#  Feasibility of measuring the magnetic dipole moments of the charm baryons
at the LHC using bent crystals

A.S. Fomin<EMAIL_ADDRESS>LAL (Laboratoire de l’Accélérateur Linéaire),
Université Paris-Sud/IN2P3, Orsay, France  NSC Kharkiv Institute of Physics
and Technology, 61108 Kharkiv, Ukraine  V.N. Karazin Kharkiv National
University, 61022 Kharkiv, Ukraine  A.Yu. Korchin<EMAIL_ADDRESS>NSC
Kharkiv Institute of Physics and Technology, 61108 Kharkiv, Ukraine  V.N.
Karazin Kharkiv National University, 61022 Kharkiv, Ukraine  A. Stocchi
<EMAIL_ADDRESS>LAL (Laboratoire de l’Accélérateur Linéaire), Université
Paris-Sud/IN2P3, Orsay, France  O.A. Bezshyyko  Taras Shevchenko National
University of Kyiv, 01601 Kyiv, Ukraine  L. Burmistrov  LAL (Laboratoire de
l’Accélérateur Linéaire), Université Paris-Sud/IN2P3, Orsay, France  S.P.
Fomin  NSC Kharkiv Institute of Physics and Technology, 61108 Kharkiv, Ukraine
V.N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine  I.V.
Kirillin  NSC Kharkiv Institute of Physics and Technology, 61108 Kharkiv,
Ukraine  V.N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine  L.
Massacrier  IPNO (Institut de Physique Nucléaire), Université Paris-Sud/IN2P3,
Orsay, France  A. Natochii  Taras Shevchenko National University of Kyiv,
01601 Kyiv, Ukraine  LAL (Laboratoire de l’Accélérateur Linéaire), Université
Paris-Sud/IN2P3, Orsay, France  P. Robbe  LAL (Laboratoire de l’Accélérateur
Linéaire), Université Paris-Sud/IN2P3, Orsay, France  W. Scandale  LAL
(Laboratoire de l’Accélérateur Linéaire), Université Paris-Sud/IN2P3, Orsay,
France  CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland  INFN Sezione di Roma, Piazzale Aldo Moro 2, 00185 Rome, Italy
N.F. Shul’ga  NSC Kharkiv Institute of Physics and Technology, 61108 Kharkiv,
Ukraine  V.N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine

(May 9, 2017)

###### Abstract

In this paper we revisit the idea of measuring the magnetic dipole moments of
the charm baryons and, in particular, of $\Lambda_{c}^{+}$ by studying the
spin precession induced by the strong effective magnetic field inside the
channels of a bent crystal. We present a detailed sensitivity study showing
the feasibility of such an experiment at the LHC in the coming years.

###### pacs:

13.30.Eg, 13.40.Em, 13.88+e, 14.20.Lq, 61.85.+p

##  I  Introduction

The magnetic dipole moment (MDM) of a particle is its fundamental
characteristic that determines the torque which particle experiences in an
external magnetic field. The MDMs of many particles are presently known
PDG:2014 . For electron the QED prediction agrees with experimentally measured
value up to very high precision. For muon the measurement of the BNL E821
experiment Bennett:2006fi  disagrees with the Standard Model prediction by 3–4
standard deviations, which may suggest physics beyond the Standard Model. The
disagreement for the muon $g-2$ is the subject of many studies (see, e.g.,
review Jegerlehner:2009 ). The MDM of the $\tau$-lepton has not been measured
so far and is of great interest for testing calculations in the Standard Model
Eidelman:2007 .

For hadrons, the MDMs are measured for the baryon octet with
$J^{P}={\tfrac{1}{2}}^{+}$. Historically, reasonable agreement between the
measured MDM and predictions of the quark model was important to substantiate
the constituent quark models of the hadrons.

In general, the MDM of the spin-$\tfrac{1}{2}$ particle is expressed as

$\vec{\mu}=\frac{2\mu}{\hbar}\vec{S},\qquad\quad\mu=\frac{q\hbar}{2mc}\,\frac{g}{2},$
(1)

where $\vec{S}=\tfrac{\hbar}{2}\vec{\sigma}$, $m$ is the particle mass, $q$ is
the particle electric charge, $g$ is the gyromagnetic factor. The value $g=2$
corresponds to a Dirac particle without magnetic moment anomaly. Usually, the
MDM of baryons is measured in units of the nuclear magneton $\mu_{N}\equiv
e\hbar/(2m_{p}c)$ PDG:2014 , where $m_{p}$ is the proton mass and $e$ is the
elementary charge.

It would be very important to measure the MDM of the charm baryons
$\Lambda_{c}^{+}(udc)$ and $\Xi_{c}^{+}(usc)$, which have not been measured so
far because of their very short lifetime of the order of $10^{-13}$ s.

There has been many calculations of the MDM of the charm baryons in various
models of their structure Franklin:1981 ; Barik:1984 ; Savage:1994 ;
SilvestreBrac:1996 ; Zhu:1997 ; Aliev:2002 ; Julia-Diaz:2004 ; Albertus:2006 ;
Kumar:2005 ; Faessler:2006 ; Karliner:2006ny ; Patel:2008 ; Majethiya:2008 ;
Aliev:2008_1 ; Aliev:2008_2 ; Sharma:2010 ; Bernotas:2013 . As for the
$\Lambda_{c}^{+}$ baryon, majority of the calculations predict the MDM and
$g$-factor in the ranges

$\frac{\mu({\Lambda_{c}^{+}})}{\mu_{N}}=0.37\text{--}0.42,\qquad
g({\Lambda_{c}^{+}})=1.80\text{--}2.05.$ (2)

Thus, an experimental study of the MDM of heavy baryons can be useful to
distinguish between different theoretical approaches.

One of the motivations for measurement of the MDM of the heavy baryons is also
studying the MDM of the charm quark. If this quark behaves as a point-like
Dirac particle, then the corresponding gyromagnetic factor $g_{c}$ is equal or
close to 2, while if the charm quark has a composite structure we can expect a
sizable deviation from this value.

In the quark model the MDM of the heavy baryon is expressed in terms of the
MDMs of the heavy and light quarks. In particular, for the charm baryons, the
spin and flavor structure of the ground-state baryons $\Lambda_{c}^{+}$ and
$\Xi_{c}^{+}$ implies that (see, e.g., Ref. Franklin:1981 )

$\mu({\Lambda_{c}^{+}})=\mu_{c},\qquad\mu({\Xi_{c}^{+}})=\frac{1}{3}\left(2\mu_{u}+2\mu_{s}-\mu_{c}\right).$
(3)

MDMs in Eqs. (3) depend on the MDM of the charm quark. Let us consider
$\Lambda_{c}^{+}$ and take “effective” mass of the $c$-quark $m_{c}=1.6$ GeV
as suggested from the charmonia spectroscopy Franklin:1981 . Keeping
explicitly the $g$-factor of the charm quark we can write

$\frac{\mu({\Lambda_{c}^{+}})}{\mu_{N}}=0.39\frac{g_{c}}{2},\qquad
g({\Lambda_{c}^{+}})=1.91\frac{g_{c}}{2}.$ (4)

For $g_{c}=2$ these values are consistent with Eqs. (2).

For $\Xi_{c}^{+}$ one needs to specify also the masses of the light
constituent quarks. Choosing $m_{u}=336$ MeV and $m_{s}=509$ MeV, which
reproduce MDMs of the baryon octet Perkins:2000 , one obtains from (3)

$\frac{\mu({\Xi_{c}^{+}})}{\mu_{N}}=0.83-0.13\frac{g_{c}}{2},\quad
g({\Xi_{c}^{+}})=4.37-0.69\frac{g_{c}}{2},$ (5)

where the first numbers in each quantity in (5) come from the $u$ and $s$
quarks, and the second — from the $c$ quark.

The combined measurements of MDMs of $\Lambda_{c}^{+}$ and $\Xi_{c}^{+}$ may
help to obtain information on the $g$-factor of the charm quark.

In the present paper we discuss the feasibility of the MDM measurement for the
positively charged charm baryons $\Lambda_{c}^{+}$ and $\Xi_{c}^{+}$ at the
LHC. This extends the proposal of the UA9 collaboration Burmistrov:2016 .

##  II  Principle of measurement

The experimental results on MDM are all obtained by a well-assessed method
that consists of measuring the polarization vector of the incoming particles
and the precession angle when the particle is traveling through an intense
magnetic field. The polarization is evaluated by analyzing the angular
distribution of the decay products. No measurement of magnetic moments of
charm or beauty baryons (and $\tau$ lepton) has been performed so far. The
main reason is that the lifetimes of charm/beauty baryons are too short to
measure the magnetic moment by standard techniques.

One proposal to meet the challenge of measuring the magnetic moments of
baryons with heavy flavored quarks is to use the strong effective magnetic
field inside the channels of a bent crystal instead of the conventional
magnetic field to induce the precession of the polarization vector and measure
the magnetic moment. Some theoretical aspects of this phenomenon with possible
applications to the LHC have recently been discussed in Baryshevsky:2016 ,
where the author carried out the preliminary estimations of the possibilities
to measure MDMs of the short-lived particles, in particular, charmed baryons
at the LHC energies. In Ref. Botella:2016  the authors suggested to use this
method for studying the electric dipole moments (EDM) of the strange $\Lambda$
baryon and the charm baryons.

The theoretical formalism of the precession of the polarization vector of
spin-$\tfrac{1}{2}$ particle in external electric, $\vec{E}$, and magnetic,
$\vec{H}$, fields has been known for a long time Thomas:1926 ; Thomas:1927 ;
Bargmann:1959 ; Hagedorn:1963 ; Beresteckii:1982 ; Jackson:1999 . In Refs.
Baryshevsky:1979 ; Lyuboshits:1980 ; Kim:1983 ; Biryukov ; Akhiezer ;
grininko1991 ; Greenenko:1992ef  this formalism was applied to the case of the
bent crystals.

In the planned fixed-target experiment at the LHC, the high-energy proton beam
produces the polarized charm baryons by interacting with nuclei of a target-
converter

$p+A\to\Lambda_{c}^{+}(\Xi_{c}^{+})+X,$ (6)

which are directed into the bent crystal. The initial polarization vector
$\vec{\xi}_{i}$ of the charm baryon is perpendicular to the reaction plane
spanned by the proton and baryon momenta, $\vec{q}$ and $\vec{p}$,
respectively, because of the space-inversion symmetry of the strong
interaction.

When falling on a bent crystal, a small fraction of baryons gets in the regime
of planar channeling (see, e.g., Tsyganov ; Biryukov ; Akhiezer ). Note that
only positively charged particles can be efficiently deflected by a bent
crystal using planar channeling phenomenon. The planar channeling of
negatively charged particles is very unstable due to the enhancement of their
multiple scattering on lattice atoms (see, e.g., fomin1997 ). However, the
negatively charged particle can be also deflected using the so-called
stochastic mechanism of multiple scattering by atomic strings of a bent
crystal. This mechanism was proposed in grininko1991 . The possibility to use
it for the MDM measurement was considered in Greenenko:1992ef .

The motion of channeled relativistic baryons in the inter-plane electric field
of a bent crystal imitates the particle motion in a strong magnetic field
directed along the crystal bending axis (axis $Oy$ in Fig. 1). The MDM vector
of baryon rotates around this axis. The gradient of the inter-plane electric
field of a silicon crystal reaches the maximum value about 5 GeV/cm that
corresponds to the value of the induction of effective magnetic field of
thousands of tesla in the rest frame of a TeV baryon. The initial value of the
3D polarization vector can be determined using the non-channeled baryons. The
absolute value of the polarization can be also measured as a by-product of
this experiment. Various aspects of this analysis will be discussed later.

The first experimental realization of such method was carried out in Fermilab
Chen:1992  at the 800 GeV proton beam. The strange $\Sigma^{+}(uus)$ baryons
(with lifetime $0.8\times 10^{-10}\,$s) produced on the Cu target had average
momentum 375 GeV/c and the absolute value of polarization $(12\,\pm\,1)$ %.
After passing 4.5 cm of the bent silicon single crystal the polarization
vector precessed by about $60^{\circ}$. This new technique allowed to obtain
the MDM of the $\Sigma^{+}$ hyperon $\mu=(2.40\pm 0.46_{stat}\pm
0.40_{syst})\,\mu_{N}$ which was consistent with the world-average value.

The proposed experiment at the LHC is much more difficult because the
lifetimes of the charm baryons $\Lambda_{c}^{+}$ and $\Xi_{c}^{+}$ are three
orders of magnitude less than the lifetime of $\Sigma^{+}$. In order to
measure the angle of MDM precession with sufficient accuracy and
correspondingly extract the MDM at the LHC energies, it is necessary to carry
out the optimization of target-converter and bent crystal parameters by means
of detailed computer simulation as well as to study the properties of charm
baryons as it is discussed in detail later.

###  II.1  Spin precession in a bent crystal.
Master formulas

Because of the extremely short lifetime of charmed baryons in comparison with
the $\Sigma^{+}$ hyperon, in our case it is not possible to prepare a beam of
polarized baryons in advance and to measure the degree of their initial
polarization, as was done in the Fermilab experiment Chen:1992 . In our case,
as explained below, the crystal could be used as a beam collimator.

To be captured into the channeling regime, the incoming particle must have a
very small angle $\theta_{x}$ between its momentum and the crystal plane of
the chosen channel, namely, $|\theta_{x}|<\theta_{\rm{L}}$, where $\theta_{\rm
L}$ is the Lindhard angle Lindhard :

$\theta_{\rm L}=\sqrt{\frac{4\pi\,n\,d\,a_{\rm TF}\,Z\,e^{2}}{\varepsilon}},$
(7)

where $n$ is the crystal atomic density, $d$ is the distance between
neighboring planes, $a_{\rm TF}$ is the Thomas-Fermi screening radius, Z$|e|$
is the charge of atomic nucleus, $\varepsilon$ is the energy of incoming
particle. The Lindhard angle is the critical angle of planar channeling for an
ideal crystal case. The axis $Ox$ is perpendicular to the channel plane (see
Fig. 1).

The $\Lambda_{c}^{+}$ baryons emitted from the amorphous target-converter are
polarized and isotropically distributed over the azimuthal angle around the
direction of the initial proton beam. The polar angle $\theta$ that determines
the characteristic cone of the relativistic $\Lambda_{c}^{+}$ baryon emission
in the laboratory frame has a value of the order of $\gamma^{-1}$, where
$\gamma=\varepsilon/m$ is the Lorentz factor of the $\Lambda_{c}^{+}$,
$\varepsilon$ and $m$ are its energy and mass, respectively. In the conditions
of the LHC experiment $\theta\approx 10^{-3}$ rad.

The critical angle of planar channeling (7) for particles with the energy of
several TeV in a silicon crystal is about several microradians, that is at
least two orders of magnitude smaller than a characteristic angular width of
the $\Lambda_{c}^{+}$ beam $\theta$ after the target-converter. Therefore,
only a small part of this beam can be captured in the channeling regime when
entering the crystal. For all channeled particles the angle $\theta_{x}$ is
limited in the interval $(-\theta_{\rm{L}},\,+\theta_{\rm{L}})$. At the same
time, there are no limitations on the value of $\theta_{y}$ of the
$\Lambda_{c}^{+}$ to be channeled.

Thus, the conditions for the particle capture into the planar channeling
regime pick out by themselves the region in the phase space of the
$\Lambda_{c}^{+}$ momentum with a certain direction of the polarization
vector, namely, across the channeling plane (up or down in Fig. 1).

After passing the bent crystal the polarization vector rotates by the angle
Lyuboshits:1980 ; Kim:1983

$\Theta_{\mu}=\gamma\left(\frac{g}{2}-1-\frac{g}{2\gamma^{2}}+\frac{1}{\gamma}\right)\Theta\approx\gamma\left(\frac{g}{2}-1\right)\Theta,$
(8)

with respect to the direction of the initial polarization vector. Here
$\Theta=L/R$ is the deflection angle of the channeled baryon momentum after
passing the bent crystal, $L$ and $R$ are the length and bending radius of the
crystal. A simple derivation of Eq. (8) is presented in Appendix A.

In the conditions of the LHC the Lorentz factor $\gamma$ can be quite big, of
the order of $10^{3}$. In this case the approximate equality in (8) holds
(unless incidentally the relation $g=2$ happens).

The schematic layout of the experiment is shown in Fig. 1. To simplify the
following formulae and for better understanding the experiment layout, here we
consider $\Lambda_{c}^{+}$ baryons to be parallel to the $z$ axis. In our
further calculations we take into account the proper angular distribution of
baryons at the entrance into the crystal.

In this frame the components of the proton momentum $\vec{q}$, baryon initial
$\vec{p}_{i}$ and final $\vec{p}_{f}$ momenta, effective electric field
$\vec{E}_{i}$ and $\vec{E}_{f}$ in the crystal, rotation axis along
$\vec{E}\times\vec{p}$, and the initial $\vec{\xi}_{i}$ and final
$\vec{\xi}_{f}$ polarization vectors are:

$\displaystyle\vec{q}$ $\displaystyle=$ $\displaystyle(0,\,q_{y},\,q_{z}),$
$\displaystyle\vec{p}_{i}$ $\displaystyle=$ $\displaystyle
p\,(0,\,0,\,1),\qquad\vec{p}_{f}=p\,(-\sin\Theta,\,0,\,\cos\Theta),$
$\displaystyle\vec{E}_{i}$ $\displaystyle=$ $\displaystyle
E\,(-1,\,0,\,0),\quad\vec{E}_{f}=E\,(-\cos\Theta,\,0,\,-\sin\Theta),$
$\displaystyle\vec{E}\times\vec{p}$ $\displaystyle=$ $\displaystyle
E\,p\,(0,\,1,\,0),$ $\displaystyle\vec{\xi}_{i}$ $\displaystyle=$
$\displaystyle\xi\,(1,\,0,\,0),\qquad\vec{\xi}_{f}=\xi\,(\cos\Theta_{\mu},\,0,\,\sin\Theta_{\mu}).$
(9)

The absolute value of polarization $\xi=|\vec{\xi}|$ stays constant and is
determined by the process (6).

Figure 1: Schematic layout of experiment. Effective electric field $\vec{E}$
is orthogonal to the momentum $\vec{p}$. The figure shows the case $g>2$.

###  II.2  Basic principles of the angular analysis

The orientation of the baryon polarization vector after the crystal can be
determined from the angular distribution of its decay products. For the weak
decays of the spin-$\tfrac{1}{2}$ baryon into the two-particle final states of
baryon and meson ($\tfrac{1}{2}\to\tfrac{1}{2}+0$,
$\tfrac{1}{2}\to\tfrac{1}{2}+1$, $\tfrac{1}{2}\to\tfrac{3}{2}+0$) the
following relation holds

$\frac{1}{N}\frac{dN}{d\cos\vartheta}=\frac{1}{2}(1+\alpha\,\xi\cos\vartheta),$
(10)

in the rest frame of the baryon (see Appendix B). Here $N$ is the number of
events, $\vartheta$ is the angle between the direction of final baryon
(analyzer) and the polarization vector $\vec{\xi}_{f}$. The weak-decay
parameter $\alpha$ characterizes parity violation in the decay.

From the angular analysis one can obtain the expression for the absolute
statistical error of the measured $g$-factor:

$\Delta g=\frac{1}{\ \alpha\,|\xi|\,\gamma\,\Theta~{}}\
\sqrt{\frac{12}{~{}N_{\Lambda_{c}^{+}}~{}}},$ (11)

where $N_{\Lambda_{c}}$ is the number of reconstructed $\Lambda_{c}^{+}$
deflected by a bent crystal. Note that Eq. (11) is obtained for a fixed value
of boost $\gamma$.

The values of absolute polarization $|\xi|$ and weak-decay parameter $\alpha$
are crucial, since the $g$-factor error $\Delta g$ is inversely proportional
to these values.

Figure 2: Polarization of $\Lambda_{c}^{+}$ as a function of its transverse
momentum. Experimental data: red crosses E791 , orange rectangular area
PolLambdac ; dashed red curves — experimental data fit by the normal
distribution; solid red curve — theoretical prediction by the so-called hybrid
model Goldstein  for the process $\pi^{-}p\to\Lambda_{c}^{+}X$. Channeled
baryons distribution over transverse momentum: blue histogram (simulation
results obtained using Pythia8).

The polarization of the $\Lambda_{c}^{+}$ baryons has been measured in the
reaction of 230 GeV/c protons with a Copper target and gives
P($\Lambda_{c}^{+}$) = $-0.65\,^{+0.22}_{-0.18}$ at transverse momentum
$p_{t}>1.2$ GeV/c PolLambdac  (the sign of the polarization is defined with
respect to the normal to the production plane, $\vec{q}\times\vec{p}_{i}$).
The E791 experiment E791  finds evidence for an increasingly negative
polarization of $\Lambda_{c}^{+}$ as a function of $p_{t}^{2}$, in agreement
with the model dha ; Goldstein . These data are shown in Fig. 2 together with
fitted curves.

In the same plot we show the theoretical prediction in the so-called hybrid
model Goldstein  (for the process $\pi^{-}p\to\Lambda_{c}^{+}X$) describing
the $\Lambda_{c}^{+}$ polarization as a function of transverse momentum.

Using simulation code Pythia version 8.1 (Pythia8) Pythia  we show the
transverse momentum distribution of channeled $\Lambda_{c}^{+}$ baryons (see
blue histogram in Fig. 2).

By convolving the transverse momentum distribution and polarization curve as a
function of transverse momentum we obtain the mean square value of
$\Lambda_{c}^{+}$ polarization around -0.37 and $-0.40\pm 0.05$ for the
theoretical prediction and experimental data, respectively.

No such measurements exist for the $\Xi_{c}^{+}$ baryons. It is also important
to mention that the absolute polarizations of $\Lambda_{c}^{+}$ and of
$\Xi_{c}^{+}$ as a function of transverse momentum could be measured by the
proposed experiment.

In addition, they could also be measured by using the available data on beam
gas interaction at the LHCb (SMOG data SMOG ).

The weak-decay parameter $\alpha$ is the decay-channel-dependent quantity and
it is compiled for various decay channels in case of the $\Lambda_{c}^{+}$
baryon in Table 1.

For the decay channels containing $\Lambda$ or $\Sigma^{+}$ in the final
states, the parameter $\alpha$ has been measured. The decay channel
$\Lambda_{c}^{+}\to p\,K^{-}\,\pi^{+}$ has a large branching fraction and it
would be interesting to use this decay mode for the MDM measurement. The E791
experiment E791  reports measurements of the amplitudes for $\Lambda_{c}^{+}$
decay into nonresonant $p\,K^{-}\,\pi^{+}$ and to
$p\,\overline{K}^{*}(890)^{0}$, $\Delta^{++}(1232)\,K^{-}$, and
$\Lambda(1520)\,\pi^{+}$ modes. Using the measured amplitudes the values of
the weak parameter $\alpha$ can be extracted with large errors as in
Botella:2016 . It would be extremely important to perform this analysis using
the LHCb data. On the other hand, no measurement of the $\alpha$ parameters
exists in case of $\,\Xi_{c}^{+}$, and it would be important to measure these
parameters in the LHCb experiments.

Table 1: Branching fractions and weak-decay parameters $\alpha$ for different decay modes of $\Lambda_{c}^{+}$. Channel | Fraction ($\Gamma_{j}/\Gamma$) | $\alpha$ | Source
---|---|---|---
$\Lambda_{c}^{+}\to\Lambda\pi^{+};\,\,\Lambda\to p\pi^{-}$ |  $(1.07\pm 0.28)\,\%$ $\times$ $(63.9\pm 0.5)\,\%$ | $-0.91\pm 0.15$ | PDG:2014
$\Lambda_{c}^{+}\to\Lambda e^{+}(\mu^{+})\nu_{e(\mu)};\,\,\Lambda\to p\pi^{-}$ |  $(2.0\pm 0.6)\,\%$ $\times$ $(63.9\pm 0.5)\,\%$ | $-0.86\pm 0.04$ | PDG:2014
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ | $(5.0\pm 1.3)\,\%$ | – | PDG:2014
$\Lambda_{c}^{+}\to\Delta(1232)^{++}K^{-};\,\,\Delta(1232)^{++}\to p\pi^{+}$ |  $(0.86\pm 0.3)\,\%$ $\times$ 99.4 % | $-0.67\pm 0.30$ | Botella:2016
$\Lambda_{c}^{+}\to p\,\overline{K}^{*}(892)^{0};\,\,\overline{K}^{*}(892)^{0}\to K^{-}\pi^{+}$ | (1.6 $\pm$ 0.5) % $\times$ 100 % | -0.545 $\pm$ 0.345 | Botella:2016
$\Lambda_{c}^{+}\to\Lambda(1520)\pi^{+};\,\,\Lambda(1520)\to pK^{-}$ | (1.8 $\pm$ 0.6) % $\times$ (45 $\pm$ 1) % | -0.105 $\pm$ 0.604 | Botella:2016

##  III  The sensitivity studies

In this paper we have performed a sensitivity study for measuring the MDM of
$\Lambda_{c}^{+}$ produced by the strong interaction of high-energy proton
beam impinging into a target-converter of a dense material. For this analysis
we decide to consider only the $\Lambda_{c}^{+}$ baryons which decayed after
having passed the full length of the crystal.

The number of reconstructed $\Lambda_{c}^{+}$ that were deflected by a bent
crystal can be expressed as follows:

$N_{\Lambda_{c}}=\Phi\ t\ \eta_{\rm{det}}\ \frac{\Gamma_{j}}{\Gamma}\
N_{\rm{tar+crys}},$ (12)

where $N_{\rm{tar+crys}}$ is the number of deflected $\Lambda_{c}^{+}$ per
proton:

$N_{\rm{tar+crys}}=\int\frac{\partial N_{\rm tar}}{\partial\varepsilon}\
\eta_{\rm{def}}\ e^{-\frac{L_{\rm{crys}}}{c\tau\gamma}}\,d\varepsilon.$ (13)

Here $\frac{\partial N_{\rm{\rm{tar}}}}{\partial\varepsilon}$ is the
$\Lambda_{c}^{+}$ energy distribution after the target:

$\frac{\partial
N_{\rm{\rm{tar}}}}{\partial\varepsilon}=\rho\,N_{\rm{A}}\,\sigma_{\Lambda_{c}}\,\frac{A_{\rm{tar}}}{M_{\rm{tar}}}\,\frac{\partial
N}{\partial\varepsilon}\,\int\limits_{0}^{L_{\rm{tar}}}e^{-\frac{L}{c\tau\gamma}}\
dL.$ (14)

Then, taking into account the energy distribution of $\Lambda_{c}^{+}$, we
obtain the expression for the absolute statistical error of measured
$g$-factor:

$\Delta g=\frac{1}{\ \alpha\,|\xi|\,\Theta\ }\ \sqrt{\frac{12}{\ \Phi\ t\
\eta_{\rm det}\ \frac{\Gamma_{j}}{\Gamma}\ \int\ \frac{\partial N_{\rm
tar+crys}}{\partial\varepsilon}\,\gamma^{2}\ d\varepsilon\ }}.$ (15)

The definitions of different terms in Eqs. (12)–(15) and their values are
given in Table 2 and discussed in the following sections.

Table 2: List of notations in Eqs. (12)–(15).  Terms in Eqs. (12)–(15) | Values | Units
---|---|---
Proton flux, $\Phi$ | $5\times 10^{8}$ | s-1
Time of data taking, $t$ | $\sim 10^{6}$ | s
Detection efficiency, $\eta_{\rm{det}}$ | 0.002–0.03 | -
Deflection efficiency, $\eta_{\rm{def}}$ | (see Sec. III.3) | -
Crystal length, $L_{\rm{crys}}$ | 4–12 | cm
$\Lambda_{c}^{+}$ decay length, $c\tau$ | 60.0 |  $\mu$m
Lorentz factor of $\Lambda_{c}^{+}$, $\gamma$ | 500–2000 | -
Normalized production spectra, $\frac{\partial N}{\partial\cal E}$ | (see Fig. 3) | TeV-1
Cross section ($p$+$N$$\rightarrow$$\Lambda_{c}^{+}$+$\dots$), $\sigma_{\Lambda_{c}}$ | $13.6\pm 2.9$ |  $\mu$b
Target density, $\rho$ | 19.25 | gr/cm3
Avogadro number, $N_{\rm{A}}$ | $6.022\times 10^{23}$ | mol-1
Nucleon number of target, $A_{\rm{tar}}$ | 183.84 | -
Molar mass of target, $M_{\rm{tar}}$ | 183.84 | gr/mol
Target thickness, $L_{\rm{tar}}$ | 0.5–2 | cm

###  III.1  $\Lambda_{c}^{+}$ production cross section: $\sigma_{\Lambda_{c}}$

The center-of-mass energy for the fixed target experiment at the 7 TeV LHC
proton beam is $\sqrt{s}$ = 115 GeV and no measurements of the
$\sigma(\Lambda_{c})$ cross section exist at this center-of-mass energy. For
this study the $\Lambda_{c}$ cross section has been estimated from the total
charm production cross section or explicitly from the $\Lambda_{c}$ cross
section measured at different center-of-mass energies.

The PHENIX experiment in proton-proton collisions at $\sqrt{s}$ = 200 GeV
measured the total charm cross section to be 567 $\pm$ 57 (stat) $\pm$ 224
(syst) $\mu$b PHENIX  which is compatible with their previous measurement
$\sigma_{c\bar{c}}$ = 920 $\pm$ 150 $\pm$ 540 $\mu$b in Ref. Adler:2005fy  and
the one derived from the analysis of Au-Au collisions Adler:2004ta
($\sigma_{c\bar{c}}$ = 622 $\pm$ 57 $\pm$ 160 $\mu$b). If we rescale the cross
sections at $\sqrt{s}$ = 115 GeV assuming a linear energy dependence, we
obtain $\sigma_{c\bar{c}}$ = 326 $\pm$ 33 $\pm$ 129 $\mu$b,
$\sigma_{c\bar{c}}$ = 529 $\pm$ 86 $\pm$ 311 $\mu$b and $\sigma_{c\bar{c}}$ =
358 $\pm$ 33 $\pm$ 92 $\mu$b, respectively. In the following, we considered
the weighted average of the three experimental results: $\sigma_{c\bar{c}}$ =
357 $\pm$ 77 $\mu$b. The results from the linear interpolation are in
agreement within 1.7$\,\sigma$ with the c$\bar{c}$ cross section obtained with
the Helaconia MC generator Shao:2012iz  in Ref. Massacrier:2015qba .

The $\Lambda_{c}$ fragmentation function (7.6 $\pm$ 0.7 ($\pm$ 2 %)) has been
taken from Ref. Gladilin:1999pj , as the average of the results from the CLEO
($f_{c\rightarrow{\Lambda_{c}}}$ = 8.1 $\pm$ 1.2 $\pm$ 1.4 %), ARGUS
($f_{c\rightarrow{\Lambda_{c}}}$ = 7.3 $\pm$ 1.0 $\pm$ 1.0 %), ALEPH
($f_{c\rightarrow{\Lambda_{c}}}$ = 7.8 $\pm$ 0.8 $\pm$ 0.4 %), DELPHI
($f_{c\rightarrow{\Lambda_{c}}}$ = 8.6 $\pm$ 1.8 $\pm$ 1.0 %) and OPAL
($f_{c\rightarrow{\Lambda_{c}}}$ = 4.8 $\pm$ 2.2 $\pm$ 0.8 %) experiments.
Predictions from Pythia8 ($f_{c\rightarrow{\Lambda_{c}}}$ = 7.21 $\pm$ 0.04 %)
and models in Ref. fragm  ($f_{c\rightarrow{\Lambda_{c}}}$ = 5.88 $\%$ (L0)
and 5.74 $\%$ (NLO)) are in agreement within the large uncertainties. Finally,
we get $\sigma(\Lambda_{c})$ = 27.1 $\pm$ 9.5 $\mu$b.

On the other hand, we can use the LHCb $\Lambda_{c}$ cross section measurement
in pp collisions at $\sqrt{s}=$ 7 TeV Aaij:2013mga . In this case the cross
section is reported in specific rapidity $y$ and transverse momentum $p_{\rm
t}$ ranges. It is equal to $\sigma_{\Lambda_{c}}\,$(2.0$\,<y<\,$4.5,
0$\,<p_{\rm t}<\,$8 GeV/c) = $233\pm 77$ $\mu$b. We used Pythia8 to
interpolate the cross section to the full $p_{\rm t}$ and rapidity range. The
correction factor is found to be 19.2 $\pm$ 0.3 $\%$. We then extrapolate
linearly the total $\Lambda_{c}$ cross section to the energy of $\sqrt{s}$ =
115 GeV. We obtain $\sigma(\Lambda_{c})$ = 19.9 $\pm$ 6.6 $\mu$b.

Finally, we can use the measurements of the D mesons cross section performed
in pA collisions at HeraB at a center-of-mass energy of $\sqrt{s}$ = 42 GeV
Abt:2007zg . The measurement of the $D^{0}$, $D^{+}$ and $D^{+}_{s}$ were used
to calculate the total charm cross section which is found to be
$\sigma_{c\bar{c}}$ = 49.1 $\pm$ 4.6 $\pm$ 7.4 $\mu$b. After energy
extrapolation, the total charm cross section at $\sqrt{s}=115$ GeV is
$\sigma_{c\bar{c}}$ = 134.4 $\pm$ 12.6 $\pm$ 20.3 $\mu$b. Assuming the
fragmentation function for the $\Lambda_{c}$ given previously, one gets
$\sigma(\Lambda_{c})$ = 10.2 $\pm$ 3.4 $\mu$b.

These three evaluations are compatible within less than 1.7 standard
deviations. The spread of the values is explained by the poorly known total
charm cross section, the poorly known $\Lambda_{c}$ fragmentation function and
the lack of experimental open charm data close to $\sqrt{s}$ = 115 GeV. For
the sensitivity study we took the weighted mean of the three values,
$\sigma(\Lambda_{c})$ = 13.6 $\pm$ 2.9 $\mu$b.

###  III.2 $\Lambda_{c}^{+}$ energy distribution: $\frac{\partial
N_{\rm{\rm{tar}}}}{\partial\varepsilon}$

The $\Lambda_{c}^{+}$ produced in the target-converter will have a wide energy
spectrum from zero to the energy of the incident proton. Low-energy
$\Lambda_{c}^{+}$, constituting a majority of the produced particles, can not
be deflected by a bent crystal at a sufficiently large angle to be used for
measuring MDM, due to their rapid decay. The normalized energy distributions
of baryons produced by a 7 TeV proton in a tungsten target of zero thickness
are shown in Fig. 3. These results are obtained using Pythia8.

Figure 3:  Energy distribution of $\Lambda_{c}^{+}$ baryons produced by 7 TeV
protons in $p\,$-$\,N$ collision in a fixed target normalized to one produced
$\Lambda_{c}^{+}$ baryon. Solid blue curve is for the initial distribution
$(L\,$=$\,0)$, dashed curves are for different distances from the production
point (listed on the right).

The simulation gives also the angular distribution of produced
$\Lambda_{c}^{+}$, which is important for the determination of the
$\Lambda_{c}^{+}$ beam fraction that could be captured in the channeling
regime in a bent crystal. For the energies higher than 0.5 GeV the
distribution is very close to the normal one with a standard deviation
$\approx\frac{1}{2}\ \gamma^{-1}$, that in the case of $\Lambda_{c}^{+}$
baryon energies of several TeV is of the order of milliradians.

Figure 4 shows the $\Lambda_{c}^{+}$ differential energy distribution after
the target (see Eq. (14)) for different target thicknesses with the parameters
listed in Table 2 and the normalized spectra given in Fig. 3 for $L=0$.

Figure 4:  Spectra of $\Lambda_{c}^{+}$ baryons right after the tungsten
targets of different thicknesses $L_{\rm{tar}}$ (listed on the right).

At high energies the number of $\Lambda_{c}^{+}$ is proportional to the target
thickness. Furthermore, the specific ionization losses of TeV baryons in a
tungsten target are about 40 MeV/cm and therefore can be neglected as well as
the multiple scattering of the $\Lambda_{c}^{+}$ in the target, that gives a
correction of the order of percent of the value of the characteristic angular
width of $\Lambda_{c}^{+}$ production $\gamma^{-1}$. The main limitation would
come from secondary particle production in the target. This should be
carefully evaluated. For the present study we decide to use $L_{\rm{tar}}=1$
cm.

###  III.3  Deflection efficiency: $\eta_{\rm def}$

The efficiency of particle deflection $\eta_{\rm def}$ is the ratio of the
number of particles which are captured in the channeling regime and deflected
by the full angle $\Theta$ to the total number of particles impinging into the
crystal. It can be expressed as:

$\eta_{\rm def}=\eta_{\rm acc}\,\left(1-\eta_{\rm dech}\right)\ $ (16)

where $\eta_{\rm acc}$ is the acceptance factor which describes the capture of
impinging particle into the channeling regime at the crystal entrance,
$\eta_{\rm{dech}}$ is the dechanneling probability inside the crystal.

The acceptance factor $\eta_{\rm acc}$ is defined first of all by the angular
acceptance factor $\eta_{\rm ang}$ which is the fraction of particles produced
in the target-converter in the narrow interval of angles with respect to the
crystal plane ($zy$). The detailed description on how we have obtained these
parameters is presented in Appendix C.

Figure 5:  Angular acceptance factor $\eta_{\rm ang}$ (dotted blue curves),
acceptance factor $\eta_{\rm acc}$ (dashed red curves), deflection efficiency
of 8 cm bent crystal $\eta_{\rm def}$ (solid black curves) as functions of
channeled particle energy in germanium (on the left) and silicon (on the
right) crystals. Curvature radius is 7.5 m for all crystals.

The results of calculations of the angular acceptance factor $\eta_{\rm ang}$
and acceptance factor $\eta_{\rm acc}$ as functions of $\Lambda_{c}^{+}$
energy are presented by the dotted blue and dashed red curves in Fig. 5,
respectively. Note that these factors have a quite different dependence on
particle energy.

Solid black curves represent the deflection efficiency $\eta_{\rm def}$ of the
crystal of length $L_{\rm crys}=8$ cm. The difference between the solid black
and dashed red curves in Fig. 5 is caused by the dechanneling effect.

Figure 5 shows that a germanium crystal has better efficiency with respect to
a silicon one and allows one to keep more energetic $\Lambda_{c}^{+}$ which,
in addition, are more efficient for the precession of the MDM measurement, see
Eq. (15).

###  III.4  Crystal parameters optimization

To obtain the optimal crystal parameters and to compare the efficiencies of
silicon and germanium crystals we introduce the relative efficiency $\eta_{\rm
rel}$ of the MDM precession measurement with respect to the efficiency of
silicon crystal with $L_{crys}=8$ cm and $R=22$ m (further, the default
crystal). This parameter corresponds to the ratio of data taking times needed
to measure the $g$-factor with the same absolute error $\Delta g$ (see Eq.
(15)) for two different crystals:

$\eta_{rel}=\frac{t_{0}}{t}=\frac{\ \Theta^{2}\ \int\frac{\partial N_{\rm
tar}}{\partial\varepsilon}\ \eta_{\rm def}\ \gamma^{2}\
e^{-\frac{L_{\rm{crys}}}{c\tau\gamma}}\,d\varepsilon}{\ \Theta_{0}^{2}\
\int\frac{\partial N_{\rm tar}}{\partial\varepsilon}\ \eta_{\rm def,0}\
\gamma^{2}\ e^{-\frac{L_{\rm crys,0}}{c\tau\gamma}}\,d\varepsilon}.$ (17)

Here quantities with index “0” correspond to the default crystal.

In Fig. 6 the upper plot represents $\eta_{\rm{rel}}$ for silicon and
germanium crystals at room temperature and for germanium cooled down to
80${}^{\circ}\,$K as a function of crystal length $L_{\rm crys}$ calculated
for the optimal curvature radius $R$ (shown in the bottom plot).

Figure 6:  Relative efficiency of MDM precession measurement $\eta_{\rm rel}$
with respect to the efficiency of default crystal as a function of crystal
length $L_{\rm crys}$ (upper plot). Optimal curvature radius $R$ as a function
of crystal length $L_{\rm crys}$ (bottom plot).

The positions of maxima of curves in Fig. 6 (upper plot) correspond to the
optimal crystal lengths. The bottom plot shows the optimal curvature radius
$R$ as a function of crystal length $L_{\rm crys}$.

Note that $\eta_{\rm rel}$ depends only on target and crystal properties as
well as the baryon energy distribution and decay time. Thus, the optimal
crystal parameters can be found by maximizing this term for all decay channels
at once. The applicability limit for this approach is that the detector
efficiency $\eta_{\rm det}$ should not have a strong dependence on the
$\Lambda_{c}^{+}$ baryon energy. In the opposite case decay parameters
$\alpha$ and $\Gamma_{j}$ and the detection efficiency $\eta_{\rm det}$ should
be integrated together with the terms in Eq. (17) over the energy.

In Table 3 we give the results for the relative efficiency of the MDM
precession measurement $\eta_{\rm rel}$ for three values of $L_{\rm crys}$,
both for silicon and germanium crystals.

Table 3: Optimal crystal parameters |  $L_{\rm crys}$ |  $R$ | $N_{\rm tar+crys}$ | $\eta_{\rm rel}$
---|---|---|---|---
Si @ 293∘K | 4 cm |  18 m |  $3.2\times 10^{-8}~{}$ |  0.5
8 cm |  22 m |  $1.6\times 10^{-8}~{}$ |  1.0
12 cm |  25 m |  $0.9\times 10^{-8}~{}$ |  1.2
Ge @ 293∘K | 4 cm |  12 m |  $4.0\times 10^{-8}~{}$ |  1.5
8 cm |  15 m |  $1.9\times 10^{-8}~{}$ |  2.5
12 cm |  18 m |  $1.1\times 10^{-8}~{}$ |  2.8
Ge @ 80∘K | 4 cm |  10 m |  $4.8\times 10^{-8}~{}$ |  2.5
8 cm |  13 m |  $2.5\times 10^{-8}~{}$ |  4.4
12 cm |  16 m |  $1.5\times 10^{-8}~{}$ |  4.8

In the table we also give the value for the number of deflected
$\Lambda_{c}^{+}$ per incident proton $N_{\rm tar+crys}$, which can be
obtained by plugging $\eta_{\rm def}$, $\partial N_{\rm
tar}/\partial\varepsilon$ and the decay factor in Eq. (13). Note that there is
no direct relation between $N_{\rm tar+crys}$ and $\eta_{\rm rel}$ as
$\eta_{\rm rel}$ is also proportional to square of the deflection angle
$\Theta^{2}$ and square of Lorentz factor $\gamma^{2}$ of $\Lambda_{c}^{+}$.
It is important to notice that the value $N_{\rm tar+crys}$ is typically of
the order of $10^{-8}$.

For the sensitivity analysis we choose a silicon crystal at room temperature
with $L_{\rm crys}=8$ cm and $R=22$ m.

As follows from Table 3, the use of germanium crystal at room temperature
increases the efficiency by a factor 2.5 (for a germanium crystal cooled down
to 80${}^{\circ}\,$K this factor is 4.4).

###  III.5  Detector efficiency: $\eta_{\rm det}$

Many decay channels of the $\Lambda_{c}^{+}$ could be used:
$\Lambda(p\pi^{-})\,\pi^{+}$, $\Lambda\ell^{+}\nu_{\ell}$,
$p\,\overline{K}^{*0}(890)$, or $\Delta^{++}(1232)\,K^{-}$. For the first two
decay modes the weak-decay parameters $\alpha$ have been measured with a
reasonable accuracy, while only preliminary measurement of the branching
fractions and evaluations of the weak-decay parameter values are available for
the other decay modes. A specific analysis should be performed for evaluating
the detector efficiency for each of these channels. For the sensitivity
studies we have decided to select two of these decay modes:
$\Lambda(p\pi^{-})\,\pi^{+}$ and $\Delta^{++}(1232)\,K^{-}$.

For a preliminary evaluation of the detector efficiency we take the LHCb as a
reference detector, by considering typical trigger, acceptance, tracking and
vertex reconstruction efficiency. In particular, due to the very energetic
spectrum, the reconstruction of $\Lambda$ baryon is rather complicated. In
fact, the $\Lambda$ present in the final states, can be very difficult to be
detected since most of them could decay after passing the detector tracking
volume. The efficiency of the $\Lambda(p\pi^{-})\,\pi^{+}$ decay channel has
been evaluated to be in the range
$\eta_{\rm{det}}(\Lambda(p\pi^{-})\,\pi^{+})=(1\text{--}3)\times 10^{-3}$. On
the other hand, the decay mode $\Delta^{++}(1232)\,K^{-}$ seems to be more
promising and a preliminary evaluation of the efficiency gives
$\eta_{\rm{det}}(\Delta^{++}(1232)\,K^{-})=(2\text{--}4)\,\%$. The other
channels could be also used and a more precise evaluation of the detector
efficiency should be the object of dedicated studies.

###  III.6  Results of the sensitivity studies

The results of the sensitivity studies have been obtained by generating the
$\Lambda_{c}^{+}$ baryons using Pythia8 and ad hoc parametric Monte Carlo for
taking into account the correlation between the kinematic effects and the
efficiency of the channeling processes. As an example, the number of
reconstructed $\Lambda_{c}^{+}$ as a function of their energy after 40 days of
data taking with a proton flux $\Phi=5\times 10^{8}$ s-1 is shown in Fig. 7.
The red histogram shows the deflected fraction of $\Lambda_{c}^{+}$ produced
by the 7 TeV proton beam in the tungsten target of thickness $L_{\rm{tar}}=1$
cm and channeled through the silicon crystal at room temperature of length
$L_{\rm crys}=8$ cm and radius of curvature $R=22$ m. The total number of
reconstructed $\Lambda_{c}^{+}$ in this case is expected to be about 6000.

Figure 7: The spectrum of reconstructed $\Lambda_{c}^{+}$ after 40 days of
data taking with proton flux $\Phi=5\times 10^{8}$ s-1. The dotted blue curve
shows the spectra of $\Lambda_{c}^{+}$ right after the 1 cm thick tungsten
target-converter.The red histogram shows the spectrum of channeled
$\Lambda_{c}^{+}$ after the same target and silicon crystal at room
temperature with $L_{\rm crys}=8$ cm and $R=22$ m.

The initial polarization of the $\Lambda_{c}^{+}$ is supposed to be known with
high precision using the large sample of the non-channeled $\Lambda_{c}^{+}$;
the polarization in the three spatial coordinates is evaluated by using the
angular analysis as described by Eq. (10). An example of the spin rotation is
given in Fig. 8.

The initial polarization is only on the transverse plane, specifically along
the direction of the $Ox$ axis (see Fig. 1). After $\Lambda_{c}^{+}$ have
passed through the crystal, the polarization acquires also a longitudinal
component (along the $Oz$ axis). The value of the $g$-factor is obtained from
Eq. (8) using variation of the polarization components and values of the boost
and bending angle.

Figure 8: Angular distribution of the polarized $\Lambda_{c}^{+}$ decay
products as a function of cos $\theta_{x}$, cos $\theta_{y}$, cos $\theta_{z}$
(see Eq. (9)). The distributions on the top are for an initial polarization
$\xi_{y}$=$\xi_{z}$=0 and $\xi_{x}$=$-$0.40. The same distributions obtained
for the $\Lambda_{c}^{+}$ after having passed through the crystal are shown at
the bottom.

The polarization angle for $g=1.9$ and the parameters used for this simulation
is of the order $\Theta_{\mu}\sim 0.2$ rad.

In Fig. 9 we show the result in the plane $\Phi\times\eta_{det}$ as a function
of days of data taking to reach a precision on $g$-factor of $\pm$ 0.1 for the
two decay modes which we have considered. The bands display different choice
of absolute $\Lambda_{c}^{+}$ polarization, $\alpha$ parameters and
$\Lambda_{c}^{+}$ cross section according to values and accuracy given in
Tables 1 and 2. As it can be noted, the bands are quite large and depending on
the values of several parameters entering this evaluation, the difference in
terms of data taking time can be very significant. It is important to
emphasize that the width of these bands is mainly coming from the two factors:
the value and the uncertainty of the $\alpha$ parameters and the
$\Lambda_{c}^{+}$ polarization. Thus, it is extremely important to measure
more accurately these parameters using, for instance, the existing LHCb data.
In Fig. 9 the results are shown for silicon crystal at room temperature. The
horizontal lines in the two plots correspond to a value for proton flux of
$\Phi=5\times 10^{8}$ s-1 and the detector efficiency in the range
$(1\text{--}3)\times 10^{-3}$ for the $\Lambda(p\pi^{-})\,\pi^{+}$ decay mode
and $(2\text{--}4)\,\%$ for the $\Delta^{++}(1232)\,K^{-}$ decay mode.

Figure 9: Flux times detection efficiency $\Phi\times\eta_{\rm{det}}$ as a
function of data taking time for two $\Lambda_{c}^{+}$ decay modes to obtain
an absolute error on the gyromagnetic factor $g$ of $\pm$ 0.1. Considering a
flux of proton of 5 $\times$ 108 s-1, the areas between horizontal lines:
$(0.5\text{--}2)\times 10^{6}$ and $(1\text{--}2)\times 10^{7}$ correspond to
$\eta_{\rm det}=(1\text{--}3)\times 10^{-3}$ (typical for the
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ decay mode) and $(2\text{--}4)\times$ 10-2
(typical for the $\Lambda_{c}^{+}\to\Delta^{++}K^{-}$ decay mode),
respectively.

The most promising channel is $\Lambda_{c}^{+}\to\Delta^{++}(1232)\,K^{-}$.
Using this mode a precision on $g$-factor of $\pm\,$0.1 can be obtained within
the time from a few to 60 days.

In Fig. 10 we show the evolution of the error on the $g$-factor using the
$\Delta^{++}(1232)\,K^{-}$ decay mode once the detector efficiency has been
fixed to a value: $\eta_{\rm det}=2\times 10^{-3}$. The data taking time
needed to reach the certain precision ranges in a quite large interval due to
the uncertainty on the polarization, $\alpha$ parameters and the
$\Lambda_{c}^{+}$ cross section.

As explained in Section III.4 and shown in Table 3, the data taking time can
be reduced by about a factor $(2.5\text{--}4.8)$, if germanium crystal could
be used.

Figure 10: Error of the gyromagnetic factor $g$ as a function of data taking
time $t$ for the $\Delta^{++}(1232)\,K^{-}$ decay mode.

##  IV  Possible experimental setup for performing this experiment

In the last decade the UA9 Collaboration has developed the technology and more
recently used it to demonstrate that bent silicon crystals can efficiently
steer the diffusive halo surrounding the circulating beam in the LHC, up to
6.5 TeV energy Scandale:2016krl .

A scenario to deflect the halo particles in the vicinity of an interaction
region of LHC is currently under study. The deflected particles should be kept
in the vacuum pipe and will follow trajectories well distinct from those of
the circulating beam core. Inserting a target in the pipe, the deflected halo
can be efficiently used for fixed-target physics. An additional absorber
should intercept halo particles not interacting with the target, thereby
allowing the possibility of fixed-target operation in parasitic mode. In
particular, by directing the deflected halo into another bent crystal tightly
packed with a short and dense target, located in the LHC pipe just before an
existing detector, living baryons should be produced and their polarization
may be measured from the analysis of the decay products. As an example, a
preliminary optical layout compatible with the existing installations in IR8
is presented talkScandale ; talkStocchi  and it is suggested to use the
interaction zone close to the LHCb detector. The LHCb detector will be
particularly well suited to perform this experiment and preliminary
discussions are undergoing.

In addition an Expression of Interest Burmistrov:2016  has been presented in
October 2016 at SPSC proposing to perform preliminary studies of the double
crystal setup in SPS. In March 2017 this proposal has been accepted by SPSC
for the next two years and the experiment will be performed in 2017 and 2018.

##  V  Conclusions

In this paper we have revisited the possibility of a measurement of the
magnetic dipole moment of the charm baryons and in particular of
$\Lambda_{c}^{+}$. As shown, the experimental setup would consist of using the
primary protons in the halo of one of the LHC beams, deflecting them by a bent
crystal into the target-crystal pack, just upstream of one of the existing
detectors of LHC. This experiment is extremely challenging but the recent
success of crystal-collimation tests of the UA9 Collaboration Scandale:2016krl
may provide the necessary technical know-how for such a complex task. The
sensitivity studies presented in this paper show that a precision of $\pm$ 0.1
on the $g$-factor could be reached within data taking time from a few days to
about one month. The uncertainty on the needed data taking time could be
significantly reduced by measuring more precisely the $\alpha$ parameters and
the absolute value of $\Lambda_{c}^{+}$ polarization.

## Acknowledgments

This research was partially conducted in the scope of the IDEATE International
Associated Laboratory (LIA). The research of S.P.F, I.V.K. and A.Yu.K. was
partially supported by the Ministry of Education and Science of Ukraine
(projects no. 0117U004866 and 0115U000473).

##  Appendix A  Aspects of formalism of the polarization precession

The 4-vector of the polarization $a=(0,\,\vec{\xi})$ of the
spin-$\tfrac{1}{2}$ particle is defined in its rest frame in which the
particle 4-momentum is $p=(m,\,0)$. In this frame the axial vector $\vec{\xi}$
is an average of the particle spin,
$\vec{\xi}=\tfrac{2}{\hbar}\langle\,\vec{S}\,\rangle$ Beresteckii:1982 .

After transforming to the frame, in which the particle 4-momentum is
$p=(\varepsilon,\,\vec{p})$, it looks as

$a=(a^{0},\,\vec{a})=(a^{0},\,\vec{a}_{\perp},\,a_{\parallel})=(\gamma
v\xi_{\parallel},\,\vec{\xi}_{\perp},\,\gamma\xi_{\parallel}),$ (18)

where $\vec{v}=\vec{p}/\varepsilon\,$ is the particle velocity,
$\gamma=\varepsilon/m\,$ is the Lorentz factor, and perpendicular and parallel
components of the 3-vectors are defined with respect to the direction of
motion. Apparently, $a\cdot p=0$ in any frame.

The polarization vector has the clear physical meaning in the rest frame of
the particle, therefore the precession of vector $\vec{\xi}$ is usually
considered. In the instantaneous rest frame the polarization vector obeys the
classical equation Beresteckii:1982

$\frac{d\vec{\xi}}{d\tau}=-\frac{eg}{2m}\vec{H}^{\star}\times\vec{\xi},$ (19)

where $\vec{H}^{\star}$ is the magnetic field in this frame and $\tau$ is the
proper time 111Velocity of light is set to unity.. In Eq. (19) the term with a
possible electric dipole moment of the particle is not included (see, for
example, Refs. Bargmann:1959 ; Botella:2016  in which such contribution is
discussed).

One way to extend Eq. (19) to the laboratory frame is to transform the
magnetic field and the time to the laboratory frame, and include the Thomas
correction Thomas:1926 ; Thomas:1927 . Another commonly used way is based on
the explicitly covariant approach Bargmann:1959  which is analyzed in detail
in Refs. Beresteckii:1982 ; Jackson:1999 . The corresponding equations can be
written as

$\displaystyle\frac{d\vec{\xi}}{dt}=\vec{\omega}\times\vec{\xi},$ (20)
$\displaystyle\vec{\omega}=\vec{\omega}_{\vec{H}}+\vec{\omega}_{\vec{E}},$
$\displaystyle\vec{\omega}_{\vec{H}}=-\frac{e}{m}\left[\left(\frac{g}{2}-1+\frac{1}{\gamma}\right)\,\vec{H}-\left(\frac{g}{2}-1\right)\,\frac{\gamma}{1+\gamma}\,\vec{v}\,(\vec{H}\,\vec{v})\right],$
$\displaystyle\vec{\omega}_{\vec{E}}=-\frac{e}{m}\left(\frac{g}{2}-\frac{\gamma}{1+\gamma}\right)\,\vec{E}\times\vec{v},$

where the electric, $\vec{E}$, and magnetic, $\vec{H}$, fields are defined in
the laboratory frame and $\vec{\omega}$ is the angular velocity of the
polarization precession.

For the purpose of the present paper it is sufficient to keep only the
electric field and choose $\vec{E}\,\vec{v}=0$ at any moment of time, since
the effective electric field in the bent crystal is orthogonal to the particle
momentum. In this case the equations of motion imply that

$\frac{d\vec{v}}{dt}=\frac{e}{m\gamma}\,\vec{E},\qquad\quad\frac{dv}{dt}=0.$
(21)

Choosing vector $\vec{E}$ in the $(xz)$ plane it is seen that the particle
rotates around the axis $Oy$ with the constant velocity (neglecting movement
along the $Oy$ axis). From (21) one obtains the corresponding angular velocity
and the rotation radius

$\omega_{0}=\frac{eE}{m\gamma v},\;\qquad R=\frac{v}{\omega_{0}}=\frac{m\gamma
v^{2}}{eE}.$ (22)

The polarization vector, as it is seen from Eqs. (20), also rotates around the
axis $Oy$ with the angular velocity

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\frac{evE}{m}\left(\frac{g}{2}-\frac{\gamma}{1+\gamma}\right)$
(23) $\displaystyle=$
$\displaystyle\gamma\left(\frac{g}{2}-1-\frac{g}{2\gamma^{2}}+\frac{1}{\gamma}\right){\omega}_{0}$

We can integrate (23) and arrive at Eq. (8) connecting the angles of
polarization precession and velocity rotation.

Note that Eq. (23) was derived earlier Lyuboshits:1980  for the arbitrary
electric field. It was also re-derived in Kim:1983  using a more elaborate
method.

##  Appendix B  Asymmetry parameter for decay of polarized $\Lambda_{c}^{+}$
to $\Delta(1232)^{++}K^{-}$

Formalism for the polarization effects in the decay
$\Lambda_{c}^{+}\to\Lambda\,\pi^{+}$
($\tfrac{1}{2}^{+}\to\tfrac{1}{2}^{+}+0^{-}$) is well-known Commins:1983 ,
sec. 6.5 (see also PDG:2014 , p. 1515). If $\Lambda_{c}^{+}$ is polarized and
polarization of $\Lambda$ baryon is not measured, then the angular
distribution is given by Eq. (10).

One of the important modes for measuring polarization of $\Lambda_{c}^{+}$
after passing the crystal is the decay
$\Lambda_{c}^{+}\to\Delta(1232)^{++}K^{-}$. This decay involves the transition
$\tfrac{1}{2}^{+}\to\tfrac{3}{2}^{+}+0^{-}$, and we briefly discuss below the
angular distribution and asymmetry parameter.

The amplitude for the decay $\Lambda_{c}^{+}\to\Delta(1232)^{++}K^{-}$ can be
written as (assuming that $\Delta^{++}$ is produced on-mass-shell)

${\cal M}=\bar{u}^{\mu}(p)\,T_{\mu}\,u(Q)\,\varphi_{K}^{*},$ (24)

where $Q$ ($p$) is the 4-momentum of the initial (final) baryon, $u(Q)$ is the
Dirac spinor, $u^{\mu}(p)$ is the Rarita-Schwinger vector-spinor, such that
$p_{\mu}u^{\mu}(p)=0$ and $\gamma_{\mu}u^{\mu}(p)=0$ (see, e.g.
Beresteckii:1982 , sec. 31), and $\varphi_{K}$ is wave function of the kaon.

In Eq. (24) $T_{\mu}$ is the transition operator which has general form
Commins:1983  (sec. 4.7): $T_{\mu}=(B-A\gamma^{5})\,Q_{\mu}$, where constants
$B$ and $A$ generate parity-conserving and parity-violating amplitudes,
respectively.

The amplitude squared and summed over the final baryon polarizations is

$\displaystyle\overline{|{\cal M}|^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm
Tr}\big{[}(p/+m_{\Delta})\,S^{\nu\mu}(p)\,T_{\mu}\,$ (25)
$\displaystyle\times\,(Q/+M_{\Lambda_{c}})(1+\gamma^{5}a/)\,\gamma^{0}T_{\nu}^{\dagger}\gamma^{0}\big{]},$

where $a$ is the 4-vector of $\Lambda_{c}^{+}$ polarization in Eq. (18),
$a/=a^{\sigma}\gamma_{\sigma}$, and tensor $S^{\nu\mu}(p)$ is

$S^{\nu\mu}(p)=-g^{\nu\mu}+\frac{1}{3}\gamma^{\nu}\gamma^{\mu}+\frac{2p^{\nu}p^{\mu}}{3m_{\Delta}^{2}}+\frac{p^{\mu}\gamma^{\nu}-p^{\nu}\gamma^{\mu}}{3m_{\Delta}}.$
(26)

From (25) one obtains

$\displaystyle\overline{|{\cal M}|^{2}}$ $\displaystyle=$
$\displaystyle\overline{|{\cal
M}_{0}|^{2}}\,\Big{(}1-\alpha\,\frac{M_{\Lambda_{c}}a\cdot p}{[(p\cdot
Q)^{2}-m_{\Delta}^{2}M_{\Lambda_{c}}^{2}]^{1/2}}\Big{)}$ (27) $\displaystyle=$
$\displaystyle\overline{|{\cal
M}_{0}|^{2}}\,\big{(}1+\alpha\,|\vec{\xi}|\cos\vartheta\big{)}$

in the rest frame of $\Lambda_{c}^{+}$, where $a=(0,\vec{\xi})$ and $a\cdot
p=-|\vec{p}||\vec{\xi}|\cos\vartheta$. The asymmetry parameter $\alpha$ reads

$\alpha=\frac{2\,{\rm
Re}(AB^{*})\,|\vec{p}|}{|A|^{2}(E-m_{\Delta})+|B|^{2}(E+m_{\Delta})},$ (28)

and the amplitude squared for the unpolarized $\Lambda_{c}^{+}$ is

$\overline{|{\cal
M}_{0}|^{2}}=\frac{4M_{\Lambda_{c}}^{3}\,\vec{p}\,^{2}}{3m_{\Delta}^{2}}\left[\,|A|^{2}(E-m_{\Delta})+|B|^{2}(E+m_{\Delta})\,\right].$
(29)

Here $E=(m_{\Delta}^{2}+\vec{p}\,^{2})^{1/2}$ is the energy of $\Delta^{++}$
in the rest frame of $\Lambda_{c}^{+}$.

The analogous consideration applies to the decay
$\Lambda_{c}^{+}\to\Lambda(1520)\,\pi^{+}$
($\tfrac{1}{2}^{+}\to\tfrac{3}{2}^{-}+0^{-}$) with interchange of $A$ and $B$.

Actually, Eqs. (27) are general and valid for other decay modes as well, in
particular, for $\Lambda_{c}^{+}\to\Lambda\,\pi^{+}$
($\tfrac{1}{2}^{+}\to\tfrac{1}{2}^{+}+0^{-}$) and $\Lambda_{c}^{+}\to
p\,\overline{K}^{*}(892)^{0}$ ($\tfrac{1}{2}^{+}\to\tfrac{1}{2}^{+}+1^{-}$).
Of course, for these decays the baryon traces differ from (25), but they are
linear in the polarization vector and the amplitude squared $\overline{|{\cal
M}|^{2}}$ is always linear in $a\cdot p$. The asymmetry parameter in (27)
depends on a specific form of the transition operator $T_{\mu}$.

##  Appendix C  Details on deflection efficiency: $\eta_{\rm ang},\ \eta_{\rm
acc},\ \eta_{\rm def}$

Angular acceptance factor $\eta_{\rm ang}$ is defined as the fraction of
$\Lambda_{c}^{+}$ baryons that are produced in the narrow interval of angles
with respect to the crystal plane ($zy$):

$\theta_{x}\in(-\theta_{\rm acc},+\theta_{\rm acc}).$ (30)

As the initial angular distribution of baryons is very close to the normal one
with a standard deviation $1/2\,\gamma^{-1}$, the angular acceptance factor
can be expressed as follows:

$\eta_{\rm ang}=\rm{erf}\left(\sqrt{2}\ \theta_{\rm{acc}}\,\gamma\right)\ $
(31)

where erf($x$) is the error function.

The acceptance angle $\theta_{\rm acc}$ is the maximal value of the angle
between the $\Lambda_{c}^{+}$ momentum and the crystal plane, at which the
particle can be captured into the channeling regime.

Figure 11: Acceptance angle as a function of energy of channeled particle in
germanium (thick curves) and silicon crystals. Solid blue curves are for
straight crystals, dashed red and dotted green curves are for bent crystals
with radii of curvature $R=$ 7.5 m and 1.5 m, respectively.

This angle is analogous to the Lindhard angle (see Eq. (7)) but with taking
into account thermal vibrations of lattice atoms and the crystal curvature.
The value $\theta_{\rm{acc}}$ is defined by the effective potential well of
plane channel of bent crystal. The form of this potential well is defined by
averaging the lattice atom potentials along the chosen crystal plane (see,
e.g., Lindhard ; Biryukov ; Akhiezer ). The dependence of acceptance angle on
the particle energy for silicon and germanium crystals is presented in Fig.
11.

As germanium has a rather small value of Debye temperature, cooling down the
crystal leads to a significant decrease of a thermal oscillation amplitude of
atoms in crystal nodes. Through this effect, reduction of the temperature to
liquid nitrogen temperature noticeably gains the deflection efficiency. For
this reason, we also present the results for germanium crystal cooled down to
$80^{\circ}$K (see upper limit of thick curves in Fig. 5 and Fig. 11)

Actually, the fulfillment of condition (30) is not sufficient for particles to
be captured into the channeling regime. It is also necessary for the channeled
particle to have the negative energy of transverse motion with respect to
interplanar potential $U(x)$ (see, e.g., Lindhard ; Biryukov ; Akhiezer ):

$\varepsilon_{t}(\theta_{x},x)=\frac{\ \varepsilon\ \theta_{x}^{2}\ }{2}+\
U_{\rm eff}(x)<0,$ (32)

where

$U_{\rm eff}=U(x)+\frac{\varepsilon}{R}\ x,\ \ \
(-\frac{d}{2}<x<\frac{d}{2}),$ (33)

where $x$ is the impact parameter with respect to the planar channel (see
e.g., Biryukov ). The second summand in Eq. (33) is centrifugal term which
describes the distortion of interplanar potential caused by the crystal
curvature.

As the characteristic width of baryon angular distribution $\gamma^{-1}$ is at
least two orders of magnitude greater than channeling acceptance angle
$\theta_{\rm acc}$, we can consider the angular distribution of channeled
baryons over $\theta_{x}$ as uniform. It is clear that the distribution over
impact parameter $x$ is uniform as well. Thus, the acceptance factor can be
written in the following form:

$\eta_{\rm acc}=\frac{\eta_{\rm ang}}{2\,d\,\Theta_{\rm acc}}\
\int\limits_{-\theta_{\rm acc}}^{\theta_{\rm acc}}\
\int\limits_{-d/2}^{d/2}\Theta_{\rm
H}\left(-\varepsilon_{t}(\theta_{x},x)\right)\ d\theta_{x}\,dx,$ (34)

where $\Theta_{\rm H}$ is the Heaviside function.

The dechanneling probability $\eta_{\rm dech}$ was calculated by means of
Monte-Carlo simulation of particle passage through the crystal using binary
collision model of incident particle interaction with atoms of crystal lattice
(see, e.g., Kudrin ; Andersen ; FominThesis ). The potential of a single atom
was taken as Moliere potential of screened Coulomb field. The multiple
scattering on electron subsystem of crystal lattice was taken into account
using the aggregate collisions model aggregate ; Bazylev:1986gc . The model
was verified by comparing its results with the experimental data Forster .

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