# High-precision determination of $g$ factors and masses of 20Ne9+ and 22Ne9+

F. Heiße<EMAIL_ADDRESS>M. Door<EMAIL_ADDRESS>T.
Sailer  P. Filianin  J. Herkenhoff  C. M. König  K. Kromer  D. Lange  J.
Morgner  A. Rischka  Ch. Schweiger  B. Tu  Max-Planck-Institut für Kernphysik,
Heidelberg, Germany  Y. N. Novikov  Kurchatov Institute-PNPI, 188300 Gatchina,
Russia  Saint Petersburg State University, 199034 Saint Petersburg, Russia  S.
Eliseev  S. Sturm  K. Blaum  Max-Planck-Institut für Kernphysik, Heidelberg,
Germany

###### Abstract

We present the measurements of individual bound electron $g$ factors of 20Ne9+
and 22Ne9+ on the relative level of $0.1\,\text{parts}$ per billion. The
comparison with theory represents the most stringent test of bound-state QED
in strong electric fields. A dedicated mass measurement results in
$m\left({}^{20}\text{Ne}\right)=19.992\,440\,168\,77\,(9)\,\text{u}$, which
improves the current literature value by a factor of nineteen, disagrees by
$4$ standard deviations and represents the most precisely measured mass value
in atomic mass units. Together, these measurements yield an electron mass on
the relative level of $0.1\,\text{ppb}$ with
$m_{\text{e}}=5.485\,799\,090\,99\,(59)\times 10^{-4}\,\text{u}$ as well as a
factor of eight improved
$m\left({}^{22}\text{Ne}\right)=21.991\,385\,098\,2\,(26)\,\text{u}$.

The theory of quantum electrodynamics (QED) successfully describes a broad
scope of phenomena ranging from elementary particle physics to atoms and
molecules Tomonaga (1946); Schwinger (1948); Feynman (1949). Since QED is a
blueprint for all quantum field theories, it is of utmost importance to verify
its sophisticated calculation methods as precisely as possible Fan  _et al._
(2023); Aoyama  _et al._ (2019). In particular, the deviation between the
experimental results and the theoretical prediction for muon $g-2$ Abi  _et
al._ (2021), muonic deuterium hyperfine splitting Kalinowski  _et al._ (2018)
as well as the fine structure anomaly in heavy muonic systems Valuev  _et al._
(2022) currently challenge our understanding of QED. That calls for additional
high-precision QED tests. Furthermore, based on such comparisons fundamental
constants can be extracted, e.g. the fine structure constant $\alpha$ Fan  _et
al._ (2023), the Rydberg constant $R_{\infty}$ Parthey  _et al._ (2011);
Antognini  _et al._ (2013) as well as the electron, proton, deuteron, muon and
charged pion masses Sturm  _et al._ (2014); Alighanbari  _et al._ (2020);
Patra  _et al._ (2020); Kortunov  _et al._ (2021); Alighanbari  _et al._
(2023); Liu  _et al._ (1999); Trassinelli  _et al._ (2016).

The determination of magnetic moments and Lamb shifts of bound leptons allows
tests of bound-state (BS) QED under extreme conditions, such as strong
electric and magnetic fields Sturm  _et al._ (2011a); Gumberidze  _et al._
(2005); Beiersdorfer  _et al._ (2005); Ullmann  _et al._ (2017); Beltrami  _et
al._ (1986). In this letter, we present a BS-QED test with highest precision
by comparing state-of-the-art bound-electron $g$-factor calculations of 20Ne9+
and 22Ne9+ with corresponding experimental results. The electron $g$ factor is
a dimensionless constant relating the electron’s magnetic moment to its spin.
The energy splitting of the Zeeman levels in a magnetic field $B$ is given by:
$h\nu_{\text{L}}=h\frac{g}{2}\frac{e}{2\pi m_{\text{e}}}B$ Beier (2000). Here,
$\nu_{\text{L}}$ is the Larmor spin precession frequency of an electron, $h$
is the Planck’s constant and $\frac{e}{m_{\text{e}}}$ is the electron’s
charge-to-mass ratio. Additionally, the free-space cyclotron frequency of an
ion in the magnetic field can be expressed by
$\nu_{\text{c}}=\frac{1}{2\pi}\frac{q_{\text{ion}}}{m_{\text{ion}}}B$, where
$\frac{q_{\text{ion}}}{m_{\text{ion}}}$ is the charge-to-mass ratio of the
ion. Combining both equations yields

$g_{\text{exp}}=2\frac{\nu_{\text{L}}}{\nu_{\text{c}}}\frac{m_{\text{e}}}{m_{\text{ion}}}\frac{q_{\text{ion}}}{e}=2\,\Gamma\frac{m_{\text{e}}}{m_{\text{ion}}}\frac{q_{\text{ion}}}{e}\quad.$
(1)

In this work, the $\Gamma$ measurement is performed at the Alphatrap setup
Sturm  _et al._ (2019), while the mass of ${}^{20}\text{Ne}^{10+}$ is measured
at the Pentatrap experiment Repp  _et al._ (2012), due to the insufficient
precision of the current literature value. The charge ratio
$\nicefrac{{q_{\text{ion}}}}{{e}}$ is a known integer number and the electron
mass as well as the mass of ${}^{22}\text{Ne}^{9+}$ are determined by other
experiments Sturm  _et al._ (2014); Huang  _et al._ (2021).

The mass of ${}^{20}\text{Ne}^{10+}$ is determined by measuring the cyclotron-
frequency ratio
$\mathcal{R}^{\text{CF}}=\nicefrac{{\nu_{\text{c}}\left({}^{20}\text{Ne}^{10+}\right)}}{{\nu_{\text{c}}\left(\text{${}^{12}$C${}^{6+}$}\right)}}$.
Given both $\nu_{\text{c}}$ measurements are carried out in the same $B$
field, the ${}^{20}\text{Ne}^{10+}$ mass can be expressed by:
$m\left({}^{20}\text{Ne}^{10+}\right)=\frac{10}{6}\,m\left(\text{${}^{12}$C${}^{6+}$}\right)/\mathcal{R}^{\text{CF}}$.
The ion masses are linked to $m\left({}^{20}\text{Ne}^{9+}\right)$ and the
neutral masses via the mass of the missing electrons Tiesinga  _et al._ (2021)
and binding energies Kramida  _et al._ (2021), which are known to sufficient
precision.

The Pentatrap and Alphatrap experiments are both cryogenic Penning-trap setups
placed in two superconducting magnets located at the Max Planck Institute for
Nuclear Physics in Heidelberg. Their trap and detection electronics are cooled
to the temperature of liquid helium $\left(4\,\text{K}\right)$. The highly
charged ions are produced externally in Heidelberg compact electron beam ion
traps (HC-EBIT) Micke  _et al._ (2018); Schweiger  _et al._ (2019),
transferred via room temperature beamlines to the respective trap setups and
cleaned from possible contamination ions via magnetron cleaning Heiße  _et
al._ (2019).

Figure 1: Sectional views of the cylindrical Penning-trap setups of Alphatrap
Sturm  _et al._ (2019) (left) and Pentatrap Repp  _et al._ (2012) (right).
Both trap towers are built from gold-plated copper ring electrodes which are
isolated by sapphire and quartz rings. The connected superconducting tank
circuits, the amplifiers for the axial detection systems and the RF excitation
lines are shown. At the Alphatrap experiment, $\nu_{\text{c}}$ is measured
simultaneously to the irradiation of the microwave drive in the Precision
trap. The CoFe alloy central ring electrode in the Analysis trap produces a
strong magnetic bottle, allowing a non-destructive spin-state detection with
close to $100\%$ fidelity. Here, a different spin orientation changes the
axial force resulting from the magnetic moment inside the inhomogeneous
magnetic field leading to axial frequency jumps of the ion (20,22Ne9+ yield
jumps of $1.8\,\text{Hz}$ and $1.6\,\text{Hz}$, respectively) Hermanspahn  _et
al._ (2000). At the Pentatrap setup, both ions’ $\nu_{\text{c}}$ are measured
successively in each measurement trap, resulting in one cyclotron frequency
ratio per trap. The ion species are effectively swapped in the measurement
traps by adiabatically transporting all ions to the respective neighbouring
trap and back (position 1 and 2). The different trapping potential depths
($\mathrm{\phi}$) in the two measurement traps are due to different resonance
frequencies of the axial detection circuits.

Due to the superposition of the static electric and magnetic field, the
cyclotron frequency splits into three individual eigenfrequencies: the axial
frequency $\nu_{\text{z}}$ on the order of hundreds of kHz, the magnetron
frequency $\nu_{-}$ on the order of single kHz and the modified cyclotron
frequency $\nu_{+}$ on the order of tens of MHz. Combined, these frequencies
can be utilized to calculate the free-space cyclotron frequency via the
invariance theorem
$\nu_{\text{c}}=\sqrt{\nu_{+}^{2}+\nu_{\text{z}}^{2}+\nu_{-}^{2}}$ Brown and
Gabrielse (1986). Due to the strong hierarchy of the eigenfrequencies,
$\nu_{+}$ has to be determined with highest precision, followed by
$\nu_{\text{z}}$. The magnetron frequency is determined only occasionally
throughout the campaign due to its stability and reduced influence on the
total uncertainty on $\nu_{\text{c}}$.

In both experiments the ion’s eigenfrequencies are measured non-destructively.
The ion’s axial oscillations induce fA image currents in high-impedance
cryogenic tank circuits Dehmelt (1969); Wineland and Dehmelt (1975) leading to
a measurable voltage signal, see Fig. 1. Interaction with the tank circuit,
typically referred to as resistive cooling, causes a temperature of the axial
motion near $4\,\text{K}$. Here, the ion effectively shorts the ambient
Johnson-Nyquist noise Johnson (1928); Nyquist (1928), leading to the
characteristic dip in the spectrum. The axial frequency can be directly
measured by fitting the well-known dip-lineshape to the spectrum. The radial
frequencies are measured via radio-frequency (RF) sideband coupling to the
axial motion, resulting in a double-dip spectrum to extract the initial
parameters for the more precise phase sensitive Ramsey-type measurements
Cornell  _et al._ (1989, 1990); Sturm  _et al._ (2011b).

The Pentatrap experiment Repp  _et al._ (2012); Roux  _et al._ (2012); Rischka
_et al._ (2020) is designed for high-precision mass-ratio determinations of
ions, see Fig. 1. The measurement scheme, similar to previous measurements at
the Pentatrap experiment Rischka  _et al._ (2020); Schüssler  _et al._ (2020);
Filianin  _et al._ (2021); Kromer  _et al._ (2022), starts with three single
ions of two species $\left({}^{20}\text{Ne}^{10+},^{12}\text{C}^{6+}\right)$
loaded in alternating order into the three central traps. The traps 2 and 3
are used in parallel for the determination of $\mathcal{R}^{\text{CF}}$, while
the traps 1 and 4 are exclusively used for storage. The properties of the
traps are summarized in Table 1. At Pentatrap $\nu_{\text{z}}$ is
simultaneously determined during the longest phase-evolution time of the
Ramsey-type $\nu_{+}$ measurement Rischka  _et al._ (2020). This reduces the
effect of electric field jitter compared to sequential measurements and allows
for shorter overall measurement times. Daily reloading of the ions, as a
result of the limited ion lifetime in Pentatrap, reduces the impact of
potential eigenfrequency shifts on $\mathcal{R}^{\text{CF}}$ due to
hypothetically unnoticed ion contaminations.

Table 1: Trap properties of the Alphatrap and Pentatrap experiments. The magnetic field inhomogeneities are defined via $B(z)=B_{0}+B_{1}z+B_{2}z^{2}$, where $z$ is the axial position. The different $\nu_{+}$ for PT and AT are for 20Ne9+ and ${}^{22}\text{Ne}^{9+}$, respectively. For Pentatrap the corresponding eigenfrequencies of ${}^{20}\text{Ne}^{10+}$ and ${}^{12}\text{C}^{6+}$ are given. | Alphatrap | Pentatrap
---|---|---
| PT | AT | Trap 2 | Trap 3
$B_{0}\left[\text{T}\right]$ | $\approx 4$ | $\approx 3.9$ | $\approx 7$
$B_{1}\left[\frac{\text{mT}}{\text{m}}\right]$ | $2.64\,(3)$ | $\genfrac{}{}{0.0pt}{1}{<}{>}\pm 10^{4}$ | $1.41\,(27)$ | $-1.49\,(16)$
$B_{2}\left[\frac{\text{mT}}{\text{m}^{2}}\right]$ | $64\,(3)$ | $43.1\,(1)\times 10^{6}$ | $64\,(5)$ | $22\,(5)$
$\nu_{+}\left[\text{MHz}\right]$ | $\approx 28/25$ | $\approx 27/24$ | $\approx 54$
$\nu_{\text{z}}\left[\text{kHz}\right]$ | $\approx 651$ | $\approx 334$ | $\approx 736$ | $\approx 502$
$\nu_{-}\left[\text{kHz}\right]$ | $\approx 8$ | $\approx 2$ | $\approx 5$ | $\approx 2$
$\nu_{\text{L}}\left[\text{GHz}\right]$ | $\approx 113$ | $\approx 108$ | – | –
$T_{\text{z}}\left[\text{K}\right]$ | $5.7\,(3)$ | $7\,(2)$

Pentatrap has an active stabilization system for the helium gas pressure and
liquid helium level in the cold bore of the magnet, resulting in a very stable
magnetic field. The temporal variation is dominated by a constant relative
degradation of about $0.3\,\frac{\text{ppb}}{\text{h}}$. Therefore, it is
sufficient to measure only one full set of unwrapping phases for different
evolution times at the beginning and merely the phase of the shortest
$\left(\phi^{\text{ref}}\right)$ and longest evolution time
$\left(\phi^{\text{precision}}\right)$ in all subsequent measurement cycles to
determine $\nu_{+}$.

After the determinations of ten $\phi^{\text{ref}}$ and ten
$\phi^{\text{precision}}$ the ion positions are swapped by transporting the
ions to the adjacent traps and the phase measurements are repeated. Each
cycle, including forty phase measurements, ion cooling and transports, takes
about $22$ minutes. An example of measurement data is shown in Fig. 2. To
accommodate for the slow magnetic field loss of our magnet, each cycle’s
$\mathcal{R}^{\text{CF}}$ is determined using interpolated data of the free
cyclotron frequencies of the two ions to a common time in each trap Filianin
_et al._ (2021).

Figure 2: (a) Exemplary $\nu_{\text{c}}$ data from traps 2 and 3. (b)
Determined $\mathcal{R}^{\text{CF}}$ for different
$\left(\rho_{+}^{\text{exc}}\right)^{2}$ including the corresponding least-
square fits and $1\,\sigma$ confidence bands with $\chi_{\text{red}}^{2}=1.11$
and $0.65$ for traps 2 and 3, respectively. Higher-order shifts
$\propto\left(\rho_{+}^{\text{exc}}\right)^{3}$ and
$\left(\rho_{+}^{\text{exc}}\right)^{4}$ are estimated to be negligible. The
$\mathcal{R}^{\text{CF}}$ of each trap is extracted from the fit at
$\left(\rho_{+}^{\text{exc}}\right)^{2}=0\,\mu\text{m}^{2}$. This
extrapolation reduces the relativistic shift and all other shifts that are
proportional to $\left(\rho_{+}^{\text{exc}}\right)^{2}$.

The statistical shot-to-shot jitter of $\mathcal{R}^{\text{CF}}$ (about
$7\times 10^{-11}$) originates mainly from the ion’s thermally distributed
initial radius of the modified cyclotron motion, which averages io
approximately $2~{}\,\mu\text{m}$ and takes a slightly different value after
each cooling cycle. This distribution persists after the excitation pulse for
the Ramsey-type measurement scheme and results in a variation of the
relativistic frequency shift on $\nu_{+}$. These fluctuations lead to a phase
jitter that increases linearly with the phase evolution time and thus limits
the phase evolution time to $20\,\text{s}$. The excitation radii were varied
between $\rho_{+}^{\text{exc}}=15-50\,\mu\text{m}$ to extrapolate
$\mathcal{R}^{\text{CF}}$ to virtually zero excitation for each trap Heiße
_et al._ (2017, 2019), see Fig. 2.

All systematic effects are summarized in Table 2. The main systematic
uncertainties are caused by the dip-fit lineshape Rau  _et al._ (2020), the
non-linear phase readout effect Filianin  _et al._ (2021) and by the trapping-
potential depth difference to match the two ions axial frequencies with the
detection systems resonance Rischka  _et al._ (2020).

Table 2:  Relative systematic corrections and their uncertainties on ${\mathcal{R}}^{\text{CF}}$ at Pentatrap after the extrapolation to zero-excitation amplitude for both traps. All values are given in parts per trillion ($10^{-12}$). The systematic shifts are assumed to be uncorrelated between the traps. Effect (parameters) | Trap 2 | Trap 3
---|---|---
$T_{z}$ ($B_{2}$, $C_{3}$, $C_{4}$, $C_{6}$) | $-0.8\,(0.3)$ | $-1.7\,(0.5)$
Magnetron frequency | $0.0\,(0.5)$ | $0.0\,(0.5)$
Non-linear phase read-out Filianin  _et al._ (2021) | $0.0\,(1.0)$ | $0.0\,(1.9)$
Position difference Rischka  _et al._ (2020) | $0.0\,(1.0)$ | $0.0\,(0.8)$
Dip lineshape Rau  _et al._ (2020) | $0.0\,(1.0)$ | $0.0\,(1.8)$
Image charge shift Van Dyck  _et al._ (1989); Van Dyck  _et al._ (2006); Schuh  _et al._ (2019) | $19.5\,(1.0)$ | $19.5\,(1.0)$
Total | $18.7\,(2.1)$ | $17.8\,(3.0)$

Given these effects, the determined $\mathcal{R}^{\text{CF}}$ for both traps
are

$\begin{split}\mathcal{R}^{\text{CF}}_{\text{Trap
2}}=&1.000\,378\,141\,801\,(5)(2)\,,\\\ \mathcal{R}^{\text{CF}}_{\text{Trap
3}}=&1.000\,378\,141\,807\,(7)(3)\,,\end{split}$

with the first and the second bracket showing the statistical and systematic
uncertainty, respectively. The weighted mean of the two traps yields
$\mathcal{R}^{\text{CF}}_{\text{mean}}=1.000\,378\,141\,802\,(4)$. To
determine $m\left({}^{20}\text{Ne}\right)$, we added the masses of the missing
electrons and their corresponding binding energies Sturm  _et al._ (2014);
Kramida  _et al._ (2021) with an additional uncertainty of $2\,\times
10^{-11}\,\text{u}$ resulting in

$m\left({}^{20}\text{Ne}\right)\,=\,19.992\,440\,168\,77\,(9)\,\text{u}\,.$

This represents the most precise mass value ever measured in atomic mass units
and deviates by $4\,\sigma$ from the current literature value Huang  _et al._
(2021); DiFilippo  _et al._ (1994). The possibility to achieve such a
precision for atomic masses paves the way for order of magnitude improved
$m\left({}^{133}\text{Cs}\right)$ and $m\left({}^{87}\text{Rb}\right)$, which
is of utmost importance for the determination of $\alpha$ via photon-recoil
experiments Parker  _et al._ (2018); Morel  _et al._ (2020). For the latter,
the mass uncertainty currently represents the second largest contribution
within the whole uncertainty budget.

The $g$-factor measurements were carried out in the Alphatrap setup Sturm  _et
al._ (2019); Arapoglou  _et al._ (2019); Egl  _et al._ (2019). A cryo-valve
separates the beamline vacuum and the trap vacuum. This ensures a vacuum of
better than $10^{-16}\,\text{mbar}$ in the trap section, enabling ion
lifetimes on the order of months.

The setup includes an Analysis trap (AT) enabling the detection of the
electron’s spin-state orientation with respect to the magnetic field via the
continuous Stern-Gerlach effect Dehmelt (1986); Hermanspahn  _et al._ (2000)
as well as a spectroscopic Precision trap (PT), distinguished by a harmonic
electric and a homogeneous magnetic field, see Fig. 1. In the PT the spin-flip
transition is probed to highest precision by irradiating a microwave (MW)
pulse $\nu_{\text{MW}}$, while simultaneously monitoring the magnetic field
via $\nu_{\text{c}}$.

The double-trap measurement sequence is similar to Sturm  _et al._ (2014);
Köhler  _et al._ (2015). Each measurement cycle begins by placing the
respective single neon ion in the AT and determining the electron spin-state
via microwave irradiation at the corresponding Larmor frequency of the AT, see
Tab. 1. Next, it is transported into the PT to measure $\nu_{\text{z}}$ and
$\nu_{+}$ via the dip and double-dip method followed by the phase-senstive
measurement of $\nu_{+}$ Sturm  _et al._ (2011b). During the longest phase-
evolution time of $\nu_{+}$ ($8\,\text{s}$ in this work), a MW pulse is
simultaneously irradiated around the expected $\nu_{\text{L}}$. Together with
$\nu_{\text{c}}$ this yields one
$\Gamma_{\text{i}}=\nicefrac{{\nu_{\text{MW}}}}{{\nu_{\text{c}}}}$. Finally,
the ion is transported back into the AT for the spin-state determination,
resulting in a cycle-length of 25 minutes.

For each cycle with a spin-state change in the PT, the corresponding
$\Gamma_{\text{i}}$ is assigned with $1$, whereas in the other case with $0$.
This boolean information of different probed $\Gamma_{\text{i}}$’s results in
a probability distribution (resonance), see Fig. 3. The resonance width is
mainly given by the intrinsic magnetic field fluctuations during the
$8~{}\text{s}$ probe-time of $\Gamma$.

Figure 3: $\Gamma$ resonance for 20Ne9+ (top) and 22Ne9+ (bottom). The upper
resonance contains 425 cycles, whereas the lower one contains 585 cycles. Both
resonances are not power saturated and fitted by the Maximum-likelihood method
with a three-parameter Gaussian lineshape (red line), since asymmetric effects
shift the resonance center far below the ppt level Brown and Gabrielse (1986);
Verdú  _et al._ (2004); Köhler  _et al._ (2015). Here, the amplitude, the
resonance width and the centre of the resonance are the free parameters. The
$1-\sigma$ confidence band of the fit is displayed in grey. The black binned
data points with binomial error bars are presented to guide the eye. The grey
shadows at the top are observed spin flips in the PT for the respective
$\Gamma$, whereas the ones at the bottom are no spin flips.

The main systematic correction of
$\Gamma\left(\text{${}^{20}$Ne${}^{9+}$}\right)$ and
$\Gamma\left(\text{${}^{22}$Ne${}^{9+}$}\right)$ is due to the special
relativity effect caused by the excited cyclotron radii of
$\rho_{+}^{\text{exc}}\approx 20~{}\mu$m for both isotopes. It amounts to a
relative shift of $-5.3(11)\times 10^{-11}$. The image charge shift correction
is $-2.5(1)\times 10^{-11}$ Van Dyck  _et al._ (1989); Van Dyck  _et al._
(2006); Schuh  _et al._ (2019). Corrections and uncertainties due to magnetic
inhomogeneities, electric field anharmonicities and dip lineshapes are on or
below the $5\times 10^{-12}$ level.

After finishing all measurements, we identified an unexpected systematic shift
originating from technical peculiarity of the cryogenic RF switches during the
phase sensitive detection of $\nu_{+}$. Test measurements set an upper limit
of a potential systematic shift of $\Gamma$ at relative $3.0\times 10^{-11}$,
for more details see Sailer (2022). This effect is diminished by not using the
switches for future measurements. The final results of both
$\Gamma$(20,22Ne9+), corrected for systematic effects, are summarized in Table
3 and are limited by statistics.

Table 3: The measured $\Gamma$’s with the Alphatrap apparatus for 20,22Ne9+ are given including their statistic and systematic uncertainties. Measured and theoretical $g$ factors are shown with combined uncertainties. For the $g$ factors the current CODATA $m_{\text{e}}$, the AME $m\left({}^{22}\text{Ne}\right)$ and $m\left({}^{20}\text{Ne}\right)$ determined in this work are used. The individual BS-QED $g$-factor contributions are only listed below $g\left(\text{${}^{20}$Ne${}^{9+}$}\right)$ since they are equal for both isotopes. The nuclear size and recoil contribution are listed individually. The BS 2-loop uncertainties of $\mathcal{O}\left(\left(Z\alpha\right)^{6}\right)$ are dominating. | ${}^{22}\text{Ne}^{9+}$ | ${}^{20}\text{Ne}^{9+}$
---|---|---
$\Gamma$ | $4450.460\,396\,87\,(42)(14)$ | $4045.837\,341\,56\,(34)(13)$
$g_{\text{exp}}$ | $1.998\,767\,262\,20(174)$ | $1.998\,767\,276\,99(19)$
$g_{\text{theo}}$ | $1.998\,767\,263\,64(12)$ | $1.998\,767\,277\,11(12)$
Dirac (relativistic point nucl.) Breit (1928) | 1.996 445 170 90
Free-electron QED Tiesinga  _et al._ (2021) | 0.002 319 304 35
BS 1-loop Self energy Yerokhin  _et al._ (2004); Yerokhin and Harman (2017); Yerokhin  _et al._ (2013) | 0.000 002 717 05
BS 1-loop Vacuum polarisation Sailer  _et al._ (2022) | \- 0.000 000 063 22
BS 2-loop Sailer  _et al._ (2022) | \- 0.000 000 003 17(12)
BS $3$-loop Grotch (1970) | 0.000 000 000 03
Recoil Sailer  _et al._ (2022) | 0.000 000 133 10 | 0.000 000 146 41
Size Sailer  _et al._ (2022) | 0.000 000 004 60 (1) | 0.000 000 004 76 (1)

The comparison of the experimental $g\left(\text{${}^{20}$Ne${}^{9+}$}\right)$
with theory Sailer  _et al._ (2022) yields the most stringent test of BS-QED
in strong electric fields to date at a level of $84~{}\text{ppm}$, slightly
surpassing the test via $g\left({}^{12}\text{C}^{5+}\right)$ and
$g\left({}^{28}\text{Si}^{13+}\right)$ Sturm  _et al._ (2014, 2013); Czarnecki
_et al._ (2018); Zatorski  _et al._ (2017).

The QED binding corrections to the free-electron $g$ factor are calculated in
the $\alpha$ expansion and include one-, two- and three-loop contributions
Beier (2000). The one-loop correction is calculated in all orders of
$Z\alpha$, whereas for the two-loop correction the perturbation series
expansion has been calculated up to
$\mathcal{O}\left(\left(Z\alpha\right)^{5}\right)$ Czarnecki  _et al._ (2020).
The estimated uncertainty from the uncalculated terms of
$\mathcal{O}\left(\left(Z\alpha\right)^{6}\right)$ is currently limiting all
$1s~{}g$-factor calculation for elements with $Z\geq 4$ Pachucki  _et al._
(2005); Czarnecki  _et al._ (2018). The three-loop correction is currently
calculated up to $\mathcal{O}\left(\left(Z\alpha\right)^{2}\right)$ Grotch
(1970). The relative theory precision of
$g\left(\text{${}^{20}$Ne${}^{9+}$}\right)$ is $6\times 10^{-11}$ and
comparable with our experimental uncertainty.

In this work, the one-loop self-energy corrections are tested at an
unprecedented level, verifying the corresponding calculations Yerokhin and
Harman (2017); Yerokhin  _et al._ (2013). In contrast, the vacuum polarisation
contributions are best tested in muonic atoms Beltrami  _et al._ (1986). Here,
the two-loop contributions are tested at the level of $7\,\%$, confirming the
$Z\alpha$ expansion for intermediate-$Z$. Additionally, the total nuclear
recoil contribution of the $g$ factor is precisely verified. This is
complementary to the $\Delta g$ measurement Sailer  _et al._ (2022), where the
differential nuclear recoil contributions have been confirmed without testing
any QED correction calculated within the $Z\alpha$ expansion.

Although the previously measured
$g_{\text{exp}}\left({}^{28}\text{Si}^{13+}\right)$ is more precise compared
to $g_{\text{exp}}\left({}^{20}\text{Ne}^{9+}\right)$, the uncalculated
$\mathcal{O}\left(\left(Z\alpha\right)^{6}\right)$ contributions of
$g_{\text{theo}}\left({}^{28}\text{Si}^{13+}\right)$ limit a more stringend
QED test in that case. Tests based on the measured
$g\left({}^{12}\text{C}^{5+}\right)$ Sturm  _et al._ (2014) are possible via
an independent $m_{\text{e}}$ determination based on the ro-vibrational
transition energies of $\text{HD}^{+}$Alighanbari  _et al._ (2020); Patra  _et
al._ (2020); Kortunov  _et al._ (2021); Alighanbari  _et al._ (2023); Karr and
Koelemeij (2023) combined with the proton’s and deuteron’s atomic masses Huang
_et al._ (2021); Fink and Myers (2021). In fact the precision of
$g_{\text{exp,theo}}\left({}^{12}\text{C}^{5+}\right)$ and
$m_{\text{e}}\left(\text{HD}^{+}\right)$ is higher compared to
$g\left({}^{20}\text{Ne}^{9+}\right)$, but the corresponding BS-QED
contributions are reduced as well due to their $Z\alpha$ scaling.

Our presented value together with the so far measured high-precision $g$
factors (${}^{12}\text{C}^{5+}$, ${}^{28}\text{Si}^{13+}$ and
${}^{118}\text{Sn}^{49+}$ Morgner  _et al._ (2023)) constrain uncalculated
higher order contributions and eventually experimentally enhance them in the
future Czarnecki  _et al._ (2018).

Additionally, the agreement of the experimental and theoretical $g$ factors
provides a consistency check for $m\left({}^{20}\text{Ne}\right)$ determined
in this work, which deviates from the literature value Huang  _et al._ (2021).
Furthermore, this agreement confirms the limits set by
$g\left({}^{28}\text{Si}^{13+}\right)$ on the $y_{\text{e}}y_{\text{n}}$
coupling constant present in the Higgs-relaxion mixing scenario Debierre  _et
al._ (2020, 2022). Together with the direct $\Delta g$ measurement Sailer  _et
al._ (2022) these $g$-factor results yield
$\left[g\left(\text{${}^{20}$Ne${}^{9+}$}\right)-g\left(\text{${}^{22}$Ne${}^{9+}$}\right)\right]-\Delta
g=1.3(1.7)\times 10^{-9}$. This could be interpreted as a successful 1-2-3
test at the $200\,\text{Hz}$ level ($1.7\times 10^{-9}\times 113\,\text{GHz}$)
in the microwave regime to test quantum mechanics Weinberg (2014, 2016);
Raizen  _et al._ (2022).

Additionally, Eq. (1) can be rearranged to express the electron mass via the
theoretical value $g_{\text{theo}}$:

$m_{\text{e}}=\frac{1}{2}g_{\text{theo}}\frac{m_{\text{ion}}}{\Gamma}\frac{e}{q_{\text{ion}}}=5.485\,799\,090\,99\,(59)\times
10^{-4}\,\text{u}\,,$ (2)

which is limited by the uncertainty of
$\Gamma\left({}^{20}\text{Ne}^{9+}\right)$. Our result is in agreement with
the current CODATA value $\left(0.6\,\sigma\right)$ and a factor of three less
precise Tiesinga  _et al._ (2021).

If we solve Eq. (1) for $m\left(\text{${}^{22}$Ne${}^{9+}$}\right)$ and
correct for the masses of the missing electrons Tiesinga  _et al._ (2021) and
the corresponding binding energies Kramida  _et al._ (2021) the result yields
a factor of eight improved mass uncertainty:

$m\left({}^{22}\text{Ne}\right)=21.991\,385\,098\,2\,(26)\,\text{u}\,,$

which is in accordance with the literature value ($0.8\,\sigma$) Huang  _et
al._ (2021).

In conclusion we determined two experimental $g$ factors of 20,22Ne9+ and the
atomic mass of 20Ne to highest precision. Together with the corresponding
theoretical $g$ factors this yields the most precise test of BS-QED in strong
electric fields. The atomic masses of 20,22Ne are improved by a factor of
sixteen and eight, respectively. Additionally, the electron mass is determined
with a relative precision of $1\times 10^{-10}$.

This work is supported by the Max-Planck-Gesellschaft (MPG), the International
Max-Planck Research Schools for Precision Tests of Fundamental Symmetries
(IMPRS-PTFS) and for Quantum Dynamics in Physics, Chemistry and Biology
(IMPRS-QD). The project received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme under Grant agreement number 832848 - FunI and by the DFG (German
Research Foundation) – Project-ID 273811115 – SFB 1225 ISOQUANT. Furthermore,
we acknowledge funding and support by the Max Planck, RIKEN, PTB Center for
Time, Constants and Fundamental Symmetries. This work comprises parts of the
PhD thesis work of M.D. to be submitted to Heidelberg University, Germany.
F.H., M.D. and T.S. contributed equally to this work.

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# Search for R-parity violating Supersymmetry using the CMS detector

Fedor Ratnikov for the CMS collaboration
Karlsrihe Institute of Technology, Karlsruhe, Germany
Institute of Theoretical and Experimental Physics, Moscow, Russia
<EMAIL_ADDRESS>talk presented at the LHCP 2013 Conference in
Barcelona, Spain, May 13-18th, 2013

###### Abstract

In this talk, the latest results from CMS on R-parity violating Supersymmetry
are reviewed. We present results using up to 20/fb of data from the 8 TeV LHC
run of 2012. Interpretations of the experimental results in terms of
production of squarks, gluinos, charginos, neutralinos, and sleptons within RP
violating susy models are presented.

##  1 Introduction

Supersymmetry (SUSY) [1, 2] is an attractive extension of the Standard Model.
It provides natural coupling unification, dynamic electroweak symmetry
breaking and a solution to the hierarchy problem. R-parity is assigned to
fields as $R_{p}=(-1)^{3B+L+s}$ where $B$, $L$, and $s$ are baryon and lepton
numbers, and spin of the particle respectively. In models with conserved
R-parity superpartners may only be produced in pairs, and the lightest
superpartner (LSP) is stable. However R-parity conservation is not a universal
property of SUSY models. The most general gauge-invariant and renormalizable
superpotential consists of the R-parity conserving (RPC) main part, and may
also contain extra R-parity violating (RPV) terms [3]:

$\displaystyle W_{\Delta L=1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}+\lambda^{\prime}_{ijk}L_{i}Q_{j}\bar{d}_{k}+\mu^{\prime}_{i}L_{i}H_{u}$
(1) $\displaystyle W_{\Delta B=1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{\prime\prime}_{ijk}u_{i}d_{j}\bar{d}_{k}$
(2)

The presence of non-vanishing RPV terms leads to the the LSP becoming
unstable, decaying to standard model (SM) particles. Therefore many SUSY
analyses, which are based on the expectation of high missing transverse energy
in SUSY events from non-observed stable LSPs, are not sensitive to RPV SUSY
models.

Recent CMS analyses [5, 4, 6] are focused on studying the lepton number
violating terms $\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}$ and
$\lambda^{\prime}_{ijk}L_{i}Q_{j}\bar{d}_{k}$, which cause specific signatures
involving leptons in events produced in pp collisions at LHC. Section 3
discusses the search for resonant production and the following decay of
$\tilde{\mu}$ which is caused by $\lambda^{\prime}_{211}\neq 0$. Section 4
addresses a search for multi-lepton signatures caused by LSP decays due to
various $\lambda$ and $\lambda^{\prime}$ terms. Finally in Section 5 we
discuss the possibility of the generic model independent search for RPV SUSY
in 4-lepton events.

##  2 Detector, trigger, and object selection

The central feature of the CMS apparatus is a superconducting solenoid, 6 m in
internal diameter, providing a magnetic field of 3.8 T. Within the field
volume there are a silicon pixel and strip tracker, a crystal electromagnetic
calorimeter, and a brass-scintillator hadron calorimeter. Muons are measured
in gas-ionization detectors embedded in the steel return yoke. Extensive
forward calorimetry complements the coverage provided by the barrel and endcap
detectors. A more detailed description of the CMS detector can be found in
Ref. [7].

Events from pp interactions must satisfy the requirements of a two-level
trigger system. The first level performs a fast selection for physics objects
(jets, muons, electrons, and photons) above certain thresholds. The second
level performs a full event reconstruction. The principal trigger used for
these analyses requires presence of at least two light leptons, electrons or
muons. Detailed trigger conditions and off-line event selections are described
in the corresponding Ref. [5, 4, 6].

##  3 Search for resonant second generation slepton production

Figure 1: Resonant smuon (left) and sneutrino (right) production and typical
decay chain into a final state with two same-sign muons and two jets. The
R-parity violating vertices are marked by a red dot.

Figure 2: Invariant mass of the two muons and jets before (left) and after (right) applying the b-tag veto and same-sign muon requirement. Data are compared to the expectation from the simulation (left) and measured backgrounds (right). Signal distributions are shown for three different kinematic configurations for a coupling value of $\lambda^{\prime}_{211}=0.01$.  Table 1: Event yields with systematic uncertainties after selection requirements, broken down in individual Standard Model background contributions, with observed 95% C.L. limits on the number of signal events $N_{sig}$ in total and for each signal region. process | totals | SR1 | SR2 | SR3
---|---|---|---|---
VVV |  0.15 $\pm$ 0.08 |  0.043 $\pm$ 0.022 |  0.054 $\pm$ 0.028 |  $<$0.001
tt+V |  0.11 $\pm$ 0.06 |  0.019 $\pm$ 0.010 |  0.038 $\pm$ 0.020 | 0
rare |  0.36 $\pm$ 0.26 |  0.32 $\pm$ 0.24 |  0.042 $\pm$ 0.042 |  $<$0.001
VV |  2.1 $\pm$ 1.1 |  0.69 $\pm$ 0.35 |  0.68 $\pm$ 0.34 |  0.003 $\pm$ 0.002
fakes |  8.2 $\pm$ 3.0 |  3.5 $\pm$ 1.6 |  1.9 $\pm$ 1.0 |  $<$0.001
$\sum$ |  10.9 $\pm$ 3.4 |  4.6 $\pm$ 1.6 |  2.7 $\pm$ 1.1 |  0.003 $\pm$ 0.002
data | 13 | 5 | 5 | 0
95% C.L. limit on $N_{sig}$ | 11.3 | 6.9 | 8.0 | 2.8
process |  | SR4 | SR5 | SR6
VVV |  |  0.036 $\pm$ 0.018 |  0.010 $\pm$ 0.005 |  0.007 $\pm$ 0.004
tt+V |  |  0.044 $\pm$ 0.023 |  0.006 $\pm$ 0.004 |  0.006 $\pm$ 0.004
rare |  |  $<$0.001 |  $<$0.001 |  $<$0.001
VV |  |  0.49 $\pm$ 0.25 |  0.15 $\pm$ 0.08 |  0.093 $\pm$ 0.050
fakes |  |  2.5 $\pm$ 1.2 |  0.22 $\pm$ 0.23 |  $<$0.001
$\sum$ |  |  3.1 $\pm$ 1.2 |  0.39 $\pm$ 0.25 |  0.11 $\pm$ 0.05
data |  | 0 | 2 | 1
95% C.L. limit on $N_{sig}$ |  | 2.9 | 6.0 | 4.6
Figure 3: Distribution of
$m_{\tilde{\mu}}=m(\text{jets},\mu_{1}^{\pm},\mu_{2}^{\pm})$ vs.
$m_{\chi}=m(\text{jets},\mu_{2}^{\pm})$ for the events selected in data
compared to the total background contribution. The crosses represent the data
points and the coloured squares show the expectation from Standard Model
backgrounds.

Figure 4: Left: observed 95% CL upper limits on $\lambda^{\prime}_{211}$ as a
function of $m_{0}$ and $m_{1/2}$ for $A_{0}=0$, sign $(\mu)=+1$ and
$\tan\beta=20$. Right: mSUGRA limits expressed in the parameter space of the
neutralino mass $m_{\tilde{\chi}^{0}_{1}}$ and smuon mass $m_{\tilde{\mu}}$.

This search which is described in details in Ref. [4], extends the results
from a previous search by the DØ collaboration [8] and is complementary to
searches for RPV SUSY performed by the LEP experiments [9]. The search
concentrates on final states with two muons and at least two jets. Fig. 1
illustrates the simplest possible Feynman diagrams leading to this final
state, which is experimentally interesting because the presence of two muons
allows to discriminate the signal from background processes. One of the muons
is expected to be produced by the resonant slepton while the other muon and
two quarks resulting in jets are expected to be produced in the subsequent
decay of the neutralino LSP. Due to the Majorana nature of the LSP, the two
muons have the same charge with about 50% probability, which allows to
discriminate further against the background. Due to the larger valence
$\mathrm{u}$-quark content of the initial state protons the configuration with
two positively charged muons is about twice as likely as the configuration
with two negatively charged muons. The kinematics of this signal is
characterized by no missing transverse energy within the detector resolution.
For the purpose of this analysis we select events with two same-sign isolated
muons with $p_{\mathrm{T}}>20$ and $p_{\mathrm{T}}>15$ GeV for the first and
second muon respectively. In addition at least two jets with
$p_{\mathrm{T}}>30$ GeV, no $\mathrm{b}$-jets, and
$E_{\mathrm{T}}^{\text{miss}}<50$ GeV are required. After this selection, two
main background components remain: low cross section backgrounds containing
two prompt same-sign leptons such as production of multiple bosons, and
backgrounds with high cross-section where leptons from semileptonic decays of
$\mathrm{c}$ or $\mathrm{b}$-hadrons or other charged particles are wrongly
identified as prompt leptons. The first contribution is estimated from the
simulation. The latter contribution is difficult to model in simulation, thus
it is estimated using data. Fig. 2 illustrates the expected backgrounds before
and after the requirement of the two same-sign muons and the $\mathrm{b}$-jets
veto.

The 13 events observed in Fig 2 (right) are further investigated using their
2D distribution in parameters
$m_{\tilde{\mu}}=m(\text{jets},\mu_{1}^{\pm},\mu_{2}^{\pm})$ vs.
$m_{\chi}=m(\text{jets},\mu_{2}^{\pm})$, where $\mu_{1}^{\pm}$ denoting the
muon with higher $p_{\mathrm{T}}$. Fig. 3 overlays the observed events with
the expected background contributions, and describes six exclusive search
regions used for the interpretation of this analysis. Table 1 presents the
observations, expected backgrounds, and respective upper limits for all search
regions. The observations are consistent with the corresponding background
estimations, therefore results are combined to put limit on
$\lambda^{\prime}_{211}$ for different mSugra models in Fig. 4.

Table 2: Observed yields for three- and four- lepton events from 19.5 fb${}^{-}1$ recorded in 2012. The channels are split by the total number of leptons (NL), the number of $\tau_{\mathrm{h}}$ candidates (Nτ), and the $S_{\mathrm{T}}$. Expected yields are the sum of simulation and estimates of backgrounds from data in each channel. SR1–SR4 require a $\mathrm{b}$-tagged jet and veto events containing $\mathrm{Z}$ bosons. SR5–SR8 contain events that either contain a $\mathrm{Z}$ boson or have no $\mathrm{b}$-tagged jet. The channels are mutually exclusive. The uncertainties include statistical and systematic uncertainties. The $S_{\mathrm{T}}$ values are given in GeV.  SR |  NL |  Nτ | $0<S_{\mathrm{T}}<300$ | $300<S_{\mathrm{T}}<600$ | $600<S_{\mathrm{T}}<1000$ | $1000<S_{\mathrm{T}}<1500$ | $S_{\mathrm{T}}>1500$
---|---|---|---|---|---|---|---
|  |  | obs | exp | obs | exp | obs | exp | obs | exp | obs | exp
SR1 | 3 | 0 | 116 |  123 $\pm$ 50 | 130 |  127 $\pm$ 54 | 13 |  18.9 $\pm$ 6.7 | 1 |  1.43 $\pm$ 0.51 | 0 |  0.208 $\pm$ 0.096
SR2 | 3 | $\geq 1$ | 710 |  698 $\pm$ 287 | 746 |  837 $\pm$ 423 | 83 |  97 $\pm$ 48 | 3 |  6.9 $\pm$ 3.9 | 0 |  0.73 $\pm$ 0.49
SR3 | 4 | 0 | 0 |  0.186 $\pm$ 0.074 | 1 |  0.43 $\pm$ 0.22 | 0 |  0.19 $\pm$ 0.12 | 0 |  0.037 $\pm$ 0.039 | 0 |  0.000 $\pm$ 0.021
SR4 | 4 | $\geq 1$ | 1 |  0.89 $\pm$ 0.42 | 0 |  1.31 $\pm$ 0.48 | 0 |  0.39 $\pm$ 0.19 | 0 |  0.019 $\pm$ 0.026 | 0 |  0.000 $\pm$ 0.021
SR5 | 3 | 0 | — | — | — | — | 165 |  174 $\pm$ 53 | 16 |  21.4 $\pm$ 8.4 | 5 |  2.18 $\pm$ 0.99
SR6 | 3 | $\geq 1$ | — | — | — | — | 276 |  249 $\pm$ 80 | 17 |  19.9 $\pm$ 6.8 | 0 |  1.84 $\pm$ 0.83
SR7 | 4 | 0 | — | — | — | — | 5 |  8.2 $\pm$ 2.6 | 2 |  0.96 $\pm$ 0.37 | 0 |  0.113 $\pm$ 0.056
SR8 | 4 | $\geq 1$ | — | — | — | — | 2 |  3.8 $\pm$ 1.3 | 0 |  0.34 $\pm$ 0.16 | 0 |  0.040 $\pm$ 0.033

##  4 Search for R-parity violating SUSY in multileptons with
$\mathrm{b}$-tagged jets

Table 3:  Kinematically allowed stop decay modes with RPV coupling $\lambda^{\prime}_{233}$. The allowed neutralino decay modes for $m_{\mathrm{t}}<m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}$ are $\widetilde{\chi}^{0}_{1}\to\mu\mathrm{t}\overline{\mathrm{b}}$ and $\nu\mathrm{b}\overline{\mathrm{b}}$.  Label | Kinematic region | Decay mode
---|---|---
A | $m_{\mathrm{t}}<m_{\widetilde{\mathrm{t}}_{1}}<2m_{\mathrm{t}},m_{\widetilde{\chi}^{0}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{t}\nu\mathrm{b}\overline{\mathrm{b}}$
B | $2m_{\mathrm{t}}<m_{\widetilde{\mathrm{t}}_{1}}<m_{\widetilde{\chi}^{0}_{1}}$ |  $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{t}\mu\mathrm{t}\overline{\mathrm{b}}$ or $\mathrm{t}\nu\mathrm{b}\overline{\mathrm{b}}$
C | $m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}<m_{\mathrm{W}^{\pm}}+m_{\widetilde{\chi}^{0}_{1}}$ |  $\widetilde{\mathrm{t}}_{1}\rightarrow\ell\nu\mathrm{b}\widetilde{\chi}^{0}_{1}$ or $\mathrm{jj}\mathrm{b}\widetilde{\chi}^{0}_{1}$
D | $m_{\mathrm{W}^{\pm}}+m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}<m_{\mathrm{t}}+m_{\widetilde{\chi}^{0}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{b}\mathrm{W}^{\pm}\widetilde{\chi}^{0}_{1}$
E | $m_{\mathrm{t}}+m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{t}\widetilde{\chi}^{0}_{1}$
Figure 5: The 95% confidence level limits in the stop and bino mass plane for
models with RPV couplings $\lambda_{122}$, $\lambda_{233}$, and
$\lambda^{\prime}_{233}$. For the couplings $\lambda_{122}$ and
$\lambda_{233}$, the region to the left of the curve is excluded. For
$\lambda^{\prime}_{233}$, the region inside the curve is excluded. The
different regions, A, B, C, D, and E, for the $\lambda^{\prime}_{233}$
exclusion result from different stop decay products as explained in Table 3.

Among modern SUSY models, “natural” supersymmetry refers to those
characterized by a relatively small fine tuning to describe particle spectra.
It requires top squarks (stops), to be lighter than about 1 TeV. The
introduction of RPV does not preclude a natural hierarchy and allows the
constraints on the stop mass to be relaxed [10].

The analysis [5] searches for pair production of top squarks with RPV decays
of the lightest sparticle, using multilepton events and $\mathrm{b}$-tagged
jets. It addresses terms $\lambda_{ijk}$ and $\lambda^{\prime}_{ijk}$ in Eqn.
1.

We select events with three or more leptons (including tau leptons) that are
accepted by a trigger required two light leptons, which may be electrons or
muons. At least one electron or muon in each event is required to have
transverse momentum of $p_{\mathrm{T}}>20$ GeV. Additional electrons and muons
must have $p_{\rm T}>10$ GeV. The majority of hadronic decays of tau leptons
($\tau_{\mathrm{h}}$) yield either a single charged track (one-prong) or three
charged tracks (three-prong), occasionally with additional electromagnetic
energy from neutral pion decays. We use one- and three-prong
$\tau_{\mathrm{h}}$ candidates that have $p_{\mathrm{T}}>20$ GeV. Leptonically
decaying taus are included with other electrons and muons. The
$E_{\mathrm{T}}^{\text{miss}}$ is not a good discriminator for RPV SUSY
search. Instead we use the $S_{\mathrm{T}}$ variable, which is the scalar sum
of $E_{\mathrm{T}}^{\text{miss}}$ and the transverse energy of jets with
$p_{\mathrm{T}}>30$ GeV and charged leptons, to provide separation between
signal and the Standard Model backgrounds.

Irreducible Standard Model backgrounds are estimated from simulation.
Contributions from fakes for electrons, muons and taus are obtained using
data-driven methods.

Observed events are classified into eight topologies according to the number
of observed light leptons and the presence of hadronic tau in event. Every
topology is further split into five search regions according to the
$S_{\mathrm{T}}$ value. Table 2 summarizes observations and expected
contributions for different search regions used in this analysis.

We generate simulated samples to evaluate models with simplified mass spectra
and the only non-zero leptonic RPV couplings $\lambda_{122}$ or
$\lambda_{233}$. The stop masses in these samples range from 700–1250 GeV in
50 GeV steps, and bino masses range from 100–1300 GeV in 100 GeV steps. In a
model with only the semi-leptonic RPV coupling $\lambda^{\prime}_{233}$, we
use stop masses 300–1000 GeV in 50 GeV steps and bino masses 200–850 GeV in 50
GeV steps. In both cases, slepton and sneutrino masses are 200 GeV above the
bino mass. Other particles are irrelevant to the results for these models.

No significant excess is observed in data. The observations from Table 2 are
combined into exclusions for the corresponding models in Fig. 5.

##  5 Generalization of Unstable LSP Search

The analysis described in details in Ref. [6] presents a new approach to a
generic interpretation of experimental results. The focus of this analysis is
the lepton number violating term $\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}$, which
causes the LSP in such a “Leptonic-RPV” (LRPV) SUSY model to decay into
leptons. SUSY particles are produced in pairs, thus a non-zero $\lambda$-term
would lead to events with 4 charged leptons produced in LSP decays. Recent
searches at the Tevatron [11] and LHC [12, 5] placed limits on $\lambda$. The
main challenge of RPV SUSY searches is that the RPV term exists on top of some
underlying RPC SUSY model, with properties which are currently barely
constrained. The analyses mentioned above resolve this problem by exploring
RPV on top of very specific RPC SUSY models. In this analysis we pursue a
significantly less model dependent approach. We require the presence of 4
isolated leptons in the event, as a direct signature of the LRPV SUSY. No
other restriction is applied, so the selection efficiency is not directly
affected by the underling SUSY event. Irreducible Standard Model backgrounds
are estimated from simulation, estimations of fakes are data-driven. The main
background for 4-lepton events is found to be ZZ production, so for every
event the variable $M_{1}$ is calculated as the invariant mass of same-flavor
opposite-sign lepton pair that is closest to the mass of Z-boson. $M_{2}$ is
then calculated as the invariant mass of the remaining lepton pair.

Table 4: Observed events and expected background contributions. $M_{1}$ and
$M_{2}$ intervals are in GeV.

| $M_{1}<76$ | $76<M_{1}<106$ | $M_{1}>106$
---|---|---|---
| all backgrounds | 1.4$\pm$0.5 | 18$\pm$4 | 0.47$\pm$0.10
$M_{2}>106$ | observed | 0 | 20 | 0
| all backgrounds | 0.52$\pm$0.30 | 153∗ | 0.16$\pm$0.06
$76<M_{2}<106$ | observed | 0 | 160 | 0
| all backgrounds | 10.4$\pm$2.0 | 35$\pm$8 | 1.0$\pm$0.2
$M_{2}<76$ | observed | 14 | 30 | 1

∗ ZZ prediction in “in Z”:“in Z” region is based on MC normalized to CMS ZZ
production cross section measurement, which is correlated with observation in
“in Z”:“in Z” region of this analysis.

Table 4 presents the observations and expected backgrounds in different
regions in $M_{1}:M_{2}$ space. Observations and expectations are consistent
in all regions. Based on the occupancy of different regions for typical ZZ
production events, the signal region is defined as “$M_{1}$ above Z” or
“$M_{1}$ below Z and $M_{2}$ above Z”. Then the upper limit on cross section
times integrated luminosity times efficiency
($\sigma\times\mathcal{L}\times\varepsilon$) for any physics process beyond
the SM contributing to this search region is 3.4 events. The expected upper
limit for this observation is 4.7 events. The leptonic decay of the pair of
LRPV neutralinos leads to 4 prompt leptons. The kinematics of these leptons
are in general driven by the momentum distribution of the decaying neutralinos
and their mass. In most scenarios the lepton momentum is well above threshold,
which results in high efficiency. However the following effects could reduce
the total efficiency:

  * •

the presence of other leptons in the event, which affects the efficiency
through the 4-lepton requirement, as well as the calculation of the $M_{1}$
and $M_{2}$ quantities;

  * •

the electron and/or muon objects reconstruction efficiency which is dependent
on $\eta$ and $p_{T}$;

  * •

the isolation efficiency, which is correlated with the occupancies around the
observed prompt leptons.

The presence of an extra lepton in the SUSY event, in addition to the 4
leptons produced from neutralino decays, could veto the event. We observe no
events containing 5 isolated leptons. Thus, the potential presence of
additional leptons in fact does not significantly affect the measurement.

To evaluate the dependency of the lepton reconstruction efficiency and the
efficiency of analysis selections from details of kinematic distributions of
decaying neutralinos, we consider two extreme cases of LRPV neutralino
production:

  * •

a simplified model with SUSY particles produced via a squark-anti-squark pair,
with the neutralino coming from a two-body decay $\tilde{q}\rightarrow
q\tilde{\chi}_{1}^{0}$, as presented in Fig. 6;

  * •

a pair of neutralinos produced in rest in the center of the CMS detector.

Figure 6:  LRPV extensions to Simplified Model [13]. The T2 RPC simplified
model is squark pair production, with $\tilde{q}\rightarrow
q\tilde{\chi}_{1}^{0}$, and $m(\tilde{g})\gg m(\tilde{q})$. The neutralinos
decay to two charged leptons and a neutrino via an LRPV term.

Figure 7:  Left: efficiency for the T2+LRPV model.
$\tilde{\chi}^{0}_{1}\rightarrow\mu^{+}\mu^{-}\nu$. Right: For every
neutralino mass the efficiency value is filled corresponding to the different
squark masses.

The first approach creates the most energetic neutralinos possible,
constrained by the relevant squark and neutralino masses. Figure 7 (left)
presents the efficiency as a function of the T2 model parameters: squark mass
and neutralino mass. This distribution illustrates that the total efficiency
of this analysis is mostly driven by the neutralino mass, while the squark
mass, which drives the neutralino spectrum, affects the efficiency only
marginally. To illustrate this further Fig. 7 (right) shows the distribution
of the efficiency for different squark masses. This distribution demonstrates,
that the variations even over a wide range of squark masses, are within
$\pm$10%.

Figure 8:  Left: efficiency of this analysis for neutralinos decaying in rest
(red points), overlaid with LRPV efficiency from Fig. 7. Right: efficiency
profiles for electrons and muons.

If the LSP is produced at the end of a long cascade of decays of SUSY
particles, the LSP $p_{T}$ spectra will be significantly softer than for LSPs
produced in two-body decays of the T2 scenario. To study the effect of soft
spectra we consider another extreme case: neutralino pairs produced in rest in
the detector frame. We generate the corresponding dataset by letting the
neutralino decay into ($e^{+},e^{-},\nu$) or ($\mu^{+},\mu^{-},\nu$). Figure 8
(left) shows the efficiency as a function of the neutralino mass overlaid with
the efficiency band obtained from the T2+LRPV model presented in Fig. 7
(right). It demonstrates that the difference between the T2+LRPV case and the
stopped neutralino case is below $\pm$10%.

Figure 9:  Isolation efficiency for 4 leptons for the set of pMSSM models
described in the text, as a function of the neutralino mass in the model. The
green and yellow bands include 68% and 95% of the model points in the
efficiency distribution respectively.

The isolation efficiency for isolated leptons from RPV decays depends on the
occupancy of the event, which in turn depends on the content of the underlying
SUSY event. To study how strong the influence of different underlying SUSY
models and different SUSY production mechanisms is, we re-use the data samples
produced in a previous CMS analysis [14]. These are MC samples for about 7300
different RPC phenomenological MSSM (pMSSM) [15] model points, each one
containing 10000 events, selected to fulfill different pre-CMS observations.
The pMSSM model is an excellent proxy for the full MSSM with a sufficiently
small number of parameters [14]. The available datasets for this set of pMSSM
models is to date the biggest sample of varying SUSY models available to us.
To evaluate the effect of different occupancies in each event of the pMSSM, we
start by extracting the generator-level information about the neutralino. Then
we generate a neutralino RPV decay into two leptons and a neutrino and finally
calculate the reconstruction level isolation around the direction of the
obtained leptons. The event is accepted if the isolation for each of the 4
charged leptons satisfies the isolation requirements for prompt leptons used
in this analysis. Figure 9 presents the efficiencies for different SUSY models
as a function of the neutralino mass in each model. Nearly all SUSY models
have a 4-lepton isolation efficiency in the range between 0.5 and 1. The green
and yellow shaded areas in the plot contain 68% and 95% of the model points
respectively.

We use the band $[0.5,1]$ as a conservative estimate for possible variations
of the analysis signal efficiency due to different types of underlying SUSY
models. We use $30\%$ uncertainty when we combine this effect with other
uncertainties.

Combining all effects, we consider the T2+LRPV model efficiency in Fig. 8
(right) to be a representative of a “best efficiency” scenario. Large hadronic
activity in the event can reduce the isolation efficiency. In line with the
pMSSM study, we conclude that the reduction of the total efficiency for this
search may be up to 50%. Therefore, we consider an efficiency band between
these two extreme cases to cover the 4-lepton efficiency for most the SUSY
models in this analysis.

Once an upper limit on $\sigma\times\mathcal{L}\times\varepsilon$ is extracted
from the observations, and the efficiency is evaluated, the corresponding
limit on the cross section, $\sigma^{SUSY}_{total}$, may be calculated.

Figure 10:  Left: 95% C.L. upper limit on total cross sections for generic
SUSY models. The band corresponds to the efficiency uncertainty as described
in the text. Right: Mass exclusions for different SUSY production mechanisms.
Left: for T2+LRPV models. Right: using a generic total RPV SUSY cross section
limit in the left plot. A $30\%$ theoretical uncertainty for NLO+NLL
calculations of SUSY production cross sections is included in the uncertainty
band.

The experimental observations together with the pMSSM based efficiency
estimation as described above drive the exclusion for the cross section of
total RPV SUSY production, which is presented in Fig. 10 (left). The bands
correspond to the 4 lepton isolation variations between 50% and 100%. Note
that this is a very generic result as this band covers RPV models with a wide
range of underlying RPC SUSY models.

To further convert the cross section limit into a mass exclusion we consider
several SUSY production mechanisms: gluino pair production, squark pair
production, and stop-quark pair production. The cross sections for these
processes as functions of the corresponding masses are NLO+NLL calculation
results of the corresponding decoupled scenarios [16]. The theoretical
uncertainties on the NLO+NLL SUSY production cross section calculations for
masses $\sim$1 TeV are about $30\%$, and are accounted for in the result.

Using these total cross sections as a function of the mass of the
corresponding SUSY particle, we convert the cross section limit bands in Fig.
10 (left) into mass exclusion bands as a function of the LSP mass. This result
is presented in Fig. 10 (right).

##  6 Conclusions

CMS developed a comprehensive program for RPV SUSY searches. In this
contribution we present the most recent results on this topic. For all
presented searches observations are consistent with expectations from the
Standard Model, thus the corresponding limits on presence of new physics are
set. We also present a new approach of generalizing physics interpretations of
experimental observations. Sampling of a big set of pMSSM models allows to
check a model dependency for obtained results, thus making more general
conclusions possible.

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