EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-210 LHCb-PAPER-2018-030 12 December 2018

Search for lepton-flavour-violating decays of Higgs-like bosons

LHCb collaboration†††Authors are listed at the end of this paper.

A search is presented for a Higgs-like boson with mass in the range 45 to
195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ decaying into a muon and a tau
lepton. The dataset consists of proton-proton interactions at a centre-of-mass
energy of 8$\mathrm{\,Te\kern-1.00006ptV}$, collected by the LHCb experiment,
corresponding to an integrated luminosity of 2$\mbox{\,fb}^{-1}$. The tau
leptons are reconstructed in both leptonic and hadronic decay channels. An
upper limit on the production cross-section multiplied by the branching
fraction at 95% confidence level is set and ranges from 22$\mathrm{\,pb}$ for
a boson mass of 45${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to
4$\mathrm{\,pb}$ for a mass of 195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

Published in Eur. Phys. J. C78 (2018) 1008

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

Decays mediated by charged-lepton flavour-violating (CLFV) processes are
forbidden in the Standard Model (SM). Their observation would be a clear sign
for physics beyond the SM. Such processes are predicted by several theoretical
models [1, 2, 3, 4, 5, 6, 7, 8], in particular those based on an effective
theory with relaxed renormalisability requirements [9], supersymmetric models
[10, 11, 12, 13, 14], composite Higgs models [15, 16], Randall-Sundrum models
[17, 18], and non-abelian flavour symmetry models [19]. Nonetheless, no
evidence for CLFV effects has been reported to date.

The LEP experiments set stringent limits on the CLFV decay of the $Z$ boson
[20, 21, 22, 23]. In the presence of CLFV couplings, the decays to $e^{\pm}$
$\mu^{\mp}$, $e^{\pm}$ $\tau^{\mp}$ and $\mu^{\mp}$ $\tau^{\mp}$ could be
mediated by a Higgs boson. At LEP2, limits on the cross-section of the
${e^{+}e^{-}}\rightarrow{e^{\pm}}{\mu^{\mp}}$,
${e^{+}e^{-}}\rightarrow{e^{\pm}}{\tau^{\mp}}$ and
${e^{+}e^{-}}\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ processes were obtained by
the OPAL collaboration for centre-of-mass energies ($\sqrt{s}$) ranging from
192 to 209$\mathrm{\,Ge\kern-1.00006ptV}$ [24]. These constraints can be
translated into limits on the Higgs CLFV decay branching fraction [9, 25],
which are on the order of $10^{-8}$ for a SM Higgs decay into an electron and
muon [25]. Recent searches for the $H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$
decay have been performed by the CMS [26] and ATLAS [27] collaborations for
the Higgs boson with $m_{H}$ = 125${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Upper limits on the branching fraction
${\mathcal{B}}{(H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}})}$ have been placed by
the two collaborations at 0.25% and 1.85%, respectively.

The possible existence of low-mass Higgs-like bosons is a feature of models
like the two-Higgs-doublet models (2HDM) [28]. Searches for such particles
have been performed by the ATLAS [29] and CMS [30] collaborations in the ditau
decay mode. Another scenario is that of a hidden gauge sector [31, 32]. In
this context, the BaBar and Belle collaborations have performed searches for a
resonance with a mass below 10${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [33,
34]. The LHCb collaboration has recently published the results of a search for
dark photons decaying into the dimuon channel, placing a stringent limit for
the production of a dimuon in the mass range from 10.6 to
70${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [35].

The LHCb detector probes the forward rapidity region which is only partially
covered by the other LHC experiments, and triggers on particles with low
transverse momenta ($p_{\mathrm{T}}$), allowing the experiment to explore
relatively small boson masses. In this paper a search for CLFV decays into a
muon and a tau lepton of a Higgs-like boson with a mass ranging from 45 to
195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is presented, using proton-
proton collision data collected at
$\sqrt{s}=8$$\mathrm{\,Te\kern-1.00006ptV}$. The Higgs-like boson is assumed
to be produced by gluon-fusion, similarly to the main production mechanism of
the SM Higgs boson at LHC [36].111The remaining Higgs production modes (e.g.,
$\sim 10\%$ from Vector-Boson Fusion) are neglected in this study. The
analysis is separated into four channels depending on the final state of the
$\tau$ lepton decay: (i) single muon
${\tau^{-}}\\!\rightarrow{\mu^{-}}{{\overline{\nu}}_{\mu}}{{\nu}_{\tau}}$,
(ii) single electron
${\tau^{-}}\\!\rightarrow{e^{-}}{{\overline{\nu}}_{e}}{{\nu}_{\tau}}$, (iii)
single charged hadron
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}({{\pi}^{0}}){{\nu}_{\tau}}$, and (iv)
three charged hadrons
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}({{\pi}^{0}}){{\nu}_{\tau}}$.
They are denoted as $\tau_{\mu}$, $\tau_{e}$, $\tau_{h1}$, and $\tau_{h3}$
respectively. The main sources of background are
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ decays,222 Throughout this note, $Z$
implies ${Z}/\gamma^{\ast}$, i.e. includes contributions from $Z$ boson
production, virtual photon production, and also their interference. heavy
flavour production from QCD processes (“QCD” in the following) and electroweak
boson production accompanied by jets (“$V\\!j$”). This analysis utilizes
reconstruction techniques and results obtained from the
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ measurement by the LHCb collaboration
[37].

##  2 Detector and simulation description

The LHCb detector [38, 39] is a single-arm forward spectrometer covering the
$2<\eta<5$ pseudorapidity range, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of $4{\mathrm{\,Tm}}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream of the
magnet. The tracking system provides a measurement of the momentum of charged
particles with a relative uncertainty that varies from 0.5% at low momentum to
1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a
track to a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/p_{\mathrm{T}}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Photons, electrons and hadrons are
identified by a calorimeter system consisting of scintillating-pad (SPD) and
preshower detectors (PS), an electromagnetic calorimeter (ECAL) and a hadronic
calorimeter (HCAL). Muons are identified by a system composed of five stations
of alternating layers of iron and multiwire proportional chambers.

Simulated data samples are used to calculate the efficiency for selecting
signal processes, to estimate the residual background level, and to produce
templates for the fit used to determine the signal yield. For this analysis,
the simulation is validated primarily by comparing ${Z}\\!\rightarrow
l^{+}l^{-}$ decays in simulation and data. The Higgs boson is generated
assuming a gluon-fusion process, and with mass values from 45 to
195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ in steps of
10${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, using Pythia 8 [40,
*Sjostrand:2006za] with a specific LHCb configuration [42]. The parton density
functions (PDF) are taken from the CTEQ6L set [43]. Decays of hadronic
particles are described by EvtGen [44], in which final-state radiation is
generated using Photos [45]. The interaction of the particles with the
detector and its response are implemented using the Geant4 toolkit [46,
*Agostinelli:2002hh] as described in Ref. [48]. Samples of
$H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ decays generated at next-to-leading
order precision by Powheg-Box [49, 50, 51, 52] with the PDF set
MMHT2014nlo68cl [53] are used for the signal acceptance determination.

##  3 Signal selection

This analysis uses data corresponding to a total integrated luminosity of
${1976\pm 23}\mbox{\,pb}^{-1}$ [54]. The data collected uses a trigger system
consisting of a hardware stage followed by a software stage. The hardware
trigger requires a muon track identified by matching hits in the muon
stations, as well as a global event cut (GEC) requiring the hit multiplicity
in the SPD to be less than 600. The software trigger selects muons or
electrons with a minimum $p_{\mathrm{T}}$ of
15${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

The $H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ candidates are identified and
reconstructed into the four channels: ${\mu}\tau_{e}$, ${\mu}\tau_{h{1}}$,
${\mu}\tau_{h{3}}$ and ${\mu}\tau_{\mu}$. The $\tau_{h3}$ candidates are
reconstructed from the combination of three charged hadrons from a secondary
vertex (SV). The ${\mu}^{\pm}\tau^{\mp}$ candidates are required to be
compatible with originating from a common PV. The muon track and the tracks
used to reconstruct the tau candidate must be in the geometrical region
$2.0<\eta<4.5$. Electron candidates are chosen amongst tracks failing the muon
identification criteria and falling into the acceptance of the PS, ECAL, and
HCAL sub-detectors. A large energy deposit, $E$, in the PS, ECAL, but not in
HCAL is required, satisfying: $E_{\text{PS}}>50\mathrm{\,Me\kern-1.00006ptV}$,
$E_{\text{ECAL}}/p>0.1$, and $E_{\text{HCAL}}/p<0.05$, where $p$ is the
reconstructed momentum of the electron candidate, after recovering the energy
of the bremsstrahlung photons [55]. Charged hadrons are required to be in the
HCAL acceptance, to deposit an energy $E_{\text{HCAL}}$ with
$E_{\text{HCAL}}/p>0.05$, and to fail the muon identification criteria. The
pion mass is assigned to all charged hadrons.

The selection criteria need to be optimised over the $m_{H}$ range used in
this analysis, from 45 to 195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Three
different sets of selection criteria are considered, dubbed L-selection,
C-selection, and H-selection. The C-selection is similar to that used for the
analysis of ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ decays [37]; as such, it is
optimised for ${m_{H}}\sim m_{Z}$. The L-selection and H-selection are
optimised for the $m_{H}$ regions below and above the $Z$ mass respectively.
All selection sets are applied in parallel to compute background estimation
and exclusion limits. Subsequently, for each $m_{H}$ hypothesis, the chosen
selection is that of L-, C-, or H-selection which provides the smallest
expected signal limit, allowing precise separation between adjacent mass
regions. As expected, it is found that the C-selection is optimal for a boson
mass of 75 and 85${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Below and above
that range the best upper limits are obtained from the L- and H-selections,
respectively. In the following discussion the requirements are applied
identically for all decay channels and selection sets unless stated otherwise.

The tau candidates are selected with
$p_{\mathrm{T}}>5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for
$\tau_{e}$,$\tau_{\mu}$, and
$p_{\mathrm{T}}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for $\tau_{h1}$. For
the $\tau_{h3}$ candidate, the charged hadrons are required to have
$p_{\mathrm{T}}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and one of them with
$p_{\mathrm{T}}>6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. They are combined to
form the tau candidates, which are required to have
$p_{\mathrm{T}}>12{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an invariant mass
in the range 0.7 to 1.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In the
H-selection, the tau candidates must have $p_{\mathrm{T}}$ in excess of
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. This requirement is not applied in
the ${\mu}\tau_{\mu}$ channel as it favours the selection of
${Z}\\!\rightarrow{\mu^{+}\mu^{-}}$ background. The muon from
$H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ decay is expected to have a
relatively large $p_{\mathrm{T}}$, thus the selection requires the muon
$p_{\mathrm{T}}$ to be greater than 20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
40${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the L-, C-, and H-selections,
respectively. A tighter requirement of
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is applied for the muon in the
${\mu}\tau_{\mu}$ channel in the H-selection due to the
${Z}\\!\rightarrow{\mu^{+}\mu^{-}}$ background. Additionally, for the
${\mu}\tau_{e}$ channel, the contribution from
$W/Z\\!\rightarrow{e}+\text{jet}$ background is suppressed by requiring the
transverse momentum of the muon to be larger than that of the $\tau_{e}$
candidate.

The relatively large lifetime of the $\tau$ lepton is used to suppress prompt
background. For the $\tau_{h3}$ candidate, a SV is reconstructed. A correction
to the visible invariant mass, $m$, computed from the three-track combination,
is obtained by exploiting the direction of flight defined from the PV to the
SV. The relation used is
$m_{\text{corr}}=\sqrt{m^{2}+p^{2}\sin^{2}\theta}+p\sin\theta$, where $\theta$
is the angle between the momentum of the $\tau_{h3}$ candidate, and its flight
direction. The $m_{\text{corr}}$ value is required to not exceed
3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. A time-of-flight variable is also
computed from the distance of flight and the partially reconstructed momentum
of the $\tau$ lepton, and a minimum value of 30$\mathrm{\,fs}$ is required.
The $m_{\text{corr}}$ and time-of-flight requirements together retain 80% of
the signal, while rejecting about 75% of the QCD background. For tau decay
channels with a single charged particle, it is not possible to reconstruct a
SV, and a selection on the particle IP is applied. A threshold of IP
$>10{\,\upmu\mathrm{m}}$ selects 85% of the $\tau_{e}$ and $\tau_{h1}$
candidates, and rejects about 50% of the $V\\!j$ background. The threshold is
increased to $50{\,\upmu\mathrm{m}}$ for $\tau_{\mu}$ candidates, in order to
suppress ${Z}\\!\rightarrow{\mu^{+}\mu^{-}}$ background. The prompt muon
instead is selected by requiring IP less than 50${\,\upmu\mathrm{m}}$,
allowing up to 50% rejection of QCD and ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$
backgrounds.

The two leptons from the Higgs decay should be approximately back-to-back in
the plane transverse to the beam. The absolute difference in azimuthal angle
of muon and tau candidates is required to be greater than 2.7 radians. This
rejects 50% of the $V\\!j$ background. The transverse momentum asymmetry of
the two particles, defined as $A_{p_{\rm
T}}={|p_{\mathrm{T}}{}_{1}-p_{\mathrm{T}}{}_{2}|}/{(p_{\mathrm{T}}{}_{1}+p_{\mathrm{T}}{}_{2}})$,
can be used to effectively suppress various background processes. The
background from the $V\\!j$ processes is suppressed by up to 60% for the
${\mu}\tau_{h{1}}$ channel by requiring $A_{p_{\rm T}}<0.4\;(0.5)$ in the
L-selection (S-selection), because of the large $p_{\mathrm{T}}$ imbalance
between the high-$p_{\mathrm{T}}$ muon from the vector boson and a hadron from
a jet. For the ${\mu}\tau_{e}$ channel, the worse momentum resolution
increases the average $A_{p_{\rm T}}$ value, hence a softer selection
$A_{p_{\rm T}}<0.6$ is used to preserve efficiency. On the contrary, for the
${\mu}\tau_{\mu}$ channel, a tighter cut is applied to suppress the dominant
background from ${Z}\\!\rightarrow{\mu^{+}\mu^{-}}$ decays. By requiring
$A_{p_{\rm T}}>0.3\ (0.4)$ in the L-selection and C-selection (H-selection),
such background is reduced by 80%, while the signal decreases to 70%.

The two leptons from the Higgs decay are required to be isolated from other
charged particles. Two particle-isolation variables are defined as $I_{p_{\rm
T}}=(\vec{p}_{\text{cone}})_{\text{T}}$ and $\hat{I}_{p_{\rm
T}}={p_{\mathrm{T}}}/{(\vec{p}+\vec{p}_{\text{cone}})}_{\text{T}}$ where
$\vec{p}$ is the momentum of the lepton candidate, the subscript T denotes the
component in the transverse plane, and $\vec{p}_{\text{cone}}$ is the sum of
the momenta of all charged tracks within a distance $R_{\eta\phi}$ = 0.5 in
the $(\eta,\phi)$ plane around the lepton candidate. The isolation requirement
$\hat{I}_{p_{\rm T}}>0.9$ is applied to the muon and tau candidates for all
decay channels and selection sets, and retain 70% of the signal candidates
while rejecting 90% of QCD events. In addition, a cut $I_{p_{\rm
T}}<2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is applied in the L-selection to
both candidates, as the lower $p_{\mathrm{T}}$ reduces the background
rejection power of the $\hat{I}_{p_{\rm T}}$ variable.

The selection criteria common or specific to each selection set and decay
channel are summarised in Table 1. The signal selection efficiencies are found
to vary from 10 to 50%. Due to the kinematic selection, the decay channels are
mutually exclusive and just one $\mu^{\pm}$ $\tau^{\mp}$ candidate per event
is found.

Table 1: Requirements for each decay channel and selection set. Selection set | Variable | ${\mu}\tau_{e}$ | ${\mu}\tau_{h{1}}$ | ${\mu}\tau_{h{3}}$ | ${\mu}\tau_{\mu}$
---|---|---|---|---|---
All |  $p_{\mathrm{T}}$($\tau$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $>5$ | $>10$ | $>12$ | $>5$
|  $p_{\mathrm{T}}{(\tau_{h3}^{\text{prong1}})}$ [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | — | — | $>1$ | —
|  $p_{\mathrm{T}}{(\tau_{h3}^{\text{prong2}})}$ [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | — | — | $>1$ | —
|  $p_{\mathrm{T}}{(\tau_{h3}^{\text{prong3}})}$ [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | — | — | $>6$ | —
|  $p_{\mathrm{T}}{(\mu)}-p_{\mathrm{T}}{(\tau)}$ [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $>0$ | — | — | —
|  $m$($\tau_{h3}$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$] | — | — | 0.7–1.5 | —
|  $m_{\text{corr}}$($\tau_{h3}$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$] | — | — | $>3$ | —
|  Time-of-flight ($\tau_{h3}$) [$\mathrm{\,fs}$] | — | — | $>30$ | —
|  IP($\tau$) [${\,\upmu\mathrm{m}}$] | $>10$ | $>10$ | — | $>50$
|  IP($\mu$) [${\,\upmu\mathrm{m}}$] | $<50$ | $<50$ | $<50$ | $<50$
|  $\Delta\phi$ [$\mathrm{\,rad}$] | $>2.7$ | $>2.7$ | $>2.7$ | $>2.7$
|  $\hat{I}_{p_{\rm T}}$($\tau$) | $>0.9$ | $>0.9$ | $>0.9$ | $>0.9$
|  $\hat{I}_{p_{\rm T}}$($\mu$) | $>0.9$ | $>0.9$ | $>0.9$ | $>0.9$
L-selection |  $p_{\mathrm{T}}$($\mu$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $>20$ | $>20$ | $>20$ | $>20$
| $A_{p_{\rm T}}$ | $<0.6$ | $<0.4$ | — | $>0.3$
|  $I_{p_{\rm T}}$($\tau$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $<2$ | $<2$ | $<2$ | $<2$
|  $I_{p_{\rm T}}$($\mu$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $<2$ | $<2$ | $<2$ | $<2$
C-selection |  $p_{\mathrm{T}}$($\mu$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $>30$ | $>30$ | $>30$ | $>30$
| $A_{p_{\rm T}}$ | — | $<0.5$ | — | $>0.3$
H-selection |  $p_{\mathrm{T}}$($\tau$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $>20$ | $>20$ | $>20$ | —
|  $p_{\mathrm{T}}$($\mu$) [${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$] | $>40$ | $>40$ | $>40$ | $>50$
| $A_{p_{\rm T}}$ | — | — | — | $>0.4$

##  4 Background estimation

Several background processes are considered:
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$, ${Z}\\!\rightarrow l^{+}l^{-}$
($l={e},{\mu})$, QCD, $V\\!j$, double bosons production ($VV$),
${t}{\overline{t}}$, and ${Z}\\!\rightarrow{{b}{\overline{b}}}$. All
backgrounds except ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ are estimated
following the procedures described in Ref. [37]. The expected yields can be
found in Table 2. The corresponding invariant-mass distributions compared with
candidates observed in the data are shown in Fig. 1. For illustration,
examples of $H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ distributions from
simulation are also superimposed.

The ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ background is estimated from the
cross-section measured by the LHCb collaboration [37] where the reconstruction
efficiency is determined from data, and the acceptance and selection
efficiency are obtained from simulation. The estimated background includes a
small amount of cross-feed from different final states of the tau decay, as
determined from simulation. The ${Z}\\!\rightarrow{\mu^{+}\mu^{-}}$ background
is dominant in the ${\mu}\tau_{\mu}$ channel. The corresponding invariant-mass
distribution is obtained from simulation and normalised to data in the $Z$
peak region, from 80 to 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In
order to suppress the potential presence of signal in this region, the muons
are required to be promptly produced. For other channels, the
${Z}\\!\rightarrow l^{+}l^{-}$ decay becomes a background source in case a
lepton is misidentified. This contribution is computed from the
${Z}\\!\rightarrow l^{+}l^{-}$ in data, and weighted by the particle
misidentification probability obtained from simulation.

The QCD and $V\\!j$ backgrounds are inferred from data using the same criteria
as for the signal but selecting same-sign $\mu^{\pm}$ $\tau^{\pm}$ candidates.
Their amounts are determined by a fit to the distribution of
$p_{\mathrm{T}}{({\mu})}-p_{\mathrm{T}}{({\tau})}$, with templates
representing each of them. The template for the QCD component is obtained from
data requiring an anti-isolation $\hat{I}_{p_{\rm T}}<0.6$ selection. The
distribution obtained from simulation is used for the $V\\!j$ component.
Factors are subsequently applied for the correction of the relative yield of
opposite-sign to same-sign candidates. For the QCD background the number of
anti-isolated opposite-sign candidates found in data is used in the
calculation of the correction factor, where it is found to be close to unity.
The factors are found consistent with the simulation. The factors for the
$V\\!j$ component are taken from simulation, and are in general larger than
unity (1.3 for ${\mu}\tau_{e}$ up to 3.1 for ${\mu}\tau_{h{1}}$, for the
L-selection). The minor contributions from $VV$, ${t}{\overline{t}}$, and
${Z}\\!\rightarrow{{b}{\overline{b}}}$ processes are estimated from
simulation.

Figure 1:  Invariant-mass distributions for the ${\mu}^{\pm}\tau^{\mp}$
candidates for the four decay channels (from top to bottom: ${\mu}\tau_{e}$,
${\mu}\tau_{h{1}}$, ${\mu}\tau_{h{3}}$, ${\mu}\tau_{\mu}$) and the three
selections (from left to right: L-selection, C-selection, H-selection). The
distribution of candidates observed (black points) is compared with
backgrounds (filled colour, stacked), and with signal hypothesis (cyan). The
signal is normalised to $\sqrt{N}$, with $N$ the total number of candidates in
the corresponding data histogram.  Table 2:  Expected number of background
candidates from each component, total background with uncertainty, and number
of observed candidates with statistical uncertainty, from each decay channel
and selection set.

Selection set | Process | ${\mu}\tau_{e}$ | ${\mu}\tau_{h{1}}$ | ${\mu}\tau_{h{3}}$ | ${\mu}\tau_{\mu}$
---|---|---|---|---|---
L-selection | ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ | 371.1$\,\pm\,$ | 26.0 | 681.7$\,\pm\,$ | 47.1 | 135.1$\,\pm\,$ | 11.7 | 137.4$\,\pm\,$ | 9.5
| ${Z}\\!\rightarrow l^{+}l^{-}$ | 8.2$\,\pm\,$ | 1.6 | 4.0$\,\pm\,$ | 1.8 | — | 155.3$\,\pm\,$ | 5.0
| QCD | 67.5$\,\pm\,$ | 10.6 | 463.6$\,\pm\,$ | 5.4 | 93.1$\,\pm\,$ | 10.9 | 19.4$\,\pm\,$ | 5.5
| $V\\!j$ | 14.5$\,\pm\,$ | 10.3 | 143.2$\,\pm\,$ | 58.6 | 40.1$\,\pm\,$ | 15.8 | 10.7$\,\pm\,$ | 5.8
| VV | 3.4$\,\pm\,$ | 0.3 | 0.9$\,\pm\,$ | 0.2 | 0.3$\,\pm\,$ | 0.1 | 0.3$\,\pm\,$ | 0.1
| ${t}{\overline{t}}$ | 1.7$\,\pm\,$ | 0.1 | 1.3$\,\pm\,$ | 0.1 | 0.7$\,\pm\,$ | 0.1 | 1.3$\,\pm\,$ | 0.2
| ${Z}\\!\rightarrow{{b}{\overline{b}}}$ | 0.2$\,\pm\,$ | 0.2 | 0.2$\,\pm\,$ | 0.2 | 0.1$\,\pm\,$ | 0.1 | 0.2$\,\pm\,$ | 0.2
| Total background | 466.6$\,\pm\,$ | 28.0 | 1294.9$\,\pm\,$ | 75.5 | 269.4$\,\pm\,$ | 20.3 | 324.5$\,\pm\,$ | 12.5
| Observed | 472.0$\,\pm\,$ | 21.7 | 1284.0$\,\pm\,$ | 35.8 | 240.0$\,\pm\,$ | 15.5 | 344.0$\,\pm\,$ | 18.5
C-selection | ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ | 200.0$\,\pm\,$ | 14.3 | 288.1$\,\pm\,$ | 20.2 | 61.3$\,\pm\,$ | 5.5 | 71.7$\,\pm\,$ | 5.2
| ${Z}\\!\rightarrow l^{+}l^{-}$ | 8.0$\,\pm\,$ | 1.7 | 4.3$\,\pm\,$ | 1.8 | — | 126.7$\,\pm\,$ | 4.5
| QCD | 10.0$\,\pm\,$ | 14.0 | 137.9$\,\pm\,$ | 14.0 | 29.9$\,\pm\,$ | 9.0 | 6.1$\,\pm\,$ | 3.6
| $V\\!j$ | 48.3$\,\pm\,$ | 17.2 | 242.9$\,\pm\,$ | 25.3 | 30.8$\,\pm\,$ | 17.6 | 7.9$\,\pm\,$ | 4.7
| VV | 3.4$\,\pm\,$ | 0.3 | 1.5$\,\pm\,$ | 0.2 | 0.3$\,\pm\,$ | 0.1 | 0.3$\,\pm\,$ | 0.1
| ${t}{\overline{t}}$ | 2.5$\,\pm\,$ | 0.1 | 1.6$\,\pm\,$ | 0.1 | 0.7$\,\pm\,$ | 0.1 | 1.5$\,\pm\,$ | 0.2
| ${Z}\\!\rightarrow{{b}{\overline{b}}}$ | 0.1$\,\pm\,$ | 0.1 | 0.1$\,\pm\,$ | 0.1 | 0.1$\,\pm\,$ | 0.1 | 0.1$\,\pm\,$ | 0.1
| Total background | 272.3$\,\pm\,$ | 17.8 | 676.4$\,\pm\,$ | 35.2 | 123.1$\,\pm\,$ | 15.0 | 214.3$\,\pm\,$ | 8.1
| Observed | 296.0$\,\pm\,$ | 17.2 | 679.0$\,\pm\,$ | 26.1 | 123.0$\,\pm\,$ | 11.1 | 235.0$\,\pm\,$ | 15.3
H-selection | ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ | 13.7$\,\pm\,$ | 1.8 | 18.4$\,\pm\,$ | 1.6 | 8.9$\,\pm\,$ | 1.1 | 2.2$\,\pm\,$ | 0.4
| ${Z}\\!\rightarrow l^{+}l^{-}$ | 4.7$\,\pm\,$ | 1.1 | 2.5$\,\pm\,$ | 1.1 | — | 33.7$\,\pm\,$ | 2.3
| QCD | — | 15.8$\,\pm\,$ | 6.3 | 9.7$\,\pm\,$ | 5.1 | —
| $V\\!j$ | 3.5$\,\pm\,$ | 2.6 | 142.6$\,\pm\,$ | 26.0 | 18.6$\,\pm\,$ | 16.5 | 7.8$\,\pm\,$ | 4.0
| VV | 1.7$\,\pm\,$ | 0.2 | 1.0$\,\pm\,$ | 0.2 | 0.1$\,\pm\,$ | 0.1 | 0.2$\,\pm\,$ | 0.1
| ${t}{\overline{t}}$ | 1.2$\,\pm\,$ | 0.1 | 0.9$\,\pm\,$ | 0.1 | 0.4$\,\pm\,$ | 0.1 | 0.8$\,\pm\,$ | 0.1
| ${Z}\\!\rightarrow{{b}{\overline{b}}}$ | 0.1$\,\pm\,$ | 0.1 | 0.1$\,\pm\,$ | 0.1 | 0.1$\,\pm\,$ | 0.1 | 0.1$\,\pm\,$ | 0.1
| Total background | 24.9$\,\pm\,$ | 3.4 | 181.2$\,\pm\,$ | 26.7 | 37.8$\,\pm\,$ | 13.6 | 44.7$\,\pm\,$ | 4.6
| Observed | 27.0$\,\pm\,$ | 5.2 | 184.0$\,\pm\,$ | 13.6 | 37.0$\,\pm\,$ | 6.1 | 39.0$\,\pm\,$ | 6.2

##  5 Results

The signal cross-section multiplied by the branching fraction is given by

$\sigma(gg\\!\rightarrow
H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}})=N_{\text{sig}}/(\mathcal{L}\cdot{\mathcal{B}}{({\tau}\\!\rightarrow
X)}\cdot\varepsilon),$ (1)

where $N_{\text{sig}}$ is the signal yield obtained from the fit procedure
described below, $\mathcal{L}$ the total integrated luminosity,
${\mathcal{B}}{({\tau}\\!\rightarrow X)}$ the tau branching fraction, and
$\varepsilon$ the detection efficiency. The latter is the product of
acceptance, reconstruction, and offline selection efficiencies. These
efficiencies are obtained from simulated samples and data for each decay
channel and selection set, following the methods developed for the
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ measurement [37]. The acceptance
obtained from the Powheg-Box generator is identical for the ${\mu}\tau_{e}$,
${\mu}\tau_{h{3}}$, and ${\mu}\tau_{\mu}$ channels, varying from 1.0% for
${m_{H}}=195{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to 3.2% for
${m_{H}}=75{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The reconstruction
efficiency, which is the product of contributions from trigger, tracking, and
particle identification, is in the range 40–70%, but only about 15% in the
case of the ${\mu}\tau_{h{3}}$ channel because of the limited tracking
efficiency for the low-momentum hadrons. With the exception of the
${\mu}\tau_{\mu}$ channel, the selection efficiency is 18–30% in the
L-selection, and 24–49% in the C-selection and H-selection. In the case of the
${\mu}\tau_{\mu}$ channel, the tighter selection on the muon $p_{\mathrm{T}}$
and impact parameter reduces the selection efficiency to 10–15%.

The systematic uncertainties are summarised in Table 3. The uncertainty on the
acceptance receives contributions from the gluon PDF uncertainty, as well as
from factorization and renormalisation scales. The uncertainties on the
reconstruction and selection efficiencies are estimated from simulation and
are calibrated using data as described in Ref. [37]. The uncertainty
associated with the invariant-mass shape is handled by selecting the weakest
expected limits among the different choices of distribution (kernel estimation
and histograms with different bin widths are used). The uncertainties on the
integrated luminosity and acceptance are fully correlated among channels,
while only a partial correlation is found for the reconstruction efficiency
uncertainties. All the other uncertainties are taken as uncorrelated.

Table 3:  Relative systematic uncertainties (in %) on the normalisation
factors in the cross-section calculation. When the uncertainty depends on
$m_{H}$ a range is indicated.

| ${\mu}\tau_{e}$ | ${\mu}\tau_{h{1}}$ | ${\mu}\tau_{h{3}}$ | ${\mu}\tau_{\mu}$
---|---|---|---|---
Luminosity | 1.16 | 1.16 | 1.16 | 1.16
Tau branching fraction | 0.22 | 0.18 | 0.48 | 0.23
PDF | 2.6–7.1 | 3.5–7.2 | 2.6–7.3 | 3.0–7.9
Scales | 0.9–1.9 | 0.8–1.7 | 0.9–1.7 | 0.9–1.9
Reconstruction efficiency | 1.8–3.6 | 1.9–5.4 | 3.3–7.1 | 1.5–3.3
Selection efficiency | 2.5–6.0 | 1.9–4.1 | 4.0–9.3 | 3.8–8.5

The signal yield is determined from a simultaneous extended likelihood fit of
the binned invariant-mass distributions of the $\mu$ $\tau$ candidates. The
distributions for signal are obtained from simulation, while distributions of
the different background sources are obtained using the method described in
Sect. 4. The amount of each background component as well as other terms in Eq.
1 containing uncertainties are treated as nuisance parameters and are
constrained to a Gaussian distribution with mean and standard deviation
corresponding to the expected value and its uncertainty, respectively.

The fit results for all $m_{H}$ values are compatible with a null signal,
hence cross-section upper limits are computed. The exclusion limits of
$\sigma(gg\\!\rightarrow H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}})$ defined at
95% confidence level are obtained from the $\text{CL}_{\text{s}}$ method [56].
As mentioned before, for each mass hypothesis the selection considered is that
providing the smallest expected limit. The $\sigma(gg\\!\rightarrow
H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}})$ exclusion limits are shown in Fig. 2,
ranging from 22$\mathrm{\,pb}$ for $m_{H}$ =
45${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to 4$\mathrm{\,pb}$ for $m_{H}$ =
195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In the particular case of
$m_{H}$ = 125${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, using the production
cross-section from Ref. [57] gives a best fit for the branching fraction of
${\mathcal{B}}{(H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}})}=-2^{+14}_{-12}\%$ and
an observed exclusion limit
${\mathcal{B}}{(H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}})}<26\%$. The
corresponding exclusion limit on the Yukawa coupling is
$\sqrt{|Y_{\mu\tau}|^{2}+|Y_{\tau\mu}|^{2}}<$1.7\text{\times}{10}^{-2}$$,
assuming the decay width
$\Gamma_{\text{SM}}=4.1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [58].

Figure 2:  Cross-section times branching fraction 95% CL limits for the
$H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ decay as a function of $m_{H}$, from
the simultaneous fit. The observed limits from individual channels are also
shown.

##  6 Conclusion

A search for Higgs-like bosons decaying via a lepton-flavour-violating process
$H\\!\rightarrow{\mu^{\pm}}{\tau^{\mp}}$ in $pp$ collisions at $\sqrt{s}$ =
8$\mathrm{\,Te\kern-1.00006ptV}$ is presented, with the tau lepton
reconstructed in leptonic and hadronic decay modes. No signal has been found.
The upper bound on the cross-section multiplied by the branching fraction, at
95% confidence level, ranges from 22$\mathrm{\,pb}$ for a boson mass of
45${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, to 4$\mathrm{\,pb}$ for
195${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The search provides information
complementary to the ATLAS and CMS collaborations.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); NSF (USA). We acknowledge the computing resources that are
provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF
(Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open-source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and
Development program of LANL (USA).

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LHCb collaboration

R. Aaij27, C. Abellán Beteta44, B. Adeva41, M. Adinolfi48, C.A. Aidala73, Z.
Ajaltouni5, S. Akar59, P. Albicocco18, J. Albrecht10, F. Alessio42, M.
Alexander53, A. Alfonso Albero40, G. Alkhazov33, P. Alvarez Cartelle55, A.A.
Alves Jr41, S. Amato2, S. Amerio23, Y. Amhis7, L. An3, L. Anderlini17, G.
Andreassi43, M. Andreotti16,g, J.E. Andrews60, R.B. Appleby56, F. Archilli27,
P. d’Argent12, J. Arnau Romeu6, A. Artamonov39, M. Artuso61, K. Arzymatov37,
E. Aslanides6, M. Atzeni44, B. Audurier22, S. Bachmann12, J.J. Back50, S.
Baker55, V. Balagura7,b, W. Baldini16, A. Baranov37, R.J. Barlow56, S.
Barsuk7, W. Barter56, F. Baryshnikov70, V. Batozskaya31, B. Batsukh61, V.
Battista43, A. Bay43, J. Beddow53, F. Bedeschi24, I. Bediaga1, A. Beiter61,
L.J. Bel27, S. Belin22, N. Beliy63, V. Bellee43, N. Belloli20,i, K. Belous39,
I. Belyaev34,42, E. Ben-Haim8, G. Bencivenni18, S. Benson27, S. Beranek9, A.
Berezhnoy35, R. Bernet44, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt44, F. Betti15,42, M.O. Bettler49, M. van Beuzekom27, Ia.
Bezshyiko44, S. Bhasin48, J. Bhom29, S. Bifani47, P. Billoir8, A. Birnkraut10,
A. Bizzeti17,u, M. Bjørn57, M.P. Blago42, T. Blake50, F. Blanc43, S. Blusk61,
D. Bobulska53, V. Bocci26, O. Boente Garcia41, T. Boettcher58, A. Bondar38,w,
N. Bondar33, S. Borghi56,42, M. Borisyak37, M. Borsato41, F. Bossu7, M.
Boubdir9, T.J.V. Bowcock54, C. Bozzi16,42, S. Braun12, M. Brodski42, J.
Brodzicka29, A. Brossa Gonzalo50, D. Brundu22, E. Buchanan48, A. Buonaura44,
C. Burr56, A. Bursche22, J. Buytaert42, W. Byczynski42, S. Cadeddu22, H.
Cai64, R. Calabrese16,g, R. Calladine47, M. Calvi20,i, M. Calvo Gomez40,m, A.
Camboni40,m, P. Campana18, D.H. Campora Perez42, L. Capriotti56, A.
Carbone15,e, G. Carboni25, R. Cardinale19,h, A. Cardini22, P. Carniti20,i, L.
Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina20, M. Cattaneo42, G.
Cavallero19,h, R. Cenci24,p, D. Chamont7, M.G. Chapman48, M. Charles8, Ph.
Charpentier42, G. Chatzikonstantinidis47, M. Chefdeville4, V. Chekalina37, C.
Chen3, S. Chen22, S.-G. Chitic42, V. Chobanova41, M. Chrzaszcz42, A.
Chubykin33, P. Ciambrone18, X. Cid Vidal41, G. Ciezarek42, P.E.L. Clarke52, M.
Clemencic42, H.V. Cliff49, J. Closier42, V. Coco42, J.A.B. Coelho7, J. Cogan6,
E. Cogneras5, L. Cojocariu32, P. Collins42, T. Colombo42, A. Comerma-
Montells12, A. Contu22, G. Coombs42, S. Coquereau40, G. Corti42, M. Corvo16,g,
C.M. Costa Sobral50, B. Couturier42, G.A. Cowan52, D.C. Craik58, A.
Crocombe50, M. Cruz Torres1, R. Currie52, C. D’Ambrosio42, F. Da Cunha
Marinho2, C.L. Da Silva74, E. Dall’Occo27, J. Dalseno48, A. Danilina34, A.
Davis3, O. De Aguiar Francisco42, K. De Bruyn42, S. De Capua56, M. De Cian43,
J.M. De Miranda1, L. De Paula2, M. De Serio14,d, P. De Simone18, C.T. Dean53,
D. Decamp4, L. Del Buono8, B. Delaney49, H.-P. Dembinski11, M. Demmer10, A.
Dendek30, D. Derkach37, O. Deschamps5, F. Desse7, F. Dettori54, B. Dey65, A.
Di Canto42, P. Di Nezza18, S. Didenko70, H. Dijkstra42, F. Dordei42, M.
Dorigo42,y, A. Dosil Suárez41, L. Douglas53, A. Dovbnya45, K. Dreimanis54, L.
Dufour27, G. Dujany8, P. Durante42, J.M. Durham74, D. Dutta56, R.
Dzhelyadin39, M. Dziewiecki12, A. Dziurda29, A. Dzyuba33, S. Easo51, U.
Egede55, V. Egorychev34, S. Eidelman38,w, S. Eisenhardt52, U. Eitschberger10,
R. Ekelhof10, L. Eklund53, S. Ely61, A. Ene32, S. Escher9, S. Esen27, T.
Evans59, A. Falabella15, N. Farley47, S. Farry54, D. Fazzini20,42,i, L.
Federici25, P. Fernandez Declara42, A. Fernandez Prieto41, F. Ferrari15, L.
Ferreira Lopes43, F. Ferreira Rodrigues2, M. Ferro-Luzzi42, S. Filippov36,
R.A. Fini14, M. Fiorini16,g, M. Firlej30, C. Fitzpatrick43, T. Fiutowski30, F.
Fleuret7,b, M. Fontana22,42, F. Fontanelli19,h, R. Forty42, V. Franco Lima54,
M. Frank42, C. Frei42, J. Fu21,q, W. Funk42, C. Färber42, M. Féo Pereira
Rivello Carvalho27, E. Gabriel52, A. Gallas Torreira41, D. Galli15,e, S.
Gallorini23, S. Gambetta52, Y. Gan3, M. Gandelman2, P. Gandini21, Y. Gao3,
L.M. Garcia Martin72, B. Garcia Plana41, J. García Pardiñas44, J. Garra
Tico49, L. Garrido40, D. Gascon40, C. Gaspar42, L. Gavardi10, G. Gazzoni5, D.
Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, D. Gerstel6, Ph.
Ghez4, S. Gianì43, V. Gibson49, O.G. Girard43, L. Giubega32, K. Gizdov52, V.V.
Gligorov8, D. Golubkov34, A. Golutvin55,70, A. Gomes1,a, I.V. Gorelov35, C.
Gotti20,i, E. Govorkova27, J.P. Grabowski12, R. Graciani Diaz40, L.A. Granado
Cardoso42, E. Graugés40, E. Graverini44, G. Graziani17, A. Grecu32, R.
Greim27, P. Griffith22, L. Grillo56, L. Gruber42, B.R. Gruberg Cazon57, O.
Grünberg67, C. Gu3, E. Gushchin36, Yu. Guz39,42, T. Gys42, C. Göbel62, T.
Hadavizadeh57, C. Hadjivasiliou5, G. Haefeli43, C. Haen42, S.C. Haines49, B.
Hamilton60, X. Han12, T.H. Hancock57, S. Hansmann-Menzemer12, N. Harnew57,
S.T. Harnew48, T. Harrison54, C. Hasse42, M. Hatch42, J. He63, M. Hecker55, K.
Heinicke10, A. Heister10, K. Hennessy54, L. Henry72, E. van Herwijnen42, M.
Heß67, A. Hicheur2, R. Hidalgo Charman56, D. Hill57, M. Hilton56, P.H.
Hopchev43, W. Hu65, W. Huang63, Z.C. Huard59, W. Hulsbergen27, T. Humair55, M.
Hushchyn37, D. Hutchcroft54, D. Hynds27, P. Ibis10, M. Idzik30, P. Ilten47, K.
Ivshin33, R. Jacobsson42, J. Jalocha57, E. Jans27, A. Jawahery60, F. Jiang3,
M. John57, D. Johnson42, C.R. Jones49, C. Joram42, B. Jost42, N. Jurik57, S.
Kandybei45, M. Karacson42, J.M. Kariuki48, S. Karodia53, N. Kazeev37, M.
Kecke12, F. Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel28, E. Khairullin37,
B. Khanji42, C. Khurewathanakul43, K.E. Kim61, T. Kirn9, S. Klaver18, K.
Klimaszewski31, T. Klimkovich11, S. Koliiev46, M. Kolpin12, R. Kopecna12, P.
Koppenburg27, I. Kostiuk27, S. Kotriakhova33, M. Kozeiha5, L. Kravchuk36, M.
Kreps50, F. Kress55, P. Krokovny38,w, W. Krupa30, W. Krzemien31, W.
Kucewicz29,l, M. Kucharczyk29, V. Kudryavtsev38,w, A.K. Kuonen43, T.
Kvaratskheliya34,42, D. Lacarrere42, G. Lafferty56, A. Lai22, D. Lancierini44,
G. Lanfranchi18, C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A.
Leflat35, J. Lefrançois7, R. Lefèvre5, F. Lemaitre42, O. Leroy6, T. Lesiak29,
B. Leverington12, P.-R. Li63, T. Li3, Z. Li61, X. Liang61, T. Likhomanenko69,
R. Lindner42, F. Lionetto44, V. Lisovskyi7, X. Liu3, D. Loh50, A. Loi22, I.
Longstaff53, J.H. Lopes2, G.H. Lovell49, D. Lucchesi23,o, M. Lucio Martinez41,
A. Lupato23, E. Luppi16,g, O. Lupton42, A. Lusiani24, X. Lyu63, F. Machefert7,
F. Maciuc32, V. Macko43, P. Mackowiak10, S. Maddrell-Mander48, O. Maev33,42,
K. Maguire56, D. Maisuzenko33, M.W. Majewski30, S. Malde57, B. Malecki29, A.
Malinin69, T. Maltsev38,w, G. Manca22,f, G. Mancinelli6, D. Marangotto21,q, J.
Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito7, M. Marinangeli43,
P. Marino43, J. Marks12, P.J. Marshall54, G. Martellotti26, M. Martin6, M.
Martinelli42, D. Martinez Santos41, F. Martinez Vidal72, A. Massafferri1, M.
Materok9, R. Matev42, A. Mathad50, Z. Mathe42, C. Matteuzzi20, A. Mauri44, E.
Maurice7,b, B. Maurin43, A. Mazurov47, M. McCann55,42, A. McNab56, R.
McNulty13, J.V. Mead54, B. Meadows59, C. Meaux6, F. Meier10, N. Meinert67, D.
Melnychuk31, M. Merk27, A. Merli21,q, E. Michielin23, D.A. Milanes66, E.
Millard50, M.-N. Minard4, L. Minzoni16,g, D.S. Mitzel12, A. Mogini8, J. Molina
Rodriguez1,z, T. Mombächer10, I.A. Monroy66, S. Monteil5, M. Morandin23, G.
Morello18, M.J. Morello24,t, O. Morgunova69, J. Moron30, A.B. Morris6, R.
Mountain61, F. Muheim52, M. Mulder27, C.H. Murphy57, D. Murray56, A. Mödden
10, D. Müller42, J. Müller10, K. Müller44, V. Müller10, P. Naik48, T.
Nakada43, R. Nandakumar51, A. Nandi57, T. Nanut43, I. Nasteva2, M. Needham52,
N. Neri21, S. Neubert12, N. Neufeld42, M. Neuner12, T.D. Nguyen43, C. Nguyen-
Mau43,n, S. Nieswand9, R. Niet10, N. Nikitin35, A. Nogay69, N.S. Nolte42, D.P.
O’Hanlon15, A. Oblakowska-Mucha30, V. Obraztsov39, S. Ogilvy18, R.
Oldeman22,f, C.J.G. Onderwater68, A. Ossowska29, J.M. Otalora Goicochea2, P.
Owen44, A. Oyanguren72, P.R. Pais43, T. Pajero24,t, A. Palano14, M.
Palutan18,42, G. Panshin71, A. Papanestis51, M. Pappagallo52, L.L.
Pappalardo16,g, W. Parker60, C. Parkes56, G. Passaleva17,42, A. Pastore14, M.
Patel55, C. Patrignani15,e, A. Pearce42, A. Pellegrino27, G. Penso26, M. Pepe
Altarelli42, S. Perazzini42, D. Pereima34, P. Perret5, L. Pescatore43, K.
Petridis48, A. Petrolini19,h, A. Petrov69, S. Petrucci52, M. Petruzzo21,q, B.
Pietrzyk4, G. Pietrzyk43, M. Pikies29, M. Pili57, D. Pinci26, J. Pinzino42, F.
Pisani42, A. Piucci12, V. Placinta32, S. Playfer52, J. Plews47, M. Plo
Casasus41, F. Polci8, M. Poli Lener18, A. Poluektov50, N. Polukhina70,c, I.
Polyakov61, E. Polycarpo2, G.J. Pomery48, S. Ponce42, A. Popov39, D.
Popov47,11, S. Poslavskii39, C. Potterat2, E. Price48, J. Prisciandaro41, C.
Prouve48, V. Pugatch46, A. Puig Navarro44, H. Pullen57, G. Punzi24,p, W.
Qian63, J. Qin63, R. Quagliani8, B. Quintana5, B. Rachwal30, J.H.
Rademacker48, M. Rama24, M. Ramos Pernas41, M.S. Rangel2, F. Ratnikov37,x, G.
Raven28, M. Ravonel Salzgeber42, M. Reboud4, F. Redi43, S. Reichert10, A.C.
dos Reis1, F. Reiss8, C. Remon Alepuz72, Z. Ren3, V. Renaudin7, S.
Ricciardi51, S. Richards48, K. Rinnert54, P. Robbe7, A. Robert8, A.B.
Rodrigues43, E. Rodrigues59, J.A. Rodriguez Lopez66, M. Roehrken42, S.
Roiser42, A. Rollings57, V. Romanovskiy39, A. Romero Vidal41, M. Rotondo18,
M.S. Rudolph61, T. Ruf42, J. Ruiz Vidal72, J.J. Saborido Silva41, N.
Sagidova33, B. Saitta22,f, V. Salustino Guimaraes62, C. Sanchez Gras27, C.
Sanchez Mayordomo72, B. Sanmartin Sedes41, R. Santacesaria26, C. Santamarina
Rios41, M. Santimaria18, E. Santovetti25,j, G. Sarpis56, A. Sarti18,k, C.
Satriano26,s, A. Satta25, M. Saur63, D. Savrina34,35, S. Schael9, M.
Schellenberg10, M. Schiller53, H. Schindler42, M. Schmelling11, T.
Schmelzer10, B. Schmidt42, O. Schneider43, A. Schopper42, H.F. Schreiner59, M.
Schubiger43, M.H. Schune7, R. Schwemmer42, B. Sciascia18, A. Sciubba26,k, A.
Semennikov34, E.S. Sepulveda8, A. Sergi47,42, N. Serra44, J. Serrano6, L.
Sestini23, A. Seuthe10, P. Seyfert42, M. Shapkin39, Y. Shcheglov33,†, T.
Shears54, L. Shekhtman38,w, V. Shevchenko69, E. Shmanin70, B.G. Siddi16, R.
Silva Coutinho44, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, N.
Skidmore12, T. Skwarnicki61, M.W. Slater47, J.G. Smeaton49, E. Smith9, I.T.
Smith52, M. Smith55, M. Soares15, l. Soares Lavra1, M.D. Sokoloff59, F.J.P.
Soler53, B. Souza De Paula2, B. Spaan10, E. Spadaro Norella21,q, P.
Spradlin53, F. Stagni42, M. Stahl12, S. Stahl42, P. Stefko43, S. Stefkova55,
O. Steinkamp44, S. Stemmle12, O. Stenyakin39, M. Stepanova33, H. Stevens10, A.
Stocchi7, S. Stone61, B. Storaci44, S. Stracka24, M.E. Stramaglia43, M.
Straticiuc32, U. Straumann44, S. Strokov71, J. Sun3, L. Sun64, K. Swientek30,
T. Szumlak30, M. Szymanski63, S. T’Jampens4, Z. Tang3, A. Tayduganov6, T.
Tekampe10, G. Tellarini16, F. Teubert42, E. Thomas42, J. van Tilburg27, M.J.
Tilley55, V. Tisserand5, M. Tobin30, S. Tolk42, L. Tomassetti16,g, D.
Tonelli24, D.Y. Tou8, R. Tourinho Jadallah Aoude1, E. Tournefier4, M.
Traill53, M.T. Tran43, A. Trisovic49, A. Tsaregorodtsev6, G. Tuci24,p, A.
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Uwer12, A. Vagner71, V. Vagnoni15, A. Valassi42, S. Valat42, G. Valenti15, R.
Vazquez Gomez42, P. Vazquez Regueiro41, S. Vecchi16, M. van Veghel27, J.J.
Velthuis48, M. Veltri17,r, G. Veneziano57, A. Venkateswaran61, T.A. Verlage9,
M. Vernet5, M. Veronesi27, N.V. Veronika13, M. Vesterinen57, J.V. Viana
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Cardona40,m, A. Vitkovskiy27, M. Vitti49, V. Volkov35, A. Vollhardt44, D. Vom
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Wang44, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden44,
C. Weisser58, M. Whitehead9, J. Wicht50, G. Wilkinson57, M. Wilkinson61, I.
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J. Wimberley60, M. Winn7, J. Wishahi10, W. Wislicki31, M. Witek29, G.
Wormser7, S.A. Wotton49, K. Wyllie42, D. Xiao65, Y. Xie65, A. Xu3, M. Xu65, Q.
Xu63, Z. Xu3, Z. Xu4, Z. Yang3, Z. Yang60, Y. Yao61, L.E. Yeomans54, H. Yin65,
J. Yu65,ab, X. Yuan61, O. Yushchenko39, K.A. Zarebski47, M. Zavertyaev11,c, D.
Zhang65, L. Zhang3, W.C. Zhang3,aa, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X.
Zhu3, V. Zhukov9,35, J.B. Zonneveld52, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
8LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14INFN Sezione di Bari, Bari, Italy
15INFN Sezione di Bologna, Bologna, Italy
16INFN Sezione di Ferrara, Ferrara, Italy
17INFN Sezione di Firenze, Firenze, Italy
18INFN Laboratori Nazionali di Frascati, Frascati, Italy
19INFN Sezione di Genova, Genova, Italy
20INFN Sezione di Milano-Bicocca, Milano, Italy
21INFN Sezione di Milano, Milano, Italy
22INFN Sezione di Cagliari, Monserrato, Italy
23INFN Sezione di Padova, Padova, Italy
24INFN Sezione di Pisa, Pisa, Italy
25INFN Sezione di Roma Tor Vergata, Roma, Italy
26INFN Sezione di Roma La Sapienza, Roma, Italy
27Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
28Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
29Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
30AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
31National Center for Nuclear Research (NCBJ), Warsaw, Poland
32Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
33Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
34Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
35Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
36Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
37Yandex School of Data Analysis, Moscow, Russia
38Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
39Institute for High Energy Physics (IHEP), Protvino, Russia
40ICCUB, Universitat de Barcelona, Barcelona, Spain
41Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
42European Organization for Nuclear Research (CERN), Geneva, Switzerland
43Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
44Physik-Institut, Universität Zürich, Zürich, Switzerland
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 27
69National Research Centre Kurchatov Institute, Moscow, Russia, associated to
34
70National University of Science and Technology ”MISIS”, Moscow, Russia,
associated to 34
71National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
34
72Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 40
73University of Michigan, Ann Arbor, United States, associated to 61
74Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xNational Research University Higher School of Economics, Moscow, Russia
ySezione INFN di Trieste, Trieste, Italy
zEscuela Agrícola Panamericana, San Antonio de Oriente, Honduras
aaSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
abPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased


# Nuclear mass and size corrections to the magnetic shielding

Krzysztof Pachucki  Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland

###### Abstract

We derive finite nuclear mass and finite nuclear size corrections to the
magnetic shielding in light ions. These corrections are important for the
accurate determination of nuclear magnetic moments. We correct several
previous formulas for the nuclear mass corrections and present improved
results for the magnetic shielding in 1H, 3He+, and 3He. Finally, we obtain an
3He atomic magnetic moment, which serves as an accurate probe to measure
magnetic fields.

##  I Introduction

The nuclear magnetic moment in atoms is partially shielded by atomic
electrons. This effect is not very significant: about $10^{-5}$ for light
elements. Nevertheless, because nuclear magnetic moments are determined from
the Zeeman shift in atomic systems, the calculation of the magnetic shielding
is necessary for their accurate determination. For example, the recent
measurement of the magnetic moment of the 3He+ ion [1] together with the
calculation of magnetic shielding [2, 3] allowed for the most accurate
determination so far of the helion magnetic moment. A similar measurement is
planned for 9Be3+, which will result in an improved determination of the 9Be
nuclear magnetic moment [4]. Moreover, the accurate values for nuclear
magnetic moments are important for determination of the atomic hyperfine
splitting (HFS), testing QED and the nuclear structure theory [5]. It is
because HFS is very sensitive to the distribution of the magnetic moment
within the nucleus.

In this work, we point out two interesting effects that are frequently
overlooked in calculations of the nuclear magnetic shielding [6]; namely,
those due to the finite nuclear mass and the finite nuclear size. Nuclear mass
corrections are as large as relativistic corrections for light atomic systems,
while finite nuclear size effects are much smaller, but they are expected to
be significant for heavier elements. These finite nuclear mass effects have
already been the subject of several works [7, 8]. Here, we rederive them
thoroughly, correct some mistakes, and update numerical values for the most
relevant cases of the H, 3He+, and 3He elements. The finite nuclear size
effects have been studied only numerically and only for hydrogen-like systems
[9, 10]. Here, we derive a compact analytic formula in terms of the charge,
magnetic, and effective Zemach nuclear radii, which accounts also for nuclear
inelastic effects.

##  II Breit-Pauli Hamiltonian with the homogenous magnetic field

To account for finite nuclear mass effects, we have to treat nuclei on an
equal footing as all electrons. Therefore, we consider a system of charged
particles, each having its own mass $m_{a}$, charge $e_{a}$, spin $s_{a}$, and
the so-called g-factor $g_{a}$, which is related to the magnetic moment by

$\displaystyle\vec{\mu}_{a}=\frac{g_{a}\,e_{a}}{2\,m_{a}}\,\vec{s}_{a}\,.$ (1)

These particles are electrons with spin $1/2$ and nuclei with an arbitrary
spin. Our derivation employs a Breit-Pauli Hamiltonian with homogenous
magnetic field and with separation of center of mass motion. It closely
follows the lines of Ref. [7]. Let us therefore introduce the total mass $M$

$\displaystyle M=\sum_{a}m_{a}\,,$ (2)

center of mass variables

$\displaystyle\vec{R}$ $\displaystyle=$
$\displaystyle\sum_{a}\frac{m_{a}}{M}\,\vec{r}_{a}\,,$ (3)
$\displaystyle\vec{P}$ $\displaystyle=$ $\displaystyle\sum_{a}\vec{p}_{a}\,,$
(4)

and relative coordinates

$\displaystyle\vec{x}_{a}$ $\displaystyle=$
$\displaystyle\vec{r}_{a}-\vec{R}\,,$ (5) $\displaystyle\vec{q}_{a}$
$\displaystyle=$ $\displaystyle\vec{p}_{a}-\frac{m_{a}}{M}\,\vec{P}\,,$ (6)

such that

$\displaystyle\bigl{[}x_{a}^{i}\,,\,q_{b}^{j}\bigr{]}$ $\displaystyle=$
$\displaystyle
i\,\delta^{ij}\,\biggl{(}\delta_{ab}-\frac{m_{b}}{M}\biggr{)}\,,$ (7)
$\displaystyle\bigl{[}R^{i}\,,\,P^{j}\bigr{]}$ $\displaystyle=$ $\displaystyle
i\,\delta^{ij}\,,$ (8) $\displaystyle\bigl{[}x_{a}^{i}\,,\,P^{j}\bigr{]}$
$\displaystyle=$ $\displaystyle\bigl{[}R^{i}\,,\,q_{a}^{j}\bigr{]}=0\,.$ (9)

The Hamiltonian of a bound system of charged particles in an external magnetic
field including leading relativistic corrections and with the separated-out
center of mass motion is [7]

$\displaystyle H_{\mathrm{in}}=$ $\displaystyle\
\sum_{a}\biggl{\\{}\frac{\vec{\pi}_{a}^{2}}{2\,m_{a}}-\frac{e_{a}}{2\,m_{a}}\,g_{a}\,\vec{s}_{a}\cdot\vec{B}-\frac{\vec{\pi}^{4}_{a}}{8\,m^{3}}+\frac{e_{a}}{8\,m^{3}_{a}}\,\Bigl{[}4\,\vec{\pi}_{a}^{2}\,\vec{s}_{a}\cdot\vec{B}+(g_{a}-2)\,\bigl{\\{}\vec{\pi}_{a}\cdot\vec{B}\,,\,\vec{\pi}_{a}\cdot\vec{s}_{a}\bigr{\\}}\Bigr{]}-\frac{e_{a}^{2}}{2}\,\chi_{a}\,\vec{B}^{2}\biggr{\\}}$
$\displaystyle\
+\sum_{a>b,b}\frac{e_{a}\,e_{b}}{4\,\pi}\,\biggl{\\{}\frac{1}{r_{ab}}-\frac{1}{2\,m_{a}\,m_{b}}\,\pi_{a}^{i}\,\biggl{(}\frac{\delta^{ij}}{r_{ab}}+\frac{r^{i}_{ab}\,r^{j}_{ab}}{r^{3}_{ab}}\biggr{)}\,\pi_{b}^{j}-\frac{2\,\pi}{3}\,\langle
r_{Ea}^{2}+r_{Eb}^{2}\rangle\,\delta^{3}(r_{ab})-\frac{2\,\pi\,g_{a}\,g_{b}}{3\,m_{a}\,m_{b}}\,\vec{s}_{a}\cdot\vec{s}_{b}\,\delta^{3}(r_{ab})$
$\displaystyle\
+\frac{g_{a}\,g_{b}}{4\,m_{a}\,m_{b}}\,\frac{s_{a}^{i}\,s_{b}^{j}}{r_{ab}^{3}}\,\biggl{(}\delta^{ij}-3\,\frac{r_{ab}^{i}\,r_{ab}^{j}}{r_{ab}^{2}}\biggr{)}\biggr{\\}}+\sum_{a,b}\frac{1}{2\,r_{ab}^{3}}\biggl{[}\frac{g_{a}}{m_{a}\,m_{b}}\,\vec{s}_{a}\cdot\vec{r}_{ab}\times\vec{\pi}_{b}-\frac{(g_{a}-1)}{m_{a}^{2}}\,\vec{s}_{a}\cdot\vec{r}_{ab}\times\vec{\pi}_{a}\bigr{)}\biggr{]}\,,$
(10)

where

$\displaystyle\vec{\pi}_{a}=$ $\displaystyle\
\vec{q}_{a}+\frac{1}{2}\,\vec{D}_{a}\times\vec{B}\,,$ (11)
$\displaystyle\vec{D}_{a}=$ $\displaystyle\
e_{a}\,\vec{x}_{a}+\frac{m_{a}}{M}\,\vec{D}\,,$ (12) $\displaystyle\vec{D}=$
$\displaystyle\ \sum_{a}e_{a}\,\vec{x}_{a}\,.$ (13)

For a point spin $s=1/2$ particle: $g=2$, $\langle
r_{E}^{2}\rangle=3/(4\,m^{2})$, and $\chi=1/(4\,m^{3})$. A similar Hamiltonian
for a system of spin $1/2$ particles was originally obtained by Hegstrom in
Ref. [11]. Our Hamiltonian in Eq. (10), however, is valid for arbitrary spin
particles and has a more compact form.

The magnetic interaction resulting from $H_{\mathrm{in}}$ neglecting the terms
quadratic in $\vec{B}$ is

$\displaystyle\delta H=$
$\displaystyle-\sum_{a}\frac{e_{a}}{2\,m_{a}}\,(\vec{x}_{a}\times\vec{q}_{a}+g_{a}\,\vec{s}_{a})\cdot\vec{B}$
$\displaystyle+\sum_{a}\,\frac{1}{4\,m_{a}^{3}}\,\Bigl{[}q_{a}^{2}\,\vec{D}_{a}\times\vec{q}_{a}\cdot\vec{B}+2\,e_{a}\,q_{a}^{2}\,\vec{s}_{a}\cdot\vec{B}$
$\displaystyle+e_{a}\,(g_{a}-2)\,\vec{q}_{a}\cdot\vec{s}_{a}\,\vec{q}_{a}\cdot\vec{B}\Bigr{]}+\sum_{a\neq
b,b}\,\frac{e_{a}\,e_{b}}{4\,\pi}\biggl{[}$
$\displaystyle-\frac{1}{4\,m_{a}\,m_{b}}\,q_{a}^{i}\biggl{(}\frac{\delta^{ij}}{r_{ab}}+\frac{r_{ab}^{i}\,r_{ab}^{j}}{r_{ab}^{3}}\biggr{)}\,(\vec{D}_{b}\times\vec{B})^{\,j}$
$\displaystyle+\frac{1}{4\,r_{ab}^{3}}\,\frac{g_{a}}{m_{a}\,m_{b}}\,(\vec{s}_{a}\times\vec{r}_{ab})\cdot(\vec{D}_{b}\times\vec{B})$
$\displaystyle-\frac{1}{4\,r_{ab}^{3}}\,\frac{(g_{a}-1)}{m_{a}^{2}}\,(\vec{s}_{a}\times\vec{r}_{ab})\cdot(\vec{D}_{a}\times\vec{B})\biggr{]}.$
(14)

This is a general interaction Hamiltonian, which is valid for an arbitrary set
of particles. In particular, one can obtain the bound electron g-factor or the
magnetic shielding in atomic and molecular systems. In the next section we
derive the atomic magnetic shielding with full account of the nuclear mass.

##  III Finite nuclear mass corrections

We will derive the magnetic shielding constant for arbitrary ions with the
vanishing orbital angular momentum $\vec{L}$. The interaction of the nuclear
spin with the magnetic field is obtained from Eq. (14)

$\displaystyle\delta H=$ $\displaystyle\
-\frac{e_{N}}{2\,m_{N}}\,g_{N}\,\vec{s}_{N}\cdot\vec{B}$
$\displaystyle+\frac{e_{N}}{4\,m_{N}^{3}}\,\Bigl{[}2\,q_{N}^{2}\,\vec{s}_{N}\cdot\vec{B}+(g_{N}-2)\,\vec{q}_{N}\cdot\vec{s}_{N}\;\vec{q}_{N}\cdot\vec{B}\Bigr{]}$
$\displaystyle\
+\sum_{b}{}^{\prime}\,\frac{e_{N}\,e}{4\,\pi}\,\frac{(\vec{s}_{N}\times\vec{r}_{Nb})}{4\,r_{Nb}^{3}}$
$\displaystyle\
\cdot\biggl{[}\frac{g_{N}}{m_{N}\,m_{e}}\,\vec{D}_{b}\times\vec{B}-\frac{(g_{N}-1)}{m_{N}^{2}}\,\vec{D}_{N}\times\vec{B}\biggr{]},$
(15)

where we assumed that for nucleus $a=N$, all other particles are electrons,
and $\sum^{\prime}$ denotes summation over electrons only. For an electronic
state with a spherical symmetry

$\displaystyle(\vec{s}_{N}\times\vec{X})\cdot(\vec{Y}\times\vec{B})=-\frac{2}{3}\,\vec{s}_{N}\cdot\vec{B}\,\vec{X}\cdot\vec{Y}\,,$
(16)

one introduces the scalar shielding constant $\sigma$

$\displaystyle\delta
H=-\frac{g_{N}\,e_{N}}{2\,m_{N}}\,\vec{s}_{N}\cdot\vec{B}(1-\sigma)\,.$ (17)

This $\sigma$ is conveniently split into two parts, consisting of the first-
and the second-order matrix elements

$\displaystyle\sigma=\sigma_{1}+\sigma_{2}\,.$ (18)

$\sigma_{1}$ results from the first-order matrix element of $\delta H$ in Eq.
(15)

$\displaystyle\sigma_{1}=$ $\displaystyle\
\frac{1}{2\,g_{N}\,m_{N}^{2}}\,\Bigl{[}2+\frac{(g_{N}-2)}{3}\Bigr{]}\,\big{\langle}q_{N}^{2}\big{\rangle}$
(19)
$\displaystyle+\frac{1}{3}\,\frac{e_{e}}{4\,\pi}\bigg{\langle}\sum_{b}{}^{\prime}\,\frac{\vec{r}_{bN}}{r_{bN}^{3}}\cdot\biggl{[}\frac{1}{m_{e}}\,\vec{D}_{b}-\frac{(g_{N}-1)}{g_{N}\,m_{N}}\,\vec{D}_{N}\biggr{]}\bigg{\rangle}.$

Because $\sum_{a}m_{a}\,\vec{x}_{a}=0$, the position $\vec{x}_{N}$ of the
nucleus with respect to mass center and the dipole operator $\vec{D}$ can be
expressed in terms of the electron coordinates only

$\displaystyle\vec{x}_{N}=$ $\displaystyle\
-\frac{m_{e}}{M}\,\sum_{a}{}^{\prime}\,\vec{r}_{aN}\,,$ (20)
$\displaystyle\vec{D}=$ $\displaystyle\
e_{e}\,\sum_{a}{}^{\prime}\vec{r}_{aN}\,\Bigl{(}1+(Z-N_{e})\,\frac{m_{e}}{M}\Bigr{)}\,,$
(21)

where $M=m_{N}+N_{e}\,m_{e}$, $N_{e}$ is the number of electrons, $Z$ is the
nuclear charge in units of the elementary charge. Consequently, the shielding
constant $\sigma_{1}$ takes the form

$\displaystyle\sigma_{1}=$ $\displaystyle\
\frac{\alpha}{3\,m_{e}}\,\biggl{\langle}\sum_{a}{}^{\prime}\,\frac{1}{r_{a}}\biggr{\rangle}+\frac{(4+g_{N})}{6\,g_{N}}\,\frac{\langle
p_{N}^{2}\rangle}{m_{N}^{2}}$ $\displaystyle\
+\frac{\alpha}{3}\,\bigg{\langle}\sum_{b}{}^{\prime}\,\frac{\vec{r}_{bN}}{r_{bN}^{3}}\cdot\sum_{a}{}^{\prime}\vec{r}_{aN}\bigg{\rangle}\,\frac{1}{g_{N}}\,\frac{m_{e}}{M}\,$
$\displaystyle\times\biggl{[}\frac{(Z-N_{e})}{M}+(1-g_{N})\,\bigg{(}\frac{1}{m_{e}}+\frac{Z}{m_{N}}\bigg{)}\biggr{]}\,,$
(22)

where $\vec{p}_{N}=-\sum_{a}{}^{\prime}\vec{p}_{a}$, and we used
$\vec{P}|\phi\rangle=0$. Consider now the following matrix element

$\displaystyle\biggl{\langle}\sum_{a}{}^{\prime}\,\frac{\vec{r}_{aN}}{r_{aN}^{3}}\cdot\sum_{b}{}^{\prime}\vec{r}_{bN}\biggr{\rangle}=$
$\displaystyle\
\frac{1}{i\,Z\,\alpha}\,\biggl{\langle}[\vec{p}_{N}\,,\,H-E]\sum_{b}{}^{\prime}\vec{r}_{bN}\biggr{\rangle}$
$\displaystyle=$ $\displaystyle\
\sum_{b}{}^{\prime}\frac{1}{i\,Z\,\alpha}\,\bigl{\langle}\vec{p}_{N}\,[H-E\,,\,\vec{r}_{bN}]\bigr{\rangle}$
$\displaystyle=$ $\displaystyle\ \langle
p_{N}^{2}\rangle\,\frac{1}{Z\,\alpha}\,\frac{M}{m_{N}\,m_{e}}\,,$ (23)

which is used to simplify $\sigma_{1}$

$\displaystyle\sigma_{1}=$ $\displaystyle\
\frac{\alpha}{3\,m_{e}}\,\bigg{\langle}\sum_{a}{}^{\prime}\,\frac{1}{r_{a}}\bigg{\rangle}+\frac{\langle
p_{N}^{2}\,\rangle}{3\,g_{N}\,m_{N}^{2}}\,\biggl{[}3-\frac{g_{N}}{2}$
$\displaystyle\
+\frac{m_{N}}{M}\,\biggl{(}1-\frac{N_{e}}{Z}\biggr{)}+(1-g_{N})\,\frac{m_{N}}{Z\,m_{e}}\biggr{]}\,.$
(24)

The $\sigma_{2}$ part is given by the second-order interaction coming from the
Hamiltonian

$\displaystyle\delta H=$ $\displaystyle\
-\sum_{a}\frac{e_{a}}{2\,m_{a}}\,\vec{x}_{a}\times\vec{q}_{a}\cdot\vec{B}\,-\frac{e_{N}\,e_{e}}{4\,\pi}\,\frac{\vec{s}_{N}}{2\,m_{N}}$
$\displaystyle\cdot\sum_{b}{}^{\prime}\frac{\vec{r}_{bN}}{r_{bN}^{3}}\times\biggl{[}g_{N}\,\frac{\vec{q}_{b}}{m_{e}}-(g_{N}-1)\,\frac{\vec{q}_{N}}{m_{N}}\biggr{]},$
(25)

namely

$\displaystyle\sigma_{2}=$
$\displaystyle\frac{2}{3}\,\biggl{\langle}\sum_{a}\frac{e_{a}}{2\,m_{a}}\,\vec{x}_{a}\times\vec{q}_{a}\,\frac{1}{(E-H)}\,\sum_{b}{}^{\prime}\,\frac{e_{e}}{4\,\pi}\,\frac{\vec{r}_{bN}}{r_{bN}^{3}}$
$\displaystyle\times\biggl{[}\frac{\vec{p}_{b}}{m_{b}}-\frac{(g_{N}-1)}{g_{N}}\,\frac{\vec{p}_{N}}{m_{N}}\biggr{]}\biggr{\rangle}\,.$
(26)

Using

$\displaystyle\sum_{a}\frac{e_{a}}{2\,m_{a}}\,\vec{x}_{a}\times\vec{q}_{a}=$
$\displaystyle\
\frac{e_{e}}{2\,m_{e}}\,\vec{L}+\biggl{(}\frac{e_{e}}{2\,m_{e}}+\frac{Z\,e_{e}}{2\,m_{N}}\biggr{)}$
$\displaystyle\times\frac{m_{e}}{M}\,\sum_{a}{}^{\prime}\vec{r}_{aN}\times\vec{p}_{N},$
(27)

we arrive at

$\displaystyle\sigma_{2}=$
$\displaystyle\frac{\alpha}{3\,M}\,\biggl{(}1+\frac{Z\,m_{e}}{m_{N}}\biggr{)}\,\biggl{\langle}\sum_{a}{}^{\prime}\vec{r}_{aN}\times\vec{p}_{N}\,\frac{1}{(E-H)}\,\sum_{b}{}^{\prime}\frac{\vec{r}_{bN}}{r_{bN}^{3}}$
$\displaystyle\times\biggl{[}\frac{\vec{p}_{b}}{m_{e}}-\frac{(g_{N}-1)}{g_{N}}\,\frac{\vec{p}_{N}}{m_{N}}\biggr{]}\biggr{\rangle}\,.$
(28)

The total shielding constant is $\sigma=\sigma_{1}+\sigma_{2}$, where
$\sigma_{1}$ is given in Eq. (24) and $\sigma_{2}$ in Eq. (28). For the
numerical calculations, it is convenient to apply the expansion in the mass
ratio, which takes the form (in a.u.):

$\displaystyle\sigma=$ $\displaystyle\ \sigma^{(2,0)}+\sigma^{(2,1)}+\ldots,$
(29) $\displaystyle\sigma^{(2,0)}=$ $\displaystyle\
\frac{\alpha^{2}}{3}\,\biggl{\langle}\sum_{a}{}^{\prime}\frac{1}{r_{a}}\biggr{\rangle}\,,$
(30) $\displaystyle\sigma^{(2,1)}=$ $\displaystyle\
\frac{\alpha^{2}}{3}\,\frac{m_{e}}{m_{N}}\biggl{[}\biggl{\langle}\sum_{a}{}^{\prime}\frac{1}{r_{a}}\frac{1}{(E-H)^{\prime}}\,p_{N}^{2}\biggr{\rangle}+\langle
p_{N}^{2}\,\rangle\frac{(1-g_{N})}{Z\,g_{N}}$ $\displaystyle\
+\biggl{\langle}\sum_{a}{}^{\prime}\vec{r}_{a}\times\vec{p}_{N}\,\frac{1}{(E-H)^{\prime}}\,\sum_{b}{}^{\prime}\,\frac{\vec{r}_{b}\times\vec{p}_{b}}{r_{b}^{3}}\biggr{\rangle}\biggr{]},$
(31)

where all matrix elements are assumed with infinite nuclear mass. The last
term in the above differs in sign from that derived previously in Ref. [7, 8],
see Table I for the updated numerical values.

For the hydrogenic ion in $nS$ state the nonrelativistic shielding constant,
using Eq. (24), is

$\displaystyle\sigma=$ $\displaystyle\
\frac{(Z\,\alpha)^{2}}{3\,n^{2}}\,\frac{m_{N}}{m_{N}+m_{e}}+\frac{(Z\,\alpha)^{2}\,m_{e}^{2}}{3\,n^{2}\,g_{N}\,(m_{N}+m_{e})^{2}}$
$\displaystyle\times$
$\displaystyle\biggl{[}3-\frac{g_{N}}{2}+\frac{m_{N}}{m_{N}+m_{e}}\,\biggl{(}1-\frac{1}{Z}\biggr{)}-(g_{N}-1)\,\frac{m_{N}}{Z\,m_{e}}\biggr{]}$
$\displaystyle=$ $\displaystyle\
\frac{(Z\,\alpha)^{2}}{3\,n^{2}\,g_{N}\,(1+x)^{2}}\,\biggl{[}\biggl{(}3-\frac{g_{N}}{2}+\frac{x}{1+x}\biggr{)}$
$\displaystyle\
+\frac{x^{2}}{Z}\,\biggl{(}\frac{1}{1+x}+g_{N}\biggr{)}\biggr{]}\,,$ (32)

where $x=m_{N}/m_{e}$. It is convenient to define $\delta
g_{N}=-g_{N}\,\sigma$

$\displaystyle\delta g_{N}=$ $\displaystyle\
\frac{(Z\,\alpha)^{2}}{3\,n^{2}\,(1+x)^{2}}\,\biggl{[}\frac{g_{N}}{2}-3-\frac{x}{1+x}$
$\displaystyle\
-\frac{x^{2}}{Z}\,\biggl{(}\frac{1}{1+x}+g_{N}\biggr{)}\biggr{]}\,,$ (33)

which is in agreement with the known formula for the electron g-factor in the
S-state of the hydrogenic ion [12, 7],

$\displaystyle\delta g_{e}=$ $\displaystyle\
\frac{(Z\,\alpha)^{2}}{3\,n^{2}\,(1+x)^{2}}\,\biggl{[}\biggl{(}\frac{g_{e}}{2}-3-\frac{x}{1+x}\biggr{)}$
$\displaystyle\ -Z\,x^{2}\,\biggl{(}\frac{1}{1+x}+g_{e}\biggr{)}\biggr{]}$
(34)

with $x=m_{e}/m_{N}$, which verifies the new formula for the magnetic
shielding in hydrogen-like ions. Its small electron mass expansion takes the
following form:

$\displaystyle\sigma=$ $\displaystyle\
\frac{Z\,\alpha^{2}}{3\,n^{2}}\,\biggl{[}1+\frac{m_{e}}{m_{N}}\,\bigg{(}\frac{1}{g_{N}}-2\bigg{)}$
$\displaystyle\
+\frac{m_{e}^{2}}{m_{N}^{2}}\,\bigg{(}\frac{4\,Z-3}{g_{N}}-\frac{Z}{2}+3\bigg{)}+\ldots\biggr{]}\,,$
(35)

where the quadratic in the mass ratio term differs from that derived
previously in Ref. [7, 3]. As seen from Table I, the largest uncertainty for
light ions comes from the relativistic recoil correction $\sigma^{(4,1)}$, but
this has not yet been studied in the literature.

##  IV Finite nuclear size corrections

Let us pass now to another nuclear correction, which is due to the finite
distribution of the charge and the magnetic moment within the nucleus. We will
study this correction for hydrogenic ions only, but generalization for an
arbitrary ion is straightforward. For light hydrogenic ions this effect is
given by

$\displaystyle\sigma_{\mathrm{fs}}=-\frac{Z\,\alpha^{2}}{3}\,\bigl{[}2\,(Z\,\alpha)^{2}\,m^{2}\,(r_{C}^{2}+r_{M}^{2})+8\,(Z\,\alpha)^{3}\,m\,\tilde{r}_{Z}\bigr{]},$
(36)

where $m=m_{e}$, $r_{C}$ is the charge radius, $r_{M}$ the magnetic radius,
and $\tilde{r}_{Z}$ the effective Zemach radius of the nucleus. This formula
is proved as follows.

The shift of nonrelativistic hydrogenic due to $r_{C}$ is given by

$\displaystyle\delta H=$ $\displaystyle\
e\,A^{0}-\frac{e}{6}\,r_{C}^{2}\,\vec{\nabla}\vec{E}$ $\displaystyle=$
$\displaystyle\
-Z\,\alpha\,\Bigl{(}\frac{1}{r}-\frac{2\,\pi}{3}\,\delta^{3}(r)\,r_{C}^{2}\Bigr{)}\,,$
(37)

where $e=e_{e}$. The finite nuclear size affects the nonrelativistic wave
function, which in turn affects the matrix elements for the nuclear magnetic
shielding

$\displaystyle\sigma_{C}=$ $\displaystyle\
2\,\frac{\alpha}{3\,m}\,\Bigl{\langle}\frac{1}{r}\,\frac{1}{(E-H)^{\prime}}\,\frac{2\,\pi}{3}\,Z\,\alpha\,r_{C}^{2}\,\delta^{3}(r)\Bigr{\rangle}$
$\displaystyle=$ $\displaystyle\
-2\,(Z\,\alpha)^{2}\,m^{2}\,r_{C}^{2}\,\frac{Z\,\alpha^{2}}{3}\,.$ (38)

To derive the contribution from the magnetic radius of the nucleus, let us
rederive the leading shielding that comes from the $e^{2}\,\vec{A}^{2}/(2\,m)$
term in the kinetic energy of the electron

$\displaystyle\delta E=$ $\displaystyle\
\frac{\alpha}{2\,m}\,\Bigl{\langle}(\vec{B}\times\vec{r})\cdot\Bigl{(}\vec{\mu}\times\frac{\vec{r}}{r^{3}}\Bigr{)}\Bigr{\rangle}$
$\displaystyle=$ $\displaystyle\
\vec{\mu}\cdot\vec{B}\,\frac{\alpha}{3\,m}\Bigl{\langle}\vec{r}\cdot\frac{\vec{r}}{r^{3}}\Bigr{\rangle}.$
(39)

The shielding $\sigma$ is thus given by

$\displaystyle\sigma=$ $\displaystyle\
-\frac{\alpha}{3\,m}\,\Bigl{\langle}\vec{r}\cdot\vec{\nabla}\Big{(}\frac{1}{r}\Big{)}\Bigr{\rangle}.$
(40)

The magnetic radius $r_{M}$ enters the magnetic interaction similarly to
$r_{C}$ in Eq. (37); therefore, the shift due to the magnetic radius is

$\displaystyle\sigma_{M}=$ $\displaystyle\
\frac{\alpha}{3\,m}\,\frac{2\,\pi}{3}\,r_{M}^{2}\,\Bigl{\langle}\vec{r}\cdot\vec{\nabla}\Bigl{(}\delta^{3}(r)\Bigr{)}\Bigr{\rangle}$
$\displaystyle=$ $\displaystyle\
\frac{\alpha}{3\,m}\,\frac{2\,\pi}{3}\,r_{M}^{2}\,(-3)\,\langle\delta^{3}(r)\rangle$
$\displaystyle=$ $\displaystyle\
-2\,(Z\,\alpha)^{2}\,m^{2}\,r_{M}^{2}\,\frac{Z\,\alpha^{2}}{3}.$ (41)

The calculation of the shift due to the Zemach radius $\tilde{r}_{Z}$ is more
complicated. $\tilde{r}_{Z}$ represents the hyperfine anomaly. If we assume
that it comes exclusively from the charge and magnetic moment distribution, it
becomes $r_{Z}$ given by Eq. (58), which can only be derived from the Dirac
equation. Let us thus start derivation from the relativistic hyperfine
splitting

$\displaystyle E_{\mathrm{hfs}}=$ $\displaystyle\
-e\,\Bigl{\langle}\psi^{\dagger}\Big{|}\vec{\alpha}\cdot\vec{A}_{\mathrm{I}}\Big{|}\psi\Bigr{\rangle},$
(42)

where, for a point nucleus

$\displaystyle\vec{A}_{\mathrm{I}}=$ $\displaystyle\
\frac{1}{4\,\pi}\,\vec{\mu}_{\mathrm{I}}\times\frac{\vec{r}}{r^{3}}.$ (43)

In the nonrelativistic limit $E_{\mathrm{hfs}}$ is given by the Fermi formula

$\displaystyle E_{F}=$ $\displaystyle\
-\frac{e}{2\,m}\,\big{\langle}\phi\big{|}\big{\\{}\vec{\sigma}\cdot\vec{p}\,,\,\vec{\sigma}\cdot\vec{A}_{\mathrm{I}}\big{\\}}\big{|}\phi\big{\rangle}$
$\displaystyle=$
$\displaystyle-\big{\langle}\phi|\vec{\mu}_{e}\cdot\vec{B}_{\mathrm{I}}|\phi\big{\rangle}$
$\displaystyle=$
$\displaystyle-\frac{2}{3}\big{\langle}\phi|\vec{\mu}_{e}\cdot\vec{\mu}_{\mathrm{I}}\,\delta^{3}(r)|\phi\big{\rangle}.$
(44)

We are now ready to consider the leading finite nuclear size correction to the
hyperfine splitting

$\displaystyle E_{Z}=$ $\displaystyle\
2\,\biggl{\langle}\\!\begin{array}[]{c}\phi^{\dagger}(0)\\\
0\end{array}\\!\bigg{|}(-e)\,\vec{\gamma}\cdot\vec{A}_{\mathrm{hfs}}\,\frac{1}{\not\\!p-m}\,e\,\gamma^{0}\,A^{0}\bigg{|}\\!\begin{array}[]{c}\phi(0)\\\
0\end{array}\\!\biggr{\rangle}$ (49) $\displaystyle=$ $\displaystyle\
2\,e^{2}\int\frac{d^{3}p}{(2\,\pi)^{3}}\,\frac{1}{\vec{p}^{\,2}}$
$\displaystyle\times$ $\displaystyle\
\biggl{\langle}\\!\begin{array}[]{c}\phi^{\dagger}(0)\\\
0\end{array}\\!\bigg{|}\vec{\gamma}\cdot\vec{A}_{\mathrm{I}}(-\vec{p})\,(\not\\!p+m)\,\gamma^{0}\,A^{0}(\vec{p})\bigg{|}\\!\begin{array}[]{c}\phi(0)\\\
0\end{array}\\!\biggr{\rangle},$ (54)

where $p^{0}=m$, and

$\displaystyle A^{0}(\vec{p})=$ $\displaystyle\
-\frac{Z\,e}{\vec{p}^{\,2}}\,G_{E}(\vec{p}^{\,2}),$ (55)
$\displaystyle\vec{A}_{\mathrm{I}}(\vec{p})=$ $\displaystyle\
-i\,\vec{\mu}_{\mathrm{I}}\times\frac{\vec{p}}{p^{2}}\,G_{M}(\vec{p}^{\,2}),$
(56)

with normalization $G_{E}(0)=G_{M}(0)=1$. $E_{Z}$ can be simplified to

$\displaystyle E_{\rm Z}=$ $\displaystyle\
\frac{2\,Z\,\alpha\,m}{\pi^{2}}\,\int\frac{d^{3}p}{p^{4}}\,\biggl{[}G_{E}(p^{2})\,G_{M}(p^{2})-1\biggr{]}\,E_{F}$
$\displaystyle=$ $\displaystyle\ -2\,Z\,\alpha\,m\,r_{\rm Z}\,E_{F}\,,$ (57)

where

$r_{\rm Z}=\int d^{3}r_{1}\int
d^{3}r_{2}\,\rho_{E}(r_{1})\,\rho_{M}(r_{2})\,|\vec{r}_{1}-\vec{r}_{2}|,$ (58)

and where $\rho_{E}$ and $\rho_{M}$ are the Fourier transforms of $G_{E}$ and
$G_{M}$. If we are about to represent complete hyperfine anomaly, then $r_{Z}$
becomes $\tilde{r}_{Z}$ in Eq. (57), because it may include the nuclear
inelastic contribution.

Let us now combine perturbation due to $r_{Z}$ and the homogenous magnetic
field

$\displaystyle\delta E=$ $\displaystyle\
2\,\Bigl{\langle}\\!\bar{\psi}_{M}\Big{|}(-e)\,\vec{\gamma}\cdot\vec{A}_{\mathrm{hfs}}\,\frac{1}{\not\\!p-e\not\\!\\!A-m}\,e\,\gamma^{0}\,A^{0}\Big{|}\psi_{M}\Bigr{\rangle},$
(59)

where

$|\psi_{M}\rangle=\biggl{(}I-\frac{1}{2\,m}\,\vec{\gamma}\,(\vec{p}-e\,\vec{A})+\frac{e}{8\,m^{2}}\,\vec{\sigma}\vec{B}\biggr{)}\left|\begin{array}[]{c}\phi_{M}(0)\\\
0\end{array}\right\rangle,$ (60)

and where $\phi_{M}$ is an eigenstate of

$\displaystyle H_{M}=$
$\displaystyle\frac{p^{2}}{2\,m}-\frac{Z\,\alpha}{r}-\frac{e}{2\,m}\,\vec{\sigma}\vec{B}\biggl{(}1-\frac{p^{2}}{2\,m^{2}}+\frac{Z\alpha}{6\,m\,r}\\!\biggr{)}.$
(61)

We claim that the $e\,\vec{A}$ terms in the propagator and in the wave
function can be neglected, because their contribution goes with the nuclear
radius to the third power. Therefore, we have only two corrections due to the
last terms in Eq. (60) and Eq. (61), namely

$\displaystyle\delta E=$ $\displaystyle\
2\,\langle\phi|H_{Z}\,\frac{e}{8\,m^{2}}\,\vec{\sigma}\vec{B}\rangle$
$\displaystyle-2\,\Big{\langle}\phi\Big{|}H_{Z}\,\frac{1}{(E-H)^{\prime}}\frac{e}{2\,m}\,\vec{\sigma}\vec{B}\biggl{(}-\frac{p^{2}}{2\,m^{2}}+\frac{Z\alpha}{6\,m\,r}\\!\biggr{)}\Big{|}\phi\Big{\rangle},$
(62)

where

$\displaystyle H_{Z}=$ $\displaystyle\
\frac{2}{3}\vec{\mu}_{e}\cdot\vec{\mu}_{\mathrm{I}}\,\delta^{3}(r)\,(2\,Z\,\alpha\,m\,r_{Z}).$
(63)

Therefore,

$\displaystyle\delta E=$ $\displaystyle\
\vec{\mu}_{\mathrm{I}}\cdot\vec{B}\,\alpha\,(Z\,\alpha)^{3}\,m\,r_{Z}\,\frac{8}{3}\,\biggl{(}\frac{1}{4}-\frac{X}{m^{2}\,(Z\,\alpha)^{3}}\biggr{)},$
(64)

where

$\displaystyle X=$ $\displaystyle\
\Big{\langle}\phi\Big{|}\pi\,\delta^{3}(r)\,\frac{1}{(E-H)^{\prime}}\biggl{(}-\frac{p^{2}}{2\,m^{2}}+\frac{Z\alpha}{6\,m\,r}\\!\biggr{)}\Big{|}\phi\Big{\rangle}$
$\displaystyle=$ $\displaystyle\ \frac{5}{4}\,(Z\,\alpha)^{3}\,m^{2}.$ (65)

Thus with $\delta E=\vec{\mu}_{\mathrm{I}}\cdot\vec{B}\,\sigma_{Z}$

$\displaystyle\sigma_{Z}=$ $\displaystyle\
-\frac{8}{3}\,\alpha\,(Z\,\alpha)^{4}\,m\,r_{Z},$ (66)

which proves Eq. (36). In addition, Yerokhin [13] verified this equation by
numerically calculating the magnetic with Dirac wave functions for various
$Z$, charge, and magnetic radii of the nucleus. The advantage of Eq. (36) over
the direct numerical calculation is the presence of $\tilde{r}_{Z}$ instead of
$r_{Z}$, which represents the sum of elastic and inelastic contributions to
HFS, and thus can be determined from the HFS anomaly.

##  V Summary

Table 1: Contributions to the shielding constant $\sigma\cdot 10^{6}$ for 1H, 3He+, and 3He using Ref. [8, 3]. New results are $\sigma^{(2,1)}$(He), $\sigma^{(2,2)}$, $\sigma^{(6)}$ and $\sigma_{\mathrm{fs}}$. Because the direct numerical calculation of QED corrections to $\sigma^{(6)}$ is not sufficiently accurate for low $Z$ [9, 10], we estimate uncertainty from QED corrections at this order by assuming that it does not exceed the known relativistic contribution to $\sigma^{6}$. $\sigma_{\mathrm{fs}}$ was calculated using: $r_{C}(p)=r_{M}(p)=0.84$ fm [14], $\tilde{r}_{Z}(p)=0.87$ fm [5], $r_{C}(h)=r_{M}(h)=1.97$ fm [15], $\tilde{r}_{Z}(h)=2.60$ fm, other physical constants are from [16]. |  |  |
---|---|---|---
|  1H |  3He+ |  3He
$\sigma^{(2,0)}$ | $17.750\,451\,5$ | $35.500\,903\,0$ | $59.936\,771\,0$
$\sigma^{(2,1)}$ | $-0.017\,603\,7$ | $-0.013\,933\,4$ | $-0.023\,020\,1$
$\sigma^{(2,2)}$ | $0.000\,014\,1$ | $0.000\,001\,4$ | $0.000\,002\,1(7)$
$\sigma^{(4,0)}$ | $0.002\,546\,9$ | $0.020\,375\,1$ | $0.052\,663\,1$
$\sigma^{(4,1)}$ | $0.000\,000\,0(28)$ | $0.000\,000\,0(74)$ | $0.000\,000\,0(192)$
$\sigma^{(5,0)}$ | $0.000\,018\,4$ | $0.000\,082\,0$ | $0.000\,096\,3$
$\sigma^{(6,0)}$ | $0.000\,000\,2(2)$ | $0.000\,006\,5(65)$ | $0.000\,012\,9(129)$
$\sigma_{\mathrm{fs}}$ | $-0.000\,000\,1$ | $-0.000\,002\,8$ | $-0.000\,005\,6(28)$
$\sigma\cdot 10^{6}$ | $17.735\,427(3)$ | $35.507\,432(10)$ | $59.966\,520(23)$
previous | $17.735\,436(3)$ | $35.507\,434(9)$ | $59.967\,029(23)$
|  |  |

The total magnetic shielding for hydrogen-like ions including contributions up
to order $\alpha^{6}$ is (cf. Eq. (25) of Ref. [2]),

$\displaystyle\sigma=$ $\displaystyle\
\frac{Z\,\alpha^{2}}{3}+\frac{97}{108}\,Z^{3}\,\alpha^{4}+\frac{289}{216}\,Z^{5}\,\alpha^{6}+\frac{8\,\alpha^{2}}{9\,\pi}\,(Z\,\alpha)^{3}$
$\displaystyle\times\biggl{[}\ln(Z\,\alpha)^{-2}+2\ln k_{0}-3\ln
k_{3}-\frac{221}{64}+\frac{3}{5}\biggr{]}$
$\displaystyle+\frac{Z\,\alpha^{2}}{3}\,\biggl{[}\biggl{(}\frac{1}{g_{N}}-2\biggr{)}\frac{m}{m_{N}}+\bigg{(}\frac{4\,Z-3}{g_{N}}-\frac{Z}{2}+3\bigg{)}\frac{m^{2}}{m_{N}^{2}}\biggr{]}$
$\displaystyle\
-\frac{Z\,\alpha^{2}}{3}\,\bigl{[}2\,(Z\,\alpha)^{2}\,m^{2}\,(r_{C}^{2}+r_{M}^{2})+8\,(Z\,\alpha)^{3}\,m\,\tilde{r}_{Z}\bigr{]},$
(67)

where [17]

$\displaystyle\ln k_{0}=$ $\displaystyle\ 2.984\,128\,556,$ (68)
$\displaystyle\ln k_{3}=$ $\displaystyle\ 3.272\,806\,545\,.$ (69)

Numerical results for all these known contributions to the magnetic shielding
in H, He+, and He are presented in Table I. The updated values are
$\sigma^{(2,1)}$ for He, where we corrected the sign error in the last term in
Eq. (31). This leading recoil correction to the magnetic shielding is about
$0.02\cdot 10^{-6}$, which is the relative $2\cdot 10^{-8}$ correction in the
determination of nuclear magnetic moments. The higher-order recoil correction,
the last term in Eq. (35), which is also corrected in this work, is much
smaller and thus is negligible at present accuracy of measurements. The same
holds for nuclear finite size effects, described by Eq. (36); they are
negligible for light elements and can safely be neglected. However, the
nuclear finite size effects can be significant for heavy elements, where they
strongly affect binding energies and hyperfine splitting.

Finally, our recommended values for the nuclear magnetic shieldings are in the
penultimate row, and they are compared to previous recommendations from Ref.
[3] in the last row. The largest change of $0.5\cdot 10^{-9}$ is for the He
atom; changes to H and He+ ion are negligible.

We can now use these new shieldings to recalculate the helion magnetic moment
from He+ measurement

$\displaystyle\mu(^{3}\mathrm{He}^{+})=$ $\displaystyle\
-4.255\,099\,606\,9(30)(17)\times\frac{\mu_{N}}{2},$ (70)

namely, it is

$\displaystyle\mu(^{3}\mathrm{He}^{++})=$ $\displaystyle\
\frac{\mu(^{3}\mathrm{He}^{+})}{1-\sigma(^{3}\mathrm{He}^{+})}$
$\displaystyle=$ $\displaystyle\ -2.127\,625\,350\,0(17)\,\mu_{N}\,,$ (71)

which differs slightly from that in Ref. [1] , while our recommended value for
the atomic 3He magnetic moment is

$\displaystyle\mu(^{3}\mathrm{He})=$ $\displaystyle\
\mu(^{3}\mathrm{He}^{+})\frac{1-\sigma(^{3}\mathrm{He})}{1-\sigma(^{3}\mathrm{He}^{+})}$
$\displaystyle=$ $\displaystyle\ -2.127\,497\,763\,7(17)\,\mu_{N}\,,$ (72)

which can serve as a reference in gaseous NMR measurements [18] because it is
the most accurately known atomic magnetic moment.

###### Acknowledgements.

We wish to thank Vladimir Yerokhin for numerical calculations of the finite
nuclear size effects in hydrogen-like ions, and Jan Kłos for participation at
the beginning of the project. This work was supported by the National Science
Center (Poland) Grant No. 2017/27/B/ST2/02459.

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