}({\bf
k})]\rangle=-\frac{e^{2}g_{w}^{2}}{64\pi^{2}M_{W}^{2}}\int
d\mathbf{q}\big{[}\rho_{s^{\prime}j}(\mathbf{x},{\bf
k})\delta_{is}-\rho_{is}(\mathbf{x},{\bf
k})\delta_{js^{\prime}}\big{]}n_{\bar{\nu}}(\mathbf{x},\mathbf{q})$ (38)
$\displaystyle\times$
$\displaystyle\bar{\mathcal{V}}_{r}(q)(1+\gamma_{5})\Big{[}q\\!\\!\\!/\epsilon_{s^{\prime}}\cdot\epsilon_{s}+\,(\epsilon\\!\\!\\!/_{s^{\prime}}\,\,q\cdot\epsilon_{s}\,+\epsilon\\!\\!\\!/_{s}\,q\cdot\epsilon_{s^{\prime}}\,)+\frac{1}{6}k\\!\\!\\!/\,(\epsilon\\!\\!\\!/_{s}\,\epsilon\\!\\!\\!/_{s^{\prime}}\,-\epsilon\\!\\!\\!/_{s^{\prime}}\,\epsilon\\!\\!\\!/_{s}\,)\Big{]}\,\mathcal{V}_{r}(q)+{O}(\frac{m_{e}^{2}}{M_{W}^{4}}).$

and for Majorana, we have

$\displaystyle(2\pi)^{3}2k^{0}\frac{d\rho_{ij}}{dt}=i\langle[H^{0}_{I},D^{0}_{ij}({\bf
k})]\rangle=-\frac{e^{2}g_{w}^{2}}{64\pi^{2}M_{W}^{2}}\int
d\mathbf{q}\big{[}\rho_{s^{\prime}j}(\mathbf{x},{\bf
k})\delta_{is}-\rho_{is}(\mathbf{x},{\bf
k})\delta_{js^{\prime}}\big{]}n_{\bar{\nu}}(\mathbf{x},\mathbf{q})$ (39)
$\displaystyle\times$
$\displaystyle\chi^{\dagger}_{r}(q)(1+\gamma_{5})\Big{[}q\\!\\!\\!/\epsilon_{s^{\prime}}\cdot\epsilon_{s}+\,(\epsilon\\!\\!\\!/_{s^{\prime}}\,\,q\cdot\epsilon_{s}\,+\epsilon\\!\\!\\!/_{s}\,q\cdot\epsilon_{s^{\prime}}\,)+\frac{1}{6}k\\!\\!\\!/\,(\epsilon\\!\\!\\!/_{s}\,\epsilon\\!\\!\\!/_{s^{\prime}}\,-\epsilon\\!\\!\\!/_{s^{\prime}}\,\epsilon\\!\\!\\!/_{s}\,)\Big{]}\,\chi_{r}(q)+{O}(\frac{m_{e}^{2}}{M_{W}^{4}}).$

As a result, in Eqs.(12 and 14), we should consider
$n^{0}_{\nu}=\sum(n^{0}_{\nu}+n^{0}_{\bar{\nu}})\approx 340\,{\rm cm^{-3}}$.

In some references such as Karl:2004bt , authors discussed various parity-odd
and parity-even effective Lagrangians that describe photon-neutrino
interaction. As can be seen, all these effective Lagrangians are dimension-8
operators or higher ones with $\alpha^{2}G_{F}^{2}$ order. As a result, their
calculations of one-loop scattering amplitude are several orders of magnitude
smaller than our results ($\alpha\,G_{F}$). In addition, in a recently
published work Dvornikov:2020olb , authors calculated the CB angle in presence
of relic neutrino-antineutrino gas using electroweak correction of vacuum
polarization of the photon. however, this effect is proportional to
$\frac{\alpha}{M_{W}^{4}}$, which is very smaller than our results.

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prop@alttheorem lem@alttheorem coroll@alttheorem defi@alttheorem
quest@alttheorem fact@alttheorem rem@alttheorem exa@alttheorem

# Normed spaces using intrinsically Lipschitz sections and Extension Theorem
for the intrinsically Hölder sections

Daniela Di Donato

###### Abstract.

The purpose of this article is twofold: first of all, we want to define two
norms using the space of intrinsically Lipschitz sections. On the other hand,
we want to generalize an Extension Theorem proved by the author in the context
of the intrinsically Hölder sections with target a topological space $Y.$ Here
our target will be $Y\times\mathbb{R}^{s}$ with $s\geq 1$ instead of $Y.$

###### Contents

  1. 1 Introduction
  2. 2 Intrinsically Hölder sections
    1. 2.1 Intrinsically Hölder sections: when $Y$ is bounded
    2. 2.2 Equivalent definition for intrinsic Hölder sections
  3. 3 Normed space for the intrinsically Lipschitz sections
    1. 3.1 Normed space: Version 1
    2. 3.2 Normed space: Version 2
  4. 4 Level sets and extensions

##  1\. Introduction

In this paper, we focus our attention on a new point of view for the
intrinsically Lipschitz graphs in the Franchi-Serapioni-SerraCassano sense
[FSSC01, FSSC03b, FSSC03a] (see also [SC16, FS16]) in metric spaces.

They introduced and analized this notion in order to establish a good notion
of rectifiability in a particular metric spaces called subRiemannian Carnot
groups [ABB19, BLU07, CDPT07].

In the usual way, Federer [Fed69] says that a subset of $\mathbb{R}^{n}$ is
countably $d$-rectifiable if it is possible to cover it with a countable union
of suitable graphs. More precisely, he considers graphs of Lipschitz maps
$f:\mathbb{R}^{d}\to\mathbb{R}^{n-d}.$ However, Ambrosio and Kirchheim [AK00]
(see also [Mag04]) proved that this definition does not work in Carnot groups
and so many mathematicians give other notions of rectifiability. The reader
can see [AM22a, AM22b, Bat21, DS91, DS93, CP06, Pau04, NY18]. As we said,
another possible solution is given by Franchi-Serapioni-SerraCassano with the
so-called "Intrinsic Lipschitz maps" in Carnot groups. The idea is similar to
Euclidean case: firstly, they introduce suitable cones called intrinsic cones
which are not equivalent with the Euclidean ones; and, then, they say that a
map $\varphi$ is intrinsic Lipschitz if it is possible to have for any point
$p$ belongs to the graph of $\varphi$ an empty intersection between a suitable
intrinsic cone with vertex $p$ and the graph of this map.

Recently, Le Donne and the author generalize this concept in metric spaces
(see [DDLD22]).

A basic difference is the following: in Franchi, Serapioni and Serra Cassano
approach, they consider intrinsically Lipschitz maps. On the other hand, in
our approach we consider the graphs and this a bit change is so important
because:

  * •

The setting are more general: the class of the metric spaces is larger than
the class of Carnot groups.

  * •

The proofs are elegantly short and simple.

  * •

We use basic mathematical tools in the proofs.

In a natural way, in [DD22c] the author introduce the notion of intrinsically
Hölder sections which extend the Lipschitz ones. Here, our setting is the
following. We have a metric space $X$, a topological space $Y$, and a quotient
map $\pi:X\to Y$, meaning continuous, open, and surjective. The standard
example for us is when $X$ is a metric Lie group $G$ (meaning that the Lie
group $G$ is equipped with a left-invariant distance that induces the manifold
topology), for example a subRiemannian Carnot group, and $Y$ is the space of
left cosets $G/H$, where $H<G$ is a closed subgroup and $\pi:G\to G/H$ is the
projection modulo $H$, $g\mapsto gH$.

######  Definition  (Intrinsic Hölder section).

Let $(X,d)$ be a metric space and let $Y$ be a topological space. We say that
a map $\varphi:Y\to X$ is a section of a quotient map $\pi:X\to Y$ if

$\pi\circ\varphi=\mbox{id}_{Y}.$

Moreover, we say that $\varphi$ is an intrinsically $(L,\alpha)$-Hölder
section with constant $L>0$ and $\alpha\in(0,1)$ if in addition

(1) $d(\varphi(y_{1}),\varphi(y_{2}))\leq
Ld(\varphi(y_{1}),\pi^{-1}(y_{2}))^{\alpha}+d(\varphi(y_{1}),\pi^{-1}(y_{2})),\quad\mbox{for
all }y_{1},y_{2}\in Y.$

Equivalently, we are requesting that

$d(x_{1},x_{2})\leq
Ld(x_{1},\pi^{-1}(\pi(x_{2})))^{\alpha}+d(x_{1},\pi^{-1}(\pi(x_{2}))),\quad\mbox{for
all }x_{1},x_{2}\in\varphi(Y).$

When $\alpha=1,$ a section $\varphi$ is intrinsic Lipschitz in the sense of
[DDLD22]; and, if in addition, $\pi$ is a Lipschitz quotient or submetry
[BJL+99, VN88], the results trivialize, since in this case being intrinsically
Lipschitz is equivalent to biLipschitz embedding, see Proposition 2.4 in
[DDLD22]. In a natural way, following the seminal papers [AGS14a, LV09, Sav22]
(see also [AGS15, AGS14b, FSS22, Stu06, Vil09]), the author introduced and
studied the link between the intrinsic sections/intrinsic Lipschitz sections
and the intrinsic Hopf-Lax semigroups [DD22b, DD22e].

The purpose of this article is twofold: first of all, we want to define two
norms using the notion of Lipschitz sections. Second, we want to generalize an
Extension Theorem with target $Y$ which is a topological space; in this paper,
our target will be $Y\times\mathbb{R}^{s}$ instead of $Y.$

More precisely, in Section 3, the main results are Theorem 3.1 and 3.2. Here,
we define two possible normed spaces using the following ingredients:

  * •

we know that there is a large class of intrinsically Hölder sections and so
Lipschitz sections that is a vector space over $\mathbb{R}$ or $\mathbb{C}$
(see Theorem 2.1);

  * •

we can define two different norms noting the following simple fact: in the
usual case, we have that $d(x,y)=d(y,x)$ for any point $x,y\in X;$ on the
other hand, in our intrinsic context, in general, we have that
$d(f(x),\pi^{-1}(y))\neq d(f(y),\pi^{-1}(x)),$ for every $x,y\in X.$

  * •

we obtain the homogeneity of our norms defined in (5) and in (9) thanks to
linearity of $\pi$ and, in particular, to Lemma 3.1.

Finally, in Section 4 the main result is Theorem 4.1 which generalizes
Extension Theorem for the intrinsically Hölder sections proved in [DD22c,
Theorem 1.3]. The main difference is that, in this project, the target space
is a topological space $Y\times\mathbb{R}^{s}$ with $s\geq 1$ instead of $Y.$
As in Vittone’s case, we build each component $f_{i}$ for $i=1,\dots,s$
separately and then join them without any additional assumptions. However, the
final step when the target space is only $Y$ does not provide a Lipschitz map
$f=(f_{1},\dots,f_{s}).$

##  2\. Intrinsically Hölder sections

###  2.1. Intrinsically Hölder sections: when $Y$ is bounded

Definition 1 it is very natural if we think that what we are studying graphs
of appropriate maps. However, in the following proposition, we introduce an
equivalent condition of (1) when $Y$ is a bounded set.

######  Proposition .

Let $\pi:X\to Y$ be a quotient map between a metric space $X$ and a
topological and bounded space $Y$ and let $\alpha\in(0,1).$ The following are
equivalent:

  1. (1)

there is $L>0$ such that

$d(\varphi(y_{1}),\varphi(y_{2}))\leq
Ld(\varphi(y_{1}),\pi^{-1}(y_{2}))^{\alpha}+d(\varphi(y_{1}),\pi^{-1}(y_{2})),\quad\mbox{for
all }y_{1},y_{2}\in Y.$

  2. (2)

there is $K\geq 1$ such that

(2) $d(\varphi(y_{1}),\varphi(y_{2}))\leq
Kd(\varphi(y_{1}),\pi^{-1}(y_{2}))^{\alpha},\quad\mbox{for all }y_{1},y_{2}\in
Y.$

We further rephrase the definition as saying that $\varphi(Y)$, which we call
the graph of $\varphi$, avoids some particular sets (which depend on
$\alpha,L$ and $\varphi$ itself):

######  Proposition .

Let $\pi:X\to Y$ be a quotient map between a metric space and a topological
space, $\varphi:Y\to X$ be a section of $\pi$, $\alpha\in(0,1)$ and $L>0$.
Then $\varphi$ is intrinsically $(L,\alpha)$-Hölder if and only if

$\varphi(Y)\cap R_{x,L}=\emptyset,\quad\mbox{for all }x\in\varphi(Y),$

where

$R_{x,L}:=\left\\{x^{\prime}\in
X\;|\;Ld(x^{\prime},\pi^{-1}(\pi(x)))^{\alpha}+d(x^{\prime},\pi^{-1}(\pi(x)))<d(x^{\prime},x)\right\\}.$

Proposition 2.1 is a triviality, still its purpose is to stress the analogy
with the intrinsically Lipschitz sections theory introduced in [DDLD22] when
$\alpha=1$. In particular, the sets $R_{x,L}$ are the intrinsic cones in the
sense of Franchi, Serapioni and Serra Cassano when $X$ is a subRiemannian
Carnot group and $\alpha=1.$ The reader can see [DD22d] for a good notion of
intrinsic cones in metric groups.

###  2.2. Equivalent definition for intrinsic Hölder sections

######  Definition  (Intrinsically Hölder with respect to a section).

Given sections $\varphi,\psi:Y\to X$ of $\pi$. We say that $\varphi$ is
intrinsically $(L,\alpha)$-Hölder with respect to $\psi$ at point $\hat{x}$,
with $L>0,\alpha\in(0,1)$ and $\hat{x}\in X$, if

  1. (1)

$\hat{x}\in\psi(Y)\cap\varphi(Y);$

  2. (2)

$\varphi(Y)\cap C_{\hat{x},L}^{\psi}=\emptyset,$

where

$C_{\hat{x},L}^{\psi}:=\\{x\in
X\,:\,d(x,\psi(\pi(x)))>Ld(\hat{x},\psi(\pi(x)))^{\alpha}+d(\hat{x},\psi(\pi(x)))\\}.$

######  Remark .

Definition 2.2 can be rephrased as follows. A section $\varphi$ is
intrinsically $(L,\alpha)$-Hölder with respect to $\psi$ at point $\hat{x}$ if
and only if there is $\hat{y}\in Y$ such that
$\hat{x}=\varphi(\hat{y})=\psi(\hat{y})$ and

(3) $d(x,\psi(\pi(x)))\leq
Ld(\hat{x},\psi(\pi(x)))^{\alpha}+d(\hat{x},\psi(\pi(x))),\quad\forall
x\in\varphi(Y),$

which equivalently means

(4) $d(\varphi(y),\psi(y))\leq
Ld(\psi(\hat{y}),\psi(y))^{\alpha}+d(\psi(\hat{y}),\psi(y)),\qquad\forall y\in
Y.$

Notice that Definition 2.2 does not induce an equivalence relation because of
lack of symmetry in the right-hand side of (4). However, following Cheeger
theory [Che99] (see also [Kei04, KM16]), in [DD22c, Theorem 4.2] we give an
equivalent property of Hölder section.

Being intrinsically Lipschitz section is equivalent to the last definition as
proved in [DD22c, Proposition 1.5]

######  Proposition .

Let $X$ be a metric space, $Y$ a topological and bounded space, $\pi:X\to Y$ a
quotient map, $L\geq 1$ and $\alpha,\beta,\gamma\in(0,1)$. Assume that every
point $x\in X$ is contained in the image of an intrinsic $(L,\alpha)$-Hölder
section $\psi_{x}$ for $\pi$. Then for every section $\varphi:Y\to X$ of $\pi$
the following are equivalent:

  1. (1)

for all $x\in\varphi(Y)$ the section $\varphi$ is intrinsically
$(L_{1},\beta)$-Hölder with respect to $\psi_{x}$ at $x;$

  2. (2)

the section $\varphi$ is intrinsically $(L_{2},\gamma)$-Hölder.

We conclude this section recall an important concept which we will be used
later.

######  Definition  (Intrinsic Hölder set with respect to $\psi$).

Let $\alpha\in(0,1]$ and $\psi:Y\to X$ a section of $\pi$. We define the set
of all intrinsically Hölder sections of $\pi$ with respect to $\psi$ at point
$\hat{x}$ as

$\displaystyle H_{\psi,\hat{x},\alpha}$ $\displaystyle:=\\{\varphi:Y\to
X\mbox{ a section of $\pi$}\,:\,\varphi\mbox{ is intrinsically
$(\tilde{L},\alpha)$-H\"{o}lder w.r.t. $\psi$ at point $\hat{x}$}$
$\displaystyle\quad\quad\mbox{ for some $\tilde{L}>0$}\\}.$

In particular, in [DD22c] we have the following statement regarding the set
$H_{\psi,\hat{x},\alpha}$.

######  Theorem 2.1 (Theorem 3.5 [DD22c]).

Let $\pi:X\to Y$ is a linear and quotient map from a normed space $X$ to a
topological space $Y.$ Assume also that $\psi:Y\to X$ is a section of $\pi$
and $\\{\lambda\hat{x}\,:\,\lambda\in\mathbb{R}^{+}\\}\subset X$ with
$\hat{x}\in\psi(Y).$

Then, for any $\alpha\in(0,1],$ the set
$\bigcup_{\lambda\in\mathbb{R}^{+}}H_{\lambda\psi,\lambda\hat{x},\alpha}\cup\\{0\\}$
is a vector space over $\mathbb{R}$ or $\mathbb{C}.$

Notice that it is no possible to obtain the statement for
$H_{\psi,\hat{x},\alpha}$ since the simply observation that if
$\psi(\hat{y})=\hat{x}$ then $\psi(\hat{y})+\psi(\hat{y})\neq\hat{x}.$

##  3\. Normed space for the intrinsically Lipschitz sections

###  3.1. Normed space: Version 1

In this section, we consider the case of intrinsically Lipschitz sections,
i.e., $\alpha=1.$

Let $\pi:\mathbb{R}^{\kappa}\to Y$ be a quotient map with
$Y\subset\mathbb{R}^{\kappa}$. Assume also that $K\subset Y$ is a compact set
and $\psi_{|K}:K\to\mathbb{R}$ is an intrinsically $L$-Lipschitz section of
$\pi$ with $L\geq 1$ and $\hat{x}=\psi(\bar{y})\in\psi(Y).$ We will use the
following notation

$ILS_{\lambda\psi_{|K},\lambda\hat{x}}:=H_{\lambda\psi_{|K},\lambda\hat{x},1}.$

Here, we show that

$(\mathbb{L},\|.\|):=\left(\bigcup_{\lambda\in\mathbb{R}^{+}}ILS_{\lambda\psi_{|K},\lambda\hat{x}}\cup\\{0\\},\|.\|\right)$

is a normed space for a suitable norm
$\|.\|=\|.\|_{ILS_{\lambda\psi,\lambda\hat{x}}}:\mathbb{L}\to\mathbb{R}^{+}$
defined as for any $\varphi\in\mathbb{L},$

(5)
$\|\varphi\|_{ILS_{\lambda\psi,\lambda\hat{x}}}:=\|\varphi\|_{\infty}+[\varphi]_{\lambda\psi,\lambda\hat{x}},$

where $\|\varphi\|_{\infty}:=\sup_{y\in K}|\varphi(y)|$ and

$[\varphi]_{\lambda\psi,\lambda\hat{x}}:=\sup_{y\in
K}d(\lambda\varphi(y),(1/\lambda\pi)^{-1}(\pi(\hat{x}))).$

Then, we are able to give the main result of this section.

######  Theorem 3.1.

Let $\pi:\mathbb{R}^{\kappa}\to Y$ be a linear and quotient map with
$Y\subset\mathbb{R}^{\kappa}$. Assume also that $\psi:K\to\mathbb{R}^{\kappa}$
is an intrinsically $L$-Lipschitz section of $\pi$ with $K\subset Y$ compact,
$L\geq 1$ and $\hat{x}\in X.$ Then, the set $\mathbb{L}$ endowed with
$\|\cdot\|_{ILS_{\psi,\hat{x}}}$ is a normed space.

We need the following lemma.

######  Lemma  (Lemma 4.7 [DD22a]).

Let $X$ be a normed space, $Y$ be a topological space and $\pi:X\to Y$ be a
linear and quotient map. Then

(6) $|\lambda|d(x_{1},\pi^{-1}(y_{2}))=d(\lambda
x_{1},(1/\lambda\pi)^{-1}(y_{2})),\quad\forall
x_{1}\in\mathbb{R}^{\kappa},y_{2}\in Y,\lambda\in\mathbb{R}-\\{0\\}.$

######  Remark .

An easy corollary of Lemma 3.1 when $Y\subset\mathbb{R}$ and
$X=\mathbb{R}^{\kappa}$ is that

$\displaystyle\lim_{h\to 0^{+}}\frac{d(h\varphi(t+h),(1/h\pi)^{-1}(t)))}{h}$
$\displaystyle=0,$ $\displaystyle\lim_{h\to
0^{+}}\frac{d(h\varphi(t),(1/h\pi)^{-1}(t+h)))}{h}$ $\displaystyle=0,$

for $t\in Y.$ Indeed, for $h>0$

$\frac{d(h\varphi(t+h),(1/h\pi)^{-1}(t))}{h}=d(\varphi(t+h),\pi^{-1}(t))\leq
d(\varphi(t+h),\varphi(t)),$

and so take to the limit for $h\to 0,$ we obtain the first limit thank to the
continuity of $\varphi$. In a similar way, it is possible to see the second
limit.

At this point, we give the proof of Theorem 3.1.

###### Proof of Theorem 3.1.

The fact $\|\varphi\|\equiv 0$ if and only if $\varphi\equiv 0$ follows
because $\|.\|_{\infty}$ is a norm. On the other hand, since $\pi$ is linear
map and thanks to Lemma 3.1, we have

$\sup_{y\in K}d(\delta\varphi(y),(1/\delta\pi)^{-1}(\pi(\hat{x})))=\sup_{y\in
K}|\delta|d(\varphi(y),\pi^{-1}(\pi(\hat{x})))$

for any $\delta\in\mathbb{R}-\\{0\\}$ and so

(7)
$\|\delta\varphi\|_{\infty}+[\delta\varphi]_{\psi,\hat{x}}=|\delta|(\|\varphi\|_{\infty}+[\varphi]_{\psi,\hat{x}}),$

for any $\varphi\in\mathbb{L}.$

Finally, the triangle inequality follows using again the linearity of $\pi.$
Indeed, if $\varphi,\eta\in\mathbb{L}-\\{0\\}$ and, in particular,
$\varphi,\eta\in
ILS_{(\lambda_{1}+\lambda_{2})\psi_{|K},(\lambda_{1}+\lambda_{2})\hat{x}}$
then for $x_{\varphi},x_{\eta}\in\mathbb{R}^{\kappa}$ such that

$\displaystyle{}_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}}$
$\displaystyle=\sup_{y\in
K}d((\lambda_{1}+\lambda_{2})\varphi(y),(1/(\lambda_{1}+\lambda_{2})\pi)^{-1}(\pi(\hat{x})))=d((\lambda_{1}+\lambda_{2})\varphi(y),x_{\varphi})$
$\displaystyle[\eta]_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}}$
$\displaystyle=\sup_{y\in
K}d((\lambda_{1}+\lambda_{2})\eta(y),(1/(\lambda_{1}+\lambda_{2})\pi)^{-1}(\pi(\hat{x})))=d((\lambda_{1}+\lambda_{2})\eta(y),x_{\eta})$

one gets

$\displaystyle{}_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}}$
$\displaystyle=\sup_{y\in
K}d((\lambda_{1}+\lambda_{2})(\varphi(y)+\eta(y)),(2/(\lambda_{1}+\lambda_{2})\pi)^{-1}(\pi(\hat{x})))$
$\displaystyle\leq\|(\lambda_{1}+\lambda_{2})\varphi(y)+(\lambda_{1}+\lambda_{2})\eta(y)-(x_{\varphi}+x_{\eta})\|$
$\displaystyle\leq\|(\lambda_{1}+\lambda_{2})\varphi(y)-x_{\varphi}\|+\|(\lambda_{1}+\lambda_{2})\eta(y)-x_{\eta}\|,$
$\displaystyle[\varphi]_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}}+[\eta]_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}},$

where in the first equality, by linearity of $\pi,$ we used the fact

$\displaystyle\pi\big{(}(\lambda_{1}+\lambda_{2})(\varphi(y)+\eta(y))\big{)}$
$\displaystyle=\pi\big{(}(\lambda_{1}+\lambda_{2})\varphi(y)\big{)}+\pi\big{(}(\lambda_{1}+\lambda_{2})\eta(y)\big{)}$
$\displaystyle=(\lambda_{1}+\lambda_{2})(\pi(\varphi(y))+(\pi(\eta(y)))$
$\displaystyle=2(\lambda_{1}+\lambda_{2})y.$

Hence,

$[\varphi+\eta]_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}}\leq[\varphi]_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}}+[\eta]_{(\lambda_{1}+\lambda_{2})\psi,(\lambda_{1}+\lambda_{2})\hat{x}},$

and this complete the proof of the statement.

∎

###  3.2. Normed space: Version 2

In the usual case, we have that $d(x,y)=d(y,x)$ for any point $x,y\in X;$ on
the other hand, in our intrinsic context, in general, we have that

$d(f(x),\pi^{-1}(y))\neq d(f(y),\pi^{-1}(x)),$

for every $x,y\in X.$ In particular, it holds

(8) $\displaystyle d(f(y),\pi^{-1}(x))-d(f(z),\pi^{-1}(x))$ $\displaystyle\leq
d(f(y),f(z)),\quad\forall x,y,z\in Y$ $\displaystyle
d(f(x),\pi^{-1}(y))-d(f(x),\pi^{-1}(z))$ $\displaystyle\nleq
d(f(y),f(z)),\quad\mbox{for some }x,y,z\in Y.$

In fact, for any fixed $x,y,z\in Y,$ if we choose $a\in\pi^{-1}(x)$ such that

$d(f(z),\pi^{-1}(x))=d(f(z),a),$

we deduce that

$\displaystyle d(f(y),\pi^{-1}(x))-d(f(z),\pi^{-1}(x))$ $\displaystyle\leq
d(f(y),a)-d(f(z),\pi^{-1}(x))$ $\displaystyle\leq
d(f(y),f(z))+d(f(z),a)-d(f(z),\pi^{-1}(x))$ $\displaystyle=d(f(y),f(z)),$

i.e., the first inequality of (8) holds. On the other hand, for the second
inequality in (8), we give the following example. let $X\subset\mathbb{R}^{2}$
the set given by the three lines with vertex $(0,8),(8,8);$ $(1,4),(8,6)$ and
$(0,3),(8,7)$ and the subset $Y$ of $\mathbb{R}^{2}$ defined as the line with
vertex $(0,0)$ and $(8,0).$ On $X$ we consider the usual distance on
$\mathbb{R}^{2}.$ Then, if we consider a continuous section $f:Y\to X$ of the
projection on the first component $\pi:X\to Y$ with
$f(x)=f((1,0))=(1,4),f(y)=f((7,0))=(8,7)$ and $f(z)=f((6,0))=(8,6),$ it is
easy to see that

$\displaystyle d(f(x),\pi^{-1}(y))-d(f(x),\pi^{-1}(z))$
$\displaystyle=\sqrt{\frac{5}{4}},$ $\displaystyle d(f(y),f(z))$
$\displaystyle=1,$

and so

$\displaystyle d(f(x),\pi^{-1}(y))-d(f(x),\pi^{-1}(z))$ $\displaystyle\nleq
d(f(y),f(z)).$

Then, it is not trivial to consider the norm $|||.|||$ defined as

(9)
$|||\varphi|||_{ILS_{\psi,\hat{x}}}:=\|\varphi\|_{\infty}+\\{\varphi\\}_{\lambda\psi,\lambda\hat{x}},$

where $\|\varphi\|_{\infty}:=\sup_{y\in K}|\varphi(y)|$ and

$\\{\varphi\\}_{\lambda\psi,\lambda\hat{x}}:=\sup_{y\in
K}d(\lambda\hat{x},(1/\lambda\pi)^{-1}(y)).$

and to prove as above the following statement.

######  Theorem 3.2.

Let $\pi:\mathbb{R}^{\kappa}\to Y$ be a linear and quotient map with $Y$ a
metric space. Assume also that $\psi:K\to\mathbb{R}^{\kappa}$ is an
intrinsically $L$-Lipschitz section of $\pi$ with $K\subset Y$ compact, $L\geq
1$ and $\hat{x}\in X.$ Then, the set $\mathbb{L}$ endowed with
$|||\cdot|||_{ILS_{\psi,\hat{x}}}$ as in (9) is a normed space.

###### Proof.

The proof follows in a similar way as in Theorem 3.1. ∎

##  4\. Level sets and extensions

In this section we prove the following theorem.

######  Theorem 4.1 (Extensions as level sets).

Let $\pi:X\to Y\times\mathbb{R}^{s}$ be a quotient map between a metric space
$X$ and a topological space $Y\times\mathbb{R}^{s}$.

Assume that $X$ is geodesic and that there exist $k\geq 1,$ $\rho:X\times
X\to\mathbb{R}$ k-biLipschitz equivalent to the distance of $X,$ and
$\tau=(\tau_{1},\dots,\tau_{s}):X\to\mathbb{R}^{s}$ k-Lipschitz and
$k$-biLipschitz on fibers such that for all $\tau_{0}\in\mathbb{R}^{s}$

  1. (1)

the set $\tau_{1}^{-1}(\tau_{0})$ is an intrinsically $k$-Lipschitz graph of a
section $\varphi_{1,\tau_{0}}:Y\times\mathbb{R}\times\\{0\\}^{s-1}\to X;$ the
set $\tau_{2}^{-1}(\tau_{0})$ is an intrinsically $k$-Lipschitz graph of a
section
$\varphi_{2,\tau_{0}}:Y\times\\{0\\}\times\mathbb{R}\times\\{0\\}^{s-2}\to X;$
…, the set $\tau_{s}^{-1}(\tau_{0})$ is an intrinsically $k$-Lipschitz graph
of a section $\varphi_{s,\tau_{0}}:Y\times\\{0\\}^{s-1}\times\mathbb{R}\to X;$

  2. (2)

for all $x_{0}\in\tau_{i}^{-1}(\tau_{0})$ the map
$X\to\mathbb{R},x\mapsto\delta_{i,\tau_{0}}(x):=\rho(x_{0},\varphi_{i,\tau_{0}}(\pi(x)))$
is $k$-Lipschitz on the set $\\{|\tau_{i}|\leq\delta_{i,\tau_{0}}\\}$.

Let $Y^{\prime}\times\mathbb{R}^{s}\subset Y\times\mathbb{R}^{s}$ a set and
$L\geq 1$. Then for every intrinsically $L$-Lipschitz section
$\varphi:Y^{\prime}\times\mathbb{R}^{s}\to\pi^{-1}(Y^{\prime}\times\mathbb{R}^{s})$
of
$\pi|_{\pi^{-1}(Y^{\prime}\times\mathbb{R}^{s})}:\pi^{-1}(Y^{\prime}\times\mathbb{R}^{s})\to
Y^{\prime}\times\mathbb{R}^{s}$, there exists a map $f:X\to\mathbb{R}^{s}$
that is $K$-Lipschitz and $K$-biLipschitz on fibers, with $K=2k(Lk+2)$, such
that

(10) $\varphi(Y^{\prime}\times\mathbb{R}^{s})\subseteq f^{-1}(0).$

In particular, each ‘partially defined’ intrinsically Lipschitz graph
$\varphi(Y^{\prime}\times\mathbb{R}^{s})$ is a subset of a ‘globally defined’
intrinsically Lipschitz graph $f^{-1}(0)$.

We need to mention that there have been several earlier partial results on
extensions of Lipschitz graphs, as for example in [FSSC06], [Vit, Theorem
1.5], [Mon14, Proposition 4.8], in the Heisenberg group with codimension
larger than one; [Vit12, Proposition 3.4], for the case of codimension one in
the Heisenberg groups; [FS16, Theorem 4.1], for the case of codimension one in
Carnot groups. Finally, for the general metric spaces the reader can see
[AP20, LN05, Oht09].

###### Proof of Theorem 4.1 (4.1.i).

It is proved in [DDLD22]. ∎

###### Proof of Theorem 4.1 (4.1.ii).

Step 1. Fix $i=1,\dots,s$ and, for simplicity, we write $\tau^{-1},f_{x_{0}}$
instead of $\tau^{-1}_{i}$ and $f_{x_{0},i}.$ Fix
$x_{0}\in\tau^{-1}(\tau_{0}).$ We consider the map $f_{x_{0}}:X\to\mathbb{R}$
defined as

(11)
$f_{x_{0}}(x)=\left\\{\begin{array}[]{lcl}2(\tau(x)-\tau(x_{0})-\alpha\delta_{\tau_{0}}(x))&&\mbox{
if }\,\,|\tau(x)-\tau(x_{0})|\leq 2\alpha\delta_{\tau_{0}}(x)\\\
\tau(x)-\tau(x_{0})&&\mbox{ if
}\,\,\tau(x)-\tau(x_{0})>2\alpha\delta_{\tau_{0}}(x)\\\
3(\tau(x)-\tau(x_{0}))&&\mbox{ if
}\,\,\tau(x)-\tau(x_{0})<-2\alpha\delta_{\tau_{0}}(x)\\\ \end{array}\right.$

where $\alpha:=kL+1.$ We prove that $f_{x_{0}}$ satisfies the following
properties:

$(i)$:

$f_{x_{0}}$ is $K$-Lipschitz;

$(ii)$:

$f_{x_{0}}(x_{0})=0;$

$(iii)$:

$f_{x_{0}}$ is biLipschitz on fibers.

where $K=\max\\{3k,2k+2\alpha k\\}=2k+2\alpha k$ because $\alpha>1.$ The
property $(i)$ derives using that $\tau,\delta_{\tau_{0}}$ are both Lipschitz
and $X$ is a geodesic space. On the other hand, $(ii)$ is true noting that
$\delta_{\tau_{0}}(x_{0})=\rho(x_{0},\varphi_{\tau_{0}}(\pi(x_{0})))=0$
because $x_{0}\in\varphi_{\tau_{0}}(Y).$

Finally, for any $x,x^{\prime}\in\pi^{-1}(y)$ we have that
$\rho(x_{0},\varphi_{\tau_{0}}(\pi(x)))=\rho(x_{0},\varphi_{\tau_{0}}(\pi(x^{\prime})))$
and so $f$ is biLipschitz on fibers because $\tau$ is so too. Hence $(iii)$
holds.

At this point, we consider the map $f:X\to\mathbb{R}$ given by

$f(x):=\sup_{x_{0}\in\varphi(Y)}f_{x_{0}}(x),\quad\forall x\in X,$

and we want to prove that it is the map we are looking for. The Lipschitz
properties are true recall that the function $\delta_{x_{0}}$ is constant on
the fibers. Consequently, the only non trivial fact to show is (10). Fix
$\bar{x}_{0}\in\varphi(Y^{\prime})$. By $(ii)$ we have that
$f_{\bar{x}_{0}}(\bar{x}_{0})=0$ and so it is sufficient to prove that
$f_{x_{0}}(\bar{x}_{0})\leq 0$ for $x_{0}\in\varphi(Y^{\prime}).$ Let
$x_{0}\in\varphi(Y^{\prime}).$ Then using in addition that $\tau$ is
$k$-Lipschitz, and that $\varphi$ is intrinsically $L$-Lipschitz, we have

$|\tau(\bar{x}_{0})-\tau(x_{0})|\leq kd(\bar{x}_{0},x_{0})\leq
Lkd(x_{0},\pi^{-1}(\pi(\bar{x}_{0})))\leq
Lkd(x_{0},\varphi_{\tau_{0}}(\pi(\bar{x}_{0})))<\alpha\delta_{\tau_{0}}(\bar{x}_{0}),$

and so

$f_{x_{0}}(\bar{x}_{0})=2(\tau(\bar{x}_{0})-\tau(x_{0})-\alpha\delta_{\tau_{0}}(\bar{x}_{0}))<0,$

i.e., (10) holds.

Step 2. We consider $f=(f_{1},\dots,f_{s})$ where each $f_{i}$ is given by
$\tau_{i}^{-1}.$ Roughly speaking, here the problem is that when we put
together $(f_{1},\dots,f_{s})$ using $\tau_{1}^{-1},\dots,\tau_{s}^{-1}$ then
$f$ can not Lipschitz. But now $f$ is Lipschitz thanks to the construction of
$Y\times\mathbb{R}^{s}$ and in particular of
$\varphi_{1,\tau_{0}}:Y\times\mathbb{R}\times\\{0\\}^{s-1}\to
X;\dots;\varphi_{s,\tau_{0}}:Y\times\\{0\\}^{s-1}\times\mathbb{R}\to X.$ ∎

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