Presented at IC Channeling 2018 |  | September 27, 2018
---|---|---
Published on ArXiv |  | October 15, 2018
Accepted by JHEP |  | March 12, 2019
Published on JHEP |  | March 26, 2019

# Feasibility of $\tau$-lepton electromagnetic dipole moments measurement
using bent crystal at the LHC

A.S. Fomin<EMAIL_ADDRESS>NSC Kharkiv Institute of Physics and
Technology, 1 Akademicheskaya St., Kharkiv, 61108 Ukraine  currently at CERN,
European Organization for Nuclear Research, Geneva 23, CH-1211 Switzerland
A.Yu. Korchin<EMAIL_ADDRESS>NSC Kharkiv Institute of Physics and
Technology, 1 Akademicheskaya St., Kharkiv, 61108 Ukraine  V.N. Karazin
Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022 Ukraine  A. Stocchi
<EMAIL_ADDRESS>LAL (Laboratoire de l’Accélérateur Linéaire), Université
Paris-Sud/IN2P3, B timent 200 — BP 34, Rue André Ampere, 91898 Orsay Cedex,
France  S. Barsuk  LAL (Laboratoire de l’Accélérateur Linéaire), Université
Paris-Sud/IN2P3, B timent 200 — BP 34, Rue André Ampere, 91898 Orsay Cedex,
France  P. Robbe  LAL (Laboratoire de l’Accélérateur Linéaire), Université
Paris-Sud/IN2P3, B timent 200 — BP 34, Rue André Ampere, 91898 Orsay Cedex,
France

###### Abstract

In this paper we discuss the possibility of measuring the anomalous magnetic
and electric dipole moments of the $\tau$ lepton. The method consists in
studying the spin precession induced by the strong effective magnetic field
inside channels of a bent crystal with a dedicated setup at the CERN Large
Hadron Collider.

PACS number: 13.20.-v, 13.35.Dx, 13.40.Em, 13.88+e, 14.60.Fg, 61.85.+p

Keywords: Fixed target experiments, Polarisation, Tau Physics

ArXiv ePrint: 1810.06699

Cite this article as:

A.S. Fomin, A.Yu. Korchin, A. Stocchi, et al. J. High Energ. Phys. 03 (2019)
156.

[arXiv:1810.06699] https://doi.org/10.1007/JHEP03(2019)156

###### Contents

  1. I  Introduction
  2. II  Principle of the measurement
    1. II.1  Initial polarisation of $\tau$ leptons from $D_{s}$ decays
    2. II.2  Final polarisation and energy of reconstructed $\tau$ leptons
    3. II.3  Experimental setup
  3. III  Sensitivity studies
    1. III.1  First crystal: spectrum of deflected $D_{s}$ mesons
    2. III.2  Second crystal: spectrum and polarisation of deflected $\tau$ leptons
    3. III.3  Preliminary study of the background from other decay channels
    4. III.4  Results of the sensitivity studies
  4. IV  Conclusions
  5. A  Polarisation of $\tau$ leptons in $D_{s}$ decays
  6. B  Optimal length for $D_{s}$ conversion to $\tau$ leptons

##  I  Introduction

The magnetic dipole moment (MDM) of many particles has been measured PDG:2018
. For example, the experimental value of the electron MDM perfectly agrees
with theoretical calculations. For muon, there is a discrepancy between the
Standard Model (SM) prediction PDG:2018  (ch. 57) and measurement by the BNL
E821 collaboration Bennett:2006fi . The deviation is about $3.5\,\sigma$ and
it can possibly originate from effects of new physics (NP). This aspect of the
muon MDM attracts considerable attention (see, e.g., Knecht:2003 ;
Passera:2004 ; Jegerlehner:2009 ).

The MDM of the $\tau$ lepton has not been measured so far and is of great
interest for a test of the SM and the search for signatures of NP.

Usually one discusses the anomalous part of the MDM,

$a_{\tau}=(g_{\tau}-2)/2,$ (1)

which is expressed through the gyromagnetic factor $g_{\tau}$ (or $g$-factor).
The value $g_{\tau}=2$ corresponds to a point-like Dirac particle ignoring
radiative corrections. Deviations from $g_{\tau}=2$ arise from the known
radiative corrections and can also stem from effects of NP. The latter are of
the order of $m_{\tau}^{2}/\Lambda^{2}$, where $m_{\tau}$ is the mass of
$\tau$ lepton and $\Lambda$ is the scale of NP. Because of the large mass of
the $\tau$ lepton the effects of NP are expected to be more noticeable than
the corresponding effects for muon.

The SM expectation for the $\tau$ MDM is given in Ref. Eidelman:2007 ,

$a_{\tau,\,\rm{SM}}=0.00117721(5).$ (2)

No measurement of the $\tau$ MDM has been performed because of the very short
lifetime of the $\tau$ lepton, $2.903(5)\,\times\,10^{-13}\,{\rm s}$. This
does not permit to apply methods used in the electron and muon $g-2$
experiments. At present the $\tau$ anomalous magnetic moment $a_{\tau}$ is
known to an accuracy of only about $10^{-2}$ PDG:2018 . This limit was
obtained by the DELPHI collaboration Abdallah:2003  from the measurement of
the $e^{+}e^{-}\to e^{+}e^{-}\tau^{+}\tau^{-}$ total cross section at LEP2.
The study of anomalous couplings of photon to $\tau$ lepton in this reaction
at LEP was proposed in Cornet:1995pw . At $95\,\%$ confidence level, the
confidence interval is PDG:2018

$-0.052<a_{\tau}<0.013.$ (3)

It is also given in Ref. Abdallah:2003  in the form $a_{\tau}=-0.018(17)$. The
authors of Ref. GonzalezSprinberg:2000  analyzed LEP1, SLD and LEP2
experiments on the $\tau$-lepton production and obtained $2\sigma$ bounds

$-0.007<a_{\tau}<0.005.$ (4)

Note that Ref. GonzalezSprinberg:2000  assumed that the electric dipole moment
(EDM) of the $\tau$ was absent, while Ref. Abdallah:2003 , when obtaining
bounds for $a_{\tau}$, assumed that EDM was equal to its SM value, which is
negligibly small.

In general, the EDM of the $\tau$ lepton, $d_{\tau}$, can take nonzero values
if both the parity $P$ and time reversal $T$ symmetries are violated
Roberts:2010 . In contrast to the interaction of MDM with the magnetic field
${\cal\vec{B}}$

$H_{M}=-\vec{\mu}_{\tau}\,{\cal\vec{B}}=-\frac{g_{\tau}}{2}\mu_{B}\,\vec{\sigma}{\cal\vec{B}},$
(5)

where $\mu_{B}=\tfrac{e\hbar}{2m_{\tau}c}$ is the Bohr magneton, the
interaction of EDM with the electric field ${\cal\vec{E}}$

$H_{E}=-\vec{d}_{\tau}\,{\cal\vec{E}}=-\frac{f_{\tau}}{2}\mu_{B}\,\vec{\sigma}{\cal\vec{E}},$
(6)

is $P$, $T$ and also $CP$ violating. This causes the interest to the EDM of
leptons. In framework of the SM the lepton EDM are extremely small as being
originated from the 4-loop diagrams and additionally because of the smallness
of $CP$ violation via the CKM matrix Commins:1999 ; Commins:2009 . For
example, the SM estimation for electron is $d_{e}\lesssim 10^{-38}$
e$\cdot$cm, while the $\tau$-lepton EDM can be larger by a factor
$\tfrac{m_{\tau}}{m_{e}}$, so that $d_{\tau}\lesssim 3.5\cdot 10^{-35}$
e$\cdot$cm, or $f_{\tau}\lesssim 1.2\cdot 10^{-20}$. The currently achieved
experimental accuracy Inami:2002  cited by PDG PDG:2018 ,

$\displaystyle-0.22<{\rm Re}\,d_{\tau}<0.45\;\,(10^{-16}\ {\rm e\cdot
cm}),\;\;{\rm or}\;-0.008<{\rm Re}\,f_{\tau}<0.0162,$ (7)
$\displaystyle-0.25<{\rm Im}\,d_{\tau}\,<0.008\;\,(10^{-16}\ {\rm e\cdot
cm}),\;\;{\rm or}\;-0.009<{\rm Im}\,f_{\tau}\,<0.00029,$ (8)

is 18 orders of magnitude worse than the SM prediction. Clearly the SM value
is hardly reachable in experiments and therefore any observation of the $\tau$
EDM will be indication of additional sources of $CP$ violation beyond the SM.

There have been various proposals to measure the MDM and EDM of the $\tau$
lepton. At the LHC, it was suggested in Refs. Hayreter:2013 ; Hayreter:2015
to obtain constraints on the $\tau$ MDM in measurements of the Drell-Yan
process. The authors of Ref. Atag:2010  discussed the possibility to set the
limits on the $\tau$ MDM by studying the two-photon production of $\tau$
leptons, $\gamma\gamma\to\tau^{+}\tau^{-}$. The process $p\,p\to
p\,\gamma^{\star}\,p\to p\,\tau^{-}\,\bar{\nu}_{\tau}\,q^{\prime}\,X$ with
almost real photon $\gamma^{\star}$ is examined in Ref. Koksal:2017nmy .

In Refs. Koksal:2018env ; Ozguven:2016rst  the potential of future colliders
for improvement of the current experimental bounds is investigated. In
particular, at the electron-positron linear collider CLIC in the processes
$\gamma\,\gamma\to\tau^{+}\,\tau^{-}$,
$\gamma\,\gamma\to\tau^{+}\,\tau^{-}\,\gamma$, and in the reactions
$e^{-}\,e^{+}\to
e^{-}\,\gamma^{\star}\,e^{+}\to\nu_{e}\,\tau^{-}\,\bar{\nu}_{\tau}\,e^{+}$
with quasi-real photon and
$e^{-}\,\gamma\to\nu_{e}\,\tau^{-}\,\bar{\nu}_{\tau}$ with Compton back-
scattering photon. The reaction $e^{-}\,p\to
e^{-}\,\gamma^{\star}\,\gamma^{\star}\,p\to e^{-}\tau^{-}\,\tau^{+}\,p$ on the
future electron-proton colliders LHeC and FCC-he with high center-of-mass
energies and high luminosities is considered in Koksal:2018xyi .

The authors of Refs. Bernabeu:2008 ; Bernabeu:2009  proposed to determine the
$\tau$ Pauli form factor $F_{2}(s)$ from the electron–positron annihilation to
a pair of tau leptons at the high-luminosity $B$ factory Super-KEKB. The
process $e^{+}e^{-}\to\tau^{+}\tau^{-}$ is planned to be studied at the
$\Upsilon$ resonances $\Upsilon(1S),\,\Upsilon(2S),\,\Upsilon(3S)$. This idea
was further reconsidered and refined in Ref. Eidelman:2016 .

A new method was proposed in Refs. Eidelman:2016 ; Fael:2016  to probe the
$\tau$ dipole moments from radiative leptonic $\tau$ decays
$\tau^{-}\to\ell^{-}\,\nu_{\tau}\,\bar{\nu}_{\ell}\,\gamma$ ($\ell=e,\,\mu$)
at the KEKB and Super-KEKB colliders. The feasibility studies Eidelman:2016
show that such a measurement at Belle II will be competitive with the current
limits for MDM given in Eq. (3).

Another method to determine the MDM of the $\tau$ lepton was discussed in Ref.
Samuel:1991 . It is based on the phenomenon of the particle spin precession in
a bent crystal. This method has been successfully used by the E761
collaboration at Fermilab Chen:1992  to measure the MDM of the $\Sigma^{+}$
hyperon. The authors of Samuel:1991  discussed the decay
$B^{+}\to\tau^{+}\nu_{\tau}$ as a source of polarised $\tau$ leptons. However,
the branching fraction of this decay is only about $10^{-4}$ which can make
realization of this idea practically difficult.

Nevertheless, the method which uses channeling of short-lived baryons in bent
crystals has recently attracted considerable attention Baryshevsky:2016 ;
Fomin:2017 ; Botella:2016 ; Bagli:2017 ; Baryshevsky:2017 . In particular, in
Ref. Fomin:2017  the feasibility of measuring the MDM of the charm baryons
$\Lambda_{c}^{+}$ and $\Xi_{c}^{+}$ at the LHC was addressed (see also details
in Ref. FominThesis ). The authors of Botella:2016 ; Bagli:2017  studied the
application of this method for future measurements of EDM and MDM of the
strange, charm and beauty baryons.

In the present paper we would like to propose the idea of measuring the MDM
and possibly EDM of the $\tau$ lepton through its spin precession in a bent
crystal. We suggest to use polarised $\tau$ leptons which come from the decay
of the $D_{s}$ mesons produced at the LHC. Because of the quark structure of
$D_{s}$ meson ($D_{s}^{+}=c\bar{s}$), the branching fraction of interest is
quite sizeable, ${\cal B}(D_{s}^{+}\to\tau^{+}\nu_{\tau})=0.0548(23)$ PDG:2018
.

The $D_{s}$ mesons are produced in $pp$ collisions with very high energies, of
a few TeV, and subsequently decay to polarised $\tau$ leptons. The leptons can
be directed into a bent crystal, get in the channeling regime, and the
direction of $\tau$ polarisation after the precession in the crystal can be
determined from the angular analysis of its decay products. Schematically, the
whole process is

$p\,p\to D_{s}^{+}X,\quad D_{s}^{+}\to\tau^{+}\nu_{\tau},\quad\tau^{+}\ {\rm
flight\ in\ the\
crystal},\quad\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau}.$ (9)

For the present study we consider the $\tau$ decay into three charged pions,
which allows to identify the $\tau$ lepton through the reconstruction of a
secondary vertex. The various aspects of this measurement are studied in the
present paper.

##  II  Principle of the measurement

###  II.1  Initial polarisation of $\tau$ leptons from $D_{s}$ decays

In $pp$ collisions the $\tau$ leptons are mainly produced in the decay
$D_{s}^{+}\to\tau^{+}\nu_{\tau}$, with a polarisation vector (see details in
Appendix A)

$\vec{\mathds{P}}=\frac{1}{M_{D}{k}_{\tau}^{*}}\left[m_{\tau}\vec{p}_{D}\,-\,E_{D}\vec{k}_{\tau}\,+\,\frac{\vec{k}_{\tau}\left(\vec{k}_{\tau}\cdot\vec{p}_{D}\right)}{m_{\tau}+\varepsilon_{\tau}}\right].$
(10)

Here $(E_{D},\,\vec{p}_{D})$ is the four-momentum of $D_{s}$ meson,
$(\varepsilon_{\tau},\,\vec{k}_{\tau})$ is the four-momentum of $\tau$ lepton
in the lab frame, and ${k}_{\tau}^{*}=(M_{D}^{2}-m_{\tau}^{2})/(2M_{D})$ is
the momentum of $\tau$ lepton in the rest frame of the $D_{s}$ meson.

It is convenient to separate the longitudinal and transverse components of the
polarisation vector with respect to the direction of motion of the $\tau$
lepton,

$\displaystyle\vec{\mathds{P}}_{\parallel}=\frac{1}{M_{D}{k}_{\tau}^{*}}\,(\varepsilon_{\tau}p_{D}\cos\theta-
E_{D}k_{\tau})\,\vec{n}_{k},$
$\displaystyle\vec{\mathds{P}}_{\perp}=\frac{m_{\tau}p_{D}}{M_{D}{k}_{\tau}^{*}}\,(\vec{n}_{p}-\vec{n}_{k}\cos\theta),$
(11)

where we defined unit vectors $\vec{n}_{k}\equiv\vec{k}_{\tau}/{k}_{\tau}$ and
$\vec{n}_{p}\equiv\vec{p}_{D}/{p}_{D}$, and
$\cos\theta=\vec{n}_{k}\vec{n}_{p}$. These components are not independent and
satisfy

$\displaystyle\vec{\mathds{P}}_{\parallel}^{2}+\vec{\mathds{P}}_{\perp}^{2}=1,$
(12)
$\displaystyle|\vec{\mathds{P}}_{\perp}|=\frac{m_{\tau}{p}_{D}}{M_{D}{k}_{\tau}^{*}}\sin\theta.$
(13)

It is seen from Eqs. (12) and (13) that if the $\tau$ lepton moves in the
direction of the $D_{s}$ meson, i.e. $\theta=0$, then the polarisation is
purely longitudinal and $|\vec{\mathds{P}}_{\parallel}|=1$. The transverse
polarisation increases with increasing $\theta$ and takes its maximal value
$|\vec{\mathds{P}}_{\perp}|=1$ at the maximal allowed angle determined from

$\sin\theta_{max}=\frac{M_{D}{k}_{\tau}^{*}}{m_{\tau}{p}_{D}}=\frac{\gamma_{\tau}^{*}v_{\tau}^{*}}{\gamma_{D}v_{D}},$
(14)

where $\vec{v}_{D}=\vec{p}_{D}/E_{D}$ is velocity of the $D_{s}$ meson in the
lab frame, $\vec{v}_{\tau}^{*}=\vec{k}_{\tau}^{*}/\varepsilon_{\tau}^{*}$ is
velocity of the $\tau$ lepton in the rest frame of the $D_{s}$ meson, and the
corresponding Lorentz factors are $\gamma_{D}=E_{D}/M_{D}$ and
$\gamma_{\tau}^{*}=\varepsilon_{\tau}^{*}/m_{\tau}$. Figure 1 shows
schematically the ellipsoid of the $\tau$ momenta and the behavior of the
$\tau$ polarisation.

Figure 1: The decay $D_{s}^{+}\to\tau^{+}\nu_{\tau}$ in the (left) $D_{s}$
rest frame and (right) Lab frame. The axis $Oz$ is chosen along the direction
of $D_{s}$ momentum. Rotational symmetry around the $Oz$ axis is implied. The
ellipsoid is actually very elongated along the axis $Oz$ because
$\gamma_{D}\gg 1$. The blue arrows show the $\tau$ momentum $\vec{k}_{\tau}$,
and the thick red arrows the $\tau$ polarisation vector, $\vec{\mathds{P}}$.
Several cases of $\tau$ polarisation in the Lab frame are shown: (1,2)
$\theta=0$, (3) $\theta=\theta_{max}$ and (4,5) an arbitrary $\theta$.

At the minimal $\tau$ energy,

$\varepsilon_{\tau}^{-}(\theta=0)=\frac{1}{M_{D}}(E_{D}\varepsilon_{\tau}^{*}-p_{D}k_{\tau}^{*})\approx\frac{m_{\tau}^{2}}{M_{D}^{2}}E_{D},$
(15)

the polarisation is longitudinal and
$\vec{\mathds{P}}_{\parallel}=+\vec{n}_{z}$, while at the maximal $\tau$
energy,

$\varepsilon_{\tau}^{+}(\theta=0)=\frac{1}{M_{D}}(E_{D}\varepsilon_{\tau}^{*}+p_{D}k_{\tau}^{*})\approx
E_{D},$ (16)

the polarisation is longitudinal and
$\vec{\mathds{P}}_{\parallel}=-\vec{n}_{z}$. At the $\tau$ energy
corresponding to the maximal angle $\theta_{max}$,

$\varepsilon_{\tau}^{+}(\theta_{max})=\varepsilon_{\tau}^{-}(\theta_{max})=\frac{2m_{\tau}^{2}}{M_{D}^{2}+m_{\tau}^{2}}E_{D},$
(17)

the polarisation is purely transverse and
$|\vec{\mathds{P}}_{\perp}(\theta_{max})|=1$.

We can rewrite the expressions for the longitudinal and transverse $\tau$
polarisations, Eq. (11), as functions of the $\tau$ and $D_{s}$ energies,
assuming $\varepsilon_{\tau}=k_{\tau}$, as we consider only energetic $\tau$
leptons,

$\displaystyle\vec{\mathds{P}}_{\parallel}(E_{D},\varepsilon_{\tau})\approx\
-\,\frac{\,\,\cfrac{M_{D}^{2}+m_{\tau}^{2}}{2\,m_{\tau}^{2}}\,-\,\cfrac{E_{D}}{\varepsilon_{\tau}}\,\,}{\cfrac{M_{D}^{2}+m_{\tau}^{2}}{2m_{\tau}^{2}}\,-\,1}\,\vec{n}_{k},$
$\displaystyle|\vec{\mathds{P}}_{\perp}(E_{D},\varepsilon_{\tau})|\approx\dfrac{\,2\,m_{\tau}^{2}}{M_{D}^{2}\,-\,m_{\tau}^{2}}\sqrt{\left(\dfrac{M_{D}^{2}}{m_{\tau}^{2}}-\dfrac{E_{D}}{\varepsilon_{\tau}}\right)\left(\dfrac{E_{D}}{\varepsilon_{\tau}}\,-\,1\right)}.$
(18)

Figure 2 presents the transverse and longitudinal polarisations of Eq. (18) of
$\tau$ leptons produced by $D_{s}$ mesons of energy $E_{D}=2\,$TeV, as a
function of the $\tau$ energy.

The blue dashed line shows the $\tau$ spectra, which in the relativistic case
is a step function. Due to a small asymmetry (see Fig. 2, left) there is a
small average longitudinal polarisation of the order of $10\,\%$ in the
direction $-\vec{n}_{z}$.

Figure 2:  (Left) Longitudinal and (right) transverse polarisations as a
function of the energy of $\tau$ leptons produced by $2\,$TeV $D_{s}$ mesons.
The red solid line is the polarisation and the blue dashed line is the $\tau$
spectrum.

Note, that due to a cylindrical symmetry, $\tau$ leptons produced from $D_{s}$
decays of the same energy at the same polar angle $\theta$ but with opposite
azimuthal angles, $\phi$ and $\phi+\pi$, would have an almost opposite
transverse polarisation,

$\vec{\mathds{P}}_{\perp}(\theta,\phi)\approx-\vec{\mathds{P}}_{\perp}(\theta,\phi+\pi).$
(19)

Thus, for the sufficient precision at the $\tau$ polarisation analysis, the
directions of the $D_{s}$ and $\tau$ momenta should be known with a precision
better than $\theta_{max}$.

###  II.2  Final polarisation and energy of reconstructed $\tau$ leptons

Let us discuss possibilities to measure the $\tau$ polarisation. In general,
it can be obtained from the angular distribution of the final particles in the
$\tau$ decays. In a two-body decay, this distribution has the form

$\frac{1}{N}\frac{dN}{d\Omega}=\frac{1}{4\pi}\left(1+\alpha\,\vec{\mathds{P}}\cdot\vec{n}\right),$
(20)

where $\vec{n}\equiv\vec{p}/{p}$ is the unit vector in the direction of
movement of the final particle-analyzer and $\alpha$ is the asymmetry
parameter. Several hadronic decay channels can be considered. They are listed
in Table 1.

Table 1: Several $\tau$ hadronic decay channels with asymmetry parameter, $\alpha$, and sensitivities. Sensitivities, $S$, are shown for the measurements made with different number of observables, $\vec{n}_{k}$ is the direction of $\tau$ momentum. Columns 2,3 and 4 are our calculations and columns 5, 6, 7 and 8 are from the analysis of Ref. Davier:1993 .  $\tau$ decay | $\alpha$ | $\alpha_{\rm eff}$ | $S(1)_{\rm eff}$ | S(1) | $S(2)$ |  S(all but $\vec{n}_{k}$) |  S(with $\vec{n}_{k}$)
---|---|---|---|---|---|---|---
$\tau^{+}\to\pi^{+}\bar{\nu}_{\tau}$ | 1 | 1 | 0.58 | 0.58 | – | 0.58 | 0.58
$\tau^{+}\to\rho^{+}\,(2\pi)\,\bar{\nu}_{\tau}$ | 0.45 | 0.49 | 0.28 | 0.26 | 0.49 | 0.49 | 0.58
$\tau^{+}\to a_{1}^{+}\,(3\pi)\,\bar{\nu}_{\tau}~{}$ |  0.021 |  0.12 – 0.21 |  0.069 – 0.12 |  0.10 |  0.23 | 0.45 | 0.58

In the second column, the parameter $\alpha$ for the two-body decay is given.
Other columns contain information on the sensitivity $S=1/(\sigma\sqrt{N})$ to
the polarisation, where $N$ is the number of events, $\sigma$ is the error on
polarization measurement at $\vec{\mathds{P}}=0$. According to Ref.
Davier:1993  if the full kinematics of the decay is reconstructed then the
maximum possible sensitivity $S=1/\sqrt{3}\approx 0.58$ is reached.

For the single-pion channel, the maximum sensitivity is achieved, even though
this channel suffers from large background. For the vector-meson channels, the
sensitivity is suppressed because of the smaller asymmetry parameter
$\alpha=({m_{\tau}^{2}-2m_{v}^{2}})/({m_{\tau}^{2}+2m_{v}^{2}})$
($m_{v}=m_{\rho},\,m_{a_{1}}$). The parameter $\alpha$ for the $a_{1}(1260)$
meson turns out to be 0.021 which leads to the low sensitivity about 0.01.
However we can estimate the influence of the large decay width of this
resonance. For this, we integrate $\alpha$ as a function of the invariant mass
$W$ of a resonance, weighted with a normalised Breit-Wigner distribution,

$\alpha_{\rm
eff}=\int_{0}^{m_{\tau}}\left(\frac{m_{\tau}^{2}-2W^{2}}{m_{\tau}^{2}+2W^{2}}\right)\,\frac{{\rm
d}W}{(W^{2}-m_{v}^{2})^{2}+m_{v}^{2}\Gamma_{v}^{2}}\,\left(\int_{0}^{m_{\tau}}\frac{{\rm
d}W}{(W^{2}-m_{v}^{2})^{2}+m_{v}^{2}\Gamma_{v}^{2}}\right)^{-1}.$ (21)

Choosing $m_{a1}=1230$ MeV and the decay width $\Gamma_{a1}=250-600$ MeV
PDG:2018 , we obtain an interval for the effective parameter for the
$a_{1}(1260)\,\nu_{\tau}$ mode in the third column of Table 1. The
corresponding sensitivity is given in the fourth column. It is seen that
accounting for the large decay width of $a_{1}(1260)$ enhances the sensitivity
considerably. For the $\rho\,\nu_{\tau}$ mode, the effect of the width is much
less important as seen also in Table 1.

The analysis of Ref. Davier:1993  shows that the sensitivity of the
polarisation measurement increases with increasing the number of observables,
and reaches the maximal value of 0.58 if, in addition, the direction of the
$\tau$ momentum is determined. Thus one can opt for the $3-$prong decay
$\tau^{+}\to
a_{1}^{+}\,\bar{\nu}_{\tau}\to\pi^{+}\pi^{+}\pi^{-}\,\bar{\nu}_{\tau}$ as it
can be separated from background processes and allows us to reconstruct the
position of the vertex. Note that various aspects of the $\tau\to
3\pi\nu_{\tau}$ decay mode are studied in Refs. Hagiwara:1989fn ; Rouge:1990 ;
Duflot:1993 .

In this paper we do not address the reconstruction of $\tau$ polarisation, but
an estimation on sensitivity. The sensitivity on polarisation in $\tau^{+}\to
a_{1}^{+}\overline{\nu}_{\tau}$ decay is very small $S\sim 0.1$. On the other
hand, if we also analyse the consequent decay
$a_{1}^{+}\to\pi^{+}\pi^{+}\pi^{-}$ the sensitivity reaches its maximal
possible value $S\sim 0.58$. Thus, a big part of sensitivity comes from
$a_{1}$, which can be fully reconstructed, so we can safely estimate that
$S>0.2$.

For this reason in the sensitivity studies (in Sec. III) we chose to use the
range for sensitivity on polarisation $(0.2<S<0.58)$.

The energy of $\tau$ can be partially reconstructed. Assuming that we fully
reconstruct $a_{1}$ from $a_{1}^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay, using the
relation between energies of $a_{1}$ and $\tau$, that is analogous to the ones
shown in Eq. (15) and (16), we obtain the relative error on $\tau$ energy

$\frac{\Delta\varepsilon_{\tau}}{\varepsilon_{\tau}}=\frac{1}{\,\sqrt{3}\,}\
\frac{\ m_{\tau}^{2}/m_{a_{1}}^{2}-1\,}{m_{\tau}^{2}/m_{a_{1}}^{2}+1}\approx
0.21.$ (22)

Of course, putting the additional silicon telescope detector in front of the
crystal could allow to fully reconstruct the decay kinematics, but this setup
would require a detailed study of the background in the environment where the
experiment will be located.

###  II.3  Experimental setup

When a relativistic $\tau$ lepton is deflected by a bent crystal, its
polarisation vector rotates by an angle which is proportional to the anomalous
MDM Baryshevsky:1979 ; Lyuboshits:1980 ; Kim:1983 ; Fomin:2017 ,

$\angle\left(\vec{\mathds{P}}_{\rm i}\vec{\mathds{P}}_{\rm
f}\right)=(1+\gamma_{\tau}\,a_{\tau})\,\Theta,$ (23)

where $\vec{\mathds{P}}_{\rm i}$ and $\vec{\mathds{P}}_{\rm f}$ are the
vectors of the initial and final polarisations of $\tau$ lepton (before and
after the crystal), $\gamma_{\tau}$ is the Lorentz factor of $\tau$ lepton
($\gamma_{\tau}\gg 1$), and $\Theta=L/R$ is the deflecting angle of the bent
crystal, $L$ and $R$ being the length and bending radius of the crystal.

Thus, to measure the anomalous MDM of $\tau$ we need to have a beam of
polarised $\tau$ leptons directed at the crystal, and to know the $\tau$
initial and final polarisations and energy. As the polarised $\tau$ leptons
are produced from the decay of $D_{s}$ mesons, the crystal should be placed at
a certain distance $L_{\rm v}$ from the target to maximise the probability of
$\tau$ production (see Appendix B).

To reconstruct the direction of initial polarisation of Eq. (11), we need to
select $\tau$ leptons coming from $D_{s}$ decays and also to know the
direction of the $D_{s}$ and $\tau$ momenta and the energy of either $D_{s}$
or $\tau$ particles. For the present study we consider the $\tau$ decay into
three charged pions, so we need a suitable setup in order to get rid of the
background processes leading to the same final state (see Sec. III.3). A
solution could be to put a silicon telescope detector between the target and
the crystal (see Fig. 3). Further optimisation of the telescope design can be
performed, however, the outlined structure is sufficient for a demonstration
of the measurement principle, which is a purpose of the present article.

Figure 3: Schematics of the setup with a silicon telescope detector.

The first problem of this setup is that the telescope detector should have a
very high angular resolution on a rather short base: the distance between the
target and the crystal should be $\sim\,10-40\,{\rm cm}$ (see Appendix B), and
the angular resolution should be better than $\theta_{max}(E_{D}=2\,{\rm
TeV})\approx 100\,\mu{\rm rad}$. The second problem is that the telescope
detector in this case is very close to the target and could suffer from the
background overload.

In this concern, we propose a two crystal setup as shown in Fig. 4. The
purpose of the first crystal is to channel $D_{s}$ mesons and so to constrain
their momentum direction. Misaligning the first crystal from the direction of
the proton beam by a small angle $\theta_{p}$ is to restrict the channeling of
the initial protons. As a result we would have a collimated beam of
$D_{s}^{+}$ mesons satisfying the condition $|\,\theta_{x}\,|<\theta_{\rm
acc}\sim 5\,\mu{\rm rad}$, where $\theta_{x}=\arcsin(p_{x}/p)$ and
$\theta_{\rm acc}$ is the angle of acceptance to the planar channeling regime
Fomin:2017 ; FominThesis .

Figure 4: Double crystal scheme.

The second crystal is for channeling the $\tau$ lepton from the $D_{s}$ decay.
Misaligning the second crystal from the direction of the $D_{s}$ beam by an
angle $\theta_{\rm v}$ would select (by channeling) only $\tau$ leptons
produced between the two crystals and would not permit the collimated $D_{s}$
beam to be channeled in the second crystal (see Fig. 4). The initial direction
of the $\tau$ polarisation is reconstructed from the $D_{s}$ and $\tau$
momenta, which are selected by channeling in the first and the second crystal,
respectively.

In the following, we will determine the optimal parameters of this setup:
lengths and bending radii of two crystals, distance between crystals and
orientation of the second crystal. The direction of the final polarisation can
be reconstructed by analysing the angular distribution of the $\tau$ decay
products, as we discussed in Sec. II.2. Finally, we obtain from (23) the value
of the anomalous MDM $a_{\tau}$.

##  III  Sensitivity studies

In this section we present a sensitivity study for measuring the MDM and EDM
of the $\tau$ lepton using the setup described in the previous section. The
absolute statistical error of the measurement of the anomalous magnetic moment
can be written in the following way,

$\Delta a_{\tau}=\sqrt{\dfrac{1}{\,N_{\rm POT}\ \eta_{\rm det}\ Br\ S^{2}\
\eta_{\rm\,setup}\,}},\qquad\eta_{\rm\,setup}=\Theta^{2}\displaystyle\int{\rm
d}\varepsilon\,\frac{\partial N_{\tau}^{\rm def}}{\partial\varepsilon}\
\gamma_{\tau}^{2}\ \,{\mathds{P}}^{2},$ (24)

where $N_{\rm POT}$ is the total number of protons on target, $\eta_{\rm det}$
is the detector efficiency, $Br$ is the branching fraction of the selected
$\tau$ decay, $S$ is the sensitivity of polarisation reconstruction by the
analysis of the $\tau$ decay, $\eta_{\rm\,setup}$ is the setup efficiency
(target plus two crystals), $\partial N_{\tau}^{\rm def}/d\varepsilon$ is the
spectrum of deflected $\tau$ leptons normalised to one initial proton,
$\gamma_{\tau}$ and
${\mathds{P}}^{2}={\mathds{P}}_{x}^{2}+{\mathds{P}}_{z}^{2}$ are the Lorentz
factor and the square of averaged polarisation (in $xz$ plane) of the
deflected $\tau$ lepton as a function of its energy. In the following
sections, we discuss these terms and give some estimates on their values.

The error of MDM connected with the precision of $\tau$ energy reconstruction
(22) can be written in the following way,

$\Delta
a^{(\gamma)}=\frac{\overline{a}+\gamma^{-1}}{1+\gamma/\Delta\gamma}\approx\frac{\overline{a}_{\tau}+0.001}{6}.$
(25)

Considering the expected value of anomalous MDM, $\overline{a}_{\tau}\sim
0.001$ and Lorentz factor, $\gamma\sim 1000$ this contribution is negligible.

Due to the MDM, the spin precession takes place only in the $xz$ plane. If
$\tau$ lepton possesses the EDM, the $y$ component of polarisation will also
be induced. In the limit $\gamma_{\tau}\gg 1$, and assuming that EDM effects
are small compared to the MDM spin precession, using formulas in Bagli:2017 ,
the ratio between absolute error on EDM and MDM is

$\frac{\Delta f_{\tau}}{\Delta
a_{\tau}}\approx\left|\frac{4\,\gamma\,\overline{a}_{\tau}}{\Theta\,(1+\gamma\overline{a}_{\tau})^{2}}\right|$
(26)

Analogous to (24) the absolute statistical error of the measurement of the EDM
can be written in the following way,

$\Delta f_{\tau}=\sqrt{\dfrac{1}{\,N_{\rm POT}\ \eta_{\rm det}\ Br\ S^{2}\
\eta^{(f)}_{\rm\,setup}\,}},\qquad\eta^{(f)}_{\rm\,setup}=\frac{\Theta^{4}}{\,16\,\overline{a}_{\tau}^{\,2}\,}\displaystyle\int{\rm
d}\varepsilon\,\frac{\partial N_{\tau}^{\rm def}}{\partial\varepsilon}\
(1+\gamma_{\tau}\,\overline{a}_{\tau})^{4}\ \,{\mathds{P}}^{2},$ (27)

###  III.1  First crystal: spectrum of deflected $D_{s}$ mesons

The spectral angular distribution of $D_{s}$ mesons produced by $7\,{\rm TeV}$
protons on a fixed target is obtained simulating minimum-bias $pp$ collisions
with the Pythia 8.2 generator Sjostrand:2007gs  and shown in Fig. 5. In the
current study we choose a tungsten target with depth $L_{\rm tar}=1\,{\rm cm}$
along the direction of the incident proton beam. As it was shown in Ref.
Fomin:2017 , the target depth is limited only by the increasing number of
secondary particles produced in deeper targets.

Figure 5: Spectral angular distribution of $D_{s}$ mesons after the first
crystal. “Deflected” describes a collimated $D_{s}$ beam deflected by a
germanium crystal by $\Theta_{D}=3\,$mrad; “Background”, $D_{s}^{+}$ produced
in the target. Purity is the ratio of deflected $D_{s}^{+}$ to all $D_{s}^{+}$
mesons (initially produced or deflected) at $\theta_{x}$. $\theta_{x}=0$
corresponds to the direction of proton beam.

Assuming no nuclear effects the production rate of $D_{s}^{+}$ mesons per one
incident proton is

$N_{D}^{\rm\,tar}=\rho\,N_{\rm{A}}\,\sigma_{D}\,\frac{A_{\rm tar}}{M_{\rm
tar}}\,L_{\rm tar},$ (28)

where $\rho$, $A_{\rm tar}$ and $M_{\rm tar}$ are the density, number of
nucleons and molar mass of the target, respectively, $N_{\rm A}$ is the
Avogadro number, and $\sigma_{D}$ is the cross section of $D_{s}^{+}$
production in $pp$ collision at center-of-mass energy $\sqrt{s}=115\,$GeV
(corresponding to the fixed target experiment at the $7\,$TeV LHC proton
beam). For the chosen target, $N_{D}^{\rm\,tar}\approx 1.4\times 10^{-4}$.

In the setup shown in Fig. 4, we propose to place a crystal with the purpose
of preparing a collimated energetic beam of $D_{s}$ mesons. The initial
direction of the crystallographic plane is chosen as $\theta_{p}=100\,\mu$rad.
Thus, after the first crystal, we can have two categories of $D_{s}$ meson
beams. The first one, referred from now on as “Background”, is composed of
$D_{s}^{+}$ mesons which are initially produced at an angle
$\theta_{x}\sim\Theta_{D}$. In this case, the $D_{s}$ mesons are poorly
collimated, and as a consequence, the average $\tau$ polarisation is close to
zero. The second category, further referred as “Deflected”, is composed of
$D_{s}^{+}$ mesons deflected by the crystal by an angle $\Theta_{D}$. These
$D_{s}$ mesons are collimated and hence can be the source of polarised $\tau$
leptons.

The purity, which is defined as the ratio between the number of deflected
$D_{s}$ mesons and the total (“Background”+“Deflected”) number of $D_{s}$
mesons, is increasing as a function of energy as can be seen on Fig. 5. For
the sensitivity analysis, we take into account this effect by introducing an
effective spectra, which is obtained by convolution of the spectra with the
purity. The background from the $D_{s}$ produced at $\Theta_{D}$ is
specifically considered, while a preliminary consideration of other sources is
addressed in section III.3.

The spectral angular distribution of $D_{s}$ mesons deflected by a bent
crystal (Fig. 5, Deflected) is obtained using a parameterisation based on the
Monte Carlo simulation of particle propagation through the crystal FominThesis
, accounting the incoherent scattering on the electron subsystem of a crystal
and on thermal vibrations of the atoms at lattice nodes, that leads to
dechanneling effect. The optimal bending radii and lengths were found for a
silicon crystal to be $R_{D}=15\,$m and $L_{D}=4.5\,$cm, and for a germanium
crystal, $R_{D}=10\,$m and $L_{D}=3\,$cm. Thus, the optimal deflection angle
is $\Theta_{D}\sim 3\,{\rm mrad}$. From our simulation, we determined that the
$D_{s}$ deflection rate per one incident proton on target by a germanium
crystal is about $1.1\times 10^{-7}$ and about $0.6\times 10^{-7}$ with a
silicon crystal.

###  III.2  Second crystal: spectrum and polarisation of deflected $\tau$
leptons

For an optimal rate of $\tau$ leptons from $D_{s}$ decays, the second crystal
should be placed at a distance $L_{\rm v}\sim 10\,{\rm cm}$ from the first one
(see Appendix B). This way, about $1.3\,\%$ of $D_{s}$ mesons will decay into
$\tau$ leptons before the entrance of the second crystal.

The momentum distribution of the $\tau$ leptons produced in the two-body decay
is defined by the momentum of the $D_{s}$ mesons. In the laboratory frame, the
$\tau$ momenta form an ellipsoid of rotation. Figure 6 represents the $\tau$
momentum distribution in the $p_{x}p_{z}$ and $p_{y}p_{z}$ phase spaces in the
laboratory frame. The two red lines at the angles $\theta_{\rm
v}\pm\theta_{\rm acc}$ from the $p_{z}$-axis highlight the area of phase space
of $\tau$ momentum that can be deflected by the second crystal. In
$p_{y}\,p_{z}$ phase space the deflected area has an elliptical shape which
results from intersection of the second crystal plane with the ellipsoid of
the $\tau$ momenta.

Figure 6: Collimation of $\tau$ momenta by the second crystal. $p_{z}$ is
chosen in the direction of the $D_{s}$ momentum. Each ellipsoid represents the
phase space of $\tau$ leptons produced by $D_{s}$ mesons of specific energy.
The red area shows the phase space of $\tau$ leptons collimated by the second
crystal. Arrows show the polarisation of $\tau$. Scale on $p_{x}$ is $\sim
10^{4}$ times smaller than on $p_{z}$ because $\gamma_{D}\gg 1$.

As can be seen from the figure, the misalignment of the two crystals leads to
a large suppression of the spectra of deflected $\tau$ leptons. For instance,
the angle between the crystals $\theta_{\rm v}=100\,\mu$rad (see Fig. 4)
matches the maximal allowed angle between the $\tau$ and $D_{s}$ momenta given
in Eq. (14) for a $D_{s}$ energy of $E_{D}=2\,$TeV, thus, $\tau$ leptons
produced by more energetic $D_{s}$ mesons would not be deflected.
Nevertheless, the deflected $\tau$ leptons produced by $2\,$TeV $D_{s}$ mesons
would be almost $100\,\%$ polarised in the $x$ direction, unlike those
produced by less energetic $D_{s}$ mesons as can be seen in Fig. 6.

The spectra of deflected $\tau$ leptons can be described by the following
expression

$\dfrac{\partial N_{\tau}^{\rm
def}}{\partial\varepsilon_{\tau}}=\int\limits_{0}^{\varepsilon_{\rm max}}{\rm
d}E_{D}\,\dfrac{\partial N_{D}}{\partial E_{D}}\ \eta\,(L_{\rm v},E_{D})\
\dfrac{\partial N_{\tau}}{\partial\varepsilon_{\tau}}(E_{D})\ \eta_{\,\rm
coll}\left(E_{D},\varepsilon_{\tau}\right)\ \eta_{\,\rm
chan}\,(\varepsilon_{\tau}),$ (29)

where $\partial N_{D}/\partial E_{D}$ is the effective spectra of deflected
$D_{s}$ mesons (accounting for the purity), $\eta(L_{\rm v},E_{D})$ is the
conversion efficiency of $D_{s}^{+}$ into $\tau^{+}$ given in Eq. (47),
$\partial N_{\tau}/\partial\varepsilon_{\tau}(E_{D})$ is the spectra of $\tau$
leptons produced by $D_{s}$ decays as a function of the $D_{s}$ energy,
$\eta_{\,\rm chan}(\varepsilon_{\tau})$ is the channeling efficiency
FominThesis  taking into account the $\tau$ decay, $\eta_{\,\rm
coll}(E_{D},\varepsilon_{\tau})=\left(\phi_{\rm max}-\phi_{\rm
min}\right)/\pi$ is the fraction of $\tau$ leptons collimated when entering
the crystal, with

$\sin\phi=\dfrac{\theta_{x}}{\theta(E_{D},\varepsilon_{\tau})},\quad\sin\phi_{\rm
max}=\dfrac{\theta_{v}\,+\theta_{\rm
acc}(\varepsilon_{\tau})}{\theta(E_{D},\varepsilon_{\tau})},\quad\sin\phi_{\rm
min}=\dfrac{\theta_{v}\,-\theta_{\rm
acc}(\varepsilon_{\tau})}{\theta(E_{D},\varepsilon_{\tau})}.$ (30)

The expressions for the projections of polarisation on the $x$ and $z$ axes in
the relativistic case are

${\mathds{P}}_{x}(E_{D},\varepsilon_{\tau},\phi)={\mathds{P}}_{\perp}(E_{D},\varepsilon_{\tau})\,\sin\phi,\qquad{\mathds{P}}_{z}(E_{D},\varepsilon_{\tau})\approx{\mathds{P}}_{\parallel}(E_{D},\varepsilon_{\tau}).$
(31)

By averaging over the $D_{s}$ spectra and taking into account the collimation
due to channeling, we get the expression of the $x$ and $z$ components of
polarisation of deflected $\tau$ leptons as a function of energy,

${\mathds{P}}_{z}(\varepsilon_{\tau})=\dfrac{1}{\partial N_{\tau}^{\rm
def}/\partial\varepsilon_{\tau}}\int{\rm d}E_{D}\ \dfrac{\partial
N_{D}}{\partial E_{D}}\ \dfrac{\partial N_{\tau}^{\rm
def}}{\partial\varepsilon_{\tau}}(E_{D})\
{\mathds{P}}_{z}(E_{D},\varepsilon_{\tau}),$ (32)

${\mathds{P}}_{x}(\varepsilon_{\tau})=\dfrac{1}{\partial N_{\tau}^{\rm
def}/\partial\varepsilon_{\tau}}\int{\rm d}E_{D}\,\dfrac{\partial
N_{D}}{\partial E_{D}}\,\dfrac{\partial
N_{\tau}}{\partial\varepsilon_{\tau}}(E_{D})\,{\mathds{P}}_{\perp}(E_{D},\varepsilon_{\tau})\,\dfrac{1}{\pi}\int\limits_{\phi_{min}(\varepsilon_{\tau})}^{\phi_{max}(\varepsilon_{\tau})}\,\sin\phi\
{\rm d}\phi.$ (33)

The integration over $\phi$ in Eq. (33) means averaging over the direction of
particles in crystal plane and over acceptance angle in the direction
perpendicular to this plane. In Eq. (32) this integration is trivial, and is
simply the term $\eta_{\rm coll}$ from Eq. (29). Due to the symmetry of the
setup the average of $y$-component of the initial polarisation is zero.

Figure 7:  Polarisation and spectra of deflected $\tau$ leptons for (left)
$\theta_{\rm v}=40\,\mu{\rm rad}$ and (right) $\theta_{\rm v}=80\,\mu{\rm
rad}$. The spectra are normalised to one initial proton. The red and green
dashed lines represents the $x$ and $z$ components of the averaged $\tau$
polarisation.

In Fig. 7 we present the spectra given in Eq. (29) and the polarisation as a
function of energy given in Eqs. (32) and (33) of deflected by 12 cm germanium
crystal $\tau$ leptons for two orientations of the second crystal, (left)
$\theta_{\rm v}=40\,\mu{\rm rad}$, $R=9\,$m, and (right) $\theta_{\rm
v}=80\,\mu{\rm rad}$, $R=7\,$m.

In the second case, the number of deflected $\tau$ leptons is smaller and the
spectra is softer, but the degree of polarisation is significantly greater.

Figure 8:  Setup efficiencies (left) for MDM measurement $\eta_{\rm setup}$
and (right) for EDM measurement $\eta^{(f)}_{\rm\,setup}$ as a function of the
second crystal length, $L$, for various values of the angle $\theta_{\rm v}$
and optimal radii of curvature $R$ (listed on the right).

By varying the orientation and dimensions of the second crystal, we find the
maximal setup efficiency for MDM measurement, $\eta_{\rm setup}\sim 1.1\times
10^{-10}$ per incident proton (see Fig. 8, left). For this case, the second
crystal parameters are the following: material — germanium, orientation angle
$\theta_{\rm v}=80\,\mu{\rm rad}$, length $L=10\,{\rm cm}$ and bending radius
$R=7\,{\rm m}$ and deflection angle $\Theta\approx 14\,{\rm mrad}$. If the
crystal length is limited to $6\,{\rm cm}$, it is better to increase the
orientation angle to $\theta_{\rm v}=100\,\mu{\rm rad}$, so that the optimal
bending radius will decrease to $R=6\,{\rm m}$, and hence the deflection angle
would be $\Theta=10\,{\rm mrad}$. In such case, the setup efficiency would be
$\eta_{\rm setup}\sim 0.9\times 10^{-10}$.

In the EDM measurement, comparing to the MDM measurement, the deflection angle
is more important than Lorentz factor and the number of deflected particles
(see Eq. (26)). For this reason the optimal bending radius is smaller (see
Fig. 8, right) and the optimal crystal length is greater. Dotted line in Fig.
8 (right) shows the EDM measurement efficiency when the crystal parameters are
optimised for the MDM measurement. Optimising the setup for EDM measurement
can gain a factor of two to the efficiency.

###  III.3  Preliminary study of the background from other decay channels

In the current study we consider the $\tau$ decay into three charged pions,

$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\,\overline{\nu}_{\tau}.$ (34)

The advantage of this decay channel is that the secondary vertex can be
reconstructed and the final $\tau$ polarisation can be mainly reconstructed
from the momenta of pions, which is also very important to maximise the value
of $S$. We now consider possible processes producing $\pi^{+}\pi^{+}\pi^{-}$
secondary vertices at the end of the second crystal.

Figure 9: Invariant mass distribution of $\pi^{+}\pi^{+}\pi^{-}$ combinations
from (red line) the main decay chain, $D_{s}^{+}\to\tau^{+}\,\nu_{\tau},\
\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\,\overline{\nu}_{\tau}$, and for the
background decay chains with $\pi^{+}\pi^{+}\pi^{-}$ in the final state and
listed at the bottom, using (left) a single-crystal setup and (right) a
double-crystal setup.

The double-crystal setup plays an important role for the background
separation. Due to a small misalignment of the second crystal $\theta_{\rm
v}\sim 100\,\mu{\rm rad}$, the particle deflected by the first crystal cannot
be involved into the channeling regime in the second one. This removes all
charged $D$ mesons decaying directly into $\pi^{+}\pi^{+}\pi^{-}$, as shown in
Fig. 9. In addition, the planar channeling in a long crystal is possible only
for positively charged particles. As a consequence, $\pi^{+}\pi^{+}\pi^{-}$
events can only come from the chain $pp\to X^{+}...\to
Y^{+}...\to\pi^{+}\pi^{+}\pi^{-}...\,$.

To study the background from these decay channels, we used the Pythia 8.2
generator. Figure 9 shows the distribution over the invariant mass of the
three reconstructed pions for the main decay channels, listed on the figures.
The simulation shows that the main source of background is the decay
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\,\overline{\nu}_{\tau}$. With a
requirement on the invariant mass $\geq 1\,{\rm GeV}$, we can reduce its
contamination to $25\,\%$. For the present study, we suppose that the $\tau$
polarisation from the $\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ decay channel is zero.
This is a pessimistic assumption since the presence of $\pi^{0}$ should not
completely break the correlation between the $\tau$ polarisation and the
angular distribution of $\pi^{+}\pi^{+}\pi^{-}$. In addition, this background
could be further reduced by the identification of the $\pi^{0}$ meson. Other
sources of background come from $B$-meson decays, but as shown in Fig. 9, they
are negligible. A detailed study could be performed to evaluate the background
coming from accidental reconstruction of three pion vertices.

###  III.4  Results of the sensitivity studies

In this section, we study the efficiency of the double-crystal setup proposed
in the previous section and determine the optimal parameters of the setup for
a $7\,{\rm TeV}$ proton beam. They are:

  * •

Target: tungsten with $L_{\rm tar}=1\,$cm;

  * •

Crystal 1: germanium with $L_{D}=3\,{\rm cm}$, $R_{D}=10\,{\rm m}$ and
$\theta_{p}=100\,\mu{\rm rad}$;

  * •

Crystal 2: germanium with $L=10\,{\rm cm}$, $R=7\,{\rm m}$ and $\theta_{\rm
v}=80\,\mu{\rm rad}$, at $L_{\rm v}=10\,$cm.

Table 2: Efficiency of the measurement: current (by process) and total (per incident proton). |  | double crystal setup | single crystal setup
---|---|---|---
Place | process (factors) |  current |  total |  current |  total
target |  $~{}p\to D_{s}^{+}$, $D_{s}^{+}$ decay | $1.1\cdot 10^{-4}$ | $1.1\cdot 10^{-4}$ | $1.1\cdot 10^{-4}$ | $1.1\cdot 10^{-4}$
crystal 1 |  $~{}D_{s}^{+}$ collimation | $0.8\cdot 10^{-2}$ | $0.9\cdot 10^{-6}$ | $-$ | $-$
crystal 1 |  $~{}D_{s}^{+}$ deflection | $1.3\cdot 10^{-1}$ | $1.1\cdot 10^{-7}$ | $-$ | $-$
void | $D_{s}^{+}\to\tau^{+}$ | $1.2\cdot 10^{-2}$ | $1.4\cdot 10^{-9}$ | $0.5\cdot 10^{-3}$ | $0.5\cdot 10^{-7}$
crystal 2 |  $\tau^{+}$ collimation | $0.5\cdot 10^{-1}$ | $~{}0.7\cdot 10^{-10}$ | $1.7\cdot 10^{-2}$ | $0.9\cdot 10^{-9}$
crystal 2 |  $\tau^{+}$ deflection | $0.3\cdot 10^{-1}$ | $~{}2.2\cdot 10^{-12}$ | $0.3\cdot 10^{-1}$ | $~{}0.3\cdot 10^{-12}$
detector | $\eta_{\rm det}\times Br$ | $0.4\cdot 10^{-1}$ | $~{}1.0\cdot 10^{-13}$ | $0.4\cdot 10^{-1}$ | $~{}1.3\cdot 10^{-12}$
reconstr. | $\eta_{\rm\,MDM}\sim\langle{\mathds{P}}^{2}\,\gamma^{2}\rangle\,\theta^{2}\,S^{2}$ |  $\sim\,$8 | $\sim 0.7\cdot 10^{-12}$ |  $\sim\,$0.5 | $\sim 0.7\cdot 10^{-12}$
reconstr. | $\eta_{\rm\,EDM}\sim\langle{\mathds{P}}^{2}\,\gamma^{2}\rangle\,\theta^{4}\,S^{2}$ |  $\sim\,$$1.5\cdot 10^{-3}$ |  $\sim\,$$1.5\cdot 10^{-16}$ | $\sim 1.1\cdot 10^{-4}$ | $\sim 1.5\cdot 10^{-16}$

In this study, we did not look in details at the detection system. We consider
the analysis of the decay channel
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\overline{\nu}_{\tau}$ ($Br=0.0899$ PDG:2018
) and suppose a detection efficiency of $\eta_{\rm det}=50\,\%$. In Table 2 we
present (column 3) the efficiencies of each process, starting from $D_{s}^{+}$
production in the target, up to reconstruction of MDM and EDM of $\tau$
lepton, and (column 4) the total efficiencies, i.e. the number of $D_{s}^{+}$
or $\tau$ after each stage of double crystal setup referred to the total
number of impinging protons on target. Last two rows in the table present
efficiencies of MDM and EDM reconstruction. Using latter, we can rewrite Eqs.
(24) and (27) in a following way

$\Delta a_{\tau}=\frac{1}{\sqrt{N_{\rm POT}\,\eta_{\rm
MDM}}}\qquad\qquad\Delta d_{\tau}=\frac{\mu_{\rm B}}{2\,\sqrt{N_{\rm
POT}\,\eta_{\rm EDM}}}$ (35)

where $\Delta d_{\tau}$ is the absolute statistical error of the measured EDM
in units [e cm] and $\mu_{\rm B}$ is the Bohr magneton.

We also estimate the efficiency of the single crystal setup presented in Fig.
3 (see columns 5 and 6 in Table 2). The numbers in the table correspond to the
following properties of the single crystal setup: germanium crystal with
$L=10\,{\rm cm}$, $R=7\,{\rm m}$ and $\theta_{\rm p}=100\,\mu{\rm rad}$ at
$L_{\rm v}=40\,$cm, from $1\,$cm tungsten target. We found that in order to
reach the reconstruction efficiency of double crystal setup the conversion
length should not be greater than $L_{\rm v}=40\,$cm. On the other hand, such
length is too short to reconstruct the $D_{s}^{+}\to\tau^{+}\,\nu_{\tau}$
vertex, i.e. to verify that the signal comes from the $\tau$ lepton produced
before the crystal, but not inside or after the one, or from other particle
with a similar final state (see Fig. 9). Due to these reasons we consider the
double crystal setup to be more efficient. Note, that due to a greater average
polarisation, the double crystal setup has a better reconstruction efficiency
of MDM and EDM (referred to one deflected $\tau$ lepton).

Figure 10: Absolute statistical error (left) of the anomalous MDM, (right) of
EDM of the $\tau$ lepton.

Figure 10 (left) shows the absolute statistical error of the measured
anomalous MDM of the $\tau$ lepton using Eq. (24) as a function of the total
number of protons on target. The blue lines show the sensitivity of the
present analysis by varying in particular the value of $S$ in the range
$0.2<S<0.58$. The maximal value corresponds to the limit on the accuracy, in
case all observables are fully reconstructed. The green lines show the limits
obtained by the DELPHI collaboration Abdallah:2003  and expected for the
BELLE$\ 2$ experiment Eidelman:2016 . The red line represents the value of the
$\tau$ anomalous MDM predicted by the SM Eidelman:2007 . Further detailed
studies will be performed to improve the current efficiency. In particular,
detailed background studies could allow us to use one prong $\tau$ decays.

For EDM measurement this setup is not very efficient. Even when optimised for
EDM measurement, the current experimental accuracy on EDM Inami:2002  could be
achieved only after about $10^{19}\,$POT, which is probably not realistic (see
Fig. 10 (right)).

##  IV  Conclusions

In this paper, we propose a method of a direct measurement of the anomalous
magnetic dipole moment (MDM) of the $\tau$ lepton by studying its spin
precession using a double-crystal setup at the LHC. Our study shows that with
this proposal, one can reach the present accuracy on the $\tau$ MDM using
$10^{16}$ protons on target. A total of $10^{18}$ protons on target are needed
to reach an accuracy equivalent to the Standard Model value. The measurement
of EDM is even more challenging and a further order of magnitude of POT is
needed to reach the present experimental accuracy. The benefit of this method
is that there is no assumption on the value of the $\tau$ electric dipole
moment. This study opens up the possibility of doing a single dedicated
experiment to measure both the MDM of charmed baryons Baryshevsky:2016 ;
Fomin:2017 ; Botella:2016 ; Bagli:2017 ; Baryshevsky:2017 ; FominThesis  and
of $\tau$ lepton at the LHC.

##  Appendix A  Polarisation of $\tau$ leptons in $D_{s}$ decays

We consider the decay of the $D_{s}$ meson

$D_{s}^{+}(p)\to\tau^{+}(k)+\nu_{\tau}(q),$ (36)

with $p=k+q$, and find the polarisation vector of the $\tau$ lepton. The four-
momentum of the $D_{s}$ meson is $p^{\mu}=(E_{D},\,\vec{p}_{D})$. We further
assume that the neutrino is massless.

To describe the polarisation of a fermion in its rest frame, one introduces
the vector $\vec{\xi}$ (${|\vec{\xi}|\leq 1}$). In the frame in which fermion
moves with the four-momentum $k^{\mu}=(\varepsilon_{\tau},\,\vec{k}_{\tau})$,
the polarisation four-vector is Beresteckii:1982

$a^{\mu}=\left(\frac{\vec{k}_{\tau}\,\vec{\xi}}{m_{\tau}},\,\,\vec{\xi}+\frac{\vec{k}_{\tau}\,(\vec{k}_{\tau}\,\vec{\xi})}{m_{\tau}(m_{\tau}+\varepsilon_{\tau})}\right),$
(37)

with the conditions $k\cdot a=0$ and $a^{2}=-\vec{\xi}^{2}$.

We can write matrix element of the $D_{s}^{+}$ decay as

${\cal
M}=\frac{g^{2}}{8M_{W}^{2}}f_{D}\,V^{*}_{cs}\,p^{\mu}\,\bar{u}(q)\gamma_{\mu}(1-\gamma_{5})v(k),$
(38)

where $f_{D}$ is the weak-decay constant of the
$D_{s}^{+}\to\tau^{+}\nu_{\tau}$ decay, $M_{W}$ is the mass of the $W$ boson,
$V_{cs}$ is the element of the Cabibbo-Kobayashi-Maskawa matrix,
$g=e/\sin\theta_{W}$, $e$ is the positron charge and $\theta_{W}$ is the weak
mixing angle.

Using the spin-density matrix of the $\tau^{+}$ lepton,

$v(k)\bar{v}(k)=\frac{1}{2}(k/-m_{\tau})\,(1+\gamma_{5}a/),$ (39)

one obtains the probability to find the $\tau^{+}$ lepton with polarisation
$a^{\mu}$. It is proportional to

$|{\cal
M}|^{2}=\frac{g^{4}m_{\tau}^{2}}{16M_{W}^{4}}|V_{cs}|^{2}f^{2}_{D}\,(k\cdot
q-m_{\tau}\,{p\cdot a}),$ (40)

with $k\cdot q=(M_{D}^{2}-m_{\tau}^{2})/2$.

Equation (40) allows one to find the $\tau$ polarisation vector
$\vec{\mathds{P}}$ which arises in the decay process. For this we write

$(k\cdot q-m_{\tau}\,{p\cdot a})\equiv k\cdot
q\,\left(1+\vec{\xi}\cdot\vec{\mathds{P}}\right),$ (41)

$\vec{\mathds{P}}=\frac{1}{k\cdot
q}\left[m_{\tau}\vec{p}_{D}\,-\,E_{D}\vec{k}_{\tau}\,+\,\frac{\vec{k}_{\tau}\,(\vec{k}_{\tau}\cdot\vec{p}_{D})}{m_{\tau}+\varepsilon_{\tau}}\right].$
(42)

The corresponding polarisation four-vector in an arbitrary frame has a form
similar to Eq. (37), namely

$S^{\mu}=\left(\frac{\vec{k}_{\tau}\cdot\vec{\mathds{P}}}{m_{\tau}},\,\,\vec{\mathds{P}}+\frac{\vec{k}_{\tau}\,(\vec{k}_{\tau}\cdot\vec{\mathds{P}})}{m_{\tau}(m_{\tau}+\varepsilon_{\tau})}\right),$
(43)

where $k\cdot S=0$ and $S^{2}=-\vec{\mathds{P}}^{2}$. One can also write
$S^{\mu}$ explicitly in terms of the four-momenta of the $D_{s}$ meson and the
$\tau$ lepton, using Eqs. (42) and (43),

$S^{\mu}=\frac{m_{\tau}}{k\cdot q}\,\bigl{(}p^{\mu}\,-\,k^{\mu}\,\frac{k\cdot
p}{m_{\tau}^{2}}\,\bigr{)},$ (44)

where $S^{2}=-1$. It follows that $\vec{\mathds{P}}^{2}=1$, so that the $\tau$
lepton is $100\,\%$ polarised.

##  Appendix B  Optimal length for $D_{s}$ conversion to $\tau$ leptons

We consider $\tau$ leptons produced in the decay of $D_{s}$ mesons and the
consequent $\tau$ decay. Thus, there are two competing processes, production
and decay of the $\tau$ lepton.

The probability of $\tau$ production in the process $D_{s}^{+}\to\tau^{+}\
\nu_{\tau}$ at the $D_{s}$ flight length $x$ is

$N_{\rm prod}={\cal B}_{i}\left(1-e^{-\,x\,/\,T_{D}}\right),$ (45)

where ${\cal B}_{i}=\Gamma_{i}/\Gamma\approx 0.055$ PDG:2018  is the branching
ratio of the current process, $T_{D}=c\,\tau_{D}\gamma_{D}$ is the decay
length of a boosted $D_{s}$ meson ($c\,\tau_{D}\approx 150\,\mu{\rm m}$). The
probability of $\tau$ decay after a flight length of $L_{\rm v}-x$ is

$N_{\rm dec}=e^{-\,(L_{\rm v}-x)\,/\,T_{\tau}},$ (46)

where $T_{\tau}=c\,\tau_{\tau}\,\gamma_{\tau}$ is the decay length of a
boosted $\tau$ lepton ($c\,\tau_{\tau}\approx 87\,\mu{\rm m}$).

The conversion efficiency, that is the ratio of the number of $\tau$ leptons
not decayed after the flight length $L_{\rm v}$, to the initial number of
$D_{s}$ mesons of energy $E_{D}$ is

$\eta(L_{\rm
v},E_{D})=\dfrac{\int\limits_{\varepsilon_{\tau}^{min}}^{\varepsilon_{\tau}^{max}}{\rm
d}\varepsilon_{\tau}\,\dfrac{\partial N_{\tau}}{\partial\varepsilon_{\tau}}\
\,\int\limits_{0}^{L_{\rm v}}{\rm d}x\,\dfrac{\partial N_{\rm prod}}{\partial
x}\ N_{\rm dec}\,(L_{\rm
v}-x)}{\int\limits_{\varepsilon_{\tau}^{min}}^{\varepsilon_{\tau}^{max}}{\rm
d}\varepsilon_{\tau}\,\dfrac{\partial N_{\tau}}{\partial\varepsilon_{\tau}}},$
(47)

where $\partial N_{\tau}\,/\,\partial\varepsilon_{\tau}$ is the $\tau$
spectrum. In a two-body decay, the $\tau$ spectrum is defined by the energy of
the $D_{s}$ meson, but in our experimental setup, $\tau$ leptons are later
captured into a channeling regime, and hence are collimated in angle and
energy in a quite complicated way (see Fig. 6,7). Thus, in this case it is
useful to find the extreme values of the conversion efficiency $\eta(L_{\rm
v},E_{D})$ that correspond to the extreme values of the $\tau$ energy.

The energy range of $\tau$ leptons produced in a two-body decay of $D_{s}$
mesons in the relativistic case is

$\varepsilon_{\tau}\in E_{D}\left[\,\frac{m_{\tau}^{2}}{M_{D}^{2}}\,,\
1\,\right].$ (48)

Substituting Eqs. (45) and (46) into Eq. (47) and considering a mono-energetic
$\tau$ spectrum of energy $\varepsilon_{\tau}^{*}$, we obtain

$\eta^{*}(L_{\rm v},E_{D},\varepsilon_{\tau}^{*})=Br_{i}\ \frac{e^{\,-L_{\rm
v}\,/\,T_{D}}-e^{\,-L_{\rm v}\,/\,T_{\tau}}}{T_{D}\,/\,T_{\tau}-1}.$ (49)

Figure 11 presents the conversion efficiency as a function of the path length
for three $D_{s}$ energies, listed in the figure. The width of the curves
represents energy spread of the $\tau$ lepton. The upper and lower edges
correspond to upper and lower limits of the $\tau$ energy given in Eq. (48).

Figure 11: $D_{s}^{+}\to\tau^{+}$ conversion efficiency as a function of the
flight length.

The optimal length for the $D_{s}$ conversion to $\tau$ leptons strongly
depends on the $D_{s}$ spectrum. For the energy range $1\,{\rm
TeV}<E_{D}<2\,{\rm TeV}$, the optimal length is around $10\,{\rm cm}$.
Considering conversion lengths $L_{\rm v}>40\,{\rm cm}$ would essentially
reduce the part of the spectra to $E_{D}<2\,$TeV.

## Acknowledgments

This research was partially conducted in the scope of the IDEATE International
Associated Laboratory (LIA). A.Yu.K. and A.S.F. acknowledge partial support by
the National Academy of Sciences of Ukraine (project KPKVK 6541230, and
project no. Ts-3/53-2018) and the Ministry of Education and Science of Ukraine
(project no. 0117U004866).

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# Transfer Learning-Enhanced Instantaneous Multi-Person Indoor Localization by
CSI
††thanks: This work was supported by the National Natural Science Foundation
of China (Grant No. 62071192) and the National Key Research and Development
Program of China (Grant No. 2020YFB1806904).

1st Zhiyuan He, 2nd Ke Deng, 3rd Jiangchao Gong, 4th Yi Zhou, 5th Desheng Wang
Wuhan National Laboratory for Optoelectronics
Huazhong University of Science and Technology
Wuhan, China
1st<EMAIL_ADDRESS>2nd<EMAIL_ADDRESS>3rd
<EMAIL_ADDRESS>
4th<EMAIL_ADDRESS>5th<EMAIL_ADDRESS>
###### Abstract

Passive indoor localization, integral to smart buildings, emergency response,
and indoor navigation, has traditionally been limited by a focus on single-
target localization and reliance on multi-packet CSI. We introduce a novel
Multi-target loss, notably enhancing multi-person localization. Utilizing this
loss function, our instantaneous CSI-ResNet achieves an impressive $99.21\%$
accuracy at $0.6m$ precision with single-timestamp CSI. A preprocessing
algorithm is implemented to counteract WiFi-induced variability, thereby
augmenting robustness. Furthermore, we incorporate Nuclear Norm-Based Transfer
Pre-Training, ensuring adaptability in diverse environments, which provides a
new paradigm for indoor multi-person localization. Additionally, we have
developed an extensive dataset, surpassing existing ones in scope and
diversity, to underscore the efficacy of our method and facilitate future
fingerprint-based localization research.

###### Index Terms:

Wireless sensing, Channel state information(CSI), WIFI localization,
Fingerprinting, Signal Processing

##  I Introduction

Amidst the technological evolution from 5G towards 6G, passive indoor
localization technologies independent of specialized portable equipment have
attracted substantial attention within the academic and industrial spheres.
These technologies underpin a wide array of location-based services and
applications, extending to elderly monitoring[1], intrusion detection[2], and
real-time human tracking[3]. Despite certain advancements made by vision-based
methods[4] in this domain, they confront dual challenges posed by lighting
conditions and privacy concerns. Similarly, prevalent infrared positioning
suffers from accuracy issues influenced by environmental temperatures[13]. In
contrast, traditional radar-based systems are often prohibitive in cost for
widespread deployment[5]. In this context, WiFi-based positioning systems,
leveraging their extensive availability and cost-effectiveness, have become a
preferred solution to address these challenges[6].

Although some scholars concentrate on utilizing Angle of Arrival (AoA) and
Time of Flight (ToF) for localization, the resolution of these techniques is
often constrained due to the limited bandwidth and antenna count of commercial
WiFi devices[5, 7], thereby narrowing their practical applicability. To
surmount these challenges, recent scholarly efforts have pivoted towards WiFi
fingerprint-based positioning methods[9]. In earlier phases of research, the
emphasis was on utilizing Received Signal Strength (RSS) for fingerprint
characterization. However, in Non-Line-of-Sight (NLOS) conditions, RSS
demonstrates increased stochasticity due to complex interactions involving
reflections, diffractions, and refractions[8]. In contrast, phase measurements
exhibit periodic variations correlated with propagation distances, thus
providing a more robust metric. Additionally, the phase information within
wireless links encapsulates pertinent data in channel responses, making the
combined use of phase and amplitude information a richer and more reliable
approach for localization purposes[10].

In this research, we focus on developing an instantaneous, passive indoor
localization methodology that integrates both amplitude and phase information.
Employing 802.11ac devices, which provide rich carrier and antenna
dimensionality, we collect Channel State Information (CSI). A meticulous
analysis of the amplitude-phase characteristics of CSI is conducted for
efficient data preprocessing, followed by constructing an effective network
for localization, which is capable of accurately locating multi-people. The
key contributions of this study are outlined as follows:

  * •

We reconceptualize the localization task as a multi-label prediction problem,
introducing an advanced CSI preprocessing strategy to mitigate errors from
WiFi restarts, thereby elevating precision.

  * •

we introduce a CSI-ResNet for localization, leveraging single-timestamp CSI
inputs for enhanced accuracy and efficiency in real-time deployment. The
network preserves CSI integrity, achieving $99.21\%$ accuracy with a $0.6m$
precision. A nuclear norm-based transfer learning approach is implemented,
ensuring adaptability across varied scenarios with minimal initial data.

  * •

A robust dataset was curated, comprising four experimental scenarios and
multi-target data from 60 unique locations. This dataset, excelling in scale
and label diversity, significantly advances the groundwork for intricate CSI-
based location prediction analyses.

This paper is structured as follows: Section II reviews related research and
the study’s rationale. Section III describes the localization system and CSI
data preprocessing, including the CSI-ResNet and transfer learning approach.
Section IV presents experimental methodologies and analyses. Section V
concludes with a summary of findings and contributions.

##  II Related work

###  II-A Indoor Localization

In the realm of WiFi-based indoor localization, two primary technologies
prevail: model-based and fingerprint-based localization[14]. Model-based
approaches typically employ geometric measurements to determine the position
relative to several known Access Points (APs). This category includes methods
like centroid localization and techniques based on Angle of Arrival (AoA) and
Time of Arrival (ToA)[15, 16]. However, these approaches generally necessitate
users to carry communication devices capable of connecting with known APs,
which may not always be feasible in certain indoor scenarios.

In contrast to model-based approaches, fingerprint-based methods in WiFi
indoor localization incorporate advanced neural network algorithms. Early
studies extensively used RSS due to its ease of access in hardware. Despite
initial advancements like the Radar system using RSS, the fluctuating nature
of RSS values, influenced by multipath fading and shadowing, limits the
precision of RSS-based localization, impacting the overall effectiveness of
these systems[17, 18, 19].

Compared to traditional RSS methods, CSI exhibits significant advantages in
indoor localization. CSI provides richer, subcarrier-level information crucial
for capturing signal propagation characteristics in complex indoor scenarios.
Wu et al.[20] first applied CSI to fine-grained indoor positioning,
establishing a model linking CSI with physical distances and achieving precise
positioning through trilateration. Furthermore, Zhou et al. expanded this
domain with the HRC algorithm, combining RSS and CSI granularity to create a
comprehensive fingerprint database to reduce data dimensions for efficient
position estimation[21].

Zhang et al.[22] were pioneers in applying Long Short-Term Memory (LSTM)
networks to CSI-based indoor localization, leveraging the temporal features of
CSI for precise positioning. Following this, Wang et al.[10] enhanced the
efficiency and accuracy of utilizing temporal features by integrating
Convolutional Neural Networks (CNNs) with Bidirectional Gated Recurrent Units
(BGRUs). Additionally, Tian et al.[23] employed Bidirectional LSTM (BLSTM)
networks for multidimensional recognition of location and activities, thereby
increasing the precision of localization.

###  II-B Motivation

Current indoor localization research, predominantly single-user oriented,
neglects multi-user environment complexities, which leads to accuracy issues
in conventional systems due to user characteristic similarities. Furthermore,
calibrating phase offsets in WiFi RF chains across diverse deployments remains
challenging[24, 25]. Moreover, high-precision positioning algorithms, reliant
on temporal CSI signals, face accuracy challenges due to inherent data
instability, such as packet loss[23]. Additionally, transferability issues are
the key to fingerprinting methods.

Our work integrates an automated phase compensation method with a dual-channel
CSI-ResNet framework, redefining multi-user positioning into a multi-label
prediction task and boosting accuracy. We address temporal CSI instability by
using single-packet CSI, reducing disturbances[23]. Additionally, a nuclear
norm-based pre-training strategy is implemented, enhancing our model’s swift
adaptability and consistent performance in new scenarios.

Figure 1: Proposed localization system architecture.

##  III System model

The proposed localization system is illustrated in Fig. 1. During the offline,
CSI data are collected and preprocessed to create a fingerprint database,
which is then used to train a CSI-ResNet for online multi-person localization.

CSI signals in wireless channels, as outlined in [23], are a blend of static
and dynamic multipath elements. Static components stem from structural
reflections, while dynamic ones arise from human movements. Our study
classifies user characteristics into positional attributes and limb movements.
Fig. 2 demonstrates that while CSI data reliably indicate position, limb
motion reflections are discernible only during activity, emphasizing the
distinct effects of human presence and motion on CSI.

Figure 2: CSI amplitude in two different scenarios. (a) CSI Measured in an
Empty Room. (b) CSI Measured in a Room with Occupants.

Consequently, the received CSI can be mathematically represented as:

$H(f,t)=H_{s}(f,t)+H_{m}(f,t)+H_{n}(f,t)$ (1)

Where $H_{s}(f,t)$ denotes the CSI measurements transmitted by static paths
such as ceiling/wall reflections, termed as static CSI component. $H_{m}(f,t)$
and $H_{n}(f,t)$ represent the CSI measurements transmitted via body
reflection paths and limb reflection paths, respectively. $H(f,t)$ can also be
expressed as:

$H(f,t)=\|H(f,t)\|\cdot e^{i\theta}$ (2)

Where $\|H(f,t)\|$ and $e^{i\theta}$ symbolize the amplitude and phase of the
transmitted signal, respectively. Thus, the received CSI can be rewritten as:

$\begin{split}\|H(f,t)\|\cdot e^{j\theta}=\|H_{s}(f,t)\|\cdot
e^{j\theta_{s}}+\|H_{m}(f,t)\|\cdot e^{j\theta_{m}}\\\ +\|H_{n}(f,t)\|\cdot
e^{j\theta_{n}}\end{split}$ (3)

However, the phase information in CSI is often affected by factors like
carrier frequency offset and sampling frequency offset, leading to significant
errors in CSI phase information. Also, the amplitude information in CSI
changes with device restarts. Hence, the actual measured CSI is:

$\begin{split}\|H_{R}(f,t)\|\cdot e^{j\theta_{R}}=\alpha(\|H_{s}(f,t)\|\cdot
e^{j\theta_{s}}\\\ +\|H_{m}(f,t)\|\cdot e^{j\theta_{m}}+\|H_{n}(f,t)\|\cdot
e^{j\theta_{n}})e^{j\theta_{e}}\end{split}$ (4)

Where $\|H_{R}(f,t)\|$ is the actual received amplitude, $\theta_{R}$ is the
actual received phase, $\alpha$ is the amplitude offset, and $\theta_{e}$ is
the intrinsic phase difference between receiving antennas. Therefore, the
received CSI requires calibration.

###  III-A CSI preprocessing

As delineated in research[24, 25], frequent rebooting of transceivers impacts
oscillator configuration, leading to random phase differences between
receiving antennas. For precise intrinsic phase difference acquisition between
receiving antennas, we followed the connection scheme depicted in Fig. 3 for
transceivers and conducted a series of detailed experimental analyses.

After restarting the transceivers 20 times, the phase differences between the
receiving antennas were meticulously recorded. By aggregating data across all
carriers and time packets, a probability distribution was obtained, as
illustrated in Fig. 4.

Figure 3: Experiment for CSI calibration. (a) Phase Offset Measurement
Experiment Setup. (b) Spliter/combiner. (c) Attenuator. (d) Schematic Diagram
of Phase Offset Measurement Experiment.

Experiments conducted with the RT-AC86U router demonstrate a distinctive
pattern in the phase difference between receiving antennas, which is shown in
Fig. 4.

Figure 4: Distribution of phase difference between Rx 2,3,4 with Rx 1.

Specifically, an integrated analysis of data and fitting the distribution
using the von Mises distribution in the angular domain reveals a stable phase
difference pattern: between antennas 1 and 2, it’s around 0.19 radians
(concentration 27.63), between 1 and 3, -0.87 radians (concentration 19.61),
and between 1 and 4, -1.74 radians (concentration 17.46). These differences
are attributed to inherent antenna properties. Stability is maintained even
after system reboots. Consequently, we adjust the CSI for consistent angular
offsets and normalize amplitude. The calibrated CSI is represented as:

$H_{\text{cali}}(M,N)=\frac{|H_{R(M,N)}(f,t)|\cdot e^{j\theta_{R}}\cdot
e^{j\Delta\theta_{N}}}{\frac{1}{M}\sum_{m=1}^{M}|H_{R(M,N)}(f,t)|}$ (5)

Where $M$ and $N$ denote the carrier and receiving antenna, respectively, and
$\mathrm{\Delta}\theta_{N}$ is the compensating phase for the corresponding
antenna.

To extract static features related to positioning from CSI data and
effectively filter out low-frequency interferences caused by human activities
such as walking and breathing, we implemented a preprocessing strategy on CSI
data. According to the literature[26], the frequency of human walking is
approximately 1.9Hz, and the frequency of breathing ranges between 1 to
1.67Hz. To effectively filter out these low-frequency signals, a high-pass
filter approach was adopted. Specifically, we applied a Fast Fourier Transform
(FFT) to the CSI data in the time domain, followed by the application of a
high-pass filter with a 2Hz cutoff. Subsequently, the data was converted back
to its original time domain using an Inverse Fast Fourier Transform (IFFT).
This process ensures that features extracted from CSI data are focused on
reflecting the spatial information and user position characteristics of the
positioning scenario, thereby excluding potential interferences from human
dynamic activities. The filtered CSI is represented as:

$\text{HPF}_{>2Hz}(H_{\text{cali}}(f,t))=H_{s}(f,t)+H_{m}(f,t)$ (6)

where ${HPF}_{>2Hz}(H_{\text{cali}}(f))$ denotes the high-pass filtered CSI.
Additionally, to further enhance the data’s accuracy and stability, a Hampel
filter was applied to eliminate potential outliers. The preprocessed data
serves as the offline input to the network.

###  III-B CSI-ResNet

CSI analysis reveals that spatial attributes, stable in contrast to the
transient nature of human movement data[23], align closely with the structured
nature of visual data. This similarity underpins the deployment of CSI-ResNet,
an advanced adaptation of the ResNet framework, proficient in pinpointing
human locational indicators within CSI. ResNet, with its residual learning
approach, is particularly effective in spatial feature discernment, overcoming
the vanishing gradient challenge in deep neural networks. Meanwhile, GRU and
LSTM networks, as discussed in[10, 23], are tailored more towards temporal
data, making them less ideal for the consistent spatial traits inherent to
CSI.

Our enhanced ResNet model employs a bifurcated convolutional input layer,
enabling distinct processing of the real and imaginary components of complex
CSI data, which adeptly captures amplitude and phase nuances, reconfiguring
CSI into a dual-layered framework to better analyze spatial characteristics.
By concentrating on the dimensions of transmit-receive antennas, our model
gains heightened sensitivity to phase variances, enriching the extraction of
locational features. Utilizing CSI’s full data range, extending past RSSI,
markedly improves the accuracy of indoor positioning. Additionally, the
integration of deep residual learning within the ResNet framework refines its
capacity for detailed spatial analysis.

Figure 5: Experimental results. (a) Metrics of different algorithms for
single-person positioning. (b) Metrics and epoch of different algorithms for
multi-people positioning. (c) Accuracy Comparison Across Different Transfer
Scenarios.

###  III-C Multi-target loss

To adapt traditional single-person indoor positioning techniques for multi-
person scenarios, a key innovation lies in the optimization of the loss
function. Traditional systems typically employ single-label classification
methods, identifying one target category from $N$ potential classes. The
conventional loss function integrates softmax and cross-entropy, expressed as:

$L=-\log\frac{e^{s_{p}}}{\sum_{j=1}^{N}e^{s_{j}}}$ (7)

Here, $s_{p}$ is the score of the target category, while $s_{j}$ represents
the scores of all candidate categories. This method aims to ensure the highest
score for the target category among all classes. However, multi-person indoor
localization faces a unique challenge: the discrete nature of individuals in
the environment leads neural networks to predict ’empty’ scenarios, i.e., all
potential target locations as unoccupied. This is primarily due to the
prevalence of non-target (unoccupied) samples over target (occupied) samples
in training data, causing a bias towards non-target predictions.

In response to the inherent limitations of traditional single-target
localization systems in multi-user scenarios, our research innovates by
implementing a multi-label classification paradigm. This allows individual
samples to be categorized into multiple classes. A novel loss function, termed
”Multi-target loss,” is designed to ensure that the score of each target
category surpasses the score of any non-target category. This methodology
effectively recalibrates the predictive balance between target and non-target
categories, crucial for accurately capturing sparsely distributed multi-person
location data. The loss function is mathematically expressed as:

$\begin{split}L_{Multi}=\sum_{i\in P}\max(0,\gamma-s_{i})+\sum_{j\in
N}\max(0,s_{j}-\gamma)\\\ \approx\sum_{i\in P}\log(1+e^{\gamma-
s_{i}})+\sum_{j\in N}\log(1+e^{s_{j}-\gamma})\end{split}$ (8)

Where $P$ and $N$ denote the sets of positive and negative classes,
respectively. This innovative approach not only enhances the predictive
accuracy for target positions but also diminishes the likelihood of falsely
predicting unoccupied states, thereby significantly improving the robustness
and precision of multi-person localization. The parameter $\gamma$ acts as a
critical threshold delineating presence or absence at specific locations,
enabling the network to discern between target and non-target positions more
sensitively. This advanced loss function enables our model to learn positional
information in multi-person scenarios effectively, extending its capabilities
beyond predicting a singular most likely position to identifying multiple
potential occupant locations, thus substantially enhancing the accuracy and
resilience of localization.

Furthermore, to address the challenge of evaluating model performance in
multi-target localization scenarios, where predictions may encompass several
location labels, we introduce SubACC[32] as an assessment metric. SubACC is a
rigorously exacting measure, typically employed to evaluate a model’s
proficiency in capturing interdependencies among labels. Its formulation is as
follows:

$\text{SubACC}=\frac{1}{N}\sum_{n=1}^{N}I(\hat{y}_{n}=y_{n})$ (9)

Where $y_{n}$ represents the actual set of position labels for the nth
prediction, and $\hat{y}_{n}$ is the corresponding predicted set. The
indicator function $I$ yields a value of 1 if the predicted set $\hat{y}_{n}$
precisely aligns with the true set $y_{n}$ and 0 otherwise. This metric
stringently evaluates the congruence between predicted and actual positional
datasets, providing a comprehensive assessment of the model’s performance in
multi-label prediction scenarios.

###  III-D Nuclear Norm-Based Transfer Pre-Training

To enhance robustness and generalizability in diverse indoor localization
scenarios through deep CSI data analysis, we implement a novel pre-training
paradigm. At the heart of this methodology is the incorporation of nuclear
norm regularization into the pre-training loss [35], which is:

$L_{Pre}=L_{Multi}+{\lambda\parallel W\parallel}_{\ast}$ (10)

where $L_{Multi}$ is the multi-target loss mentioned in Section III-C,
${\parallel W\parallel}_{\ast}$ represents the nuclear norm of the network
weight matrix $W$, and $\lambda$ is the regularization coefficient controlling
the influence of the nuclear norm. In multi-person indoor localization, the
model’s effectiveness hinges on precise identification and differentiation of
individual positions. The nuclear norm, a robust tool for low-rank
representation, encourages our network to learn streamlined, low-rank
features, thus highlighting key localization attributes. By guiding the
network towards low-rank solutions, the model focuses on critical features for
accurate multi-person position prediction.

On the other hand, the benefits of low-rank representation in adapting to new
scenarios and generalizing are particularly significant in multi-person
localization. Simplifying the learned feature space, our model more accurately
captures and differentiates human positions, relying on essential signal
characteristics. This approach not only accelerates the model’s adaptation to
new environments but also enhances its versatility in diverse settings, like
different office layouts and public spaces. The model’s ability to swiftly
adapt and optimize using low-rank features streamlines deployment, making it
highly practical for dynamic real-world applications.

##  IV Experiment

Our experimental setup encompassed various scenarios: a university lab (Fig.
6(a), $21m^{2}$), an outdoor area for model validation (Fig. 6(b), $40m^{2}$),
and two conference rooms (Fig. 6(c)-(d), $35m^{2}$ each). A pair of Asus RT-
AC86U routers, modified by Nexmon CSI tool [33] and controlled via a wired
Ubuntu 16.04 host, served as transceivers, situated 1.2 meters high and 8
meters apart. These routers operated on an 80MHz bandwidth at $5.2GHz$, with
each featuring a four-antenna array (9 dB each, 3 cm spacing). Data collection
was performed at $50Hz$, collecting 550 packets per $0.6m$ spaced reference
point, aligning with indoor user spatial considerations [34].

Figure 6: Four experimental scenarios. (a) Laboratory. (b) Rooftop. (c)
Meeting room 1. (d) Meeting room 2.

Proposed CSI-ResNet was evaluated against six state-of-the-art indoor
localization algorithms: Random Forest [29], MLP [30], VGG [31], XGBoost [28],
KNN [27], and BiLSTM [23]. For single-target detection, we used accuracy and
Macro F1 score, while SubACC was the metric for multi-person detection. These
assessments were based on our extensive dataset, with results presented in
Fig. 5.

In single-person localization, CSI-ResNet excelled with a $99.14\%$ accuracy
and Macro F1 score, surpassing RF and KNN, as Fig. 5(a) shows. It marginally
outperformed other deep learning models like VGG and BiLSTM. XGBoost, despite
its potential, fell short in multi-person scenarios due to its lack of multi-
label processing. For multi-person localization, CSI-ResNet achieved a
$99.21\%$ SubACC, outshining others with just 5 epochs for optimal
performance, indicating its rapid deployment efficacy, as depicted in Fig.
5(b). Comparatively, in scenarios utilizing CSI, phase, and RSSI, Fig. 5(b)
indicates that while RSSI and phase systems delivered notable results, the
CSI-based system reached near $100\%$ accuracy, highlighting CSI’s superiority
in providing detailed indoor localization data.

Furthermore, CSI-ResNet’s transferability was assessed by applying it as a
pre-trained model in the scenarios of Fig. 6(b)-(d), initially trained in Fig.
6(a) scenario, using only $1\%$ of the original dataset in new scenarios for
initialization. Three transfer training approaches were compared: raw-data-
based transfer, pre-trained transfer, and Nuclear Norm-based transfer.

In Fig. 5(c), the results reveal a clear distinction in transfer methods for
novel scenarios. Training with raw data fell short, with accuracy under $50\%$
in every scenario, demonstrating incompetence at localization tasks. Pre-
trained transfer approach fared better, peaking at $89.22\%$ in a similar
setting but dropping to $76.02\%$ in the diverse scenario. However, the
Nuclear Norm-based method consistently surpassed $85\%$ accuracy across
different scenarios, demonstrating its superior adaptability and
transferability in different scenarios. This method’s effectiveness in high
localization accuracy, only requiring merely $1\%$ of the dataset for initial
setup, firmly establishes it as an optimal transfer approach in multi-people
localization tasks.

##  V Conclusion

Our research heralds a new era in indoor localization with the introduction of
a dual-channel CSI-ResNet. This innovative method uniquely converts multi-
target challenges into precise multi-label predictions, achieving remarkable
precision by utilizing just single-packet CSI data. Incorporating high-pass
filtering, phase compensation, and amplitude normalization, we significantly
enhance accuracy. The Nuclear Norm-based pre-training strategy ensures
outstanding adaptability and generalization, surpassing existing deep learning
models with reduced data and training time requirements. Future endeavors will
concentrate on advancing user-specific tracking in intricate scenarios,
thereby establishing a new standard in indoor localization technology.

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