#  Progress towards the first measurement of charm baryon dipole moments

S. Aiola  INFN Sezione di Milano, Milan, Italy  L. Bandiera  INFN Sezione di
Ferrara, Ferrara, Italy  G. Cavoto  INFN Sezione di Roma, Rome, Italy
Università di Roma “La Sapienza”, Rome, Italy  F. De Benedetti  INFN Sezione
di Milano, Milan, Italy  J. Fu  INFN Sezione di Milano, Milan, Italy
Università degli Studi di Milano, Milan, Italy  V. Guidi  INFN Sezione di
Ferrara, Ferrara, Italy  Università degli Studi di Ferrara, Ferrara, Italy  L.
Henry  INFN Sezione di Milano, Milan, Italy  Università degli Studi di Milano,
Milan, Italy  IFIC, Universitat de València-CSIC, Valencia, Spain  D.
Marangotto  INFN Sezione di Milano, Milan, Italy  Università degli Studi di
Milano, Milan, Italy  F. Martinez Vidal  IFIC, Universitat de València-CSIC,
Valencia, Spain  V. Mascagna  INFN Sezione di Milano Bicocca, Milan, Italy
Università degli Studi dell’Insubria, Como, Italy  J. Mazorra de Cos  IFIC,
Universitat de València-CSIC, Valencia, Spain  A. Mazzolari  INFN Sezione di
Ferrara, Ferrara, Italy  A. Merli  INFN Sezione di Milano, Milan, Italy
Università degli Studi di Milano, Milan, Italy  N. Neri  INFN Sezione di
Milano, Milan, Italy  Università degli Studi di Milano, Milan, Italy  M. Prest
INFN Sezione di Milano Bicocca, Milan, Italy  Università degli Studi
dell’Insubria, Como, Italy  M. Romagnoni  INFN Sezione di Ferrara, Ferrara,
Italy  Università degli Studi di Milano, Milan, Italy  J. Ruiz Vidal  IFIC,
Universitat de València-CSIC, Valencia, Spain  M. Soldani now at INFN Sezione
di Ferrara, Ferrara, Italy INFN Sezione di Milano Bicocca, Milan, Italy
Università degli Studi dell’Insubria, Como, Italy  A. Sytov  INFN Sezione di
Ferrara, Ferrara, Italy  V. Tikhomirov  Institute for Nuclear Problems and
Belarusian State University, Minsk, Belarus  E. Vallazza  INFN Sezione di
Milano Bicocca, Milan, Italy

###### Abstract

Electromagnetic dipole moments of short-lived particles are sensitive to
physics within and beyond the Standard Model of particle physics but have not
been accessible experimentally to date. To perform such measurements it has
been proposed to exploit the spin precession of channeled particles in bent
crystals at the LHC. Progress that enables the first measurement of charm
baryon dipole moments is reported. In particular, the design and
characterization on beam of silicon and germanium bent crystal prototypes, the
optimization of the experimental setup, and advanced analysis techniques are
discussed. Sensitivity studies show that first measurements of
${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ baryon dipole moments can be performed
in two years of data taking with an experimental setup positioned upstream of
the LHCb detector.

###### pacs:

14.20.Lq 13.40.Em

##  I Introduction

Electromagnetic dipole moments are static properties of particles that are
sensitive to physics within and beyond the Standard Model (SM) of particle
physics. For particles like the proton, neutron, muon and electron, such
measurements provide among the most stringent tests of the SM
Schneider:2017lff ; Sahoo:2016zvr ; Afach:2015sja ; Bennett:2006fi ;
Bennett:2008dy ; Hanneke:2008tm ; Baron:2013eja . In classical physics, the
magnetic dipole moment (MDM) measures the strength and orientation of the
magnetic field generated by the motion of electric charges or by the intrinsic
magnetism of matter. In particle physics, the MDM is proportional to the
particle spin-polarization vector ${\boldsymbol{s}}$ and for spin-1/2
particles is given by ${\boldsymbol{\mu}}=g\mu_{\textrm{B}}{\boldsymbol{s}}/2$
(Gaussian units), where $g$ is the dimensionless gyromagnetic factor,
$\mu_{\textrm{B}}=e\hslash/(2mc)$ is the particle magneton, and $m$ its mass.
The spin polarization is a unit vector defined as
${\boldsymbol{s}}=2\langle{\boldsymbol{S}}\rangle/\hslash$, where
${\boldsymbol{S}}$ is the spin operator. The measurement of the baryon MDM
provides experimental anchor points for low energy models of strong
interactions, while for leptons can be confronted with precise calculations
for a stringent SM test Dainese:2019xrz ; Miller:2007kk . The electric dipole
moment (EDM) of a system measures the separation of the positive and negative
electric charge distribution. The particle EDM is proportional to its spin-
polarization vector and is defined as
${\boldsymbol{\delta}}=d\mu_{\textrm{B}}{\boldsymbol{s}}/2$, where $d$ is the
gyroelectric factor. The EDM is expected to be minuscule for baryons and
leptons in the SM and any observation would imply the existence of physics
beyond the SM Chupp:2017rkp ; Beacham:2019nyx ; Unal:2020ezc . There are no
direct measurements to date of such properties for charm and beauty baryons,
and also for the $\tau$ lepton, due to the difficulties imposed by their short
lifetimes. For the charm quark (chromo-)EDM only indirect limits exist based
on the experimental bounds on the neutron EDM Sala:2013osa ; Gisbert:2019ftm .

Recently it has been proposed to measure charm baryon MDM/EDM at a fixed-
target experiment to be installed at the Large Hadron Collider (LHC)
Baryshevsky:2016cul ; Burmistrov:2194564 ; Botella:2016ksl ; Bagli:2017foe ;
Bezshyyko:2017var , following an experiment on the $\Sigma^{+}$ hyperon MDM
Chen:1992wx ; SAMSONOV1996271  and other proposals at Fermilab Baublis:1994ku
; Khanzadeev:1996ag . By exploiting the phenomenon of particle channeling in
bent crystals, the electric and magnetic dipole moments of short-lived
particles can be measured by studying the spin precession induced by the
intense electric field between the crystal atomic planes, firstly proposed by
Baryshevsky in 1979 Baryshevsky:1979 . In a crystal, the strong electric field
experienced by a positively charged particle in the proximity of the atomic
planes exerts a strong force and the particle trajectory becomes confined
within two crystalline planes. This phenomenon, called planar channeling, can
occur if the entrance angle between the particle trajectory and a crystal
plane is lower than the Lindhard critical angle for channeling,
$\theta_{L}=\sqrt{2U_{0}/(p\beta c)}$, where $U_{0}$ is the potential-well
depth, $p$ the particle momentum and $\beta$ its velocity Lindhard ;
Biryukov1997 . For a 1$\mathrm{\,Te\kern-1.00006ptV}$ charged particle
$\theta_{L}\approx 6.3\mathrm{\,\mu rad}$ ($\theta_{L}\approx 7.7\mathrm{\,\mu
rad}$) in a silicon (germanium) crystal.

Figure 1:  Deflection of a particle with initial and final momentum
$\boldsymbol{p}$ and $\boldsymbol{p}^{\prime}$, and spin-polarization
precession in the $yz$ and $xz$ plane induced by the MDM and the EDM,
respectively. The solid (blue) arrow represents the initial spin-polarization
vector ${\boldsymbol{s}}$, aligned along the $y$ axis in this example, normal
to the crystal plane at entry. The spin is rotated of an angle $\varPhi$ to
${\boldsymbol{s}^{\prime}}$ after channeling in a crystal with bending angle
$\theta_{C}$. The dashed (red) arrow indicates the (magnified)
$s^{\prime}_{x}$ spin polarization component proportional to the particle EDM.
The ${\boldsymbol{B}^{*}}$ (${\boldsymbol{E}^{*}}$) indicates the magnetic
(electric) field in the particle rest frame.

The ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ charm baryons, produced by
interactions of the 7$\mathrm{\,Te\kern-1.00006ptV}$ LHC proton beam on a
fixed target, are allowed to have initial polarization perpendicular to the
production plane, due to parity symmetry conservation in strong interactions.
The generation occurs within a cone with aperture $1/\gamma\approx
1\mathrm{\,mrad}$, where $\gamma$ is the Lorentz factor of the charm baryon.
After the target, a bent crystal is placed to act as a special type of
spectrometer deflecting channeled particles of a constant angle $\theta_{C}$
within the detector acceptance. The upgraded LHCb detector is particularly
suited for this experiment thanks to its forward geometry and excellent
performance for the reconstruction of heavy hadrons Aaij:2014jba ;
Bediaga:2012uyd . However, only a small fraction of the produced charm
baryons, entering the crystal at an angle within the critical angle, is
channeled and deflected. A sizeable spin precession, induced by the intense
electromagnetic field between crystal atomic planes, allows to probe for
electromagnetic dipole moments, as illustrated in Fig. 1. For ultra-
relativistic channeled particles with $\gamma\gg 1$, the spin precession angle
induced by the MDM in the $yz$ plane is

$\varPhi\approx\frac{g-2}{2}\gamma\theta_{C},$ (1)

where $\theta_{C}=L/\rho$ is the crystal bending angle, $L$ is the length of
the crystal arc and $\rho$ the curvature radius. The presence of a non-zero
EDM, in the limit of $d\ll g-2,$ introduces a spin-polarization component
$s^{\prime}_{x}$ perpendicular to the plane of the crystal bending
Botella:2016ksl ,

$s^{\prime}_{x}\approx s_{0}\frac{d}{g-2}(1-\cos\varPhi),$ (2)

where $s_{0}$ is the initial polarization. According to previous studies, a
germanium crystal allows for enhanced sensitivity to MDM and EDM with respect
to the silicon crystal Bagli:2017foe ; Fomin:2019wuw  due to the higher
electric field between crystal atomic planes.

In this article, the design and characterization on beam of the first silicon
and germaniun bent crystal prototypes for charm baryon spin precession are
reported in Sec. II. Advanced experimental techniques and sensitivities
studies for optimal future measurements of charm baryon dipole moments are
discussed in Sec. III and Sec. IV, respectively. The main improvements with
respect to previous studies Bagli:2017foe  include: a full amplitude analysis
for optimal determination of baryon polarization, a more realistic
polarization model Bezshyyko:2017var , an optimized target thickness,
CRYSTALRAD Monte Carlo simulations for particle channeling also for cryogenic
temperatures Sytov:2014jfa , and additional charm baryon decay modes.

##  II Bent crystal prototypes and test on beam

Crystal-assisted efficient steering of high-energy particle beams requires
bent crystals with a tight control over their deformational state and an
exceptionally low number of crystalline defects. Non-uniformities of the
deformational state of the crystal and crystalline defects would indeed lead
particles to be lost from channeling condition, lowering the efficiency of the
steering process Biryukov1997 ; Bagli:2015jga .

Crystals are manufactured through a revisitation of a protocol
Baricordi:2007zz ; Germogli:2015gja ; Mazzolari:2018hsu  already assessed for
manufacturing of crystals of few $\mathrm{\,mm}$ length suitable for steering
of particle beams circulating at the LHC Scandale:2016krl . The prime
materials are a (111) oriented $5\mathrm{\,mm}$ thick silicon wafer and a
(110) oriented $1\mathrm{\,mm}$ thick germanium wafer. With the aim to
maintain an optimal steering efficiency, wafers with less than
$1/\mathrm{\,cm}^{2}$ dislocation density were selected from a large stock of
wafers.

Mechanical dicing was applied to wafers to obtain a $50\times 5\times
80\mathrm{\,mm}^{3}$ crystal and a $35\times 1\times 55\mathrm{\,mm}^{3}$
crystal made of silicon and germanium, respectively. The first value indicates
the dimension along the beam and in both cases was chosen according to Ref.
Bagli:2017foe . Each crystal was mechanically bent along this longer dimension
through a bender properly shaped to impart to the crystal the nominal bending
radius of $5.0\mathrm{\,m}$ for silicon and $3.6\mathrm{\,m}$ for germanium,
respectively. Given the nominal bending radius, channeled particles are
deflected at an angle equal to $16\mathrm{\,mrad}$ for silicon and
$15\mathrm{\,mrad}$ for germanium, see Fig. 2. The uniformity of the crystal
deformation plays a key role for obtaining the expected steering efficiency:
to enhance uniformity of crystal curvature, the nominal shape of the surfaces
of the bender in contact with the crystal have been optimized through finite
element models to free-form surfaces. After bending, the crystal deformational
state was characterized GERMOGLI2017308  by means of high-resolution
diffraction of a $8\mathrm{\,ke\kern-1.00006ptV}$ X-ray beam.

Figure 2: (a) Sketch of an assembly made of crystal (red part) and associated
bender (grey parts). Surfaces in contact with the crystal (green color) have
been machined to a free-form surface. As the crystal is clamped in the bender,
it bends around the $x$ axis. A particle beam, initially propagating along the
$z$ axis is channeled between bent atomic planes and deflected. (b) Sketch of
the experimental setup at the CERN H8 beam line. A
$180\mathrm{\,Ge\kern-1.00006ptV}$ positive hadron beam is directed to the
crystal. The trajectory of each particle is reconstructed before and after
interaction with the crystal by a tracking telescope (grey boxes). The crystal
(red rectangle) is mounted on a high-resolution goniometer (blue circle). As
the crystal is oriented in order to channel the incoming beam between its
atomic planes, beam steering occurs and a fraction of the beam is deflected
(green arrow). Distances between elements of the setup are expressed in
$\mathrm{\,mm}$.

At the H8 external beam line of the Super Proton Synchrotron (SPS) at CERN,
the crystal/bender assemblies were mounted on a goniometer capable of
rotations with accuracy of $1\mathrm{\,\mu rad}$ and aligned to a
$180\mathrm{\,Ge\kern-1.00006ptV}$ positively charged hadron beam in order to
measure particle channeling. The beam angular divergence ($26\mathrm{\,\mu
rad}$) resulted to be wider than $\theta_{L}$ for
$180\mathrm{\,Ge\kern-1.00006ptV}$ particles ($\approx 15\mathrm{\,\mu rad}$
for silicon and $\approx 17\mathrm{\,\mu rad}$ for germanium), also the
dimension along the $y$ coordinate ($\approx 8\mathrm{\,mm}$) was larger than
the geometrical size of the crystal. To select particle trajectories
intercepting the crystal and impinging on bent atomic planes at angle
comparable to $\theta_{L}$, a tracking telescope based on four microstrips
detectors was used. Two detectors were placed before and two were placed after
the crystal to reconstruct particle trajectories before and after the
interaction. Monte Carlo simulations of the beam test setup, based on the
Geant4 toolkit Agostinelli2003250 , are used to determine the angular
uncertainty on particle trajectories before and after the interaction with the
crystal, which resulted to be $\approx 7\mathrm{\,\mu rad}$ and $\approx
50\mathrm{\,\mu rad}$, respectively.

---

Figure 3:  (a) Angular distributions of a $180\mathrm{\,Ge\kern-1.00006ptV}$
hadron beam after interaction with the silicon (black lines) and germanium
(red lines) crystals. Experimental data (solid curves) are compared to results
of Monte Carlo simulations (dashed curves). The maxima located on the right
side of the plot correspond to particles that are deflected, being channeled
over the entire crystal length. The oscillations between the two peaks are due
to the discrete structure of the strip detectors. By selecting particles
impinging on the crystal within $\pm 30\mathrm{\,\mu rad}$, steering
efficiencies of $(8.9\pm 0.5)\%$ and $(10.8\pm 0.5)\%$ are measured for
silicon and germanium crystals, respectively. (b) Monte Carlo simulation
results for the steering efficiency of the two crystals as a function of the
energy of the beam, assuming uniform bending radius and a particle beam with
uniform angular distribution at the crystal entrance.

The angular distribution of the beam after interaction with the crystal is
shown in Fig. 3(a). The peak of the distribution on the left is populated by
particles which are not channeled between atomic planes at the crystal entry
face TARATIN1987425  and by particles which are initially channeled but with
an impact parameter with respect to atomic planes smaller than the thermal
vibration amplitude of the atoms of the crystal. For such particles, single
scattering events with the inner shell electrons of the atoms or with the
atomic nuclei result in a drastic change of their trajectory, leading them out
of channeling regime soon after being channeled Scandale:2015zea . The peaks
on the right, populated by channeled particles over the entire length of the
crystal, are centered at angles of $15988\pm 5\mathrm{\,\mu rad}$ and
$14670\pm 2\mathrm{\,\mu rad}$ for the case of silicon and germanium crystals,
respectively. Given the fact that in the final application the crystal would
interact with a particle beam whose divergence is much larger than the
critical angle for channeling, the steering efficiency of the crystal is
measured for particles reaching the crystal at a nominal angle of $\pm
30\mathrm{\,\mu rad}$, corresponding to an angular window about two times
larger than the critical angle for channeling, and with deflection angle in a
window of $\pm 500\mathrm{\,\mu rad}$ around the peak of maximum deflection.
Within that angular window, the efficiencies are measured according to Ref.
ROSSI2015369 .

| Length ($\mathrm{\,mm}$) | Thickness ($\mathrm{\,mm}$) |  Deflection angle ($\mathrm{\,mrad}$) | Steering efficiency (%)
---|---|---|---|---
X-Ray | Channeling | Measured | Simulation
Germanium | 55 | 1 | $14.5\pm 0.8$ | $14.670\pm 0.002$ | $10.8\pm 0.5$ | $12.3\pm 0.5$
Silicon | 80 | 5 | $16.1\pm 0.8$ | $15.988\pm 0.005$ | $8.9\pm 0.5$ | $9.9\pm 0.5$
Table 1: Geometrical parameters of silicon and germanium crystals used in the
experiment, and recorded channeling efficiencies compared to expected from
Monte Carlo simulations. The length (thickness) represents the longitudinal
(transverse) dimension of the crystal with respect to the beam direction. The
uncertainties on the steering efficiency results take into account both
statistical and systematic effects.

Table 1 summarizes the most important features of the crystal prototypes and
the measured channeling efficiencies. Using Monte Carlo simulations for the
interaction between particle beam and crystal (for details see Appendix B.1),
the steering efficiency as a function of the energy of the beam is determined
for the two crystals assuming a uniform bending radius. Steering efficiency
results are shown in Fig. 3(b).

##  III Experimental techniques

Charm baryon MDM and EDM measurements require advanced analysis techniques
discussed below. Combining these with an optimized experimental setup, the
sensitivity to the observables of interest is extracted. To that end, large
samples of pseudoexperiments are used, where the kinematics of
${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ baryons produced in the target are
generated according to their momentum spectra as obtained from Pythia 8.244
event generator Sjostrand:2014zea  tuned to NNPDF3.1sx NNLO NLLx LUXQED
$\alpha_{s}(m_{Z})=0.118$ parton distribution functions Ball:2017nwa ;
Bertone:2017bme  including LHCb charm production data Bertone:2018dse .

As a case study, the ${{\Lambda}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
baryon decay is discussed. In the helicity formalism Jacob:1959at  the
transition amplitude between an initial state with spin projection
$m_{{\Lambda}^{+}_{c}}$ along the quantization axis and a final state with
proton helicity $m_{p}$ is indicated as ${\cal
A}_{m_{{\Lambda}^{+}_{c}},m_{p}}(\boldsymbol{\xi})$, as a function of the
phase space variables $\boldsymbol{\xi}$. The decay distribution can be
written as

$W(\boldsymbol{\xi}|s)=f(\boldsymbol{\xi})+sg(\boldsymbol{\xi}),$ (3)

where $s$ is the spin-polarization magnitude along a given axis, and the
functions $f(\boldsymbol{\xi})$ and $g(\boldsymbol{\xi})$ are determined by
the decay amplitudes ${\cal
A}_{m_{{\Lambda}^{+}_{c}},m_{p}}(\boldsymbol{\xi})$ as

$\displaystyle f(\boldsymbol{\xi})$ $\displaystyle=$ $\displaystyle
W(\boldsymbol{\xi}|s=0)$ $\displaystyle\propto$ $\displaystyle\sum_{m_{p}=\pm
1/2}\left[\left|{\cal A}_{1/2,m_{p}}(\boldsymbol{\xi})\right|^{2}+\left|{\cal
A}_{-1/2,m_{p}}(\boldsymbol{\xi})\right|^{2}\right],$ $\displaystyle
g(\boldsymbol{\xi})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}W(\boldsymbol{\xi}|s=1)-W(\boldsymbol{\xi},s=-1)\big{]}$
$\displaystyle\propto$ $\displaystyle\sum_{m_{p}=\pm 1/2}\left[\left|{\cal
A}_{1/2,m_{p}}(\boldsymbol{\xi})\right|^{2}-\left|{\cal
A}_{-1/2,m_{p}}(\boldsymbol{\xi})\right|^{2}\right],$

and satisfy the normalization conditions

$\int f(\boldsymbol{\xi})d\boldsymbol{\xi}=1,\quad\quad\quad\int
g(\boldsymbol{\xi})d\boldsymbol{\xi}=0.$ (5)

The value of the average event information $S^{2}$ represents the sensitivity
to the polarization $s$ that can be computed as Davier:1992nw

$S^{2}=\int\frac{g^{2}(\boldsymbol{\xi})}{f(\boldsymbol{\xi})+s_{0}g(\boldsymbol{\xi})}d\boldsymbol{\xi},$
(6)

where $s_{0}$ is the best estimate for the polarization whose variance is
$\sigma_{s}^{2}=\left(NS^{2}\right)^{-1}$, with $N$ the number of signal
events. The ${{\Lambda}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay
amplitude is dominated by the sum of quasi two-body $pK^{*}$,
$\Lambda^{*}{{\pi}^{+}}$, and $\Delta^{++}{{K}^{-}}$ resonant contributions
Tanabashi:2018oca ; Aitala:1999uq . The average event information for
$s_{0}=0$, based on a pseudoexperiment, is shown in Fig. 4 as a function of
the Dalitz plot position, defined by the squared invariant masses
$m^{2}({p}{{K}^{-}})$ and $m^{2}({{K}^{-}}{{\pi}^{+}})$, with an average value
$S^{2}\approx 0.145$ Marangotto:2020tzf .

Figure 4:  Average event information $S^{2}$ to the ${\Lambda}^{+}_{c}$ baryon
polarization as a function of the Dalitz plot position defined by the squared
invariant masses $m^{2}({p}{{K}^{-}})$ and $m^{2}({{K}^{-}}{{\pi}^{+}})$, as
obtained from a large statistics pseudoexperiment.

This result can be compared with a two-body decay, where $S^{2}$ for $s_{0}=0$
is related to the effective decay asymmetry parameter $\alpha_{\rm eff}$,
which characterizes parity violation, $S^{2}=\alpha_{\rm eff}^{2}/3$, and it
is found to be compatible to that estimated for the
${{\Lambda}^{+}_{c}}\rightarrow\Delta^{++}{{K}^{-}}$ decay using the known
value $\alpha_{\Delta^{++}{{K}^{-}}}=-0.67\pm 0.30$ Botella:2016ksl , affected
by large uncertainty. Hence, the full amplitude analysis of the
${{\Lambda}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay enables the
measurement of the ${\Lambda}^{+}_{c}$ spin precession using three-body decay
modes Marangotto:2020tzf ; Marangotto:2020ead , which effectively increases
the statistics for the dipole moments measurement by a factor ${\cal
B}({{\Lambda}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}})/{\cal
B}({{\Lambda}^{+}_{c}}\rightarrow\Delta^{++}{{K}^{-}})\approx 6$ with respect
to that assumed in previous studies Botella:2016ksl ; Bagli:2017foe ;
Bezshyyko:2017var ; Fomin:2019wuw .

Additional ${\Lambda}^{+}_{c}$ three- and four-body decays can be exploited
for the measurement of the dipole moments in order to further increase the
signal yield up to an additional factor of three, assuming similar average
event information $S^{2}$ for all decays. This can be done analogously for
$\Xi^{+}_{c}$ baryon decays. In Tables 2 and 3 are listed the decay modes
considered for the sensitivity studies with corresponding effective branching
fractions, taking into account the ability to reconstruct the final state
charged particles with the LHCb detector. Due to the high momentum of charm
baryons produced in the fixed-target interactions, a large fraction of the
long-lived ${\Sigma^{+}}$, ${\Sigma^{-}}$ and $\Xi^{-}$ strange baryons
traverse the entire LHCb tracking system before decaying, thus their
trajectories can be reconstructed as for stable charged particles. The crystal
bending angle and length determine the momentum spectrum of channneled charm
baryons and consequently the fraction of strange baryons reconstructed as
stable charged particles. This requires that final state particles travel more
than $9.4\mathrm{\,m}$ from their production point positioned
$1.2\mathrm{\,m}$ upstream of the nominal $p$ $p$ interaction region. Despite
the lower production rate of the $\Xi^{+}_{c}$ compared to the
${\Lambda}^{+}_{c}$ baryon, estimated to be about 70% Bagli:2017foe , and its
smaller effective branching ratio, similar signal yields are expected for both
baryon types decaying after the bent crystal due to their different lifetimes,
about a factor of two longer for the $\Xi^{+}_{c}$ baryon.

When there is a ${{\pi}^{0}}$ meson in the final state, a partial
reconstruction of the decay can be performed based on charged particles only
(for details see Appendix B.3). The charm baryon invariant mass is
reconstructed using the corrected mass Abe:1997sb , the charm hadron flight
length and direction are determined by reconstructing the production and the
decay vertex positions, and its momentum can be estimated without bias by
reconstructing the decay kinematics. Similarly, the average charm baryon spin-
polarization vector can also be measured with no bias employing the
reconstructed kinematics, and ultimately using a technique based on templates
for full positive and negative polarization, similar to that for the
measurement of the polarization of the $\tau$ leptons where the neutrinos are
not reconstructed Davier:1992nw ; Fu:2019utm .

${\Lambda}^{+}_{c}$ final state |  ${\cal B}$ (%) | $\epsilon_{\textrm{3trk}}$ |  ${\cal B}_{\textrm{eff}}$ (%)
---|---|---|---
${p}{{K}^{-}}{{\pi}^{+}}$ | $6.28\pm 0.32$ | 0.99 | 6.25
${\Sigma^{+}}{{\pi}^{-}}{{\pi}^{+}}$ | $4.50\pm 0.25$ | 0.54 | 2.43
${\Sigma^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | $1.87\pm 0.18$ | 0.71 | 1.33
${p}{{\pi}^{-}}{{\pi}^{+}}$ | $0.461\pm 0.028$ | 1.00 | 0.46
$\Xi^{-}{{K}^{+}}{{\pi}^{+}}$ | $0.62\pm 0.06$ | 0.73 | 0.45
${\Sigma^{+}}{{K}^{-}}{{K}^{+}}$ | $0.35\pm 0.04$ | 0.51 | 0.18
${p}{{K}^{-}}{{K}^{+}}$ | $0.106\pm 0.006$ | 0.98 | 0.11
${\Sigma^{+}}{{\pi}^{-}}{{K}^{+}}$ | $0.21\pm 0.06$ | 0.54 | 0.11
${p}{{K}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ | $4.46\pm 0.30$ | 0.99 | 4.43
${\Sigma^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ | $3.20$ | 0.54 | 1.72
${\Sigma^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{0}}$ | $2.1\pm 0.4$ | 0.71 | 1.49
${\Sigma^{+}}[{p}{{\pi}^{0}}]{{\pi}^{-}}{{\pi}^{+}}$ | $2.32$ | 0.46 | 1.06
${\Sigma^{+}}[{p}{{\pi}^{0}}]{{K}^{-}}{{K}^{+}}$ | $0.18$ | 0.46 | 0.08
${\Sigma^{+}}[{p}{{\pi}^{0}}]{{\pi}^{-}}{{K}^{+}}$ | $0.11$ | 0.46 | 0.05
All | - | - | 20.2
Table 2: List of ${\Lambda}^{+}_{c}$ baryon decay modes and corresponding absolute branching fraction ${\cal B}$ Tanabashi:2018oca , efficiency for LHCb reconstructibility of the three-charged particles $\epsilon_{\textrm{3trk}}$, and the effective branching fraction ${\cal B}_{\textrm{eff}}={\cal B}\times\epsilon_{\textrm{3trk}}$. The efficiency and effective branching fraction depend on the crystal parameters, here for germanium at room temperature with deflection angle of 16$\mathrm{\,mrad}$ and 10$\mathrm{\,cm}$ length.  $\Xi^{+}_{c}$ final state | ${\cal RB}$ |  ${\cal B}$ (%) | $\epsilon_{\textrm{3trk}}$ |  ${\cal B}_{\textrm{eff}}$ (%)
---|---|---|---|---
$\Xi^{-}{{\pi}^{+}}{{\pi}^{+}}$ | 1 | $2.86\pm 1.27$ | 0.64 | 1.84
${\Sigma^{+}}{{K}^{-}}{{\pi}^{+}}$ | $0.94\pm 0.10$ | - | 0.42 | 1.14
${\Sigma^{+}}{{\pi}^{-}}{{\pi}^{+}}$ | $0.48\pm 0.20$ | - | 0.44 | 0.60
${p}{{K}^{-}}{{\pi}^{+}}$ | $0.21\pm 0.04$ | - | 0.99 | 0.60
${\Sigma^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | $0.18\pm 0.09$ | - | 0.61 | 0.31
${\Sigma^{+}}{{K}^{-}}{{K}^{+}}$ | $0.15\pm 0.06$ | - | 0.41 | 0.18
$\Omega^{-}{{K}^{+}}{{\pi}^{+}}$ | $0.07\pm 0.04$ | - | 0.42 | 0.08
$\Sigma^{+}[{p}{{\pi}^{0}}]{{K}^{-}}{{\pi}^{+}}$ | 0.48 | - | 0.57 | 0.79
$\Sigma^{+}[{p}{{\pi}^{0}}]{{\pi}^{-}}{{\pi}^{+}}$ | 0.25 | - | 0.57 | 0.40
$\Sigma^{+}[{p}{{\pi}^{0}}]{{K}^{-}}{{K}^{+}}$ | 0.08 | - | 0.59 | 0.13
All | - | - | - | 6.1
Table 3: List of $\Xi^{+}_{c}$ baryon decay modes and corresponding relative
branching absolute ${\Xi^{+}_{c}}\to\Xi^{-}{{\pi}^{+}}{{\pi}^{-}}$ branching
fraction ${\cal B}$ Li:2019atu , estimate of the efficiency for LHCb
reconstructibility of the three-charged particles $\epsilon_{\textrm{3trk}}$,
and the effective branching fraction ${\cal B}_{\textrm{eff}}={\cal
RB}\times{\cal
B}({\Xi^{+}_{c}}\to\Xi^{-}{{\pi}^{+}}{{\pi}^{-}})\times\epsilon_{\textrm{3trk}}$
or ${\cal B}_{\textrm{eff}}={\cal B}\times\epsilon_{\textrm{3trk}}$ where
${\cal B}$ measurement is available. The reported $\epsilon_{\textrm{3trk}}$
and ${\cal B}_{\textrm{eff}}$ are for a 16$\mathrm{\,mrad}$ bent,
10$\mathrm{\,cm}$ long germanium crystal at room temperature.

The charm baryon polarization is perpendicular to the production plane,
defined by the momenta of the impinging proton and that of the outgoing
${\Lambda}^{+}_{c}$ baryon, and is parameterized as a function of its
transverse momentum $p_{\mathrm{T}}$ with respect to the direction of the
impinging proton as

$s_{0}(\mbox{$p_{\mathrm{T}}$})\approx A\left(1-e^{-Bp^{2}_{T}}\right),$ (7)

with $A\approx-0.9$ and $B\approx 0.4$
$({\mathrm{\,Ge\kern-1.00006ptV\\!/}c})^{-2}$. The parametrization is based on
a phenomenological dependence Barlag:1994bk  used to describe the experimental
results Aitala:1999uq .

The ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ baryon polarization versus
$p_{\mathrm{T}}$ can be measured precisely in fixed-target collisions at LHCb
using the SMOG system Aaij:2014ida ; Bursche:2649878  to further improve the
polarization model.

##  IV Sensitivity studies and setup optimization

The sensitivity to dipole moments is studied using pseudoexperiments and
Pythia simulations for different angles between the impinging proton direction
and the crystal orientation. The crystal reference frame $(x,y,z)$ is rotated
at different angles $\theta_{y,C}$ around the $x_{L}$ axis in the
$(x_{L},y_{L},z_{L})$ laboratory frame Fomin:2019wuw , with $x$ parallel to
the $x_{L}$ axis and $z_{L}$ parallel to the impinging proton direction. For
non-zero $\theta_{y,C}$ values an initial polarization along the $x$ and $y$
axes is induced,

${\boldsymbol{s}}=(s_{x},s_{y},0)\approx\frac{s_{0}(\mbox{$p_{\mathrm{T}}$})}{\mbox{$p_{\mathrm{T}}$}}\left(-p_{y_{L}},p_{x_{L}},0\right),$
(8)

where $p_{x_{L}}$ and $p_{y_{L}}=p\sin\theta_{y,C}$ are the transverse
momentum components along the laboratory $x_{L}$\- and $y_{L}$-axis,
respectively, and $p$ is the total momentum of the charm baryon. This is a
consequence of parity conservation in strong interactions that forces the
${\Lambda}^{+}_{c}$ polarization vector to be perpendicular to the production
plane, defined by the proton and the ${\Lambda}^{+}_{c}$ baryon momenta. A
probability density function based upon the quasi-two body approximation for
the decay rate,

${\cal W}\propto 1+\alpha_{\rm
eff}{\boldsymbol{s}}^{\prime}\cdot\hat{\boldsymbol{k}},$ (9)

is used in order to effectively reproduce the fit procedure for data, where
$\hat{\boldsymbol{k}}$ is the direction of the resonant intermediate state in
the charm baryon helicity frame and ${\boldsymbol{s}}^{\prime}$ the spin-
polarization vector after spin precession in the crystal,

$\displaystyle s^{\prime}_{x}$ $\displaystyle\approx
s_{y}\frac{a^{\prime}d^{\prime}}{{a^{\prime}}^{2}_{d}}(1-\cos\varPhi)+s_{x}\left(\frac{{a^{\prime}}^{2}}{{a^{\prime}}_{d}^{2}}+\frac{{d^{\prime}}^{2}}{{a^{\prime}}^{2}_{d}}\cos\varPhi\right),$
$\displaystyle s^{\prime}_{y}$ $\displaystyle\approx
s_{y}\left(\frac{{d^{\prime}}^{2}}{{a^{\prime}}^{2}_{d}}+\frac{{a^{\prime}}^{2}}{{a^{\prime}}_{d}^{2}}\cos\varPhi\right)+s_{x}\frac{a^{\prime}d^{\prime}}{{a^{\prime}}^{2}_{d}}(1-\cos\varPhi),$
$\displaystyle s^{\prime}_{z}$ $\displaystyle\approx-
s_{y}\frac{a^{\prime}}{{a^{\prime}}_{d}}\sin\varPhi+s_{x}\frac{d^{\prime}}{{a^{\prime}}_{d}}\sin\varPhi,$
(10)

where $a^{\prime}=a+1/(1+\gamma)$, with $a=(g-2)/2$ the anomalous magnetic
moment, $d^{\prime}=d/2$,
$a^{\prime}_{d}=\sqrt{{a^{\prime}}^{2}+{d^{\prime}}^{2}}$, and
$\varPhi=\gamma\theta_{C}a^{\prime}_{d}$. These expressions hold at ${\cal
O}(10^{-2})$ precision and follow by solving the T-BMT equation Thomas:1926dy
; Thomas:1927yu ; Bargmann:1959gz  for initial polarization along the $x$ and
$y$ crystal frame directions, according to the procedure described elsewhere
Botella:2016ksl ; Bagli:2017foe . The expected precession angle $\varPhi$ and
the initial polarization-vector ${\boldsymbol{s}}$ can be estimated on event-
by-event basis using the measured boost factor $\gamma$ and the transverse
momentum of the charm baryon.

Following this approach, an enhanced sensitivity is obtained while retaining
the complete dependence on $g$, $d$ and $\gamma$, reducing significantly the
need of a large crystal orientation angle $\theta_{y,C}$ for the EDM case
Fomin:2019wuw , and solving discrete ambiguities in the simultaneous
extraction of the dipole moments Bagli:2017foe . Pseudoexperiments have been
generated assuming a proton flux on target of about $10^{6}~{}{p}/\sec$, which
corresponds to $4.3\times 10^{10}$ protons on target ($\mathrm{\,PoT}$) in ten
hours operations of the LHC Mirarchi:2019vqi , about $1.37\times 10^{13}$
$\mathrm{\,PoT}$ can be delivered by the accelerator in two years with an
operational efficiency of about 55%. In Fig. 5(a) the optimal configuration
for the measurement of the MDM alone, i.e. using Eq. (10) with $d=0$, is
obtained at $\theta_{y,C}=0$, which corresponds to the maximum initial
polarization along the $y$ axis, perpendicular to the magnetic field
${\boldsymbol{B}^{*}}$. In Fig. 5(b), the optimal configuration for the
combined measurement of MDM and EDM, i.e. using Eq. (10) with $d$ as free
parameter, is obtained at $\theta_{y,C}$ different from zero. For the MDM the
difference with respect to Fig. 5(a) is due to limited $\mathrm{\,PoT}$ sample
and to correlations between the $a$ and $d$ parameters. The sensitivity to EDM
requires a polarization component perpendicular to the electric field
${\boldsymbol{E}^{*}}$, e.g. initial polarization along the $x$ axis. However,
as illustrated in Fig. 5(b), for small $a\approx-0.03$ Dainese:2019xrz , the
decrease of EDM sensitivity with $\theta_{y,C}\approx 0.3\ (0)\mathrm{\,mrad}$
with respect to the maximal sensitivity at $0.5\mathrm{\,mrad}$ amounts to
about 20% (100%), with marginal impact on the MDM (for further details see
Appendix B.4).

A smaller crystal orientation angle reduces potential systematic uncertainties
due to initial polarization along the crystal $x$ axis, mimicking EDM effects.
Nevertheless, simulation studies show that a precise determination of the
polarization model allows such systematic uncertainty to be kept under
control. The study of charm baryons with opposite polarization and the use of
up- and down-bending crystals, inducing opposite spin precession, offer
complementary tools to prove the robustness of the results and control
systematic uncertainties PhysRevAccelBeams.22.081004 . For large
$a\approx-0.76$ Dainese:2019xrz  the choice of crystal orientation has no
impact on the sensitivities.

Figure 5: Regions of miniminal uncertainty of the $g$ (red curves) and $d$
(black) factors as a function of the crystal orientation angle $\theta_{y,C}$
and the target thickness $T$, for (a) MDM measurement alone and (b) combined
MDM and EDM measurement, with magnetic anomalous moment $a\approx-0.03$ and
${{\Lambda}^{+}_{c}}\to{p}{{K}^{-}}{{\pi}^{+}}$ decays. The markers and values
represent the minimum uncertainty on the $g$ and $d$ factors relative to
$1.37\times 10^{13}$ $\mathrm{\,PoT}$, corresponding to two years of data
taking at a rate of $10^{6}~{}{p}/\mathrm{s}$ Mirarchi:2019vqi , using a
16$\mathrm{\,mrad}$ bent, 10$\mathrm{\,cm}$ long germanium crystal at room
temperature. The curves are the regions whose uncertainties are increased by
20%, 50% and 100% with respect to the minimum. A similar behaviour is observed
for other considered crystal configurations.

In addition, in Fig. 5 the $\rm W$ target thickess is optimized considering
the inelastic interactions of charm baryons within the target, regulated by
the nuclear interaction length and the survival probability due to its
lifetime Tanabashi:2018oca  (for details see Appendix B.2). A $\rm W$ target
thickness of $2\mathrm{\,cm}$, about a factor of two thinner than that where
maximal ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ yields at the target exit are
reached, is considered as operational value. With this choice the charm baryon
dipole moment sensitivies are reduced by about 20% with respect to the optimal
thickness reached around 5 $\mathrm{\,cm}$, as illustrated in Fig. 5 for
${\Lambda}^{+}_{c}$ baryons, while detector occupancies are largely mitigated,
as required for detector operation and safety.

$S^{2}$ | $\approx 0.145$
---|---
$\epsilon_{\textrm{DET}}\ (\%)$ | $0.20$
Crystal configuration | Ge | Si | Ge $77\mathrm{\,K}$ | Ge ${\rm S_{2}}$
Crystal angle ($\mathrm{\,mrad}$) | 16 | 16 | 16 | 7
Crystal length ($\mathrm{\,cm}$) | 10 | 10 | 10 | 7
${\Lambda}^{+}_{c}$ baryon
$\sigma$ (${\mathrm{\,\mu b}}$/nucleon) | $10.6$
$a=(g-2)/2$ | $\approx-0.03~{}[-0.76]$
$d$ | $0$
$\mathcal{B}_{\textrm{eff}}\ (\%)$ | $20.2$ | $19.5$ | $20.6$ | $20.6$
$\epsilon_{\textrm{CH}}\ (\times 10^{-4})$ | $4.5$ | $2.7$ | $8.7$ | $11.1$
$\epsilon_{\textrm{DF}}\ (\times 10^{-2})$ | $3.3$ | $1.7$ | $4.9$ | $13.1$
$N_{\textrm{rec}}$ | $586$ | $181$ | $1748$ | $5879$
$\langle\gamma\rangle$ | $709$ | $573$ | $834$ | $855$
$\langle\mbox{$p_{\mathrm{T}}$}\rangle\ ({\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})$ | $0.79$ | $0.71$ | $0.86$ | $0.87$
$s_{x}\ (\%)$ | $11.8$ | $8.6$ | $14.1$ | $15.6$
$s_{y}\ (\%)$ | $-15.3$ | $-14.2$ | $-16.1$ | $-16.1$
$\sigma_{\boldsymbol{\mu}}\ (\times 10^{-2}\ {\boldsymbol{\mu}_{\bf N}})$ | $1.6$ | $3.4$ | $0.8$ | $0.9$
$\sigma_{\boldsymbol{\delta}}\ (\times 10^{-16}\ e\mathrm{\,cm})$ | $2.2~{}[9.8]$ | $5.6~{}[17.1]$ | $0.9~{}[5.7]$ | $1.0~{}[2.9]$
$\Xi^{+}_{c}$ baryon
$\sigma$ (${\mathrm{\,\mu b}}$/nucleon) | $7.5$
$a=(g-2)/2$ | $\approx 0.05~{}{[-0.47]}$
$d$ | $0$
$\mathcal{B}_{\textrm{eff}}\ (\%)$ | $6.1$ | $5.8$ | $6.2$ | $6.2$
$\epsilon_{\textrm{CH}}\ (\times 10^{-4})$ | $5.7$ | $3.8$ | $10.4$ | $12.2$
$\epsilon_{\textrm{DF}}\ (\times 10^{-2})$ | $10.7$ | $7.6$ | $13.0$ | $24.6$
$N_{\textrm{rec}}$ | $627$ | $284$ | $1429$ | $3146$
$\langle\gamma\rangle$ | $514$ | $433$ | $576$ | $588$
$\langle\mbox{$p_{\mathrm{T}}$}\rangle\ ({\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})$ | $0.74$ | $0.69$ | $0.78$ | $0.79$
$s_{x}\ (\%)$ | $8.6$ | $6.7$ | $10.2$ | $10.5$
$s_{y}\ (\%)$ | $-15.7$ | $-14.8$ | $-16.3$ | $-16.5$
$\sigma_{\boldsymbol{\mu}}\ (\times 10^{-2}\ {\boldsymbol{\mu}_{\bf N}})$ | $1.8$ | $3.1$ | $1.0$ | $1.5$
$\sigma_{\boldsymbol{\delta}}\ (\times 10^{-16}\ e\mathrm{\,cm})$ | $3.0~{}[5.1]$ | $5.9~{}[6.8]$ | $1.5~{}[3.5]$ | $2.4~{}[2.1]$
Table 4: Relevant parameters for ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ baryons
considered for sensitivity studies: the average event information $S^{2}$
(assumed to be similar for all modes), the detector efficiency
$\epsilon_{\textrm{DET}}$, which includes trigger, reconstruction and
selection criteria of signal events based upon the $3h$ system, the baryon
production cross section $\sigma$, the anomalous magnetic moment $a$ (two
values are used from Ref. Dainese:2019xrz , the second is indicated in squared
brackets), the gyroelectric factor $d=0$, the effective branching ratio
$\mathcal{B}_{\textrm{eff}}$, the channeling efficiency
$\epsilon_{\textrm{CH}}$, which includes the efficiency of the particle to be
trapped into channeling regime, the decay flight efficiency
$\epsilon_{\textrm{DF}}$ of the baryon within the crystal length, the number
of reconstructed charm baryons $N_{\textrm{rec}}$, the average boost
$\langle\gamma\rangle$ and transverse momentum
$\langle\mbox{$p_{\mathrm{T}}$}\rangle$, the initial spin-polarization
$s_{x}$, and the initial spin-polarization $s_{y}$ for events with positive
$p_{x_{L}}$ ($-s_{y}$ for negative $p_{x_{L}}$). Uncertainties on the MDM and
EDM, $\sigma_{\boldsymbol{\mu}}$ and $\sigma_{\boldsymbol{\delta}}$,
respectively, are relative to $1.37\times 10^{13}$ $\mathrm{\,PoT}$. At LHCb,
silicon and germanium bent crystals at room temperature, and germanium at
$77\mathrm{\,K}$, are considered. For comparison, the $\rm S_{2}$ scenario is
evaluated with germanium at room temperature. The corresponding crystal
parameters are reported in the table. The target thickness is 2
$\mathrm{\,cm}$ in all cases. All three- and four-body modes from Tables 2 and
3 are considered.

|
---|---
|

Figure 6:  Initial polarizations $s_{x}$ (hatched blue bands) and $s_{y}$ for
events with positive $p_{x_{L}}$ ($-s_{y}$ for negative $p_{x_{L}}$, solid
green) as a function of the ${\Lambda}^{+}_{c}$ baryon momentum for (a)
silicon $293\mathrm{\,K}$, (b) germanium $293\mathrm{\,K}$, (c) germanium
$77\mathrm{\,K}$, and (d) germanium $293\mathrm{\,K}$ for $\rm S_{2}$, with
parameters reported in the text. The bands represent one standard deviation
regions from the pseudoexperiments. The histograms, normalized to $1.37\times
10^{13}$ $\mathrm{\,PoT}$, show the spectra of channeled and reconstructed
particles.

|
---|---
|

Figure 7:  Same as Fig. 6 for $\Xi^{+}_{c}$ baryons. The oscillations in the
initial polarization bands (particularly visible for $s_{y}$) are due to the
limited sample size of the simulations.

Relevant parameters for the sensitivity studies are reported in Table 4, along
with the expected signal yields and uncertainties on the MDM and EDM in two
years of data taking with the LHCb detector, for different crystal
configurations. Figures 6 and 7 show the corresponding reconstructed
${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ momentum spectra, along with the
estimated $s_{x}$ and $s_{y}$ initial polarizations as a function of the
baryon momentum. The latter are determined by the convolution of Eq. (8) with
the transverse momentum distribution of channeled baryons. How the
sensitivities evolve as a function of $\mathrm{\,PoT}$ is illustrated in Fig.
8. Silicon and germanium crystals with deflection angle of 16$\mathrm{\,mrad}$
and 10$\mathrm{\,cm}$ length are considered for compatibility with the LHCb
detector acceptance and operations, while maximizing experimental sensitivity.
For the case of germanium, sensitivity results at $77\mathrm{\,K}$ in addition
to room temperature are shown. The scenario of a future dedicated experiment
at the LHC, indicated as $\rm S_{2}$, is also studied using a setup based on a
germanium crystal at room temperature of $7\mathrm{\,mrad}$ bending angle and
$7\mathrm{\,cm}$ length, which would allow effective separation of channeled
charm baryons from forwardly produced background particles, to be kept outside
the detector acceptance. Identical detector performance and signal
reconstruction efficiency to the LHCb apparatus are assumed, but extended in
the forward direction. Nevertheless, in the $\rm S_{2}$ scenario channeled
baryons are deflected at smaller angles with high momentum, extending up to
about 4 $\mathrm{\,Te\kern-1.00006ptV}$, and would require an advanced
detector design with a long lever arm and intense magnetic field for precise
particle momentum measurements, and high granularity to be able to operate and
reconstruct the events under high background levels in the very forward
region.

Compared to the configuration of germanium at room temperature, the
significantly higher yields with germanium $\rm S_{2}$, close to a factor of
ten (five) for ${\Lambda}^{+}_{c}$ ($\Xi^{+}_{c}$) baryons, and the 20% (15%)
harder momentum spectrum, reflect in an increase of MDM sensitivity equivalent
to a data sample about three times larger (same size). This is a consequence
of the different bending angles at LHCb and $\rm S_{2}$, i.e. 16 and 7
$\mathrm{\,mrad}$, respectively. For germanium at $77\mathrm{\,K}$, with
yields higher by a factor of three (two) and momentum spectra similar to
germanium $\rm S_{2}$, the uncertainties are reduced by an equivalent data
sample close to a factor of four (three) larger. Silicon provides
significantly lower yields and softer momentum spectra, resulting in
sensitivities whose equivalent data sample size is a factor of four (three)
smaller. EDM sensitivities strongly depend on the anomalous magnetic moment
$a$, although they present similar features. Thus, improved sensitivities with
respect to the LHCb detector based configurations are expected at $\rm S_{2}$
if much higher $\mathrm{\,PoT}$ would be achievable.

Figure 8:  Uncertainties on the MDM and EDM of ${\Lambda}^{+}_{c}$ (a) and
$\Xi^{+}_{c}$ (b) baryons as a function of $\mathrm{\,PoT}$ at LHCb and at a
dedicated experiment ($\rm S_{2}$) with increased forward acceptance. The
sensitivities at LHCb for room temperature $293\mathrm{\,K}$ silicon and
germanium bent crystals are compared with a germanium crystal cooled at
$77\mathrm{\,K}$, with parameters reported in the text. All three- and four-
body ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ decays from Tables 2 and 3 are
considered, with anomalous magnetic moment $a$ assumed to be $\approx-0.03$
and $\approx 0.05$, respectively. The vertical long-dashed lines refer to
$1.37\times 10^{13}$ $\mathrm{\,PoT}$, corresponding to two years of data
taking.

##  V Conclusions

In summary, progress towards the first measurement of charm baryons dipole
moments is reported. An experimental setup based on bent crystals and a $\rm
W$ target placed upstream of the LHCb detector is studied. Silicon and
germanium long bent crystals have been tested on a
$180~{}\mathrm{\,Ge\kern-1.00006ptV}$ hadron beam with relatively high
channeling efficiency measured for both prototypes. The germanium crystal
provides enhanced sensitivity to MDM and EDM compared to the silicon crystal,
especially when cooled down at $77\mathrm{\,K}$ Bezshyyko:2017var . Advanced
analysis techniques have been developed for three- and four-body charm baryon
decays, providing enhanced sensitivity to the measurements. For a baseline
configuration with $1.37\times 10^{13}$ $\mathrm{\,PoT}$ impinging on 2
$\mathrm{\,cm}$ $\rm W$ target and with a germanium crystal at room
temperature, similar sensitivities for the MDM (EDM) of ${\Lambda}^{+}_{c}$
and $\Xi^{+}_{c}$ baryons below $2\times 10^{-2}\ {\boldsymbol{\mu}_{\bf N}}$
($3\times 10^{-16}~{}e\mathrm{\,cm}$) are achievable, which corresponds to a
relative precision on the gyromagnetic factors $g$ of about 4%. A germanium
crystal cooled at 77 $\mathrm{\,K}$ would improve the sensitivity by a factor
of two. These results enable a unique program of MDM and EDM measurements of
charm baryons at LHCb, capable to test advanced low energy models of strong
interactions and to search for physics beyond the SM. A future dedicated
experiment with significantly higher $\mathrm{\,PoT}$ would offer the
possibility to improve the sensitivity to charm baryon dipole moments, and to
explore beauty baryons and ultimately the $\tau$ lepton Fomin:2018ybj ;
Fu:2019utm .

##  Appendix A Manufacturing of crystal and its bender

Silicon and germanium crystals are prepared starting from wafers available
from commercial suppliers. The thickness of the wafers are $5\mathrm{\,mm}$
and $1\mathrm{\,mm}$ for the silicon and germanium wafers, respectively.
Wafers with a dislocation density lower than $1/\mathrm{\,cm}^{2}$ over the
entire region interacting with the particle beam are selected from a stock of
wafers. The density of dislocations is characterized through the etch pit
density Sirtl:1961 ; Aragona:1972  and the X-ray topography techniques. The
miscut angle, i.e. the angle between the optical surface of the wafer surface
and the atomic planes, is measured using a high-resolution X-ray
diffractometer (Panalytical X’Pert3 MRD XL) coupled to an autocollimator.
Subsequently, the miscut is reduced to less than $0.01^{\circ}$ through
polishing with a Logitech PM5 equipment. The wafers are diced with a dicing
machine (Disco DAD3220) to rectangular crystals, then cleaned in hot bath of
acetone under ultrasonic agitation. Bending of the crystal occurs as
consequence of clamping of the crystal between surfaces of a properly machined
bender. Assemblies describing crystal benders are modeled through finite
element modelling (Ansys R18), and the shape of the surface in contact with
the crystal are properly modelled to maximize uniformity of the deformation of
the crystal. Bender are manufactured through milling and electro discharge
machining of a block of stainless steel 316LN. This material are chosen for
its compatibility with the environment of LHC, where the devices are supposed
to operate. After machining, the bender is cleaned in acetone bath under
ultrasonic agitation. To avoid interference of dust which might deposit on the
surfaces of the bender or of the crystal, the assembly of crystal on the
bender is accomplished in a ISO-4 clean room.

##  Appendix B Simulation studies

###  B.1 Simulation of interaction between particle beam and crystal

A precise treatment of scattering of channeled particles in the crystal is
fundamental for the design of experiments aiming to study spin precession of
short lifetime particles. Scattering from nuclei and inner shell electrons
play an important role in describing the dynamics of channeled particles,
rendering the description of this process more difficult. Propagation of
particles in the crystal is described by the Monte Carlo simulation CRYSTALRAD
Sytov:2014jfa ; Sytov:2019gad . State of the art simulations are based on the
solution of the equation of motion of a charged particle interacting with a
crystalline lattice and are quantum-mechanically grounded, including refined
treatments of both large and small-angle scattering. Recently observed
experimentally Mazzolari:2019fus  incoherent scattering modification is also
part of modern simulation codes. Bent crystals cooled to cryogenic
temperatures are more efficient than at room temperature due to the lower
vibration amplitude of the atoms Forster:1989eh . According to channeling
simulations the improvement is more significant for germanium than for silicon
crystals. Steering efficiencies for different crystal configurations
considered in the sensitivity studies, as obtained from these simulations, are
shown in Fig. 9.

Figure 9:  Monte Carlo simulation results for the steering efficiency vs
energy of particles impinging parallel to the atomic planes, assuming uniform
bending radius, of germanium and silicon crystals with deflection angle
16$\mathrm{\,mrad}$ and 10$\mathrm{\,cm}$ length (LHCb scenario), and
germanium 7$\mathrm{\,mrad}$ bent and 7$\mathrm{\,cm}$ length ($S_{2}$), at
room temperature. For germanium in the LHCb scenario results with cooling at
$77\mathrm{\,K}$ are also shown.

###  B.2 Evaluation of the $\boldsymbol{{{\Lambda}^{+}_{c}}}$ and
$\boldsymbol{{\Xi^{+}_{c}}}$ spectra after the tungsten target and the crystal

The variation per unit length of the number of protons, $N_{p}$, due to the
interaction with the tungsten target is given by

$\frac{{\textrm{d}}N_{p}}{{\textrm{d}}z}=-\frac{N_{p}}{\lambda_{\rm W}},$ (11)

where $\lambda_{\rm W}=A_{T}/(\rho N_{A}A_{\rm
W}^{\textrm{part}}\sigma_{{p}N})\approx 8.87\mathrm{\,cm}$ is the $\rm W$
nuclear interaction length at $\sqrt{s}\approx
115\mathrm{\,Ge\kern-1.00006ptV}$, with $\rho$ and $A_{T}$ the $\rm W$ density
and atomic mass respectively, $N_{A}$ the Avogadro number,
$A_{\textrm{W}}^{\textrm{part}}$ the number of participant nucleons calculated
on the basis of the Glauber model Miller_2007 ; Loizides:2016djv , and
$\sigma_{{p}N}$ the proton-nucleon inelastic cross-section at $\sqrt{s}\approx
115\mathrm{\,Ge\kern-1.00006ptV}$ Tanabashi:2018oca .

The variation per unit length of the number of ${\Lambda}^{+}_{c}$ (and
similarly $\Xi^{+}_{c}$) baryons $N_{{\Lambda}^{+}_{c}}$, is determined by the
disappearance of protons in the $p$ $\rm W$ interaction, and by the decay and
nuclear interaction in the target of the produced ${\Lambda}^{+}_{c}$ baryons,

$\frac{{\textrm{d}}N_{{{\Lambda}^{+}_{c}}}}{{\textrm{d}}z}=\frac{N_{\mathrm{\,PoT}}}{\lambda_{{\rm
W},{{\Lambda}^{+}_{c}}}}e^{-z/\lambda_{\rm
W}}-\frac{N_{{{\Lambda}^{+}_{c}}}}{\lambda^{\prime}},$ (12)

where $1/\lambda^{\prime}=1/\lambda_{\rm
W}^{({{\Lambda}^{+}_{c}})}+1/(\beta\gamma c\tau)$, $\beta\gamma$ is the
${\Lambda}^{+}_{c}$ Lorentz boost factor, $\tau$ its lifetime, $c$ the speed
of light, $N_{\mathrm{\,PoT}}$ the number of protons hitting the target,
$\lambda_{\rm W}^{({{\Lambda}^{+}_{c}})}\approx\lambda_{\rm W}$ the
${\Lambda}^{+}_{c}$ interaction length, and $\lambda_{{\rm
W},{{\Lambda}^{+}_{c}}}$ is the mean free path for ${\Lambda}^{+}_{c}$
production. The latter is estimated as $\lambda_{{\rm
W},{{\Lambda}^{+}_{c}}}=A_{T}/(\rho
N_{A}A_{N}\sigma_{{{\Lambda}^{+}_{c}}})\approx 81.35\mathrm{\,m}$, where
$A_{N}$ is the $\rm W$ atomic mass number and $\sigma_{{{\Lambda}^{+}_{c}}}$
the proton-nucleon to ${{\Lambda}^{+}_{c}}$ cross-section at $\sqrt{s}\approx
115\mathrm{\,Ge\kern-1.00006ptV}$, obtained by scaling linearly the measured
proton-nucleon to ${c}{\overline{c}}$ cross-section at
$\sqrt{s}=86.6\mathrm{\,Ge\kern-1.00006ptV}$ Aaij:2018ogq  and using the
${\Lambda}^{+}_{c}$ fragmentation fraction Lisovyi:2015uqa ; Gladilin:2014tba
. For $\Xi^{+}_{c}$ baryons the fragmentation fraction is estimated to be
about $0.7$ times smaller than for ${\Lambda}^{+}_{c}$, assuming it to be
similar for $\Xi^{+}_{c}$, $\Xi^{0}_{c}$ and $\Omega^{0}_{c}$ baryons
Bagli:2017foe , leading to $\lambda_{{\rm W},{\Xi^{+}_{c}}}\approx
114.72\mathrm{\,m}$. Integration of Eq. (12) for a target of thickness $z$ and
fixed $\beta\gamma$ yields

$N_{{{\Lambda}^{+}_{c}}}(z,\beta\gamma)=\frac{N_{\mathrm{\,PoT}}}{\lambda_{{\rm
W},{{\Lambda}^{+}_{c}}}}\beta\gamma c\tau\left(e^{-z/\lambda_{\rm
W}}-e^{-z/\lambda^{\prime}}\right).$ (13)

The number of ${\Lambda}^{+}_{c}$ baryons after traversing a thickness $z$ of
the target, $N_{{{\Lambda}^{+}_{c}}}(z)$, is then given by the convolution of
Eq. (13) with the normalized ${\Lambda}^{+}_{c}$ baryon momentum spectra,
$r(\beta\gamma)=1/\sigma_{{\Lambda}^{+}_{c}}{\textrm{d}}\sigma_{{\Lambda}^{+}_{c}}(\boldsymbol{p})/{\textrm{d}}^{3}\boldsymbol{p}$,
estimated using Monte Carlo simulations based on Pythia. Figure 10 shows the
number of ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ baryons exiting the target as
a function of the target thickness for the case of
$N_{\mathrm{\,PoT}}=1.37\times 10^{13}$ protons.

Figure 10:  Number of produced ${\Lambda}^{+}_{c}$ and $\Xi^{+}_{c}$ baryons
exiting the $\rm W$ target as a function of the target thickness. The case of
$1.37\times 10^{13}$ $\mathrm{\,PoT}$, corresponding to two years running in
LHCb, is shown.

The number of ${\Lambda}^{+}_{c}$ baryons produced in a target of thickness
$T$ and traversing a bent crystal of length $L$ is given by the convolution of

$N_{{{\Lambda}^{+}_{c}},\textrm{cry}}(T,L,\beta\gamma)=N_{{{\Lambda}^{+}_{c}}}(T,\beta\gamma)\epsilon_{\textrm{CH}}(\beta\gamma)\epsilon_{\textrm{DF}}(L,\beta\gamma)$
(14)

with $r(\beta\gamma)$, where $\epsilon_{\textrm{CH}}(\beta\gamma)$ is the
steering efficiency of the crystal, which includes the efficiency of the
${\Lambda}^{+}_{c}$ particle to be trapped into channeling regime, and
$\epsilon_{\textrm{DF}}(T,L,\beta\gamma)=e^{-L/\beta\gamma c\tau}$ accounts
for the particle decay flight, i.e. survival probability within the crystal
length.

###  B.3 Reconstruction of $\boldsymbol{{{\Lambda}^{+}_{c}}}$ and
$\boldsymbol{{\Xi^{+}_{c}}}$ decays with a $\boldsymbol{{{\pi}^{0}}}$ in the
final state

The momentum in the laboratory frame of a ${\Lambda}^{+}_{c}$ (and similarly
for $\Xi^{+}_{c}$) baryon decaying into final states containing an undetected
neutral pion can be reconstructed, up to a two-fold ambiguity, using kinematic
information of the remaining charged-particle decay products ($3h$). The
ambiguity arises from two different configurations in the baryon rest frame
which are indistinguishable in the laboratory frame. Either a relatively low
momentum ${\Lambda}^{+}_{c}$ charm baryon decays into the $3h$ system with a
small angle with respect to the ${\Lambda}^{+}_{c}$ flight direction, or the
$3h$ system is produced with a large angle from a higher momentum
${\Lambda}^{+}_{c}$. The ambiguity vanishes for events in which the angle
$\theta$ between the flight direction of the baryon and the $3h$ system
reaches its maximum allowed value,

$\theta_{\textrm{max}}=\arcsin\frac{\sqrt{\left(m^{2}-m_{3h}^{2}-m_{{{\pi}^{0}}}^{2}\right)^{2}-4m_{3h}^{2}m_{{{\pi}^{0}}}^{2}}}{2mp_{3h}},$
(15)

leading to a single solution

$p=\frac{p_{3h}\left(m^{2}+m_{3h}^{2}-m_{{{\pi}^{0}}}^{2}\right)\cos\theta_{\textrm{max}}}{2\left(m_{3h}^{2}+p_{3h}^{2}\sin^{2}\theta_{\textrm{max}}\right)},$
(16)

which is used as estimate of the ${\Lambda}^{+}_{c}$ momentum magnitude. Here,
$p_{3h}$ and $m_{3h}$ are the momentum magnitude and invariant mass of the
$3h$ system, respectively, $m$ is the nominal charm baryon mass, and
$m_{{{\pi}^{0}}}$ the neutral pion mass. This procedure does not impose the
additional kinematical contraint when the ${\pi}^{0}$ meson is produced via
$\Sigma^{+}$ decays. The ${\Lambda}^{+}_{c}$ three-momentum then can be
determined from the ${\Lambda}^{+}_{c}$ flight direction, itself obtained from
the knowledge of the production (primary), $\boldsymbol{x}_{\textrm{PV}}$, and
the ${\Lambda}^{+}_{c}$ decay, $\boldsymbol{x}_{{\Lambda}^{+}_{c}}$, vertex
positions, as $\boldsymbol{p}=p\boldsymbol{u}$, where $\boldsymbol{u}$ is a
unit vector along the ${\Lambda}^{+}_{c}$ flight direction,

$\boldsymbol{u}=\frac{\boldsymbol{x}_{{\Lambda}^{+}_{c}}-\boldsymbol{x}_{\textrm{PV}}}{|\boldsymbol{x}_{{\Lambda}^{+}_{c}}-\boldsymbol{x}_{\textrm{PV}}|}.$
(17)

The average initial spin-polarization vector can then be inferred from Eq.
(8).

Figure 11(a)(b) illustrates the difference between the estimated momentum and
spin-polarization magnitudes using the kinematical recovery technique and the
true ${\Lambda}^{+}_{c}$ momentum, compared to the case when the
${\Lambda}^{+}_{c}$ three-momentum and polarization vectors are estimated
using the $3h$ system only. Whereas the former provides unbiased estimates
with resolutions about $150{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$0.05$, respectively, the latter is clearly biased. Estimated vertex
resolutions around $100{\,\mu\mathrm{m}}$ and $10\mathrm{\,mm}$ in the
transverse and longitudinal directions, respectively, along with an invariant
mass resolution of about $15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
Bagli:2017foe , do not impact significantly these distributions.

|  |
---|---|---

Figure 11:  Distributions of missing (a) momentum and (b) spin-polarization
magnitudes using the kinematical reconstruction technique for
${\Lambda}^{+}_{c}$ decays with an undetected neutral pion (red) in the final
states listed in Table 2. As a comparison, the missing momentum and
polarization magnitudes for the reconstructed $3h$ system (black), composed by
a proton combined with charged kaons and pions, are shown. The relative
contributions from the different final states are given by the effective
branching fractions ${\cal B}_{\textrm{eff}}$. (c) Corrected mass distribution
for the same decays. As a comparison the invariant mass of the reconstructed
charged $3h$ system, composed by a proton or a $\Sigma^{\pm}$ hyperon combined
with charged kaons and pions, is shown. The ${\Lambda}^{+}_{c}$ known mass is
reported as a vertical dashed line. The estimated invariant mass resolution at
LHCb, about 15 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, is negligible in
comparison to the width of the corrected mass distribution. All distributions
are normalized to unity.

To characterize efficiently and with low background signal ${\Lambda}^{+}_{c}$
decays with undetected neutral pions, the corrected mass Abe:1997sb , defined
as

$m_{\textrm{corr}}=\sqrt{m^{2}_{3h}+p^{2}_{\perp}}+p_{\perp},$ (18)

is used, where $m_{3h}$ is the reconstructed invariant mass of the three
charged tracks in the final state and $p_{\perp}$ is the momentum of the $3h$
system transverse to the charm baryon flight direction, estimated as

$p_{\perp}=|\boldsymbol{p}_{3h}\times\boldsymbol{u}|.$ (19)

Figure 11(c) shows the $m_{\textrm{corr}}$ distribution for the mixtusre of
three- and four-body ${\Lambda}^{+}_{c}$ decay modes listed in Table 2 with a
${\pi}^{0}$ in the final state, and compared with the invariant mass of the
$3h$ system and the known ${\Lambda}^{+}_{c}$ baryon mass Tanabashi:2018oca .

###  B.4 Sensitivity studies and crystal setup optimization

Sensitivity studies for the MDM and EDM of ${\Lambda}^{+}_{c}$ and
$\Xi^{+}_{c}$ baryons are based on Pythia simulations and pseudoexperiments
where the number of events is estimated according to the branching fractions
reported in Table 2 and Table 3, respectively. The values for the production
cross sections, the average event information $S^{2}$, and the gyromagnetic
factors are reported in Table 4. Other relevant parameters used for the
sensitivity studies, also reported, are evaluated for each different setup
configuration according to simulations, in particular: the channeling
efficiency $\epsilon_{\textrm{CH}}$, the decay flight efficiency
$\epsilon_{\textrm{DF}}$, the number of recontructed charm baryons
$N_{\textrm{rec}}$, the average boost $\langle\gamma\rangle$ and transverse
momentum $\langle\mbox{$p_{\mathrm{T}}$}\rangle$, and the initial average
spin-polarization vector ${\boldsymbol{s}}$. For the pseudoexperiments, Eq.
(10) is used for spin-polarization projections after precession in the
crystal.

Regions of minimal uncertainty of $d$ and $g$ factors are explored for
different crystal configurations and target thickness, with deflection angles
of 16 and $7\mathrm{\,mrad}$. These steering angles are chosen for
compatibility with the LHCb detector acceptance and to allow effective
separation of channeled ${\Lambda}^{+}_{c}$ baryons from forwardly produced
background particles in the $\rm S_{2}$ scenario Bagli:2017foe , respectively.
Crystal lengths around 10 and $7\mathrm{\,cm}$ are chosen to minimize material
for detector operation and safety while accepting a 50% and below 20% increase
of uncertainties, respectively. The crystal orientation angle and target
thickness are chosen to be $0.3\mathrm{\,mrad}$ and $2\mathrm{\,cm}$,
respectively.

To assess the benefits of using event-by-event information, Fig. 12(a) shows
sensitivity regions as a function of the crystal orientation angle and the
target thickness for combined MDM and EDM measurement, using average event
information. These regions and minimal uncertainties are to be compared to
those illustrated in Fig. 5(b). A sensitivity improvement equivalent to a data
sample size larger than a factor of three (of two for MDM measurement alone)
is observed when using event-by-event estimates of the precesion angle
$\varPhi$ and the initial polarization-vector ${\boldsymbol{s}}$ with respect
to the case when these are averaged over all events.

Figure 12: Regions of miniminal uncertainty of the $g$ (red curves) and $d$
(black) factors as a function of the crystal orientation angle $\theta_{y,C}$
and the target thickness $T$, with $a\approx-0.03$ and
${{\Lambda}^{+}_{c}}\to{p}{{K}^{-}}{{\pi}^{+}}$ decays, (a) for combined MDM
and EDM measurement using average estimates of the precesion angle $\varPhi$
and the initial polarization-vector ${\boldsymbol{s}}$, and (b) from first
order, analytical estimates as given by Eq. (20). The markers and values
represent the minimum uncertainty on the $g$ and $d$ factors, relative to
$1.37\times 10^{13}$ $\mathrm{\,PoT}$ and a 16$\mathrm{\,mrad}$ bent,
10$\mathrm{\,cm}$ long germanium crystal at room temperature. The curves are
the regions whose uncertainties are increased by 20%, 50% and 100% with
respect to the minimum.

First order, analytical approximation to the dipole moment sensitivities is
given by

$\displaystyle\sigma_{g}$ $\displaystyle\approx\frac{2}{S\langle
s_{y}\rangle\langle\gamma\rangle\theta_{C}}\frac{1}{\sqrt{N_{\rm rec}}},$
$\displaystyle\sigma_{d}$
$\displaystyle\approx\frac{g-2}{S}\frac{1}{\sqrt{\langle
s_{y}\rangle^{2}\eta^{2}+2\langle s_{x}\rangle^{2}\eta}}\frac{1}{\sqrt{N_{\rm
rec}}},$ (20)

with $\eta=1-\cos\langle\varPhi\rangle$, which follows from Eq. (10) in the
limit $a\gg d,\ 1/\gamma$,

$\displaystyle s^{\prime}_{x}$ $\displaystyle\approx
s_{y}\frac{d}{g-2}(1-\cos\varPhi)+s_{x},$ $\displaystyle s^{\prime}_{y}$
$\displaystyle\approx s_{y}\cos\varPhi+s_{x}\frac{d}{g-2}(1-\cos\varPhi),$
$\displaystyle s^{\prime}_{z}$ $\displaystyle\approx-
s_{y}\sin\varPhi+s_{x}\frac{d}{g-2}\sin\varPhi,$ (21)

where $\varPhi$ is given by Eq. (1). In this case the uncertainties are
estimated separately for events with positive and negative $p_{x_{L}}$, and
then combined. Figure 12(b) shows the corresponding sensitivity regions.
Differences with respect to Fig. 12(a) are related to the approximations in
Eq. (21) and the correlations between $g$ and $d$ in the combined measurement,
neglected in the analytical estimates. The dependence of the EDM sensitivity
with the crystal orientation angle is more evident in Fig. 12(b) due to an
incomplete and approximated use of the spin precession information.

## Acknowledgements

We express our gratitude to our colleagues of the LHCb collaboration, in
particular to M. Ferro-Luzzi, M. Palutan, C. Parkes, P. Robbe N. Tuning, and
from CERN, in particular A. Fomin, D. Mirarchi, S. Redaelli for stimulating
discussions and very useful feedback. We would like to thank S. Forte and J.
Rojo for useful discussions on parton distribution functions. A. Sytov thanks
V. Haurylavets and A. Leukovich for providing help with Geant4 simulations. We
acknowledge support from INFN (Italy), MICINN and GVA (Spain), the ERC
Consolidator Grants SELDOM G.A. 771642 and CRYSBEAM G.A. 615089. In addition,
we acknowledge the CINECA award under the ISCRA initiative for the
availability of high performance computing resources and support.

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