# Abelian charge inspired by family number

Phung Van Dong  Corresponding author<EMAIL_ADDRESS>Tran
Ngoc Hung<EMAIL_ADDRESS>Duong Van Loi
<EMAIL_ADDRESS>Phenikaa Institute for Advanced Study and
Faculty of Basic Science, Phenikaa University, Yen Nghia, Ha Dong, Hanoi
100000, Vietnam

###### Abstract

Quark has an electric charge either $-1/3$ or $2/3$ and a baryon number $1/3$,
where the divisions $3$’s match the color number. Although the electric charge
and the baryon number have a nature distinct from the color charge, the
matching is necessary for the standard model or a relevant $B-L$ extension
consistent at quantum level, since the relevant anomaly
$[SU(2)_{L}]^{2}U(1)_{A}$ for $A=Y$ or $B-L$ must vanish. If elementary
particles have a new $U(1)$ charge differently from $A$, such anomaly is not
cancelled for each family. However, if we demand that the anomaly is cancelled
over all families, this relates the color number to the family number instead
of the electric charge and baryon number, and interestingly the family number
guides us to a novel $U(1)$ theory. We will discuss the implication of this
theory for neutrino mass, recent $W$-boson mass anomaly, FCNC, and particle
colliders.

##  I Motivation

The weak isospin symmetry $SU(2)_{L}$, which forms isodoublets
$l_{L}=(\nu_{L},e_{L})$ and $q_{L}=(u_{L},d_{L})$ while the right-handed
partners $e_{R}$, $u_{R}$, and $d_{R}$ are isosinglets, is not completed. The
electric charge $Q$, which is $\mathrm{diag}(0,-1)$ for $l_{L}$ and
$\mathrm{diag}(2/3,-1/3)$ for $q_{L}$, is neither commuted nor closed
algebraically with $SU(2)_{L}$, because of $[Q,T_{1}\pm iT_{2}]=\pm(T_{1}\pm
iT_{2})\neq 0$ and $\mathrm{Tr}(Q)\neq 0$. Since $[T_{3},T_{1}\pm
iT_{2}]=\pm(T_{1}\pm iT_{2})$ is similar to $Q$, a new abelian charge $Y\equiv
Q-T_{3}$ arises, such that $[Y,T_{1}\pm iT_{2}]=0=[Y,T_{3}]$. This defines a
complete group, $SU(2)_{L}\otimes U(1)_{Y}$, which establishes the standard
electroweak model [1, 2, 3]. The model successfully works at quantum level
because every anomaly vanishes for which the most important one is
$[SU(2)_{L}]^{2}U(1)_{Y}\sim
3Y_{q_{L}}+Y_{l_{L}}=3\times\frac{1}{6}-\frac{1}{2}=0$. This cancellation
occurring for each family is due to the fact that the color number $3$ relates
the electric charge of quark to that of lepton. Besides the hypercharge $Y$,
the model contains an accident gauge symmetry of baryon number minus lepton
number $B-L$, if a right-handed neutrino $\nu_{R}$ is simply imposed for each
family. In this case, every anomaly is cancelled too for which the most
important one is $[SU(2)_{L}]^{2}U(1)_{B-L}\sim
3[B-L]_{q_{L}}+[B-L]_{l_{L}}=3\times\frac{1}{3}-1=0$. This cancellation is due
to the fact that the color number $3$ relates the $B-L$ number of quark to
that of lepton, analogous to the case of electric charge. Who arranges such
matchings, since the electric charge and the $B-L$ number have a nature
distinct from the color number?

This work does not directly answer such question. We guess that at high energy
the partial interactions are unified in a simple group, well-known as grand
unification. The unification group would contain the color charge and weak
isospin, besides neutral generators that determine the hypercharge and even
$B-L$. This work does not search for a specific unification group. Instead, we
investigate possible neutral charges that lead to the hypercharge. The
hypothetical symmetry of $B-L$ although mentioned will not be addressed.
Obviously, an individual neutral charge, called $X$, does not necessarily have
the above property as of the hypercharge, i.e. $3X_{q_{L}}+X_{l_{L}}\neq 0$
for a single family. The anomaly $[SU(2)_{L}]^{2}U(1)_{X}$ is cancelled,
needed to a correlative contribution with exotic particles or between
families. The latter case would make a constraint on the number of families.
The minimal grand unifications such as $SU(5)$ [4], $SO(10)$ [5], and Pati-
Salam model [6] do not contain $X$, since they leave only the hypercharge
and/or $B-L$, $T_{3R}$. The trinification [7] and $E_{6}$ [8] as well as their
variants [10, 9] would provide such a $X$.

In what follows, we search for such neutral charges and propose the model of
interest. We discuss the new physics insights that the new proposal reveals.
Interestingly, this approach provides a potential explanation to neutrino
mass, $W$-boson mass deviation, compelling phenomena for flavor physics, as
well as hints for particle colliders.

##  II Proposal of the model

Instead of $Y,B-L$ charges and even their combination, we look for an entirely
alternative charge $X$ such that the relation $3X_{q_{L}}+X_{l_{L}}=0$ does
not hold for each family. That said, the new electroweak group takes the form,

$SU(2)_{L}\otimes U(1)_{X}\otimes U(1)_{N},$ (1)

where $3X_{q_{L}}+X_{l_{L}}\neq 0$ for each family, and the charge $N$ is
necessarily included such that $U(1)_{X}\otimes U(1)_{N}$ is broken down to
$U(1)_{Y}$ at high energy, i.e. $N=Y-X$. The $N$ charge obeys
$3N_{q_{L}}+N_{l_{L}}=-(3X_{q_{L}}+X_{l_{L}})\neq 0$ as $X$ does, for each
family. Thus $X,N$ have a nature different from $Y$ (and $B-L$ too), but the
breaking of $X,N$ explains $Y$ and the matching.

Given a grand unification, $X$ and $N$ would be unified with $SU(2)_{L}$ in a
higher isospin group. Hence, the magnitudes of $X,N$ can be fixed as the
neutral charges of the large group. Obviously, $X$ (or $N$) is not originated
from a charge, such as $T_{3R}$ or $B-L$, as in minimal left-right symmetry,
Pati-Salam group, and $SO(10)$. Looking for trinification, we find that
$T_{8L,R}$ have a property of $X$, namely $X\sim T_{8L}+T_{8R}$, while $N$ is
followed by $N=Y-X$. Note that fermion doublets may be arranged in
trinification triplets/antitriplets. Hence, we restrict ourselves by assigning
$X_{l_{L}}=x$ for all lepton doublets, $X_{q_{L}}=x$ for a number of quark
doublets, while $X_{q_{L}}=-x$ for the rest of quark doublets. Here the
normalization of $x$ is left arbitrarily. The condition
$3X_{q_{L}}+X_{l_{L}}\neq 0$ for each family simply requires $x\neq 0$.
Summing over $N_{f}$ families, the anomaly $[SU(2)_{L}]^{2}U(1)_{X}\sim
3(m-n)x+N_{f}x$ vanishes if $N_{f}=3(n-m)$, where $m$ and $n=N_{f}-m$ denote
numbers of $q_{L}$ doublets that possess $x$ and $-x$, respectively. It is
clear that the similar anomaly for $N$, i.e. $[SU(2)_{L}]^{2}U(1)_{N}$,
vanishes as a result. Hence, the family number $N_{f}=3(n-m)$ is an integer
multiple of the color number, $3$, for which fixing $N_{f}=3$ leads to $n=2$
and $m=1$. We define two kinds of family indices, like $a,b,...$ run over
1,2,3 according to $N_{f}=3$ and $\alpha,\beta,...$ run over 1,2 according to
$n=2$. We then assign $(X,N)$ charges to fermion representations as in Table
1, where the color and isospin quantum numbers are also included for
completeness.

Field | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{X}$ | $U(1)_{N}$
---|---|---|---|---
$l_{aL}=\begin{pmatrix}\nu_{aL}\\\ e_{aL}\end{pmatrix}$ | 1 | 2 | $x$ | $-1/2-x$
$\nu_{aR}$ | 1 | 1 | $x$ | $-x$
$e_{aR}$ | 1 | 1 | $x$ | $-1-x$
$q_{\alpha L}=\begin{pmatrix}u_{\alpha L}\\\ d_{\alpha L}\end{pmatrix}$ | 3 | 2 | $-x$ | $1/6+x$
$u_{\alpha R}$ | 3 | 1 | $-x$ | $2/3+x$
$d_{\alpha R}$ | 3 | 1 | $-x$ | $-1/3+x$
$q_{3L}=\begin{pmatrix}u_{3L}\\\ d_{3L}\end{pmatrix}$ | 3 | 2 | $x$ | $1/6-x$
$u_{3R}$ | 3 | 1 | $x$ | $2/3-x$
$d_{3R}$ | 3 | 1 | $x$ | $-1/3-x$
$H=\begin{pmatrix}H^{+}_{1}\\\ H^{0}_{2}\end{pmatrix}$ | 1 | 2 | 0 | $1/2$
$\Phi=\begin{pmatrix}\Phi^{+}_{1}\\\ \Phi^{0}_{2}\end{pmatrix}$ | 1 | 2 | $-2x$ | $1/2+2x$
$\chi$ | 1 | 1 | $-2x$ | $2x$
Table 1:  Field representation under the gauge symmetry.

That said, the third quark doublet has a $X$-charge as of lepton doublets,
while the first and second quark doublets have an opposite $X$-charge. Hence,
$3X_{q_{L}}+X_{l_{L}}=4x$ for the third family, while
$3X_{q_{L}}+X_{l_{L}}=-2x$ for the first and second families. The anomaly
$[SU(2)_{L}]^{2}U(1)_{X}\sim 3X_{q_{L}}+X_{l_{L}}$ vanishes when summing over
three families. It is verified that all the other anomalies, say
$[SU(3)_{C}]^{2}U(1)_{X}$, $[\mathrm{gravity}]^{2}U(1)_{X}$, those for
$X\rightarrow N$, $[U(1)_{X}]^{2}U(1)_{N}$, $U(1)_{X}[U(1)_{N}]^{2}$,
$[U(1)_{X}]^{3}$, and $[U(1)_{N}]^{3}$, vanish too. The anomaly cancellation
is due to the fact that the family number matches the color number. Last, but
not least, the right-handed neutrinos are presented, required for anomaly
cancellation. Additionally, for symmetry breaking and mass generation, we have
introduced the scalar fields: (i) $H$ is identical to that of the standard
model, (ii) $\Phi$ couples the two kinds of quarks $\bar{q}_{\alpha}$ and
$q_{3}$ necessary for recovering CKM matrix, and (iii) $\chi$ couples to
right-handed neutrinos $\nu_{R}\nu_{R}$ responsible for neutrino mass
generation.

Up to the gauge fixing and ghost terms, the total Lagrangian takes the form,

$\mathcal{L}=\mathcal{L}_{\mathrm{kin}}+\mathcal{L}_{\mathrm{Yuk}}-V.$ (2)

The kinetic term is

$\displaystyle\mathcal{L}_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle\sum_{F}\bar{F}i\gamma^{\mu}D_{\mu}F+\sum_{S}(D^{\mu}S)^{\dagger}(D_{\mu}S)$
(3)
$\displaystyle-\frac{1}{4}G_{p\mu\nu}G^{\mu\nu}_{p}-\frac{1}{4}A_{j\mu\nu}A^{\mu\nu}_{j}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}C_{\mu\nu}C^{\mu\nu},$

where $F$ and $S$ run over fermion and scalar multiplets, respectively. The
covariant derivative and field strength tensors are given by

$\displaystyle
D_{\mu}=\partial_{\mu}+ig_{s}t_{p}G_{p\mu}+igT_{j}A_{j\mu}+ig_{X}XB_{\mu}+ig_{N}NC_{\mu},$
$\displaystyle
G_{p\mu\nu}=\partial_{\mu}G_{p\nu}-\partial_{\nu}G_{p\mu}-g_{s}f_{pqr}G_{q\mu}G_{r\nu},$
$\displaystyle
A_{j\mu\nu}=\partial_{\mu}A_{j\nu}-\partial_{\nu}A_{j\mu}-g\epsilon_{jkl}A_{k\mu}A_{l\nu},$
$\displaystyle B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu},\hskip
14.22636ptC_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu},$

where $(g_{s},g,g_{X},g_{N})$, $(t_{p},T_{j},X,N)$, and
$(G_{p\mu},A_{j\mu},B_{\mu},C_{\mu})$ denote the coupling constants, the
generators, and the gauge bosons of the
$(SU(3)_{C},SU(2)_{L},U(1)_{X},U(1)_{N})$ groups, respectively. A kinetic
mixing term between the two $U(1)$ gauge fields is suppressed, due to its
small effect. The Yukawa Lagrangian and the scalar potential are given by

$\displaystyle\mathcal{L}_{\mathrm{Yuk}}$ $\displaystyle=$ $\displaystyle
h^{e}_{ab}\bar{l}_{aL}He_{bR}+h^{\nu}_{ab}\bar{l}_{aL}\tilde{H}\nu_{bR}+\frac{1}{2}f^{\nu}_{ab}\bar{\nu}^{c}_{aR}\nu_{bR}\chi$
(4) $\displaystyle+h^{d}_{\alpha\beta}\bar{q}_{\alpha L}Hd_{\beta
R}+h^{u}_{\alpha\beta}\bar{q}_{\alpha L}\tilde{H}u_{\beta
R}+h^{d}_{33}\bar{q}_{3L}Hd_{3R}+h^{u}_{33}\bar{q}_{3L}\tilde{H}u_{3R}$
$\displaystyle+h^{\prime d}_{\alpha 3}\bar{q}_{\alpha L}\Phi d_{3R}+h^{\prime
u}_{3\beta}\bar{q}_{3L}\tilde{\Phi}u_{\beta R}+h^{\prime
d}_{3\beta}\bar{q}_{3L}\Phi^{\prime}d_{\beta R}+h^{\prime u}_{\alpha
3}\bar{q}_{\alpha L}\tilde{\Phi}^{\prime}u_{3R}$ $\displaystyle+H.c.$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu^{2}_{1}H^{\dagger}H+\mu^{2}_{2}\Phi^{\dagger}\Phi+\mu^{2}_{3}\chi^{\dagger}\chi+\mu_{4}[(\Phi^{\dagger}H)\chi+H.c.]$
(5)
$\displaystyle+\lambda_{1}(H^{\dagger}H)^{2}+\lambda_{2}(\Phi^{\dagger}\Phi)^{2}+\lambda_{3}(\chi^{\dagger}\chi)^{2}$
$\displaystyle+\lambda_{4}(H^{\dagger}H)(\chi^{\dagger}\chi)+\lambda_{5}(\Phi^{\dagger}\Phi)(\chi^{\dagger}\chi)$
$\displaystyle+\lambda_{6}(H^{\dagger}H)(\Phi^{\dagger}\Phi)+\lambda_{7}(H^{\dagger}\Phi)(\Phi^{\dagger}H).$

Above, $f^{\nu}$, $h$’s, and $\lambda$’s are dimensionless, while
$\mu_{1,2,3,4}$ have a mass dimension. Additionally, we define
$\tilde{H}=i\sigma_{2}H^{*}$ and $\tilde{\Phi}=i\sigma_{2}\Phi^{*}$. The last
two terms in (4) before the $H.c.$ are radiatively induced by one-loop
diagrams despited in Fig. 1 respectively, through $\Phi,H$ exchanges and their
couplings to quarks, and that the scalar trilinear coupling
$\Phi^{\dagger}H\chi$ is set by $\mu_{4}$. Computing these diagrams, we
identify $\Phi^{\prime}\equiv H\chi^{*}(\mu_{4}/16\pi^{2}M^{2})$ and
$\tilde{\Phi}^{\prime}\equiv\tilde{H}\chi(\mu_{4}/16\pi^{2}M^{2})$, where
$1/16\pi^{2}$ is the loop factor, and $M$ is the characteristic large mass of
heavy scalar field running in the loop; additionally, the Yukawa couplings
induced are $h^{\prime d}_{3\beta}\equiv h^{\prime
u}_{3\delta}h^{*u}_{\gamma\delta}h^{d}_{\gamma\beta}$ and $h^{\prime
u}_{\alpha 3}\equiv h^{\prime d}_{\alpha 3}h^{*d}_{33}h^{u}_{33}$. For
convenience, in the following we will match
$\langle\Phi^{\prime}\rangle=\langle
H\rangle\langle\chi\rangle^{*}(\mu_{4}/16\pi^{2}M^{2})$ with
$\langle\Phi\rangle$ by rescaling $h^{\prime d}_{3\beta}$ and $h^{\prime
u}_{\alpha 3}$ appropriately.

Figure 1:  Yukawa couplings of $\Phi^{\prime}$ and $\tilde{\Phi}^{\prime}$
types generated by corresponding Feynman diagrams (left and right) that
identify $\Phi^{\prime}\sim H\chi^{*}$ and
$\tilde{\Phi}^{\prime}\sim\tilde{H}\chi$, respectively.

At this point, there are two extra models alternative to the current model,
given in order

  1. 1.

A generic model may be built by introducing two new fundamental Higgs doublets
in addition to the usual $H$, i.e. $\Phi\sim(2,-2x,1/2+2x)$ and
$\Phi^{\prime}\sim(2,2x,1/2-2x)$, which transform under the symmetry (1) as
$H\chi$ and $H\chi^{*}$, respectively. This model is not significantly
different from the current model for phenomenological aspects interested in
this work, thus not considered.

  2. 2.

A minimal model can be built by suppressing both $\Phi$ and $\Phi^{\prime}$.
Instead, their Yukawa interactions in (4) are given by corresponding effective
interactions by substituting $\Phi=H\chi/M^{\prime}$ and
$\Phi^{\prime}=H\chi^{*}/M^{\prime}$, where $M^{\prime}$ is a cutoff scale set
by a more fundamental theory. Interestingly, this model does not encounter any
tree-level FCNC coupled to scalars in contrast to the above models. However,
it presents the phenomenological aspects interested in this work similar to
the current model, thus skipped too.

The gauge symmetry is broken as follows

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$
---
$\downarrow\Lambda$
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$
$\downarrow v_{1,2}$
$SU(3)_{C}\otimes U(1)_{Q}$

Here the vacuum expectation values (VEVs) are given by

$\displaystyle\langle\chi\rangle=\Lambda/\sqrt{2},$ (6) $\displaystyle\langle
H\rangle=(0,v_{1}/\sqrt{2}),$ (7)
$\displaystyle\langle\Phi\rangle=(0,v_{2}/\sqrt{2}),$ (8)

and we require $\Lambda\gg v_{1,2}$ and $v^{2}_{1}+v^{2}_{2}=(246\
\mathrm{GeV})^{2}$ for consistency with the standard model. As stated, the
combined field $\Phi^{\prime}$ gets a value matching that of $\Phi$, i.e.
$\langle\Phi^{\prime}\rangle=(0,v_{2}/\sqrt{2})$.

##  III Fermion mass

Substituting the VEVs to the Yukawa Lagrangian, we obtain fermion masses.
First, charged leptons gain a mass via $h^{e}$ coupling to be

$[m_{e}]_{ab}=-h^{e}_{ab}\frac{v_{1}}{\sqrt{2}}.$ (9)

Diagonalizing this mass matrix, we get the masses of electron, muon, and tau,
as usual.

The neutrinos $\nu_{L,R}$ gain a Dirac mass via $h^{\nu}$ coupling,

$[m_{D}]_{ab}=-h^{\nu}_{ab}\frac{v_{1}}{\sqrt{2}},$ (10)

while $\nu_{R}$ obtains a Majorana mass via $f^{\nu}$ coupling,

$[m_{M}]_{ab}=-f^{\nu}_{ab}\frac{\Lambda}{\sqrt{2}}.$ (11)

The total mass matrix of neutrinos takes the form,

$\displaystyle\mathcal{L}_{\mathrm{Yuk}}\supset-\frac{1}{2}(\bar{\nu}_{aL},\bar{\nu}^{c}_{aR})\begin{pmatrix}0&[m_{D}]_{ab}\\\
[m^{T}_{D}]_{ab}&[m_{M}]_{ab}\end{pmatrix}\begin{pmatrix}\nu^{c}_{bL}\\\
\nu_{bR}\end{pmatrix}+H.c.$

Because of $v_{1}\ll\Lambda$, i.e. $m_{D}\ll m_{M}$, the active neutrinos
$\sim\nu_{L}$ receive a small mass via the canonical seesaw,

$[m_{\nu}]_{ab}=-[m_{D}m^{-1}_{M}m^{T}_{D}]_{ab}\sim v^{2}_{1}/\Lambda.$ (12)

Taking $v_{1}\sim 100$ GeV and $\Lambda\sim 5$–10 TeV, the observed values
$m_{\nu}\sim 0.1$ eV require $h^{\nu}\sim 10^{-5}$, given that $f^{\nu}\sim
1$. The heavy neutrinos $\sim\nu_{R}$ obtain a mass $m_{M}$ at $\Lambda$
scale.

Concerning quarks, their masses are given by

$[m_{d}]_{\alpha\beta}=-h^{d}_{\alpha\beta}\frac{v_{1}}{\sqrt{2}},\hskip
14.22636pt[m_{d}]_{33}=-h^{d}_{33}\frac{v_{1}}{\sqrt{2}},$ (13)
$[m_{d}]_{\alpha 3}=-h^{\prime d}_{\alpha 3}\frac{v_{2}}{\sqrt{2}},\hskip
14.22636pt[m_{d}]_{3\beta}=-h^{\prime d}_{3\beta}\frac{v_{2}}{\sqrt{2}},$ (14)

for down-type quarks, while

$[m_{u}]_{\alpha\beta}=-h^{u}_{\alpha\beta}\frac{v_{1}}{\sqrt{2}},\hskip
14.22636pt[m_{u}]_{33}=-h^{u}_{33}\frac{v_{1}}{\sqrt{2}},$ (15)
$[m_{u}]_{\alpha 3}=-h^{\prime u}_{\alpha 3}\frac{v_{2}}{\sqrt{2}},\hskip
14.22636pt[m_{u}]_{3\beta}=-h^{\prime u}_{3\beta}\frac{v_{2}}{\sqrt{2}},$ (16)

for up-type quarks. Hence, the quark masses are governed by two weak scales
$v_{1,2}$ as well as two kinds of couplings $h$ and $h^{\prime}$. The small
mixing between the third quark family and the first two quark families can be
understood by $h^{\prime}<h$ and $v_{2}<v_{1}$. Of course, diagonalizing the
quark mass matrices, we get the masses of $u,d,c,s,t,b$ and CKM matrix, as
expected.

##  IV Gauge boson mass

Substituting the VEVs to the scalar kinetic term, i.e.
$\sum_{S}(D^{\mu}S)^{\dagger}(D_{\mu}S)$, we get gauge boson masses. First
notice that the gluons are massless since $SU(3)_{C}$ is conserved, unbroken.
Concerning the electroweak sector, the charged gauge boson $W^{\pm}=(A_{1}\mp
iA_{2})/\sqrt{2}$ is a physical field by itself with mass,

$m^{2}_{W}=\frac{g^{2}}{4}(v^{2}_{1}+v^{2}_{2}).$ (17)

Whereas, the neutral gauge bosons $A_{3}$, $B$, and $C$ mix via a mass matrix,

$\mathcal{L}_{\mathrm{kin}}\supset\frac{1}{2}(A_{3},B,C)M^{2}\begin{pmatrix}A_{3}\\\
B\\\ C\end{pmatrix},$ (18)

where

$M^{2}=\begin{pmatrix}\frac{g^{2}}{4}(v^{2}_{1}+v^{2}_{2})&xgg_{X}v^{2}_{2}&-\frac{gg_{N}}{4}[(1+4x)v^{2}_{2}+v^{2}_{1}]\\\
xgg_{X}v^{2}_{2}&4x^{2}g^{2}_{X}(\Lambda^{2}+v^{2}_{2})&-g_{X}g_{N}[x(1+4x)v^{2}_{2}+4x^{2}\Lambda^{2}]\\\
-\frac{gg_{N}}{4}[(1+4x)v^{2}_{2}+v^{2}_{1}]&-g_{X}g_{N}[x(1+4x)v^{2}_{2}+4x^{2}\Lambda^{2}]&g^{2}_{N}[4x^{2}\Lambda^{2}+\frac{v^{2}_{2}}{4}(1+4x)^{2}+\frac{v^{2}_{1}}{4}]\end{pmatrix}.$
(19)

The mass matrix of neutral gauge bosons has an exact, zero eigenvalue (i.e.
photon mass) corresponding to the exact mass eigenstate (i.e. photon field),
such as

$\frac{A}{e}=\frac{A_{3}}{g}+\frac{B}{g_{X}}+\frac{C}{g_{N}},$ (20)

which is normalized by

$\frac{1}{e^{2}}=\frac{1}{g^{2}}+\frac{1}{g^{2}_{X}}+\frac{1}{g^{2}_{N}}.$
(21)

The photon field properly couples to the electric charge
$Q=T_{3}+Y=T_{3}+X+N$, since the above solution can be obtained by
substituting each generator by corresponding gauge field over coupling, say
$Q$ by $A/e$, $T_{3}$ by $A_{3}/g$, $X$ by $B/g_{X}$, and $N$ by $C/g_{N}$.
The electromagnetic coupling is identified as $e=gs_{W}$ through the sine of
the Weinberg’s angle, while the tan of this angle is given by $t_{W}=g_{Y}/g$,
where the hypercharge coupling is obtained by
$g_{Y}=g_{X}g_{N}/\sqrt{g^{2}_{X}+g^{2}_{N}}=g_{X}s_{\theta}=g_{N}c_{\theta}$,
thus $t_{\theta}=g_{N}/g_{X}$. We rewrite

$A=s_{W}A_{3}+c_{W}(s_{\theta}B+c_{\theta}C).$ (22)

We choose two other fields: the usual $Z$ field,

$Z=c_{W}A_{3}-s_{W}(s_{\theta}B+c_{\theta}C),$ (23)

which is given orthogonally to $A$, as usual, and the new field,

$Z^{\prime}=c_{\theta}B-s_{\theta}C,$ (24)

which is given orthogonally to the hypercharge field in parentheses.

In the new basis $(A,Z,Z^{\prime})$, the photon $A$ is decoupled as a physical
field, while the remainders $Z,Z^{\prime}$ mix through a mass matrix,

$\mathcal{L}_{\mathrm{kin}}\supset\frac{1}{2}(Z,Z^{\prime})\begin{pmatrix}m^{2}_{Z}&m^{2}_{ZZ^{\prime}}\\\
m^{2}_{ZZ^{\prime}}&m^{2}_{Z^{\prime}}\end{pmatrix}\begin{pmatrix}Z\\\
Z^{\prime}\end{pmatrix},$ (25)

where

$\displaystyle m^{2}_{Z}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4c^{2}_{W}}(v^{2}_{1}+v^{2}_{2}),$ (26)
$\displaystyle m^{2}_{ZZ^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}t_{W}}{2s_{2\theta}c_{W}}\left[s^{2}_{\theta}v^{2}_{1}+(s^{2}_{\theta}+4x)v^{2}_{2}\right],$
(27) $\displaystyle m^{2}_{Z^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}t^{2}_{W}}{s^{2}_{2\theta}}\left[s^{4}_{\theta}v^{2}_{1}+(s^{2}_{\theta}+4x)^{2}v^{2}_{2}+16x^{2}\Lambda^{2}\right].$
(28)

Diagonalizing the $Z,Z^{\prime}$ mass matrix, we obtain two physical fields,

$\displaystyle Z_{1}$ $\displaystyle=$ $\displaystyle
c_{\varphi}Z-s_{\varphi}Z^{\prime},$ (29) $\displaystyle Z_{2}$
$\displaystyle=$ $\displaystyle s_{\varphi}Z+c_{\varphi}Z^{\prime},$ (30)

where the $Z$-$Z^{\prime}$ mixing angle ($\varphi$) obeys

$t_{2\varphi}=\frac{2m^{2}_{ZZ^{\prime}}}{m^{2}_{Z^{\prime}}-m^{2}_{Z}}\simeq\frac{s_{2\theta}\left[s^{2}_{\theta}v^{2}_{1}+(s^{2}_{\theta}+4x)v^{2}_{2}\right]}{16s_{W}x^{2}\Lambda^{2}},$
(31)

and the $Z_{1,2}$ masses are given by

$\displaystyle m^{2}_{Z_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{Z}+m^{2}_{Z^{\prime}}-\sqrt{(m^{2}_{Z}-m^{2}_{Z^{\prime}})^{2}+4m^{4}_{ZZ^{\prime}}}\right]\simeq
m^{2}_{Z}-\frac{m^{4}_{ZZ^{\prime}}}{m^{2}_{Z^{\prime}}},$ (32) $\displaystyle
m^{2}_{Z_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{Z}+m^{2}_{Z^{\prime}}+\sqrt{(m^{2}_{Z}-m^{2}_{Z^{\prime}})^{2}+4m^{4}_{ZZ^{\prime}}}\right]\simeq
m^{2}_{Z^{\prime}}.$ (33)

The $Z$-$Z^{\prime}$ mixing, i.e. $\varphi$, is small as suppressed by
$(v_{1},v_{2})^{2}/\Lambda^{2}$. The field $Z_{1}$ has a mass approximating
that of $Z$, called the standard model $Z$-like boson, whereas the field
$Z_{2}$ is a new heavy gauge boson with mass proportional to $\Lambda$.

##  V $W$-boson mass deviation

The model under consideration yields a tree-level mixing of $Z$ with
$Z^{\prime}$. Because of this mixing, the observed $Z_{1}$ mass is reduced,
compared with the standard model $Z$ mass. Additionally, this reduction gives
rise to a positive contribution to $T$-parameter,

$\alpha
T=\rho-1=\frac{m^{2}_{W}}{c^{2}_{W}m^{2}_{Z_{1}}}-1\simeq\frac{m^{4}_{ZZ^{\prime}}}{m^{2}_{Z}m^{2}_{Z^{\prime}}}\simeq\frac{\left[s^{2}_{\theta}v^{2}_{1}+(s^{2}_{\theta}+4x)v^{2}_{2}\right]^{2}}{16x^{2}(v^{2}_{1}+v^{2}_{2})\Lambda^{2}},$
(34)

with the aid of $m_{W}=m_{Z}c_{W}$ and $m_{Z_{1}}$ approximated above. Since
the $Z_{1}$ mass is fixed as precisely measured, the positive value of $\alpha
T$ dominantly enhances the $W$ mass (cf. [11, 12]),

$\Delta m^{2}_{W}=\frac{c^{4}_{W}m^{2}_{Z}}{c^{2}_{W}-s^{2}_{W}}\alpha
T\simeq\frac{g^{2}c^{2}_{W}\left[s^{2}_{\theta}v^{2}_{1}+(s^{2}_{\theta}+4x)v^{2}_{2}\right]^{2}}{64x^{2}c_{2W}\Lambda^{2}}.$
(35)

Use the recent measurement $m_{W}|_{\mathrm{CDF}}=80.4335\pm 0.0094$ GeV [13],
which deviates from the standard model prediction,
$m_{W}|_{\mathrm{SM}}=80.357\pm 0.006$ GeV, at 7$\sigma$. We make a contour of
$\Delta m^{2}_{W}=m^{2}_{W}|_{\mathrm{CDF}}-m^{2}_{W}|_{\mathrm{SM}}$ as
function of $v_{1}$ and $\Lambda$ for a choice of $x=\pm\frac{1}{6}$ and
$\pm\frac{1}{2}$, as in Figure 2 (solid, color curves). Here we have used
$v_{2}=\sqrt{(246\ \mathrm{GeV})^{2}-v^{2}_{1}}$, fixed
$t_{\theta}=g_{N}/g_{X}=1$, and taken $s^{2}_{W}=0.231$ and $\alpha=1/128$. It
is clear from the figure that concerning the $W$-mass deviation, the new
physics scale $\Lambda$ depends on $x$ for each $v_{1}$. For each $|x|$-value,
$\Lambda$ is larger for positive $x$, while $\Lambda$ is smaller for negative
$x$. Additionally, $\Lambda$ is appropriate to the seesaw setup as well as the
FCNC and collider bounds (discussed below) for interested values of $x$ with
an appropriate choice of $v_{1}$. However, the scale $\Lambda$ is being
constrained by $Z$-fermion couplings, specified in the following section.

Figure 2:  CDF $W$-mass deviation contoured as function of $(v_{1},\Lambda)$
according to several values of $x$-charge parameter (solid, color curves). New
physics scale $\Lambda$ limited by the electroweak precision test on
$|\varphi|\lesssim 10^{-3}$ as correspondingly combined (dashed curves).

##  VI Interactions

Substituting the gauge fields in terms of the physical fields obtained above,
the covariant derivative takes the form,

$\displaystyle D_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+ig_{s}t_{p}G_{p\mu}+ieQA_{\mu}+ig(T_{+}W^{+}_{\mu}+H.c.)$
(36)
$\displaystyle+\frac{ig}{c_{W}}\left[c_{\varphi}(T_{3}-s^{2}_{W}Q)-\frac{s_{\varphi}s_{W}}{s_{\theta}c_{\theta}}(X-s^{2}_{\theta}Y)\right]Z_{1\mu}$
$\displaystyle+\frac{ig}{c_{W}}\left[s_{\varphi}(T_{3}-s^{2}_{W}Q)+\frac{c_{\varphi}s_{W}}{s_{\theta}c_{\theta}}(X-s^{2}_{\theta}Y)\right]Z_{2\mu},$

where $T_{\pm}=(T_{1}\pm iT_{2})/\sqrt{2}$. Substituting this covariant
derivative into the fermion kinetic term,
$\sum_{F}\bar{F}i\gamma^{\mu}D_{\mu}F$, the interactions of gauge bosons with
fermions are derived.

It is clear that gluons, photon, and $W^{\pm}$ interact with fermions as in
the standard model,

$\displaystyle\mathcal{L}_{\mathrm{int}}$ $\displaystyle\supset$
$\displaystyle-\frac{g_{s}}{2}\bar{q}\gamma^{\mu}\lambda_{p}qG_{p\mu}-eQ(f)\bar{f}\gamma^{\mu}fA_{\mu}-\frac{g}{\sqrt{2}}[(\bar{\nu}_{L}\gamma^{\mu}e_{L}+\bar{u}_{L}\gamma^{\mu}d_{L})W^{+}_{\mu}+H.c.],$
(37)

where $\lambda_{p}$ for $p=1,2,\cdots,8$ denotes the Gell-Mann matrices.
Additionally, $q$ and $f$ indicate every quark and fermion in mass eigenstate
basis, while $\nu$, $e$, $u$, and $d$ stand for neutrinos, charged leptons, up
quarks, and down quarks in gauge basis, respectively.

The interactions of $Z_{1,2}$ with fermions have the form,

$\mathcal{L}_{\mathrm{int}}\supset-\frac{g}{2c_{W}}\bar{f}\gamma^{\mu}[g^{Z_{I}}_{V}(f)-g^{Z_{I}}_{A}(f)\gamma_{5}]fZ_{I\mu},$
(38)

where $I=1,2$, and

$\displaystyle g^{Z_{1}}_{V}(f)$ $\displaystyle=$
$\displaystyle(c_{\varphi}-s_{\varphi}s_{W}t_{\theta})T_{3}(f_{L})$ (39)
$\displaystyle-2s_{W}(c_{\varphi}s_{W}-s_{\varphi}t_{\theta})Q(f)-2(s_{\varphi}s_{W}/s_{\theta}c_{\theta})X(f),$
$\displaystyle g^{Z_{1}}_{A}(f)$ $\displaystyle=$
$\displaystyle(c_{\varphi}-s_{\varphi}s_{W}t_{\theta})T_{3}(f_{L}),$ (40)
$\displaystyle g^{Z_{2}}_{V,A}(f)$ $\displaystyle=$ $\displaystyle
g^{Z_{1}}_{V,A}(f)|_{c_{\varphi}\rightarrow
s_{\varphi},s_{\varphi}\rightarrow-c_{\varphi}}.$ (41)

The $V,A$ couplings of $Z_{1,2}$ with fermions are collected in Table 2 and 3,
respectively.

$f$ | $g^{Z_{1}}_{V}(f)$ | $g^{Z_{1}}_{A}(f)$
---|---|---
$\nu_{a}$ | $\frac{c_{\varphi}}{2}-s_{W}s_{\varphi}\left(\frac{t_{\theta}}{2}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}-s_{W}s_{\varphi}t_{\theta}\right)$
$e_{a}$ | $\frac{c_{\varphi}}{2}\left(4s^{2}_{W}-1\right)-s_{W}s_{\varphi}\left(\frac{3t_{\theta}}{2}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{W}s_{\varphi}t_{\theta}-c_{\varphi}\right)$
$u_{\alpha}$ | $\frac{c_{\varphi}}{6}\left(3-8s^{2}_{W}\right)+s_{W}s_{\varphi}\left(\frac{5t_{\theta}}{6}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}-s_{W}s_{\varphi}t_{\theta}\right)$
$u_{3}$ | $\frac{c_{\varphi}}{6}\left(3-8s^{2}_{W}\right)+s_{W}s_{\varphi}\left(\frac{5t_{\theta}}{6}-\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}-s_{W}s_{\varphi}t_{\theta}\right)$
$d_{\alpha}$ | $\frac{c_{\varphi}}{6}\left(4s^{2}_{W}-3\right)-s_{W}s_{\varphi}\left(\frac{t_{\theta}}{6}-\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{W}s_{\varphi}t_{\theta}-c_{\varphi}\right)$
$d_{3}$ | $\frac{c_{\varphi}}{6}\left(4s^{2}_{W}-3\right)-s_{W}s_{\varphi}\left(\frac{t_{\theta}}{6}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{W}s_{\varphi}t_{\theta}-c_{\varphi}\right)$
Table 2:  Couplings of $Z_{1}$ with fermions. $f$ | $g^{Z_{2}}_{V}(f)$ | $g^{Z_{2}}_{A}(f)$
---|---|---
$\nu_{a}$ | $\frac{s_{\varphi}}{2}+s_{W}c_{\varphi}\left(\frac{t_{\theta}}{2}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}+s_{W}c_{\varphi}t_{\theta}\right)$
$e_{a}$ | $\frac{s_{\varphi}}{2}\left(4s^{2}_{W}-1\right)+s_{W}c_{\varphi}\left(\frac{3t_{\theta}}{2}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(s_{W}c_{\varphi}t_{\theta}+s_{\varphi}\right)$
$u_{\alpha}$ | $\frac{s_{\varphi}}{6}\left(3-8s^{2}_{W}\right)-s_{W}c_{\varphi}\left(\frac{5t_{\theta}}{6}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}+s_{W}c_{\varphi}t_{\theta}\right)$
$u_{3}$ | $\frac{s_{\varphi}}{6}\left(3-8s^{2}_{W}\right)-s_{W}c_{\varphi}\left(\frac{5t_{\theta}}{6}-\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}+s_{W}c_{\varphi}t_{\theta}\right)$
$d_{\alpha}$ | $\frac{s_{\varphi}}{6}\left(4s^{2}_{W}-3\right)+s_{W}c_{\varphi}\left(\frac{t_{\theta}}{6}-\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(s_{W}c_{\varphi}t_{\theta}+s_{\varphi}\right)$
$d_{3}$ | $\frac{s_{\varphi}}{6}\left(4s^{2}_{W}-3\right)+s_{W}c_{\varphi}\left(\frac{t_{\theta}}{6}+\frac{2x}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(s_{W}c_{\varphi}t_{\theta}+s_{\varphi}\right)$
Table 3:  Couplings of $Z_{2}$ with fermions.

Notice that $|\varphi|\ll 1$ as substantially suppressed by
$(v_{1},v_{2})^{2}/\Lambda^{2}$. In the limit $\varphi=0$, the field $Z_{1}$
becomes the standard model $Z$ boson, and the $V,A$ couplings of $Z_{1}$
become those of $Z$ in the standard model, i.e. $g^{Z_{1}}_{V,A}(f)\rightarrow
g^{Z}_{V,A}(f)$ for $\varphi\rightarrow 0$. In other words, the current theory
is consistent with the standard model at low energy. When $0\neq|\varphi|\ll
1$, the well-measured couplings of $Z$ with fermions are modified by an amount
proportional to $\varphi$,

$\displaystyle g^{Z_{1}}_{V}(f)$ $\displaystyle=$ $\displaystyle
T_{3}(f_{L})-2s^{2}_{W}Q(f)+\mathcal{O}(\varphi),$ (42) $\displaystyle
g^{Z_{1}}_{A}(f)$ $\displaystyle=$ $\displaystyle
T_{3}(f_{L})+\mathcal{O}(\varphi),$ (43)

where note that $\theta$ is always finite. With the aid of the electroweak
precision data [14], the new physics contribution is safe, given that
$|\varphi|\lesssim 10^{-3}$ which applies for most $Z^{\prime}$ models
including ours. Indeed, the electroweak precision data include low-energy weak
neutral current experiments, precision $Z$-pole physics, and others. The low-
energy weak neutral current experiments are modified by $Z^{\prime}$ exchange,
mainly sensitive to its mass. The precision $Z$-pole physics at the LEP and
SLC are mainly sensitive to $Z$-$Z^{\prime}$ mixing, which lowers the $Z$ mass
and shifts the $Z$-fermion couplings, by contrast. Notice that the $Z$-fermion
couplings will determine the various $Z$-pole observables, such as asymmetries
and partial widths. The precision data strongly constrain the $Z$-$Z^{\prime}$
mixing angle, $\varphi$, which is extracted by comparing the very precise
measurements of the coupling constants at the $Z$ pole with the standard model
predictions including radiative corrections (cf. [15]). They also provide
lower limits on the $Z^{\prime}$ mass, but these limits are generally weaker
than the LEPII and LHC, obtained later (these high-energy experiments that are
above the $Z$-pole will supply strong constraints on the $Z^{\prime}$ mass,
but generally not sensitive to small $Z$-$Z^{\prime}$ mixing). The precision
data constraint can be presented for two cases of $\rho_{0}$-paramter,
$\rho_{0}$ free for arbitrary Higgs structure and $\rho_{0}=1$ for Higgs
doublets and singlets like ours, but there is only a slight difference between
the limits given. Alternatively, performing a global fit to the full
electroweak data set necessarily gives comparable bounds as $|\varphi|\sim
10^{-3}$, see [16] for instance.

With the choice of parameters as given above, the limit $|\varphi|\lesssim
10^{-3}$ produces a lower bound on $\Lambda$ for each $v_{1}$, as despited in
Figure 2 (dashed curves) for a comparison to $W$ mass (solid, color curves).
Only the dashed curves with $x=\pm 1/2$ are suitable to the $W$-mass deviation
(solid, color curves with $x=\pm 1/2$) for $v_{1}<185$ GeV and $v_{1}<55$ GeV,
respectively. By contrast, the dashed curves with $x=\pm 1/6$ are
significantly bigger than those from the $W$-mass constraint, thus ruling out
the values $x=\pm 1/6$ for every $v_{1}$. That said, the viable parameter
space can be considered as ($x=1/2$, $\Lambda=4.5$–8.5 TeV, $v_{1}=0$–185 GeV)
and ($x=-1/2$, $\Lambda\sim 5$ TeV, $v_{1}=0$–55 GeV).

Last, but not least, in contrast to the standard model, both $Z_{1,2}$ flavor-
change when interacting with quarks, while conserving lepton flavors. This is
due to the fact that quark families transform differently under $X,N$, while
lepton families transform the same under such charges. Additionally, quark
families transform the same under $T_{3}$ and $Y$ that define $Z$. Hence, the
flavor changing effect associated with $Z_{1}$ results from a $Z$-$Z^{\prime}$
mixing to be small. Whereas, the flavor change dominantly occurs associated
with $Z_{2}$, even for $\varphi=0$. We will investigate the FCNCs coupled to
$Z_{2}$, while those with $Z_{1}$ are safe, given that $|\varphi|\lesssim
10^{-3}$.

##  VII FCNCs

The couplings of $Z_{1,2}$ with fermions arise from

$\mathcal{L}_{\mathrm{int}}\supset-\sum_{F}\bar{F}\gamma^{\mu}(gT_{3}A_{3\mu}+g_{X}XB_{\mu}+g_{N}NC_{\mu})F.$
(44)

Changing to the basis of $(A,Z,Z^{\prime})$, i.e.

$\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle s_{W}A+c_{W}Z,$ (45)
$\displaystyle B$ $\displaystyle=$ $\displaystyle
c_{W}s_{\theta}A-s_{W}s_{\theta}Z+c_{\theta}Z^{\prime},$ (46) $\displaystyle
C$ $\displaystyle=$ $\displaystyle
c_{W}c_{\theta}A-s_{W}c_{\theta}Z-s_{\theta}Z^{\prime},$ (47)

and by using $Q=T_{3}+Y$, $Y=X+N$, $e=gs_{W}$, and
$g_{X}s_{\theta}=g_{N}c_{\theta}=gt_{W}$, we obtain

$\mathcal{L}_{\mathrm{int}}\supset-\sum_{F}\bar{F}\gamma^{\mu}\left[eQA_{\mu}+\frac{g}{c_{W}}(T_{3}-s^{2}_{W}Q)Z_{\mu}+\frac{gt_{W}}{s_{\theta}c_{\theta}}(X-s^{2}_{\theta}Y)Z^{\prime}_{\mu}\right]F.$
(48)

Because $Q$, $T_{3}$, and $Y$ are universal for every repeated fermion
flavors, such as neutrinos, charged leptons, up-type quarks, down-type quarks,
the flavor change is only associated with $X$-charge. Additionally, lepton
flavors are universal under $X$. Hence, this flavor change occurs only for
quarks. The relevant Lagrangian is

$\mathcal{L}_{\mathrm{int}}\supset-\frac{gt_{W}}{s_{\theta}c_{\theta}}\bar{q}_{a}\gamma^{\mu}Xq_{a}Z^{\prime}_{\mu}=x\frac{gt_{W}}{s_{\theta}c_{\theta}}(\bar{q}_{1}\gamma^{\mu}q_{1}+\bar{q}_{2}\gamma^{\mu}q_{2}-\bar{q}_{3}\gamma^{\mu}q_{3})Z^{\prime}_{\mu},$
(49)

where $q_{a}$ denotes quarks of either up-types $(u_{\alpha},u_{3})$ or down-
types $(d_{\alpha},d_{3})$, and notice also that $q_{a}$ is vector-like under
$X$.

Changing to the mass basis,

$\displaystyle q_{aL}=[V_{qL}]_{ai}q_{iL},\hskip
14.22636ptq_{aR}=[V_{qR}]_{ai}q_{iR},$ (50)

where $q=u,d$, and $i=1,2,3$ labels mass eigenstates, $u_{i}=(u,c,t)$ and
$d_{i}=(d,s,b)$, so that the quark mass matrices $m_{u,d}$ are diagonalized,

$V^{\dagger}_{uL}m_{u}V_{uR}=\mathrm{diag}(m_{u},m_{c},m_{t}),\hskip
14.22636ptV^{\dagger}_{dL}m_{d}V_{dR}=\mathrm{diag}(m_{d},m_{s},m_{b}).$ (51)

Here, note that the CKM matrix is $V_{\mathrm{CKM}}=V^{\dagger}_{uL}V_{dL}$.
Additionally, the quark current in (49) couples only the same quark chirality,
i.e. $q\gamma^{\mu}q=q_{L}\gamma^{\mu}q_{L}+q_{R}\gamma^{\mu}q_{R}$. Using the
unitarity condition, $V^{\dagger}_{qL}V_{qL}=1=V^{\dagger}_{qR}V_{qR}$, we
rewrite (49) as

$\mathcal{L}_{\mathrm{int}}\supset
x\frac{gt_{W}}{s_{\theta}c_{\theta}}\bar{q}_{i}\gamma^{\mu}q_{i}Z^{\prime}_{\mu}-2x\frac{gt_{W}}{s_{\theta}c_{\theta}}[V^{*}_{qL}]_{3i}[V_{qL}]_{3j}\bar{q}_{iL}\gamma^{\mu}q_{jL}Z^{\prime}_{\mu}+(L\rightarrow
R),$ (52)

where $i,j$ are summed. The first term conserves quark flavors, whereas the
last two terms give rise to FCNCs for $i\neq j$,

$\mathcal{L}_{\mathrm{FCNC}}=L_{ij}\bar{q}_{iL}\gamma^{\mu}q_{jL}Z^{\prime}_{\mu}+(L\rightarrow
R),$ (53)

where we have defined,

$L_{ij}=-2x\frac{gt_{W}}{s_{\theta}c_{\theta}}[V^{*}_{qL}]_{3i}[V_{qL}]_{3j}.$
(54)

Substituting $Z^{\prime}=c_{\varphi}Z_{2}-s_{\varphi}Z_{1}$ into the FCNC
Lagrangian and integrating $Z_{1,2}$ out, we get effective interactions,

$\mathcal{H}^{\mathrm{eff}}_{\mathrm{FCNC}}=\frac{L^{2}_{ij}}{m^{2}_{Z_{2}}}(\bar{q}_{iL}\gamma^{\mu}q_{jL})^{2}+2(LR)+(RR),$
(55)

where the last two terms differ from the first one only in chiral structures.
We have neglected the contribution of $Z_{1}$, because it is more suppressed
than that of $Z_{2}$, i.e.

$\frac{s^{2}_{\varphi}}{m^{2}_{Z_{1}}}\ll\frac{c^{2}_{\varphi}}{m^{2}_{Z_{2}}},$
(56)

due to $t_{\varphi}\sim m^{2}_{Z_{1}}/m^{2}_{Z_{2}}$; additionally, we have
utilized $c_{\varphi}\simeq 1$. The effective couplings in (55) are further
approximated as

$\frac{L^{2}_{ij}}{m^{2}_{Z_{2}}}\simeq\frac{1}{\Lambda^{2}}([V^{*}_{qL}]_{3i}[V_{qL}]_{3j})^{2},$
(57)

and similar for the others, which are all independent of $\theta,x$, since
$\Lambda\gg v_{1},v_{2}$.

The effective interactions in (55) contribute to the neutral-meson mixing
amplitudes, extensively studied in the literature. Assuming the dominant new
physics effects come from the first term, $L^{2}_{ij}$, the existing data
imply

$\frac{L^{2}_{12}}{m^{2}_{Z_{2}}}<\left(\frac{1}{10^{4}\
\mathrm{TeV}}\right)^{2},\hskip
14.22636pt\frac{L^{2}_{13}}{m^{2}_{Z_{2}}}<\left(\frac{1}{500\
\mathrm{TeV}}\right)^{2},\hskip
14.22636pt\frac{L^{2}_{23}}{m^{2}_{Z_{2}}}<\left(\frac{1}{100\
\mathrm{TeV}}\right)^{2},$ (58)

according to $K^{0}$-$\bar{K}^{0}$ mixing, $B^{0}_{d}$-$\bar{B}^{0}_{d}$
mixing, and $B^{0}_{s}$-$\bar{B}^{0}_{s}$ mixing, respectively [17]. Aligning
the quark mixing to the down quark sector, without loss of generality, we have
$V_{uL}=1$, thus $V_{\mathrm{CKM}}=V_{dL}$. Since $V_{\mathrm{CKM}}$ is well
measured, we input [14]

$[V_{dL}]_{31}=0.00886,\hskip 14.22636pt[V_{dL}]_{32}=0.0405,\hskip
14.22636pt[V_{dL}]_{33}=0.99914.$ (59)

Hence, the constraints in (58) yield

$\Lambda>3.6\ \mathrm{TeV},\hskip 14.22636pt\Lambda>4.4\ \mathrm{TeV},\hskip
14.22636pt\Lambda>4\ \mathrm{TeV},$ (60)

corresponding to the mentioned meson mixing systems, compatible to the
previous bounds.

Alternatively, the effective interactions $(LR)$ and $(RR)$ also contribute to
the neutral-meson mixing amplitudes through switching on the right-handed
quark mixing matrix, $V_{qR}$. Since $V_{qR}$ is left arbitrarily as in the
standard model (where only the left-handed quark mixing is constrained by the
CKM matrix $V_{\mathrm{CKM}}=V^{\dagger}_{uL}V_{dL}$), an issue arises that
this $V_{qR}$ contribution leads to dangerous FCNCs? However, as given from
outset, our theory originates from a trinification which conserves a left-
right symmetry. We expect that the left-right symmetry is still good at the
energy scale of the current theory, i.e. $V_{qR}\sim V_{qL}$. This would
suppress the dangerous $(LR)$ and $(RR)$ contributions to neutral-meson
mixings as the CKM factor does so for the $(LL)$ coupling. To be concrete, we
first consider the contribution of all effective interactions in (55) to the
mass difference of $K^{0}$-$\bar{K}^{0}$ mixing system, such as

$\Delta m_{K}=2\mathrm{Re}\langle
K^{0}|\mathcal{H}^{\mathrm{eff}}_{\mathrm{FCNC}}|\bar{K}^{0}\rangle,$ (61)

where $K^{0}$ is constructed from two quarks $(q_{i},q_{j})=(d,s)$. Further,
we have

$\displaystyle\Delta m_{K}$ $\displaystyle=$ $\displaystyle
2\mathrm{Re}\langle
K^{0}|\frac{L^{2}_{12}}{m^{2}_{Z_{2}}}(\bar{d}_{L}\gamma^{\mu}s_{L})^{2}+2\frac{L_{12}R_{12}}{m^{2}_{Z_{2}}}(\bar{d}_{L}\gamma^{\mu}s_{L})(\bar{d}_{R}\gamma_{\mu}s_{R})+\frac{R^{2}_{12}}{m^{2}_{Z_{2}}}(\bar{d}_{R}\gamma^{\mu}s_{R})^{2}|\bar{K}^{0}\rangle$
(62) $\displaystyle\simeq$
$\displaystyle\frac{2m_{K}f^{2}_{K}}{3m^{2}_{Z_{2}}}\mathrm{Re}\left\\{L^{2}_{12}-\left[\frac{3}{2}+\left(\frac{m_{K}}{m_{d}+m_{s}}\right)^{2}\right]L_{12}R_{12}+R^{2}_{12}\right\\},$

with the aid of

$\displaystyle\langle
K^{0}|(\bar{d}_{L}\gamma^{\mu}s_{L})^{2}|\bar{K}^{0}\rangle=\langle
K^{0}|(\bar{d}_{R}\gamma^{\mu}s_{R})^{2}|\bar{K}^{0}\rangle=\frac{1}{3}m_{K}f^{2}_{K},$
(63) $\displaystyle\langle
K^{0}|(\bar{d}_{L}\gamma^{\mu}s_{L})(\bar{d}_{R}\gamma_{\mu}s_{R})|\bar{K}^{0}\rangle=-\frac{1}{2}\left[\frac{1}{2}+\frac{1}{3}\left(\frac{m_{K}}{m_{d}+m_{s}}\right)^{2}\right]m_{K}f^{2}_{K},$
(64)

in agreement with [18]. Here, the hadronic matrix elements have been
determined by the vacuum insertion approximation using PCAC [19]. Similarly
for $B^{0}_{d,s}$-$\bar{B}^{0}_{d,s}$ mixings, i.e. $(q_{i},q_{j})=(d,b)$ and
$(s,b)$, respectively, we have

$\displaystyle\Delta m_{B_{d}}$ $\displaystyle\simeq$
$\displaystyle\frac{2m_{B_{d}}f^{2}_{B_{d}}}{3m^{2}_{Z_{2}}}\mathrm{Re}\left\\{L^{2}_{13}-\left[\frac{3}{2}+\left(\frac{m_{B_{d}}}{m_{d}+m_{b}}\right)^{2}\right]L_{13}R_{13}+R^{2}_{13}\right\\},$
(65) $\displaystyle\Delta m_{B_{s}}$ $\displaystyle\simeq$
$\displaystyle\frac{2m_{B_{s}}f^{2}_{B_{s}}}{3m^{2}_{Z_{2}}}\mathrm{Re}\left\\{L^{2}_{23}-\left[\frac{3}{2}+\left(\frac{m_{B_{s}}}{m_{s}+m_{b}}\right)^{2}\right]L_{23}R_{23}+R^{2}_{23}\right\\}.$
(66)

The contribution terms of new physics in (62), (65), and (66) are equally
suppressed by an approximate left-right symmetry, $V_{dR}\sim V_{dL}$, which
implies $R_{ij}\sim L_{ij}$. Assuming the contributions $(LR)$ and $(RR)$
equivalent to $(LL)$, using the approximation (57), the $\Lambda$ bounds from
the mass differences (cf. [18]) are compatible to the ones determined in (60).

##  VIII Collider bounds

Since $Z$ and $Z^{\prime}$ slightly mix, we will omit their mixing effect in
this section, without loss of generality. That said, we set $\varphi\to 0$ and
$Z_{2}\to Z^{\prime}$ as a physical field. Because $Z^{\prime}$ interacts with
both usual quarks and leptons, it definitely presents interesting signals at
the current hadron and lepton colliders.

The LEPII experiment [20] has studied the process $e^{+}e^{-}\to f\bar{f}$
that produces a pair of ordinary fermions $(f\bar{f})$ through exchange of
$Z^{\prime}$. This process is best described by the following effective
interactions,

$\mathcal{L}^{\mathrm{eff}}_{\mathrm{LEP2}}\supset\frac{g^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}\left[\bar{e}\gamma^{\mu}(a^{Z^{\prime}}_{L}(e)P_{L}+a^{Z^{\prime}}_{R}(e)P_{R})e\right]\left[\bar{f}\gamma_{\mu}(a^{Z^{\prime}}_{L}(f)P_{L}+a^{Z^{\prime}}_{R}(f)P_{R})f\right],$
(67)

where the chiral gauge couplings
$a^{Z^{\prime}}_{L,R}(f)=\frac{1}{2}[g^{Z^{\prime}}_{V}(f)\pm
g^{Z^{\prime}}_{A}(f)]$, as usual. Considering dilepton signals, $f=\mu,\tau$,
all the leptons possess universal couplings, such that

$\mathcal{L}^{\mathrm{eff}}_{\mathrm{LEPII}}\supset\frac{g^{2}[a^{Z^{\prime}}_{L}(e)]^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}(\bar{e}\gamma^{\mu}P_{L}e)(\bar{f}\gamma_{\mu}P_{L}f)+(LR)+(RL)+(RR),$
(68)

where the explicit couplings are given, using Table 3,

$a^{Z^{\prime}}_{L}(e)=\frac{s_{W}(s^{2}_{\theta}+2x)}{s_{2\theta}},\hskip
14.22636pta^{Z^{\prime}}_{R}(e)=\frac{s_{W}(s^{2}_{\theta}+x)}{s_{\theta}c_{\theta}}.$
(69)

The LEPII experiment has searched for such chiral interactions, giving several
constraints on respective couplings, commonly achieved at order of a few TeV.
Considering the bound derived for a new $U(1)$ gauge boson like ours, it
typically obeys [21]

$\frac{g^{2}[a^{Z^{\prime}}_{L}(e)]^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}<\frac{1}{(6\
\mathrm{TeV})^{2}}.$ (70)

Using the approximation,
$m^{2}_{Z^{\prime}}\simeq(16x^{2}g^{2}t^{2}_{W}/s^{2}_{2\theta})\Lambda^{2}$,
we get

$\Lambda>1.5(2+s^{2}_{\theta}/|x|)\ \mathrm{TeV}=4.5\ \mathrm{TeV},$ (71)

for $t_{\theta}=1$ and $|x|=1/2$, as before, in agreement with the previous
bounds.

At the LHC experiment, both new physics signals of dijet [22] and dilepton
[23] have been investigated. However, since the current bound for dijet is
less sensitive than dilepton, we will only examine the latter. The cross-
section producing a final state of dilepton can be estimated, using narrow
width approximation, such as

$\sigma(pp\to Z^{\prime}\to
f\bar{f})=\frac{1}{3}\sum_{q}\frac{dL_{q\bar{q}}}{dm^{2}_{Z^{\prime}}}\hat{\sigma}(q\bar{q}\to
Z^{\prime})\mathrm{Br}(Z^{\prime}\to f\bar{f}),$ (72)

where the luminosity $dL_{q\bar{q}}/dm^{2}_{Z^{\prime}}$ can be extracted from
[24] for LHC $\sqrt{s}=13$ TeV. The partonic cross-section and branching
$\mathrm{Br}(Z^{\prime}\to f\bar{f})=\Gamma(Z^{\prime}\to
f\bar{f})/\Gamma_{Z^{\prime}}$ take the form,

$\displaystyle\hat{\sigma}(q\bar{q}\to Z^{\prime})=\frac{\pi
g^{2}}{12c^{2}_{W}}[(g^{Z^{\prime}}_{V}(q))^{2}+(g^{Z^{\prime}}_{A}(q))^{2}],$
(73) $\displaystyle\Gamma(Z^{\prime}\to
f\bar{f})=\frac{g^{2}m_{Z^{\prime}}}{48\pi
c^{2}_{W}}N_{C}(f)[(g^{Z^{\prime}}_{V}(f))^{2}+(g^{Z^{\prime}}_{A}(f))^{2}],$
(74) $\displaystyle\Gamma_{Z^{\prime}}=\frac{g^{2}m_{Z^{\prime}}}{48\pi
c^{2}_{W}}\sum_{f}N_{C}(f)[(g^{Z^{\prime}}_{V}(f))^{2}+(g^{Z^{\prime}}_{A}(f))^{2}],$
(75)

where we exclude the decay $Z^{\prime}\to\nu_{R}\bar{\nu}_{R}$, and note also
that $Z^{\prime}$ decay to other fields negligibly contributes to $Z^{\prime}$
width. In Figure 3, we plot the dilepton production cross-section for
$f=e,\mu,\tau$, commonly labelled $l$, which have the same $Z^{\prime}$
couplings, thus cross-section. The ATLAS [23] yields a negative result for new
dilepton event of high mass, thus this translates to a lower limit for
$Z^{\prime}$ mass $m_{Z^{\prime}}>4.6$ TeV and $m_{Z^{\prime}}>2.8$ TeV for
the model with $x=1/2$ and $x=-1/2$, respectively, where $t_{\theta}=1$ is
taken as above. Hence, using
$m_{Z^{\prime}}\simeq(4|x|gt_{W}/s_{2\theta})\Lambda$ leads to $\Lambda>6.4$
TeV and $\Lambda>3.9$ TeV according to the $Z^{\prime}$ mass bounds. These LHC
bounds are in agreement with the previous constraints.

Figure 3:  Dilepton production cross-section as function of $Z^{\prime}$ mass,
where dotted lines denote observed limit for different widths extracted at
dilepton invariant-mass resonance, using 36.1 fb-1 of $pp$ collision at
$\sqrt{s}=13$ TeV by ATLAS [23]. The black and gray dotted lines are
theoretical prediction according to $x=1/2$ and $-1/2$, respectively.

##  IX Conclusion

A guide from potential grand unification is that fermion families at low
energy might be not universal, unlike the conventional standard model. This
work has shown that such abelian remnants are presented, making a match of
family number to color number. Additionally, the hypercharge is a result of
their breaking, making an electrically-neutral world whose proton and electron
charges are cancelled out.

Interestingly, the Higgs fields experience the new abelian charges, causing a
mixing of the usual $Z$ boson with new $Z^{\prime}$ boson, explaining the
recently-measured $W$-boson mass anomaly. The electroweak precision data on
$Z$-$Z^{\prime}$ mixing combined with the $W$ mass measurement implies the new
physics scale $\Lambda=4.5$–8.5 TeV and a weak scale $v_{1}=0$–185 GeV for the
charge parameter $x=1/2$, while $\Lambda\sim 5$ TeV and $v_{1}=0$–55 GeV for
$x=-1/2$, where for all cases, the relative new gauge couplings set as
$t_{\theta}=g_{N}/g_{X}=1$. Additionally, the values of $x=\pm 1/6$ are
obviously ruled out.

This theory contains the right-handed neutrinos as fundamental constituents,
and the symmetry breaking leads to naturally-small neutrino masses, suppressed
by the new physics scale that sets Majorana right-handed neutrino masses also
at TeV scale.

The family nonuniversality occurs only for quarks, and this leads to FCNCs of
quarks coupled to $Z^{\prime}$. An analysis for the contribution of left
currents $(LL)$ implies a bound for new physics scale $\Lambda>$ 3.6 TeV, 4
TeV, and 4.4 TeV according to the mixings, $K^{0}$-$\bar{K}^{0}$,
$B^{0}_{s}$-$\bar{B}^{0}_{s}$, and $B^{0}_{d}$-$\bar{B}^{0}_{d}$,
respectively. The contribution of right currents such as $(LR)$ and $(RR)$ to
the mixings may be large, but safely suppressed by assuming an approximate
left-right symmetry. This kind of constraint is independent of $x,\theta$.

Considering $Z^{\prime}$ contribution to dilepton signals at LEPII gives a
bound $\Lambda>4.5$ TeV for $|x|=1/2$ and $t_{\theta}=1$, while the dilepton
production cross-section at LHC induced by $Z^{\prime}$ implies $\Lambda>$ 6.4
TeV and 3.9 TeV with respect to $x=1/2$ and $x=-1/2$, respectively, while
keeping $t_{\theta}=1$. Hence, the collider, FCNC, electroweak precision data,
and $W$-mass measurement on the new physics scale are all compatible.

If the current theory is embedded in a trinification [7] or $E_{6}$ [8], this
implies $x=\pm 1/6$, as well as the family nonuniversality is lost. An
analysis shows that our proposal can be properly embedded in a flipped
trinification [10] for arbitrary $x$ as the smallest choice of a unification
symmetry.

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# On the Limits of Single Anchor Localization: Near-Field vs. Far-Field

Don-Roberts Emenonye, Harpreet S. Dhillon, and R. Michael Buehrer D.-R.
Emenonye, H. S. Dhillon and R. M. Buehrer are with Wireless@VT, Bradley
Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg,
VA, 24061, USA. Email: {donroberts, hdhillon<EMAIL_ADDRESS>The support of
the US National Science Foundation (Grants ECCS-2030215 and CNS-2107276) is
gratefully acknowledged.

###### Abstract

It is well known that a single anchor can be used to determine the position
and orientation of an agent communicating with it. However, it is not clear
what information about the anchor or the agent is necessary to perform this
localization, especially when the agent is in the near-field of the anchor.
Hence, in this paper, to investigate the limits of localizing an agent with
some uncertainty in the anchor location, we consider a wireless link
consisting of source and destination nodes. More specifically, we present a
Fisher information theoretical investigation of the possibility of estimating
different combinations of the source and destination’s position and
orientation from the signal received at the destination. To present a
comprehensive study, we perform this Fisher information theoretic
investigation under both the near and far field propagation models. One of the
key insights is that while the source or destination’s $3$D orientation can be
jointly estimated with the source or destination’s $3$D position in the near-
field propagation regime, only the source or destination’s $2$D orientation
can be jointly estimated with the source or destination’s $2$D position in the
far-field propagation regime. Also, a simulation of the FIM indicates that in
the near-field, we can estimate the source’s $3$D orientation angles with no
beamforming, but in the far-field, we can not estimate the source’s $2$D
orientation angles when no beamforming is employed.

###### Index Terms:

6G localization, anchor uncertainty, far-field, near-field, FIM.

##  I Introduction

Recently, due to the ubiquitous deployment of multi-antenna base stations,
single-anchor localization has been proposed and studied with [1, 2, 3] and
without a reconfigurable intelligent surface (RIS) [4, 5, 6]. Localization is
usually performed under the assumption that the anchor location (position and
orientation) is perfectly known [7]. However, in practical systems, this
assumption might not hold. For example, in scenarios where unmanned aerial
vehicles (UAV) act as anchors, there could be inherent uncertainty in the
locations of the UAVs [8, 9]. Another example involves localization using
RISs. RISs are being considered to aid localization by acting as virtual
anchors; however, their ubiquitous deployment means that their locations can
change (e.g., when they are placed on movable objects), resulting in
uncertainty in their locations. Lastly, in indoor localization systems, the
locations of the indoor anchors can easily be disturbed after deployment.
Hence, in this paper, to investigate localization with anchor uncertainty, we
present a Fisher information view of estimating different combinations of a
source and destination’s position and orientation under the near and far field
propagation regimes.

###  I-A Prior Art

Prior literature on single-anchor localization involves deriving the
fundamental limits for the accuracy achievable in estimating the position and
orientation of an agent [1]. These bounds are extended to the case of $3$D
localization of an agent in [2]. In [3], the amount of information in the non-
line of sight (NLOS) paths and their usefulness for localization is analyzed.
The bounds of single-anchor localization with a RIS have been studied in [4].
These bounds are extended to account for near-field propagation in [5, 6]. In
the context of anchor state uncertainty, localization has been investigated
with and without a RIS. In [10], the positioning problem in the presence of
anchor uncertainty is studied, the resulting non-convex optimization problem
is relaxed to a second-order cone programming problem, and semidefinite
programming is applied. The authors in [11] derive the geometric dilution of
precision in the presence of anchor position uncertainty, and a trade-off is
made between range errors and position errors by applying the modified spring
mass method. The anchor position offset and the agent’s position are estimated
in [12] using the signal strength of the received signals. In [13], a rigorous
investigation of the impact of anchor uncertainty on received signal strength-
based localization techniques is presented. The bounds given in [13] serve as
lower bounds to the algorithm in [12]. In [14], multipath propagation between
the uncertain anchor and the agent is investigated, the error model of the
anchor uncertainty is assumed, and importance sampling is used to obtain the
agent’s position. Uncertainties are considered in the case of RIS-assisted
localization in [4, 5]. While the prior art primarily includes robust
algorithms to handle uncertainty in anchors’ position, a comprehensive Fisher
information-based analysis on the estimation of the anchor orientation has yet
to be studied. It is important to note the anchor orientation is particularly
important as the localization of agents is now being considered with large
antenna single anchors. Moreover, the effect of anchor location uncertainty
has not been investigated under the near-field propagation regime.

###  I-B Contributions

In this paper, through the Fisher information matrix (FIM), we present a
theoretical investigation of the limits of single-anchor localization by
determining the combinations of positions and orientations of the source and
destination nodes that can be estimated in the near and far field propagation
regimes. Further, using the FIM, we present a lower bound for the source
orientation and destination position accuracy. One key result from the FIM-
based analysis is that in the near-field, the source or destination’s $3$D
orientations can be estimated jointly with either the source or destination’s
$3$D positions. Also, in the far-field, the source or destination’s $2$D
orientations can be estimated jointly with either the source or destination’s
$2$D positions. Another result is that while the presence of a beamforming
matrix is not required in the near-field to estimate the source’s $3$D
orientation angles, a beamforming matrix is required in the far-field to
estimate the source’s $2$D orientation angles.

Notation: the transpose operator is $(\cdot)^{\mathrm{T}}$; the hermitian
transpose operator is $(\cdot)^{\mathrm{H}}$; the submatrix in the matrix
$\bm{V}$, with rows in the range, $g_{1}:v_{1}$, and the columns in the range
$g_{2}:v_{2}$ is extracted using the operation
$[\bm{V}]_{[g_{1}:v_{1},g_{2}:v_{2}]}$; $\operatorname{Tr}(\cdot)$ is the
matrix trace operator; $\left\lVert\cdot\right\rVert$ denotes the Euclidean
norm ; the positive definiteness of a matrix is characterized by $\succ$ ; the
first derivative operator is $\nabla$ ; the expectation operator with respect
to the random vector $\bm{v}$ is $\mathbb{E}_{\bm{v}}\\{\cdot\\}$.

##  II System Model

We consider a source with its centroid located at
$\bm{p}_{B}=[x_{B},y_{B},z_{B}]^{\mathrm{T}}$, and its $b^{\text{th}}$ antenna
element located at $\bm{s}_{b}=[x_{b},y_{b},z_{b}]^{\mathrm{T}}$. The location
of the centroid is defined with respect to the global origin, while the
location specified by $\bm{s}_{b}$ is defined with respect to $\bm{p}_{B}$.

Figure 1: An illustration showing a source communicating with a destination.

This point $\bm{s}_{b}$ can also be written as
$\bm{s}_{b}=\bm{Q}_{B}\tilde{\bm{s}}_{b}$, where
$\tilde{\bm{s}}_{b}=[\tilde{x}_{b},\tilde{y}_{b},\tilde{z}_{b}]^{\mathrm{T}}$
is the previously known position of the antenna coordinate with respect to
$\bm{p}_{B}$ before an orientation offset,
$\bm{\Phi}_{B}=[\alpha_{B},\psi_{B},\varphi_{B}]^{\mathrm{T}}$. The subsequent
$3$D orientation matrix is defined as $\bm{Q}_{B}$ [15]. There are $N_{B}$
antennas at the source, and each antenna can be described with respect to the
global origin as $\bm{p}_{b}=\bm{p}_{B}+\bm{s}_{b}$. The destination is
located at $\bm{p}_{U}=[x_{U},y_{U},z_{U}]^{\mathrm{T}}$, and its
$u^{\text{th}}$ antenna element is located at
$\bm{s}_{u}=[x_{u},y_{u},z_{u}]^{\mathrm{T}}$. The corresponding vectors,
$\bm{p}_{U}$, $\bm{s}_{u}$, $\tilde{\bm{s}}_{u}$ and $\bm{p}_{u}$ have similar
definitions as the corresponding source’s vectors. Note that the orientation
angles and the matrix related to the destination are denoted by
$\bm{\Phi}_{U}$ and $\bm{Q}_{U}$, respectively. The position of the
destination’s centroid located at ${\bm{p}}_{{U}}$ can be described in
relation to the position of the source’s centroid located at ${\bm{p}}_{{B}}$
as ${\bm{p}}_{{U}}={\bm{p}}_{{B}}+d_{{B}{U}}\bm{\Delta}_{{B}{U}},$ where
$d_{{B}{U}}$ is the distance from point ${\bm{p}}_{{B}}$ to point
${\bm{p}}_{{U}}$ and $\bm{\Delta}_{{B}{U}}$ is the corresponding unit
direction vector
$\bm{\Delta}_{{B}{U}}=[\cos\phi_{{B}{U}}\sin\theta_{{B}{U}},\sin\phi_{{B}{U}}\sin\theta_{{B}{U}},\cos\theta_{{B}{U}}]^{\mathrm{T}}$.
All points defined locally that describe the location of elements on the
source antenna array with respect to the source’s centroid can be written in
the matrix form as
${\bm{S}}_{B}=[{\bm{s}}_{1},{\bm{s}}_{2},\cdots,{\bm{s}}_{N_{B}}].$ Similarly,
the points defined locally that describe the location of elements on the
destination antenna array with respect to the destination’s centroid can be
written in the matrix form as
${\bm{S}}_{U}=[{\bm{s}}_{1},{\bm{s}}_{2},\cdots,{\bm{s}}_{N_{U}}].$ Matrices
$\tilde{\bm{S}}_{B}$ and $\tilde{\bm{S}}_{U}$ can be described similarly, by
collecting the appropriate vectors $\tilde{\bm{s}}_{b}$ and
$\tilde{\bm{s}}_{u}$.

###  II-A Signal Model

The communication from the source to the destination is achieved through the
transmission of $T$ symbols from the source with $N_{B}$ transmit antennas to
the destination with $N_{U}$ receive antennas. During each transmission, the
source precodes a data stream, $\bm{x}\in\mathcal{C}^{N_{D}\times 1}$, to the
$N_{B}$ transmit antennas with a beamforming matrix
$\bm{F}_{t}\in\mathcal{C}^{N_{B}\times N_{D}}$ under the constraint
$\mathbb{E}\left\\{\left\lVert\bm{x}\right\rVert^{2}\right\\}=1$. The signal
received during the $t^{\text{th}}$ transmission is

$\displaystyle\bm{y}_{t}=\bm{H}\bm{F}_{t}\bm{x}+\bm{n}_{t},=\bm{\mu}_{t}+\bm{n}_{t}.$
(1)

In the above equation, $\bm{\mu}_{t}$ is the noise-free part (useful part) of
the signal, and $\bm{n}_{t}\sim\mathcal{C}\mathcal{N}(0,N_{0})$ represents the
thermal noise local to the destination’s antenna array. The element in the
$u^{\text{th}}$ row and $b^{\text{th}}$ column of the channel matrix $\bm{H}$
is $[\bm{H}]_{[u,b]}=\beta e^{-j2\pi f_{c}\tau_{bu}}$. Here,
$\beta=\beta_{\mathrm{R}}+j\beta_{\mathrm{I}}$ is the complex path gain,
$f_{c}$ is the operating frequency, and $\tau_{bu}$ is the propagation delay
from the $b^{\text{th}}$ transmit antenna located at $\bm{p}_{b}$ on the
source’s antenna array to the receive antenna located at $\bm{p}_{u}$ on the
destination’s antenna array. Now, the signal received at the destination’s
$u^{\text{th}}$ receive antenna during the $t^{\text{th}}$ transmission is

$\displaystyle\bm{y}_{t,u}$
$\displaystyle=\sum_{b=1}^{N_{B}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}[\bm{H}]_{[u,b]}+\bm{n}_{t}.$
(2)

The definition of the delay given as
$\tau_{bu}=\frac{\left\lVert\bm{p}_{u}-\bm{p}_{b}\right\rVert}{c}$
incorporates any potential spherical curvature wavefront present in the signal
received at the destination. When the destination experiences substantial
wavefront curvature, it is said to be located within the near-field
propagation regime. It is important to note that at sufficiently larger
distances between the destination and the source, the spherical wavefront can
be approximated by a plane wave. With this plane wave approximation, the delay
can be approximated as
$\tau_{bu}=\tau_{BU}+\Delta_{BU}^{\mathrm{T}}({\bm{s}}_{u}-{\bm{s}}_{b})/c.$
When this approximation holds, the destination is said to be located within
the far-field propagation regime. The boundary that defines the near and far
field propagation regime is called the Fraunhofer distance. This Fraunhofer
distance can be computed as $d_{\mathrm{f}}=2D^{2}/\lambda$ with $\lambda$
indicating the wavelength of the signal and $D$ the maximum diameter among the
source and destination surface diameters [5]. While, (1) and (2) adequately
represent the signals received in the near-field, an approximation of signals
received in the far-field can be written as

$\displaystyle\bm{y}_{t}=\beta\bm{a}_{UB}(\Delta_{BU})\bm{a}_{BU}^{\mathrm{H}}(\Delta_{BU})e^{-j2\pi
f_{c}\tau_{BU}}\bm{F}_{t}\bm{x}+\bm{n}_{t},$ (3)

where
$\bm{a}_{BU}(\Delta_{BU})=e^{-j2\frac{\pi}{\lambda}{\bm{S}}_{B}^{\mathrm{T}}\Delta_{BU}}$
and
$\bm{a}_{UB}(\Delta_{BU})=e^{-j2\frac{\pi}{\lambda}{\bm{S}}_{U}^{\mathrm{T}}\Delta_{BU}}$.

###  II-B Source and Destination Position and Orientation Estimation

In this letter, we provide the different combinations of source and
destination position and orientation that can be estimated through the signals
received across the $N_{U}$ antennas during the $T$ transmissions. We
determine this by evaluating the FIM under the following parameterizations:
case I) $\bm{\eta}=[\bm{p}_{U},\bm{\Phi}_{U},\bm{\beta}]^{\mathrm{T}}$, case
II) $\bm{\eta}=[\bm{p}_{U},\bm{\Phi}_{B},\bm{\beta}]^{\mathrm{T}}$, case III)
$\bm{\eta}=[\bm{p}_{B},\bm{\Phi}_{U},\bm{\beta}]^{\mathrm{T}}$, and case IV)
$\bm{\eta}=[\bm{p}_{B},\bm{\Phi}_{B},\bm{\beta}]^{\mathrm{T}}$. Here,
$\bm{\beta}=[\beta_{\mathrm{R}},\beta_{\mathrm{I}}]^{\mathrm{T}}$. Note that
the location parameters for each individual case can be collected into the
vector $\bm{\zeta}$. The FIM computations are carried under three scenarios:
i) far-field model with beamforming, ii) near-field model with no beamforming,
and iii) near-field model with beamforming. Note that the case for using the
far-field model with identity beamforming matrices across the $T$
transmissions is not possible. This is because the joint estimation of the
source orientation, $\bm{\Phi}_{B}$, and $\bm{\beta}$ is not feasible under
this condition (see Appendix -A).

##  III Information in the Received signal

To analyze the amount of location information present in the received signal,
we introduce the mathematical definition of the FIM for an unknown parameter
vector, $\bm{\eta}$, in the following definition.

######  Definition 1.

Based on a set of observations $\mathbf{y}$, the Fisher information of a
parameter vector, $\bm{\eta}$, is written as

$\displaystyle\mathbf{J}_{\bm{\eta}}$
$\displaystyle\triangleq-\mathbb{E}_{\mathbf{y}{}}\left[\frac{\partial^{2}\ln\chi(\mathbf{y}|\bm{\eta})}{\partial\bm{\eta}\partial\bm{\eta}^{\mathrm{T}}}\right]$
(4)

where $\mathbb{E}_{\nu}$ is expectation taken over the random variable $\nu$,
$\chi(\mathbf{y}|\bm{\eta})$ is the likelihood of $\bm{y}$ conditioned on
$\bm{\eta}$. We note that the error covariance matrix of an unbiased estimate,
$\hat{\bm{\eta}}$, of an unknown parameter vector, ${\bm{\eta}}$ satisfies the
following information inequality
$\mathbb{E}_{\bm{y}}\left\\{(\hat{\bm{\eta}}-\bm{\eta})(\hat{\bm{\eta}}-\bm{\eta})^{\mathrm{T}}\right\\}\succeq\mathbf{J}_{\bm{\eta}}^{-1}.$

The FIM for the parameter vector
$\bm{\eta}=[\bm{p}_{U},\bm{\Phi}_{B},\bm{\beta}]^{\mathrm{T}}$ has the
following structure

$\displaystyle\mathbf{J}_{\bm{\eta}}\triangleq\left[\begin{array}[]{cccc}\mathbf{J}_{\bm{p}_{U}\bm{p}_{U}}&\mathbf{J}_{\bm{p}_{U}\bm{\Phi}_{B}}&\mathbf{J}_{\bm{p}_{U}{\bm{\beta}_{\mathrm{R}}}}&\mathbf{J}_{\bm{p}_{U}{\bm{\beta}_{\mathrm{I}}}}\\\
\mathbf{J}_{\bm{\Phi}_{B}\bm{p}_{U}}&\mathbf{J}_{\bm{\Phi}_{B}\bm{\Phi}_{B}}&\mathbf{J}_{\bm{\Phi}_{B}{\bm{\beta}_{\mathrm{R}}}}&\mathbf{J}_{\bm{\Phi}_{B}{\bm{\beta}_{\mathrm{I}}}}\\\
\mathbf{J}_{{\bm{\beta}_{\mathrm{R}}}\bm{p}_{U}}&\mathbf{J}_{{\bm{\beta}_{\mathrm{R}}}\bm{\Phi}_{B}}&\mathbf{J}_{{\bm{\beta}_{\mathrm{R}}}{\bm{\beta}_{\mathrm{R}}}}&\mathbf{J}_{{\bm{\beta}_{\mathrm{R}}}{\bm{\beta}_{\mathrm{I}}}}\\\
\mathbf{J}_{{\bm{\beta}_{\mathrm{I}}}\bm{p}_{U}}&\mathbf{J}_{{\bm{\beta}_{\mathrm{I}}}\bm{\Phi}_{B}}&\mathbf{J}_{{\bm{\beta}_{\mathrm{I}}}{\bm{\beta}_{\mathrm{R}}}}&\mathbf{J}_{{\bm{\beta}_{\mathrm{I}}}{\bm{\beta}_{\mathrm{I}}}}\end{array}\right]\in\mathcal{R}^{8\times
8}.$ (5)

The submatrices in the above matrix can be computed using
$\mathbf{J}_{\bm{\eta}_{{\mathbf{v}_{1}}}\bm{\eta}_{{\mathbf{v}_{2}}}}\triangleq\frac{2}{\sigma^{2}}\sum_{t=1}^{{T}}\Re\left\\{\frac{\partial\bm{{\mu}_{t}}^{\mathrm{H}}}{\partial{\bm{\eta}}_{\mathbf{v}_{1}}}\frac{\partial\bm{{\mu}{t}}}{\partial{\bm{\eta}}_{\mathbf{v}_{2}}}\right\\}$
where $\bm{\eta}_{\mathbf{v}_{1}}\in\bm{\eta}$,
$\bm{\eta}_{\mathbf{v}_{2}}\in\bm{\eta}$ are both dummy variables, and
$1/\sigma^{2}$ is the SNR which incorporates the pathloss and composite noise
power. The required first derivatives are presented in the following sections.

###  III-A First Derivatives under the Far-Field Model

The first derivative of the useful part of the received signal with respect to
$\nu\in[\bm{p}_{B},\bm{p}_{U}]$ under the far-field model is

$\nabla_{\nu}{\bm{\mu}_{t,u}}=\beta
e^{-j2\frac{\pi}{\lambda}{\bm{s}}_{u}^{\mathrm{T}}\Delta_{BU}}\bm{a}_{BU}^{\mathrm{H}}(\Delta_{BU})\bm{K}_{\nu}e^{-j2\pi
f_{c}\tau_{BU}}\bm{F}_{t}\bm{x},$

where $\bm{K}_{\nu}$ is expressed in (6).

$\displaystyle\bm{K}_{\nu}$
$\displaystyle=\text{diag}\Bigg{[}-\frac{j2\pi}{\lambda}\bigg{(}\bm{s}_{u}^{\mathrm{T}}\nabla_{\nu}{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}}-\nabla_{\nu}{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}}^{\mathrm{T}}\bm{S}_{B}+\nabla_{\nu}d_{BU}\bigg{)}\Bigg{]},$
(6) $\displaystyle\nabla_{{\bm{p}_{B}}}d_{BU}$
$\displaystyle=-1\times{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}},\;\;\nabla_{{\bm{p}_{B}}}{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}}=\frac{-d_{BU}-(\bm{p}_{U}-\bm{p}_{B})\nabla_{{\bm{p}_{U}}}d_{BU}}{d_{BU}^{2}},$
(7) $\displaystyle\nabla_{{\bm{p}_{U}}}d_{BU}$
$\displaystyle={\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}},\;\;\nabla_{{\bm{p}_{U}}}{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}}=\frac{d_{BU}-(\bm{p}_{U}-\bm{p}_{B})\nabla_{{\bm{p}_{U}}}d_{BU}}{d_{BU}^{2}}.$
(8)

The first derivatives of the useful part of the received signal with respect
to $\nu\in\bm{\Phi}_{B}$ and $\nu\in\bm{\Phi}_{U}$ under the far-field model
are

$\nabla_{\nu}{\bm{\mu}_{t}}=\beta\bm{\tilde{P}}_{\nu}\bm{a}_{UB}(\Delta_{BU})\bm{a}_{BU}^{\mathrm{H}}(\Delta_{BU})e^{-j2\pi
f_{c}\tau_{BU}}\bm{F}_{t}\bm{x},$ $\nabla_{\nu}{\bm{\mu}_{t,u}}=\beta
e^{-j2\frac{\pi}{\lambda}{\bm{s}}_{u}^{\mathrm{T}}\Delta_{BU}}\bm{a}_{BU}^{\mathrm{H}}(\Delta_{BU})\bm{P}_{\nu}e^{-j2\pi
f_{c}\tau_{BU}}\bm{F}_{t}\bm{x},$

respectively, where

$\displaystyle\tilde{\bm{P}}_{\nu}$
$\displaystyle=\text{diag}\Bigg{[}-\frac{j2\pi}{\lambda}(\nabla_{\nu}\bm{S}_{u})^{\mathrm{T}}{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}}\Bigg{]},$
$\displaystyle\bm{P}_{\nu}$
$\displaystyle=\text{diag}\Bigg{[}\frac{j2\pi}{\lambda}{\Bigg{[}\frac{\bm{p}_{U}-\bm{p}_{B}}{d_{BU}}\Bigg{]}}^{\mathrm{T}}\nabla_{\nu}\bm{S}_{B}\Bigg{]}.$

Also,
$\nabla_{{\bm{\Phi}_{B}}}\bm{S}_{B}=\nabla_{{\bm{\Phi}_{B}}}\bm{Q}_{B}\tilde{\bm{S}}_{B}$
and
$\nabla_{{\bm{\Phi}_{U}}}\bm{S}_{U}=\nabla_{{\bm{\Phi}_{U}}}\bm{Q}_{U}\tilde{\bm{S}}_{U}.$
Finally, the first derivative of the useful part of the received signal with
respect to complex path gain under the far-field model is

$\displaystyle\nabla_{{\bm{\beta}_{\mathrm{R}}}}\bm{\mu}_{t}$
$\displaystyle=\bm{a}_{UB}(\Delta_{BU})\bm{a}_{BU}^{\mathrm{H}}(\Delta_{BU})\bm{F}_{t}\bm{x}e^{-j2\pi
f_{c}\tau_{BU}},$
$\displaystyle\nabla_{{\bm{\beta}_{\mathrm{I}}}}\bm{\mu}_{t}$
$\displaystyle=j\bm{a}_{UB}(\Delta_{BU})\bm{a}_{BU}^{\mathrm{H}}(\Delta_{BU})\bm{F}_{t}\bm{x}e^{-j2\pi
f_{c}\tau_{BU}}.$

The above first derivatives are used to compute the submatrices with a similar
structure as that shown in (5) when the far-field model is used.

###  III-B First Derivatives under the Near-Field Model

The first derivatives of the useful part of the received signal with respect
to $\bm{\eta}$ under the near-field model are

$\displaystyle\nabla_{{\bm{p}_{U}}}{\bm{\mu}_{t,u}}$ $\displaystyle=(-j2\pi
f_{c})\beta\sum_{b=1}^{N_{B}}\nabla_{{\bm{p}_{U}}}{\tau_{bu}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}e^{-j2\pi
f_{c}\tau_{bu}},$ $\displaystyle\nabla_{{\bm{p}_{B}}}{\bm{\mu}_{t,u}}$
$\displaystyle=(-j2\pi
f_{c})\beta\sum_{b=1}^{N_{B}}\nabla_{{\bm{p}_{B}}}{\tau_{bu}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}e^{-j2\pi
f_{c}\tau_{bu}},$ $\displaystyle\nabla_{{\bm{\Phi}_{U}}}{\bm{\mu}_{t,u}}$
$\displaystyle=(-j2\pi
f_{c})\beta\sum_{b=1}^{N_{B}}\nabla_{{\bm{\Phi}_{U}}}{\tau_{bu}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}e^{-j2\pi
f_{c}\tau_{bu}},$ $\displaystyle\nabla_{{\bm{\Phi}_{B}}}{\bm{\mu}_{t,u}}$
$\displaystyle=(-j2\pi
f_{c})\beta\sum_{b=1}^{N_{B}}\nabla_{{\bm{\Phi}_{B}}}{\tau_{bu}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}e^{-j2\pi
f_{c}\tau_{bu}}.$
$\displaystyle\nabla_{{\bm{\beta}_{\mathrm{R}}}}\bm{\mu}_{t,u}$
$\displaystyle=\sum_{b=1}^{N_{B}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}e^{-j2\pi
f_{c}\tau_{bu}},$
$\displaystyle\nabla_{{\bm{\beta}_{\mathrm{I}}}}\bm{\mu}_{t,u}$
$\displaystyle=j\sum_{b=1}^{N_{B}}\sum_{d=1}^{N_{D}}[\bm{F}_{t}]_{[b,d]}[\bm{x}]_{[d]}e^{-j2\pi
f_{c}\tau_{bu}}.$

Here, $\nabla_{{\bm{p}_{B}}}{\tau_{bu}}=\nabla_{{\bm{p}_{B}}}{d_{bu}}/c$,
$\nabla_{{\bm{p}_{U}}}{\tau_{bu}}=\nabla_{{\bm{p}_{U}}}{d_{bu}}/c$,
$\nabla_{{\bm{\Phi}_{U}}}{\tau_{bu}}=\nabla_{{\bm{\Phi}_{U}}}{d_{bu}}/c$,
$\nabla_{{\bm{\Phi}_{B}}}{\tau_{bu}}=\nabla_{{\bm{\Phi}_{B}}}{d_{bu}}/c$,
$\nabla_{{\bm{p}_{B}}}{d_{bu}}=-\frac{\bm{p}_{u}-\bm{p}_{b}}{d_{bu}}$,
$\nabla_{{\bm{p}_{U}}}{d_{bu}}=\frac{\bm{p}_{u}-\bm{p}_{b}}{d_{bu}}$,
$\nabla_{{\bm{\Phi}_{B}}}{d_{bu}}=-\frac{(\bm{p}_{u}-\bm{p}_{b})^{\mathrm{T}}}{d_{bu}}(\nabla_{{\bm{\Phi}_{B}}}\bm{Q}_{B}\tilde{\bm{s}}_{b}),$
and
$\nabla_{{\bm{\Phi}_{U}}}{d_{bu}}=\frac{(\bm{p}_{u}-\bm{p}_{b})^{\mathrm{T}}}{d_{bu}}(\nabla_{{\bm{\Phi}_{U}}}\bm{Q}_{U}\tilde{\bm{s}}_{u}).$
The above first derivatives are used to compute the submatrices with a similar
structure as that shown in (5) when the near-field model is used. After
computing $\mathbf{J}_{\bm{\eta}}$, to focus on the available information
concerning the location parameters, we present a mathematical description of
the EFIM.

######  Definition 2.

If the FIM of a parameter
$\bm{\eta}=[\bm{\eta}_{1}^{\mathrm{T}}\;\;\bm{\eta}_{2}^{\mathrm{T}}]^{\mathrm{T}}$
is specified by

$\displaystyle\mathbf{J}_{\bm{\eta}}=\left[\begin{array}[]{cc}\mathbf{J}_{\bm{\eta}_{1}\bm{\eta}_{1}}&\mathbf{J}_{\bm{\eta}_{1}\bm{\eta}_{2}}\\\
\mathbf{J}_{\bm{\eta}_{1}\bm{\eta}_{2}}^{\mathrm{T}}&\mathbf{J}_{\bm{\eta}_{2}\bm{\eta}_{2}}\end{array}\right],$
(9)

where
$\bm{\eta}\in\mathbb{R}^{N},\bm{\eta}_{1}\in\mathbb{R}^{n},\mathbf{J}_{\bm{\eta}_{1}\bm{\eta}_{1}}\in\mathbb{R}^{n\times
n},\mathbf{J}_{\bm{\eta}_{1}\bm{\eta}_{2}}\in\mathbb{R}^{n\times(N-n)}$, and
$\mathbf{J}_{\bm{\eta}_{2}\bm{\eta}_{2}}\in$ $\mathbb{R}^{(N-n)\times(N-n)}$
with $n<N$, then the EFIM [5] of the parameter of interest ${\bm{\eta}_{1}}$
is given by

$\displaystyle\mathbf{J}_{\bm{\eta}_{1}}^{\mathrm{e}}$
$\displaystyle=\mathbf{J}_{{\bm{\eta}_{1}}{\bm{\eta}_{1}}}-\mathbf{J}_{{\bm{\eta}_{1}}{\bm{\eta}_{2}}}\mathbf{J}_{{\bm{\eta}_{2}}{\bm{\eta}_{2}}}^{-1}\mathbf{J}_{{\bm{\eta}_{1}}{\bm{\eta}_{2}}}^{\mathrm{T}}.$
(10)

Using Definition 2, the EFIM of the parameter vector $\bm{\eta}$ is computed
for different parameters of interest. For example, the EFIM when the parameter
of interest is $\bm{\zeta}=[\bm{p}_{U},\Phi_{U}]^{\mathrm{T}}$ is
$\mathbf{J}_{\bm{\zeta}}^{\mathrm{e}}\in\mathcal{R}^{6\times 6}$. Here, the
nuisance parameter is the complex path gain.

##  IV Results

In this section, we use numerical simulations to find out which combinations
of position and orientation parameters can be estimated - a parameter,
$\bm{\zeta}$, can be estimated if the corresponding EFIM,
$\mathbf{J}_{\bm{\zeta}}^{\mathrm{e}}$, is positive definite [5]. We also
provide numerical position error bound (PEB) and orientation error bound (OEB)
results for the case in which the source orientation and destination position
are the unknown parameters. Our simulation framework consists of a source
whose centroid is located at $\bm{p}_{B}=[1.5,1.0,4.0]^{\mathrm{T}}$ with the
orientation angles $\bm{\Phi}_{B}=[1.1,2.2,0.7]^{\mathrm{T}}$. The position
vectors are in meters, and the orientation vectors are in radians. The source
has $N_{B}=100$ antennas and the following number of transmit beams are
considered $N_{D}\in[16,32,48,64]^{\mathrm{T}}$. For each simulation, $T=20$
symbols are transmitted, and the beamforming matrix
$\bm{F}_{t}\in\mathcal{C}^{N_{B}\times N_{D}}$ changes during each of the $T$
transmit symbols. The rows of this beamforming matrix are selected from a
discrete Fourier transform-based (DFT) codebook. The destination is located at
$\bm{p}_{U}=[2.6,2.15,5.1]^{\mathrm{T}}$ with the orientation angles
$\bm{\Phi}_{U}=[0.1,0.2,0.1]^{\mathrm{T}}$. The Fraunhofer distance indicates
that the destination is experiencing near-field propagation. The incorrect
case when the far-field model is applied in this near-field simulation setup
is termed “far-field.” The correct case when the near-field model is used is
termed “near-field.”

Table I: Location estimation possibilities with near and far field models. N/A indicates the parameter is known. The terms $3$D and $2$D indicate the dimensions in which the parameter can be estimated in the propagation regimes.  | Near-field | Far-field
---|---|---
Unknown Parameters | $\bm{p}_{U}$ | $\bm{\Phi}_{U}$ | $\bm{p}_{B}$ | $\bm{\Phi}_{B}$ | $\bm{p}_{U}$ | $\bm{\Phi}_{U}$ | $\bm{p}_{B}$ | $\bm{\Phi}_{B}$
Source position and source orientation | N/A | N/A |  $3$D |  $3$D | N/A | N/A |  $2$D |  $2$D
Source position and destination orientation | N/A |  $3$D |  $3$D | N/A | N/A |  $2$D |  $2$D | N/A
Destination position and source orientation |  $3$D | N/A | N/A |  $3$D |  $2$D | N/A | N/A |  $2$D
Source position | N/A | N/A |  $3$D | N/A | N/A | N/A |  $2$D | N/A
Source orientation | N/A | N/A | N/A |  $3$D | N/A | N/A | N/A |  $2$D
Destination position |  $3$D | N/A | N/A | N/A |  $2$D | N/A | N/A | N/A
Destination orientation | N/A |  $3$D | N/A | N/A | N/A |  $2$D | N/A | N/A

With this simulation setup, we generate Table I. This table highlights
different combinations of the source and destination location that can be
estimated. The “not applicable” term is used to highlight the fact that the
parameter is known. When the term $3$D is used, it means that the $3$D version
of that parameter can be estimated, and if the $3$D version of the parameter
can be estimated, all lower dimensions can also be estimated. As evident in
Table I, it is impossible to estimate either the $3$D position coordinates or
the $3$D orientation angles with only the signal from the line of sight (LOS)
path when the far-field model is incorrectly applied to the near-field setup.
However, if the near-field setup is correctly applied, estimating the $3$D
position coordinates or the $3$D orientation angles are feasible with the LOS
signal even without a beamforming matrix. While a $2$D estimation of the
source or destination’s orientation angles is feasible when the far-field
model is used and $N_{U}>1$, it is important to note that estimating the
source orientation angles is only possible in the far-field with beamforming
(see Appendix -A). This is in contrast with the near-field setup in which the
estimation of the source’s orientation angles is possible even with no
beamforming provided that $N_{U}>1$.

(a)

(b)

Figure 2: PEB and OEB vs. $N_{U}$

In Figs. 2a and 2b, we present the PEB and OEB as a function of varying
numbers of receive antennas. Also, in these figures, the term “FF” is used to
distinguish the incorrect case when the far-field model is applied to the
study from the case when the near-field model is correctly applied to the
study. As expected, the spherical wavefront in the near-field model results in
more accurate localization. From the figures, the spherical wavefront is more
advantageous for the estimation of the orientation.

##  V Conclusion

This paper has examined the estimation of different combinations of a single-
source and single destination’s position and orientation. Through a study of
the FIM, we have shown that while the source or destination’s $3$D orientation
can be jointly estimated with the source or destination’s $3$D position in the
near-field propagation regime, only the source or destination’s $2$D
orientation can be jointly estimated with the source or destination’s $2$D
position in the far-field propagation regime. Also, while without beamforming
in the near-field, the source’s $3$D orientation can be estimated, the
source’s $2$D orientation angles can not be estimated without beamforming in
the far-field. Finally, a simulation of the PEB and OEB shows that the
spherical information present in the near-field is much more useful for
estimating orientation information.

###  -A Analysis of Joint Estimation of $[\bm{\Phi}_{B},\bm{\beta}]$ under the
Far-Field Model

We start the proof by dropping the subscript $t$ and using the identity
beamforming matrix across the $T$ transmissions. The FIM,
$\mathbf{J}_{\bm{\Phi}_{B}}$, under the parameterization
$\bm{\eta}=[\bm{\Phi}_{B},\bm{\beta}]^{\mathrm{T}}$, is obtained by using the
appropriate first derivatives in Definitions (1), and it has the following
structure

$\displaystyle\mathbf{J}_{\bm{\eta}}=\left[\begin{array}[]{ccccc}\mathbf{J}_{\bm{\Phi}_{B}}&\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{R}}]}&\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{I}}]}\\\
\mathbf{J}_{{\bm{\Phi}}_{R},{{\beta}}_{R}}^{\mathrm{T}}&\mathbf{J}_{{{\beta}}_{R}}&0\\\
\mathbf{J}_{{\bm{\Phi}}_{R},{{\beta}}_{I}}^{\mathrm{T}}&0&\mathbf{J}_{{{\beta}}_{I}}\\\
\end{array}\right],$

and the EFIM can be written as

$\displaystyle\begin{split}\mathbf{J}_{\bm{\Phi}_{B}}^{\mathrm{e}}&=\mathbf{J}_{\bm{\Phi}_{B}}-[\mathbf{J}_{{{\beta}}_{R}}]^{-1}\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{R}}]}\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{R}}]}^{\mathrm{T}}+\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{I}}]}\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{I}}]}^{\mathrm{T}},\end{split}$
(11)

and
$\mathbf{J}_{\bm{\Phi}_{B}}^{\mathrm{e}}=\mathbf{J}_{\bm{\Phi}_{B}}-[\mathbf{J}_{{{\beta}}_{R}}]^{-1}\mathbf{J}_{{{\beta}}_{R}}\mathbf{J}_{{\bm{\Phi}}_{R}},$
the second equation results from noticing
$\mathbf{J}_{{{\beta}}_{R}}\mathbf{J}_{{\bm{\Phi}}_{R}}=\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{R}}]}\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{R}}]}^{\mathrm{T}}+\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{I}}]}\mathbf{J}_{[\bm{\Phi}_{B},{\beta}_{\mathrm{I}}]}.$
The proof follows as $\mathbf{J}_{\bm{\Phi}_{B}}^{\mathrm{e}}=0$. Hence, with
no beamforming, the source orientation can not be estimated with the far-field
propagation model.

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