# Hints of Natural Supersymmetry in Flavor Anomalies?

P. S. Bhupal Dev<EMAIL_ADDRESS>Department of Physics and McDonnell Center
for the Space Sciences, Washington University, St. Louis, MO 63130, USA
Amarjit Soni<EMAIL_ADDRESS>Physics Department, Brookhaven National
Laboratory, Upton, NY 11973, USA  Fang Xu<EMAIL_ADDRESS>Department of
Physics and McDonnell Center for the Space Sciences, Washington University,
St. Louis, MO 63130, USA

###### Abstract

The recent results from the Fermilab muon $g-2$ experiment, as well as the
persisting hints of lepton flavor universality violation in $B$-meson decays,
present a very strong case for flavor-nonuniversal new physics beyond the
Standard Model. We assert that a minimal $R$-parity violating supersymmetric
scenario with relatively light third-generation sfermions (dubbed as ‘RPV3’)
provides a natural, well-motivated framework for the simultaneous explanation
of all flavor anomalies, while being consistent with a multitude of low-energy
flavor constraints, as well as with limits from high-energy collider searches.
We further propose complementary tests and distinct signatures of this
scenario in the high-$p_{T}$ searches at current and future colliders.
Specifically, we find that an sbottom in the mass range of 2–12 TeV accounts
for $R_{D^{(*)}}$ and $R_{K^{(*)}}$ flavor anomalies and it only plays a minor
role in the $(g-2)_{\mu}$ anomaly, whereas a sneutrino with mass between 0.7–1
TeV is the dominant player for $(g-2)_{\mu}$. In this context, we propose
specific collider signatures of sbottom via its decays to
$\overline{t}(t)\mu^{+}\mu^{-}$, and of sneutrino pairs with their decays
leading to a highly distinctive and spectacular four-muon final state, which
can be used to completely probe the RPV3 parameter space of interest.

##  I Introduction

This is an era of anomalies, as a growing list of experimental results,
ranging from flavor physics to neutrinos to Dark Matter, show deviations from
the Standard Model (SM) expectations at a few $\sigma$ level [1]. While most
of them might be an artifact (or a combination) of statistical fluctuations,
systematic effects, theory/background uncertainties, experimental errors or
other unknown issues that need further scrutiny, the intriguing possibility
that some anomalies might represent genuine new physics signals makes it
worthwhile studying all possible aspects, including beyond the SM (BSM)
interpretations.

Of the existing statistically significant ($\gtrsim 3\sigma$) anomalies,
particularly striking are the persistent hints of Lepton Flavor Universality
Violation (LFUV) in semileptonic $B$-meson decays, as reported by three
different experiments with completely independent datasets, namely, BaBar [2],
Belle [3, 4, 5, 6, 7] and LHCb [8, 9, 10, 11, 12], with a combined
significance of $4.5\sigma$ [13]. Moreover, the LFUV observables expressed in
term of the ratio of branching ratios (BRs) $R_{D^{(*)}}^{\tau/\ell}={\rm
BR}(B\to D^{(*)}\tau\overline{\nu})/{\rm BR}(B\to D^{(*)}\ell\overline{\nu})$
(with $\ell=e,\mu$) and $R_{K^{(*)}}^{\mu/e}={\rm BR}(B\to
K^{(*)}\mu^{+}\mu^{-})/{\rm BR}(B\to K^{(*)}e^{+}e^{-})$, are theoretically
clean, i.e., with strongly suppressed hadronic and CKM-angle uncertainties,
thus making them less vulnerable to higher-order quantum corrections [14, 15].
There are also other intriguing aspects of data that deserve attention. For
instance, as noticed in Ref. [16], there are 11 different measurements to date
on the charged-current $B$-decays $B\to D^{(*)}\ell\nu$ and $B_{c}\to J/\psi\
\ell\nu$ (with $\ell=e,\mu,\tau$), all of which have experimental central
values above the SM central value. Similarly, both $R_{K}$ [12] and
$R_{K^{*}}$ [10] measurements from LHCb are consistently below the SM
prediction of unity, which is a useful aspect of data that can be used to
discriminate candidate BSM scenarios.111For example, some BSM scenarios with
right-handed currents predict an anti-correlation between $R_{K}$ and
$R_{K^{*}}$. Also, the recently updated $R_{K}$ measurement from LHCb with
twice as much data [12] did not budge the central value from the old
measurement (up to three decimal places, 0.846) [11]. With more statistics
from LHCb and Belle-II expected in the near future, the status of the
$B$-anomalies should be further clarified.

Another long-standing anomaly that also hints at LFUV is the discrepancy
between the SM and experimental values of the muon anomalous magnetic moment
$(g-2)_{\mu}$ [17]. There has been an important recent development, as the
first result from Fermilab $g-2$ experiment [18] is found to be compatible
with the old BNL result [19] to six significant figures. When combined and
compared with the world-average of SM prediction [20], it increases the
significance of the $(g-2)_{\mu}$ anomaly to $4.2\sigma$ [18]. This does
require a very accurate theoretical prediction from the SM [20]. Over the
years a systematically improvable strategy has been developed, based on a firm
theoretical foundation of dispersion relations [21, 22, 23] and it has the
power of being data-driven (called the $R$-ratio method) [24, 25, 26, 27, 28].
In the meantime, a completely independent, non-perturbative, lattice
simulation method, originally initiated in Ref. [29], has significantly
matured over the years [30, 31, 32, 33, 34, 35, 36]; see Ref. [20] for a
review. In fact, a new lattice result from the BMW collaboration [36] already
seems to have achieved competitive accuracy and claims compatibility between
its SM calculation and the experimental result at $1.5\sigma$. However, this
result using a particular fermion-discretization method (“staggered fermions”)
disagrees with the $R$-ratio results at $\approx 3.7\sigma$ [37, 36] and is
also in $\approx 2.5\sigma$ tension [37] with another lattice/data (“window
method”) result of RBC-UKQCD collaboration [31] using domain-wall fermions
[38, 39, 40, 41]. The domain-wall method essentially preserves the chiral
symmetry of the continuum theory at any lattice spacing and therefore, domain
wall fermions behave as continuum-like fermions so the corresponding chiral
perturbation theory is very much continuum-like [42]. This is unlike the case
of staggered fermions which involves many unphysical degrees of freedom that
only decouple in the continuum limit, thus requiring a fairly cumbersome
treatment of staggered chiral perturbation theory [43, 44, 45] for
extrapolation to the continuum limit. For these reasons and many more,222For
instance, another concern raised with the BMW result is that if a change in
the hadronic vacuum polarization contribution brings the SM value of
$(g-2)_{\mu}$ closer to the experimental value as claimed by the BMW
collaboration, it might lead to other problems in global electroweak fit [46,
47, 48, 49] (see also the rebuttal in Ref. [36]).  widespread and long-
standing understanding in the lattice community is that important physics
results should be taken seriously only after there is consistency and
agreement amongst results obtained in the continuum limit after use of as many
different fermion discretization methods as possible.

Taken at face value, all three flavor anomalies, viz. $R_{D^{(*)}}$,
$R_{K^{(*)}}$, and $(g-2)_{\mu}$, provide a strong evidence for some flavor-
nonuniversal BSM physics, with a combined significance of more than $5\sigma$
[16]. With updates from Belle-II, LHCb and Fermilab expected soon, the chances
are high that at least one of these anomalies will survive the test of time.
Under such a watershed departure from the past, it is very likely that nature
is also trying to address some long-standing, persistent issue(s) with the SM.
One such basic concern with the SM is the fact that it is exceedingly fine-
tuned, i.e. unnatural due to radiative instability of the Higgs which
primarily originates from the heaviness of the top quark, a member of the
third generation. The LFUV observable $R_{D^{(*)}}$ involves $b\to c\tau\nu$,
where at least two fermions are from the third generation. Taking this cue
from experiment and keeping radiative stability issue in mind, along with
minimality, we were originally led [50] to propose a minimal $R$-parity
violating (RPV) supersymmetry (SUSY) framework with the third-generation
superpartners lighter than the first two (hence dubbed as ‘RPV3’) as a
compelling BSM candidate for the $R_{D^{(*)}}$ anomaly. In a follow-up work
[16], we presented a benchmark point where $R_{K^{(*)}}$ and $(g-2)_{\mu}$ can
also be explained together with $R_{D^{(*)}}$ within the RPV3 framework. In
this paper, we perform a comprehensive study of the minimal RPV3 parameter
space that could simultaneously address all three flavor anomalies, in light
of the recent updates and taking into account all relevant collider and flavor
constraints. Let us assert that indeed there is a plethora of existing
experimental constraints from $Z$, $W$, $\tau$, $D^{0}$, $B$, $B_{s}$ and
$B_{c}$ decays, as well as $B_{s}-\overline{B}_{s}$ mixing, that have to be
enforced on RPV3, thus making it rather restrictive. We then propose specific
collider signatures, partly based on general crossing symmetry arguments, that
could independently probe the preferred RPV3 region at the energy frontier.
Such complementarity with the intensity frontier observables makes the RPV3
framework particularly predictive and testable in the near future.333Note also
that there is significant motivation and rationale for this theoretical
scenario from an entirely different perspective [51]. Moreover, due to
renormalization group (RG) evolution effects, heavy first two generations play
a significant role in obtaining the observed 125 GeV Higgs mass, which makes
this model less fine-tuned than the constrained Minimal Supersymmetric SM
(cMSSM) [52, 53]. While the RPV3 framework has many nice features, it does
possess two appreciable caveats which we will briefly discuss towards the end.

The rest of the paper is organized as follows: In Sec. II, we review the RPV3
framework and present all relevant expressions for the RPV3 explanation of the
flavor anomalies. In Sec. III, we present a numerical scan of a 6-dimensional
RPV3 parameter space. In Sec. IV, we choose three benchmark points from our
numerical scan and illustrate the allowed regions preferred by the anomalies.
In Sec. V, we propose some collider signals that can be used an independent
test of the anomalies. In Sec. VI, we discuss possible ways to distinguish the
RPV3 model from other BSM scenarios. Our conclusions are given in Sec. VII. In
Appendix A, we give some details of the low-energy constraints used in our
analysis.

##  II RPV3 Framework

The effective number of degrees of freedom relevant to the three LFUV
observables in RPV3 is of ${\cal O}(36)$ whereas in the SM it is of ${\cal
O}(15)$. So the question might arise: What have we gained at the expense of
doubling the degrees of freedom? The answer is that it has a deeper Bose-Fermi
symmetry rationale, and many associated attractive features of SUSY
automatically in-built, such as radiative stability of the Higgs boson,
radiative neutrino masses, radiative electroweak symmetry breaking, stability
of the electroweak vacuum, gauge coupling unification, (gravitino) dark matter
and baryogenesis [54]. Moreover, as a necessary generalization of the Yang-
Mills theory [55], all the interactions allowed by the enlarged internal
symmetry of the theory must be included, which readily removes the accidental
flavor symmetry of the SM and leads naturally to LFUV. Our RPV3 framework also
has other non-trivial features that are consistent with the experimental
observations, such as having both $R_{K}$ and $R_{K^{*}}$ less than one and
the $D^{*}$ and tau polarizations essentially the same as in the SM, both of
which come automatically because RPV3 subsumes the chiral gauge couplings of
the SM.

It is worth noting here that the semileptonic $B$-meson decays in question
involve interactions of a bottom quark, a member of the third-generation, in
terms of either $b\to c\ell^{-}\overline{\nu}$ or $b\to s\ell^{+}\ell^{-}$
transitions. Analogous semileptonic decays of charmed mesons $D\to
X\ell^{+}\nu$ ($X=\pi,K,\omega,\eta,\rho$) [56, 57, 58, 59, 60], all involving
$c\to d\ell^{+}\nu$ transition, as well as the ratio of the rates of leptonic
kaon decays $K^{\pm}\to e^{\pm}\nu$ and $K^{\pm}\to\mu^{\pm}\nu$ [61, 62], and
of $\Lambda$-baryon decays $\Lambda\to pe^{-}\overline{\nu}$ and $\Lambda\to
p\mu^{-}\overline{\nu}$ [63] are all in complete agreement with the SM.
Therefore, it is conceivable that the third-generation fermions are special in
the SM, and in the same vein, we take the third-generation sfermions to be
special in RPV3 (similar to the ‘natural SUSY’ hypothesis [51, 64]).

The $R_{D^{(*)}}$ anomaly can be accommodated in RPV3 at tree-level via the
$LQD$ interactions [50, 65, 66, 67, 68, 69, 70, 71]:

$\displaystyle{\cal L}_{LQD}\ =\
\lambda^{\prime}_{ijk}\big{[}\widetilde{\nu}_{iL}\overline{d}_{kR}d_{jL}+\widetilde{d}_{jL}\overline{d}_{kR}\nu_{iL}+\widetilde{d}^{*}_{kR}\overline{\nu}^{c}_{iL}d_{jL}$
$\displaystyle\quad-\widetilde{e}_{iL}\overline{d}_{kR}u_{jL}-\widetilde{u}_{jL}\overline{d}_{kR}e_{iL}-\widetilde{d}^{*}_{kR}\overline{e}^{c}_{iL}u_{jL}\big{]}+{\rm
H.c.}$ (1)

Similarly, the $R_{K^{(*)}}$ anomaly can be explained via both tree and loop-
level $LQD$ interactions alone or together with $LLE$ interactions [72, 67,
73, 74, 68, 70, 75, 76]:

$\displaystyle{\cal L}_{LLE}\ =\ $
$\displaystyle\frac{1}{2}\lambda_{ijk}\big{[}\widetilde{\nu}_{iL}\overline{e}_{kR}e_{jL}+\widetilde{e}_{jL}\overline{e}_{kR}\nu_{iL}+\widetilde{e}_{kR}^{*}\overline{\nu}_{iL}^{c}e_{jL}$
$\displaystyle\qquad-(i\leftrightarrow j)\big{]}+{\rm H.c.}$ (2)

The muon $g-2$ gets additional contributions from both $LQD$ and $LLE$ terms
[77], but as we will see later, the $LLE$ contribution is more relevant for
our parameter space of interest.444The simultaneous presence of $\lambda$ and
$\lambda^{\prime}$ couplings is consistent with proton decay constraints, as
long as the relevant $\lambda^{\prime\prime}$ ($UDD$-type) couplings are
sufficiently suppressed [54], which can be done using a baryon triality [78,
79].

Out of the $3^{3}=27$ independent RPV couplings $\lambda^{\prime}_{ijk}$ in
Eq. (1) and the $3^{2}=9$ independent $\lambda_{ijk}$ (since it is
antisymmetric in the first two indices, i.e. $\lambda_{ijk}=-\lambda_{jik}$)
in Eq. (2), we only consider those involving third-generation sfermions in our
RPV3 framework. In what follows, we calculate the RPV3 contributions to the
flavor anomalies.

###  II.1 $R_{D}$ and $R_{D^{(*)}}$

Figure 1: RPV3 contribution to $R_{D^{(*)}}$ via sbottom exchange involving
$\lambda^{\prime}$ couplings. Here $\widetilde{\lambda^{\prime}}_{ijk}$ is
defined as $\lambda^{\prime}_{ilk}V_{jl}$ (with $V_{jl}$ being the CKM matrix
elements). The complete set of diagrams can be found in Ref. [16].

The $b\to c\ell\nu$ transition relevant for the $R_{D^{(*)}}$ anomaly gets a
BSM contribution at tree level from the $LQD$ interactions via the right-
handed sbottom ($\widetilde{b}_{R}$) exchange, as shown in Fig. 1. It gives
rise to a SM-like effective Hamiltonian

$\displaystyle{\cal H}_{\text{eff}}^{b\to c\ell\nu}\ =\ $
$\displaystyle\frac{4G_{F}}{\sqrt{2}}V_{cb}\left(1+C_{V_{L}}\right){\cal
O}_{V_{L}}+\text{H.c.}$ (3)

(where $G_{F}$ is the Fermi constant and $V_{cb}$ is the (2,3) CKM element)
with the operator ${\cal
O}_{V_{L}}=(\overline{c}_{L}\gamma^{\mu}b_{L})(\overline{\ell}_{L}\gamma_{\mu}\nu_{\ell
L})$ with a corresponding coefficient $C_{V_{L}}\simeq 0.09$, as preferred by
the $q^{2}$ and $D^{*}$ polarization data [80]. We then have [68]

$\displaystyle\frac{R_{D}}{R_{D}^{\text{SM}}}\ =\
\frac{R_{D^{*}}}{R_{D^{*}}^{\text{SM}}}=\frac{|\Delta^{c}_{31}|^{2}+|\Delta^{c}_{32}|^{2}+|1+\Delta^{c}_{33}|^{2}}{|\Delta^{c}_{21}|^{2}+|1+\Delta^{c}_{22}|^{2}+|\Delta^{c}_{23}|^{2}}~{},$
(4) $\displaystyle{\rm with}~{}\Delta_{ll^{\prime}}^{c}\ =\
\frac{v^{2}}{4m^{2}_{\widetilde{b}_{R}}}\lambda_{l^{\prime}33}^{\prime}\left(\lambda_{l33}^{\prime}+\lambda_{l23}^{\prime}\frac{V_{cs}}{V_{cb}}+\lambda_{l13}^{\prime}\frac{V_{cd}}{V_{cb}}\right)~{},$

$v=(\sqrt{2}G_{F})^{-1/2}$ being the electroweak scale. The $R_{D^{(*)}}$
anomaly can be explained for the ratio in Eq. (4) being $1.15\pm 0.04$ [16],
which dictates the $R_{D^{(*)}}$-allowed parameter space in the
$(m_{\widetilde{b}_{R}},\lambda^{\prime}_{lk3})$ plane.

From Eq. (1) we see that for the $LQD$ interactions in RPV3, the dimension-six
effective interaction for the semileptonic $B\to D^{(*)}$ decays is
essentially identical to the $(V-A)\times(V-A)$ structure of the SM effective
Hamiltonian (after the appropriate Fierz transformation) with the difference
being just in the overall coefficient. Therefore, the fact that the
experimentally observed $q^{2}$ distribution and the $D^{*}$ and $\tau$
polarizations prefer the ${\cal O}_{V_{L}}$ operator [80] is consistent with
our RPV3 scenario.

In presence of $LLE$ interactions, there is another contribution to
$R_{D^{(*)}}$ from RPV3 with left-handed stau exchange [16]; however, this
involves right-handed bottom and charged-lepton, and the corresponding
effective operator ${\cal O}_{V_{R}}$ does not provide the best-fit to the
$b\to c\ell\nu$ observables [80]. Therefore, we choose the coupling
$\lambda^{\prime}_{323}=0$ so that the stau channel is not relevant.
Similarly, for a light neutralino $\widetilde{\chi}$, there are additional
contributions involving $B\to D^{(*)}\ell\widetilde{\chi}$, which however turn
out to be sub-dominant [16].

It is also important to note that in the MSSM with two Higgs doublets, there
is a standard $R$-parity conserving (RPC) contribution to $b\to c\ell\nu$ due
to charged Higgs exchange. But this goes in the wrong direction and is much
smaller for $R_{D^{*}}$ [81, 82]. Moreover, this is in tension with LHC
mono-$\tau$ data [83] and also induces a large ${\rm
BR}(B_{c}\to\tau\nu)>50\%$ which is problematic [84, 85, 86] (see however Ref.
[87]). Therefore, one must resort to the RPV interactions given above to
explain the $b\to c\ell\nu$ anomalies within a SUSY framework.

###  II.2 $R_{K}$ and $R_{K^{(*)}}$

As for the $R_{K^{(*)}}$ anomaly involving $b\to s\ell^{+}\ell^{-}$
transitions, simultaneous RPV contributions to electron and muon final states
would be strongly constrained by lepton flavor violating (LFV) searches like
$\mu\to e\gamma$. Therefore, we only consider corrections to the muonic
channel, as preferred by recent global fits [88]. The relevant effective
Hamiltonian is

$\mathcal{H}_{\text{eff}}^{b\to
s\ell\ell}=-\frac{4G_{F}}{\sqrt{2}}V_{ts}^{*}V_{tb}\frac{e^{2}}{16\pi^{2}}\sum_{i=9,10}\left[C_{i}^{\ell}Q_{i}^{\ell}+C_{i}^{\prime\ell}Q_{i}^{\prime\ell}\right]$
(5)

with the operators
$Q_{9}^{\ell}=(\overline{s}\gamma_{\alpha}P_{L}b)(\overline{\ell}\gamma^{\alpha}\ell)$,
$Q_{10}^{\ell}=(\overline{s}\gamma_{\alpha}P_{L}b)(\overline{\ell}\gamma^{\alpha}\gamma_{5}\ell)$,
and $Q_{9,10}^{\prime}$ are obtained from $Q_{9,10}$ by replacing $P_{L}\to
P_{R}$. Global fits of all relevant data, including anugular observables or
absolute rate for $B\to K^{(*)}\mu^{+}\mu^{-}$ and also rate for
$B_{s}\to\phi\mu^{+}\mu^{-}$, prefer the Wilson coefficients
$C_{9}^{\mu}=-C_{10}^{\mu}=-0.35\pm 0.08$ [88], whereas $C_{9}^{\prime\mu}$
and $C_{10}^{\prime\mu}$ are compatible with zero at $2\sigma$ level.

In the RPV3 scenario, new contributions to $b\to s\ell\ell$ transitions arise
both at tree and loop levels. Tree-level exchange of stops gives contributions
to the wrong chirality Wilson coefficients [73], which can be translated into
an approximate $3\sigma$ confidence level (CL) upper bound on

$\left|\lambda^{\prime}_{233}\lambda^{\prime}_{232}\right|\ \lesssim\
10^{-3}\times\left(\frac{m_{\widetilde{t}_{L}}}{1\,\text{TeV}}\right)^{2}~{}.$
(6)

This can be satisfied by either making the stop relatively heavier, or by
setting one of $\lambda^{\prime}_{23k}$ (with $k=2$ or 3) small.

Figure 2: Representative box diagrams for the dominant RPV3 contributions to
$b\to s\mu^{+}\mu^{-}$ in our scenario. The complete set of diagrams can be
found in Ref. [16].

At one-loop level, there are new contributions to $b\to s\ell\ell$ with
sbottoms, stops, staus and sneutrinos in the loop [73, 74, 68, 16]. For the
minimal RPV3 case considered here, the only relevant diagrams are those shown
in Fig. 2 and the resulting Wilson coefficients (after taking into account all
possible combinations of box diagrams) are given by

$\displaystyle
C_{9}^{\mu}=-C_{10}^{\mu}=\frac{m_{t}^{2}}{m_{\widetilde{b}_{R}}^{2}}\frac{|\lambda^{\prime}_{233}|^{2}}{16\pi\alpha_{\text{em}}}-\frac{v^{2}}{m_{\widetilde{b}_{R}}^{2}}\frac{X_{bs}X_{\mu\mu}}{64\pi\alpha_{\rm
em}V_{tb}V_{ts}^{*}},$ (7)

where $X_{bs}=\sum_{i=1}^{3}\lambda^{\prime}_{i33}\lambda^{\prime}_{i23}$ and
$X_{\mu\mu}=\sum_{j=1}^{3}|\lambda^{\prime}_{2j3}|^{2}$. Requiring Eq. (7) to
match the global-fit result yields the $R_{K^{(*)}}$-allowed parameter space
in the $(m_{\widetilde{b}_{R}},\lambda^{\prime}_{lk3})$ plane. Note that the
correlation between $R_{K}$ and $R_{K^{(*)}}$, i.e. both going in the same
direction is automatically obtained in the RPV3 setup because of the same
underlying gauge structure as in the SM.

It is important to note here that in the MSSM with RPC couplings only, the
only way to get lepton-flavor non-universal contribution to $b\to s\ell\ell$
is through box diagrams with light winos (or binos) and large non-universality
in slepton masses [89, 90]. However, in order to get the required
$C_{9}^{\mu}\sim-0.35$, one needs an extremely light spectrum of winos and
smuons around 100 GeV and sbottoms around 500 GeV, which are ruled out by the
LHC data [91]. Therefore, just like in the case of $R_{D^{(*)}}$, the
explanation of the $R_{K^{(*)}}$ anomaly within SUSY requires one to invoke
RPV interactions.

###  II.3 $(g-2)_{\mu}$

Figure 3: Relevant contribution to the $(g-2)_{\mu}$ from $\lambda$ and
$\lambda^{\prime}$ couplings in our RPV3 scenario. The complete set of
diagrams can be found in Ref. [16].

The RPV3 contributions to $(g-2)_{\mu}$ can arise from both $\lambda$ and
$\lambda^{\prime}$ couplings [77], as shown in Fig. 3:

$\displaystyle\Delta a_{\mu}$ $\displaystyle\ =\
\frac{m_{\mu}^{2}}{96\pi^{2}}\sum_{k=1}^{3}\left(\frac{2(|\lambda_{32k}|^{2}+|\lambda_{3k2}|^{2})}{m^{2}_{\widetilde{\nu}_{\tau}}}\right.$
$\displaystyle\qquad\left.-\frac{|\lambda_{3k2}|^{2}}{m^{2}_{\widetilde{\tau}_{L}}}-\frac{|\lambda_{k23}|^{2}}{m^{2}_{\widetilde{\tau}_{R}}}+\frac{3|\lambda^{\prime}_{2k3}|^{2}}{m_{\widetilde{b}_{R}}^{2}}\right)\,,$
(8)

which should be compared with the observed discrepancy of $\Delta a_{\mu}^{\rm
obs}=(251\pm 59)\times 10^{-11}$ [18]. Note that the stau contributions are of
the wrong sign, and therefore, are required to be sub-dominant by making the
staus relatively heavier. As we will see later, the $\lambda$-contribution to
$(g-2)_{\mu}$ from sneutrinos is dominant over the
$\lambda^{\prime}$-contribution from sbottoms in our RPV3 scenario. There are
other diagrams, as shown in Ref. [16], which are not relevant to our
discussion. For instance, the stop-mediated diagrams cancel, so are not shown
in Fig. 3. Similarly, the diagram with right-handed stau mediator does not
contribute because one of the corresponding couplings is assumed to be zero
here.

There are additional RPC SUSY contributions to $\Delta a_{\mu}$ involving
smuons and muon sneutrinos [92, 93, 94]. However, since these second-
generation sfermions are decoupled from the low-energy theory in RPV3, we only
focus on the RPV contributions.

##  III Numerical Scan

Our aim is to find the minimum set of RPV3 model parameters that could
simultaneously address all three flavor anomalies, while being consistent with
all other low-energy flavor constraints (for a detailed discussion and
explicit expressions, see Appendix A and Ref. [16]555We found that the
parameter setup of Case 3 in Ref. [16] which could explain all flavor
anomalies is actually problematic for the LFV decays $\mu\to e\gamma$ and
$B_{s}\to e\mu$. This prompted us to look for new solutions in this work.) and
the high-energy LHC data, while at the same time giving rise to potentially
observable collider signals as an independent test of the anomalies. To this
effect, we choose to work with the following 6-dimensional parameter space
that turns out to be the most important for pursuing collider implications:

$\displaystyle\\{\lambda_{232},\lambda_{233}^{\prime},\lambda_{223}^{\prime},\lambda_{232}^{\prime},m_{\widetilde{b}_{R}},m_{\widetilde{\nu}_{\tau}}\\}$
(9)

and drop the other couplings and masses from our discussion (unless otherwise
specified).666Because the RG evolution of any RPV coupling is always
proportional to the coupling itself (at least up to two-loop level) [95], the
couplings set to zero at the input scale remain zero at all scales under the
RG flow.  As for the above choice of our couplings, note that once we choose a
nonzero $\lambda_{232}=-\lambda_{322}$ to explain the $(g-2)_{\mu}$ anomaly
[cf. Eq. (8)], other relevant $\lambda_{3ij}$ couplings cannot be large at the
same time due to the constraints from LFV decays
$\tau^{-}\to\mu^{+}\mu^{-}\mu^{-}$ and $\mu\to e\gamma$. Similarly, only some
of the $\lambda^{\prime}_{2ij}$’s are allowed to be large at the same time as
$\lambda_{232}$; if instead we chose $\lambda^{\prime}_{3ij}$, for instance,
combined with $\lambda_{232}$ and light tau-sneutrino propagator, this will
lead to strong tree-level meson decays
$\overline{d}_{i}d_{j}\to\mu^{+}\mu^{-}$. Note that due to our choice of
couplings (i.e. $\lambda^{\prime}_{3ij}=0$), we do not have the single
production of tau-sneutrino at the LHC, and therefore, only discuss their pair
production in the main text.

As for omitting the remaining third-generation sfermion masses from Eq. (9),
the right-stau mass $m_{\widetilde{\tau}_{R}}$ is irrelevant, because it only
enters in the $(g-2)_{\mu}$ expression (8), but with the coupling
$\lambda_{k23}$, which is set to zero for our benchmark points. The left-stau
term in Eq. (8) has a negative contribution and its effect can be ignored for
$m_{\widetilde{\tau}_{L}}\gtrsim{\cal O}(2\ \mathrm{TeV})$. For concreteness,
we will just set $m_{\widetilde{\tau}_{L}}=4$ TeV in the following analysis.
Similarly, the left-stop mass $m_{\widetilde{t}_{L}}$ only influences the
$C_{9}^{\prime}$ and $C_{10}^{\prime}$ [cf. Eq. (6)] and the
$B_{s}\to\mu^{+}\mu^{-}$ constraint, if both $\lambda^{\prime}_{233}$ and
$\lambda^{\prime}_{232}$ are large at the same time. When this is the case, we
can make $m_{\widetilde{t}_{L}}$ appropriately heavier using Eq. (6) without
affecting any other observable; therefore, we do not include
$m_{\widetilde{t}_{L}}$ in Eq. (9). Finally, the left-sbottom mass
$m_{\widetilde{b}_{L}}$ does not influence the anomaly observables, but is
only relevant for constraints like $b\to s\gamma$, $B\to K\nu\overline{\nu}$
and $B_{s}$-$\overline{B}_{s}$; so wherever applicable, we will just set
$m_{\widetilde{b}_{L}}=m_{\widetilde{b}_{R}}$ for simplicity.

We then perform a random scan over the 6-dimensional parameter space given in
Eq. (9) with the following ranges:

$\displaystyle|\lambda_{232}|\in[2.5,\ 3.5],\
|\lambda^{\prime}_{233}|\in[0.01,\ 3],\ $
$\displaystyle|\lambda^{\prime}_{223}|\in[0.01,\ 3],\
|\lambda^{\prime}_{232}|\in[0.01,\ 3],\ $ $\displaystyle
m_{\widetilde{b}_{R}}\in[1.2,\ 10]~{}{\rm TeV},\
m_{\widetilde{\nu}_{\tau}}\in[0.7,\ 1.2]~{}{\rm TeV},$ (10)

and look for solutions that could simultaneously explain $R_{D^{(*)}}$,
$R_{K^{(*)}}$ and $(g-2)_{\mu}$ anomalies at either $2\sigma$ or $3\sigma$,
while being consistent with all the low-energy constraints discussed above.
Out of 30 million points scanned, we only found 1570 solutions, as shown by
the scatter plots in Fig. 4. Note that the lower edges of the
$m_{\widetilde{\nu}_{\tau}}$ and $m_{\widetilde{b}_{R}}$ scan ranges
correspond to the current LHC limits, see Sec. V.

Figure 4: Scatter plots showing the correlations between various RPV3
parameters in Eq. (9). All these points can simultaneously explain
$R_{D^{(*)}}$, $R_{K^{(*)}}$ and $(g-2)_{\mu}$ anomalies at $3\sigma$ CL,
while being consistent with all the low-energy and LHC constraints. The yellow
(blue) points correspond to $|\lambda^{\prime}_{232}|<0.2$ and
$|\lambda^{\prime}_{223}|<(>)1$. The red points correspond to
$|\lambda^{\prime}_{232}|>0.2$ and
$1.5<\lambda^{\prime}_{223}/\lambda^{\prime}_{232}<5.5$. The green points
correspond to the crossover region from red to blue. The three benchmark
points (BP1, BP2, BP3) discussed in the text are taken respectively from the
densest regions of the red, yellow and blue solutions.

We divide all the obtained solutions into three characteristically different
regions, as shown by the red, blue and yellow points in Fig. 4. In particular,
from Fig. 4(c), we see that the yellow (blue) points correspond to
$|\lambda^{\prime}_{232}|<0.2$ and $|\lambda^{\prime}_{223}|<(>)1$, while the
red points correspond to $|\lambda^{\prime}_{232}|>0.2$ and
$1.5<\lambda^{\prime}_{223}/\lambda^{\prime}_{232}<5.5$, and the green points
simply correspond to the crossover region from red to blue. We can immediately
make several observations from these plots, as follows:

  1. (i)

From Figs. 4(b,c), we see that there are both yellow and blue solutions with
very small (or almost vanishing) $\lambda^{\prime}_{232}$ which means that for
these points, we can automatically satisfy the
$C_{9}^{\prime}-C^{\prime}_{10}$ constraint discussed in main text, as well as
the $B_{s}\to\mu^{+}\mu^{-}$ constraint for any value of the stop mass.

  2. (ii)

From Fig. 4(c), we see that the red points cluster around
$\lambda^{\prime}_{223}/\lambda^{\prime}_{232}\sim 3$; this helps to avoid the
$B_{s}-\overline{B}_{s}$ constraint due to an accidental cancellation.

  3. (iii)

From Figs. 4(a,b,c), we find that $\lambda^{\prime}_{233}$ and
$\lambda^{\prime}_{223}$ must have opposite signs. This is mainly needed to
make the second term of Eq. (7) negative in order to satisfy the $R_{K^{(*)}}$
anomaly. Similarly, $\lambda^{\prime}_{223}$ and $\lambda^{\prime}_{232}$ are
preferred to have the same sign to get cancellation in
$B_{s}-\overline{B}_{s}$ mixing. On the other hand, as Eq. (8) suggests and as
shown in Fig. 4(i), the sign of $\lambda_{232}$ does not matter.

  4. (iv)

According to Fig. 4(i), the different colored points are totally mixed, which
implies mutual orthogonality between the
$(m_{\widetilde{\nu}_{\tau}},\lambda)$ and
$(m_{\widetilde{b}_{R}},\lambda^{\prime})$ parameter spaces. In other words,
$(m_{\widetilde{\nu}_{\tau}},\lambda)$ mostly influences the $(g-2)_{\mu}$
solutions, whereas $(m_{\widetilde{b}_{R}},\lambda^{\prime})$ influences the
$R_{D^{(*)}}$ and $R_{K^{(*)}}$ solutions and the low-energy flavor
constraints. This is further illustrated in Figs. 5 and 6.

  5. (v)

From Fig. 4(e), we find that $|\lambda^{\prime}_{223}|\lesssim
0.57\>(m_{\widetilde{b}_{R}}/\mathrm{1\ TeV})$, which is mainly due to the
$D^{0}\rightarrow\mu^{+}\mu^{-}$ constraint. Similarly, from Fig. 4(d), we get
$|\lambda^{\prime}_{233}|\lesssim 1.0\>(m_{\widetilde{b}_{R}}/\mathrm{1\
TeV})$. These two conditions imply that the $\lambda^{\prime}$ contributions
to $(g-2)_{\mu}$ in Eq. (8) cannot be large; therefore, the bulk of the RPV3
contribution must come from the $\lambda$ sector, which requires relatively
larger $\lambda_{232}\gtrsim 2.8$ and smaller
$m_{\widetilde{\nu}_{\tau}}\lesssim 0.9$ TeV (to keep $\lambda_{232}$
perturbative) to satisfy the $(g-2)_{\mu}$ anomaly, as confirmed in Fig. 4(i).

  6. (vi)

From Fig. 4(g), we find that
$\sqrt{-\lambda^{\prime}_{223}\lambda^{\prime}_{233}}\sim(0.20-0.28)\>(m_{\widetilde{b}_{R}}/\mathrm{1\
TeV})$, which mainly comes from the $B\rightarrow K\nu\overline{\nu}$
constraint.

  7. (vii)

From Fig. 4(a), we find that for the yellow and blue points,
$|\lambda^{\prime}_{233}\lambda^{\prime}_{223}|$ is small to satisfy the
$B_{s}-\overline{B}_{s}$ mixing constraint.

  8. (viii)

From Fig. 4(f), we see that $|\lambda^{\prime}_{232}|\lesssim 1.5$. Thus,
according to Fig. 4(c), $|\lambda^{\prime}_{232}|$ should be either small
(yellow and blue) or $\sim|\lambda^{\prime}_{223}|/3$ (red). Also, from Fig.
4(d), $|\lambda^{\prime}_{233}|\gtrsim 0.20$ and from Fig. 4(e),
$|\lambda^{\prime}_{223}|\gtrsim 0.12$.

  9. (ix)

Figs. 4(d,e,f) suggest that $m_{\widetilde{b}_{R}}\gtrsim 1.44$ TeV, slightly
stronger than the direct LHC bound of 1.23 TeV [96].

  10. (x)

Fig. 4(h) gives the range of $X_{bs}X_{\mu\mu}$ for the RPV3 contribution to
$R_{K^{(*)}}$, since the second term in Eq. (7) gives the correct sign,
whereas the first term gives the wrong sign.

##  IV Benchmark Points

We will choose our benchmark points for the collider study in the next section
based on the results of our numerical scan in Fig. 4 and the above-mentioned
observations. Specifically, we choose three benchmark points (BP1, BP2, BP3),
one each from the red (BP1), yellow (BP2) and blue (BP3) solutions found
above.

  * •

BP1 (Red):
$\lambda^{\prime}_{233}=-\lambda^{\prime}_{223}=-3\lambda^{\prime}_{232}$. The
allowed region in this case is shown in Fig. 5 (a) by the red shaded region.

  * •

BP2 (Yellow): $\lambda^{\prime}_{233}=-8\lambda^{\prime}_{223}$,
$\lambda^{\prime}_{232}=0$. The allowed region in this case is shown in Fig. 5
(b) by the yellow shaded region.

  * •

BP3 (Blue): $\lambda^{\prime}_{223}=-6\lambda^{\prime}_{233}$,
$\lambda^{\prime}_{232}=0$. The allowed region in this case is shown in Fig. 5
(c) by the blue shaded region.

The size of the allowed region in each case is directly correlated with the
density of the corresponding points in Fig. 4. Therefore, our BP1 is taken
from the densest region of the red solution, in order to maximize the overlap
region in Fig. 5. For BP2 and BP3, we just choose $\lambda^{\prime}_{232}=0$
for simplicity. Since the $\lambda$ coupling and the tau-sneutrino mass are
relevant only for $(g-2)_{\mu}$, we fix $\lambda_{232}=-\lambda_{322}=2.8$ and
$m_{\widetilde{\nu}_{\tau}}=0.7$ TeV (see Fig. 6) in all three cases to
explain the $(g-2)_{\mu}$ anomaly at $3\sigma$ ($2\sigma$) CL, as shown by the
orange shaded regions with solid (dashed) boundaries in Fig. 5.

In BP1, since both $\lambda^{\prime}_{233}$ and $\lambda^{\prime}_{232}$ are
nonzero, there is a lower limit on the stop mass from the
$C_{9}^{\prime}-C^{\prime}_{10}$ constraint [cf. Eq. (6)] :
$m_{\widetilde{t}_{L}}\gtrsim(14-40)$ TeV for the overlap region.777According
to Ref. [97], stop masses lower than about 10 TeV are preferred from
naturalness point of view. For a quantitative measure, the level of fine-
tuning must be less than some fixed amount, taken there to be the arbitrary
threshold of 10%. However, this limit does not apply for BP2 and BP3, since
$\lambda^{\prime}_{232}=0$ in those cases; therefore, the stop can be as light
as the current LHC bound of $\sim 800$ GeV [91] in these cases. We have also
checked that the constraints from $B_{s}\to\mu^{+}\mu^{-}$, whose amplitude is
proportional to $C_{10}^{\mu}-C_{10}^{\prime\mu}$ [98], is easily satisfied
for all three BPs, with the RPV3 contribution to ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})\lesssim 10^{-12}$, well below the current
experimental precision: ${\rm BR}(B_{s}\to\mu^{+}\mu^{-})_{\rm
exp}=\left(2.69^{+0.37}_{-0.35}\right)\times 10^{-9}$ [99].

We should also comment on the $B_{s}-\overline{B}_{s}$ mixing constraint. For
our benchmark points, the last term in Eq. (A.2) does not contribute, as both
$\lambda^{\prime}_{332}$ and $\lambda^{\prime}_{323}$ are set to zero.888As
discussed before, a non-zero $\lambda^{\prime}_{3ij}$ combined with
$\lambda_{232}$ and light tau-sneutrino propagator will lead to strong tree-
level meson decays $\overline{d}_{i}d_{j}\to\mu^{+}\mu^{-}$. The excellent
agreement between the experimental measurement [99] and SM prediction [100] of
${\rm BR}(B_{s}^{0}\to\mu^{+}\mu^{-})$ requires an almost zero last term in
Eq. (A.2). For BP1 with
$\lambda^{\prime}_{223}/\lambda^{\prime}_{232}=3\simeq-
P_{1}^{LR}/P_{1}^{VLL}$, there is a cancellation (at the level of 5%) between
the other two terms in Eq. (A.2), thus enabling us to explain $R_{K^{(*)}}$
within $1\sigma$, while this is not the case in BP2 and BP3 where
$R_{K^{(*)}}$ can only be explained at $3\sigma$ level. Note that one can
always assume a non-zero $\lambda^{\prime}_{232}$ for BP2 and BP3
(corresponding to the yellow and blue points out of the vertical axis in Figs.
4(b,c)) to make $B_{s}-\overline{B}_{s}$ mixing constraint weaker and enlarge
the allowed parameter space for $R_{K^{(*)}}$, but this makes the $B\to
K\nu\overline{\nu}$ constraint stronger, which limits the allowed region in
BP2.

(a) BP1 (Red)

(b) BP2 (Yellow) (c) BP3 (Blue)

Figure 5:  Three RPV3 benchmark cases in the
$(m_{\widetilde{b}_{R}},\lambda^{\prime}_{233})$ parameter space explaining
the flavor anomalies. The cyan, pink and orange shaded regions with solid
(dashed) boundaries explain the $R_{D^{(*)}}$, $R_{K^{(*)}}$ and $(g-2)_{\mu}$
anomalies at $3\sigma$ ($2\sigma$) respectively. The black-shaded region is
excluded by the current LHC search for sbottoms in the bottom+neutralino
channel, whereas the dark green-shaded region is the LHC exclusion derived
from a $\mu^{+}\mu^{-}+1b$ search. The horizontal dotted line shows the
perturbativity limit of $\sqrt{4\pi}$. Other shaded regions show the relevant
low-energy flavor constrains on the parameter space from $B\to
K\nu\overline{\nu}$ (brown), $B_{s}-\overline{B}_{s}$ mixing (magenta),
$D^{0}\to\mu^{+}\mu^{-}$ (purple), $b\to s\gamma$ (grey) and
$Z\to\ell^{+}\ell^{-}$ (violet). The allowed overlap regions simultaneously
explaining the $R_{D^{(*)}}$, $R_{K^{(*)}}$ and $(g-2)_{\mu}$ anomalies are
shown by the red (top), yellow (bottom left) and blue (bottom right) shaded
regions for the three benchmark cases. The $*$ mark on the top panel gives
representative values of $m_{\widetilde{b}_{R}}$ and $\lambda^{\prime}_{233}$
in the BP1 scenario that are used in Fig. 6. The green solid, dashed and dot-
dashed contours respectively show the $2\sigma$ sensitivities of the 14 TeV
LHC, 27 TeV and 100 TeV $pp$ colliders in the $\overline{t}\mu^{+}\mu^{-}$
channel discussed in the text.

Our fit results for the best-case scenario are shown in Figs. 5 and 6 for the
mutually orthogonal parameter spaces of
$(m_{\widetilde{b}_{R}},\lambda^{\prime})$ and
$(m_{\widetilde{\nu}_{\tau}},\lambda)$ respectively. In Fig. 5, the cyan,
pink, and orange-shaded regions with solid (dashed) boundaries explain the
$R_{D^{(*)}}$, $R_{K^{(*)}}$, and $(g-2)_{\mu}$ anomalies respectively at
$3\sigma$ ($2\sigma$) CL. The black-shaded region is the 13 TeV LHC exclusion
derived from a sbottom search in the bottom+neutralino channel [96]. The dark-
green-shaded region is the 13 TeV LHC exclusion derived from a
$\mu^{+}\mu^{-}+1b$ search [101] that is also applicable to our RPV3 scenario;
see Sec. V. The horizontal dashed line shows the perturbativity limit of
$\sqrt{4\pi}$. Other shaded regions in Fig. 5 show the relevant low-energy
flavor constrains on the $(m_{\widetilde{b}_{R}},\lambda^{\prime}_{233})$
parameter space from $B\to K\nu\overline{\nu}$ (brown),
$B_{s}-\overline{B}_{s}$ mixing (magenta), $D^{0}\to\mu^{+}\mu^{-}$ (purple),
$b\to s\gamma$ (grey) and $Z\to\ell^{+}\ell^{-}$ (violet); see Appendix A for
more details.

Figure 6: The $(g-2)_{\mu}$-preferred region (orange-shaded) of the
$(m_{\widetilde{\nu}_{\tau}},\lambda_{232})$ parameter space. The purple-
shaded region is excluded by a 13 TeV LHC multi-lepton search [102], whereas
the green curve is the 14 TeV HL-LHC sensitivity. The horizontal (gray) dashed
lines show the positions of the Landau pole for given $\lambda$-couplings and
the black dashed line shows the perturbativity limit. The $*$ gives
representative values of $m_{\widetilde{\nu}_{\tau}}$ and $\lambda_{232}$ used
in Fig. 5.

Now turning to the $(m_{\widetilde{\nu}_{\tau}},\lambda_{232})$ parameter
space, the $(g-2)_{\mu}$-preferred region at $3\sigma$ ($2\sigma$) is shown by
the solid (dashed) orange contours in Fig. 6. We have fixed the other RPV3
parameters using a benchmark point from the allowed region in Fig. 5 as shown
by the $*$ mark. The purple-shaded region is excluded by recasting the results
of a recent 13 TeV LHC multi-lepton search [102], whereas the green curve is
the 14 TeV HL-LHC sensitivity; see Sec. V. The horizontal black dashed line
shows the perturbativity limit of $\sqrt{4\pi}$ as before. Because of the
orthogonality between the $(m_{\widetilde{b}_{R}},\lambda^{\prime})$ and
$(m_{\widetilde{\nu}_{\tau}},\lambda)$ parameter spaces, the position of the
$*$ in Fig. 6 will not change much for BP2 and BP3; therefore, we do not
include the corresponding figures for BP2 and BP3.

Since the required $\lambda$-couplings are fairly large in our scenario, we
also show the Landau pole positions by the horizontal gray dashed lines, which
are obtained by numerically solving the relevant one-loop RG equations (RGEs)
[95]. Because the non-zero $\lambda^{\prime}$ couplings in our scenario do not
couple to the third-generation slepton or sneutrino, the RGE for the
$\lambda_{232}$ coupling (and similarly, for the $\lambda_{322}$ coupling) is
very simple at one-loop level:

$\displaystyle\frac{\rm d}{{\rm
d}t}\lambda_{232}\simeq\frac{\lambda_{232}}{16\pi^{2}}\left(4\lambda_{232}^{2}-\frac{9}{5}g_{1}^{2}-3g_{2}^{2}\right)\approx\frac{1}{4\pi^{2}}\lambda_{232}^{3}\,,$
(11)

where $g_{1}$ and $g_{2}$ are the $U(1)_{Y}$ and $SU(2)_{L}$ gauge couplings,
respectively, both of which are much smaller than the $\lambda_{232}$ coupling
in the parameter space shown in Fig. 6. We find that the benchmark point shown
in Fig. 6 hits the Landau pole at 12.4 TeV.

One possible way out of the Landau pole issue is to consider a non-zero
$\lambda_{233}$ coupling instead of $\lambda_{232}$. This does not affect the
$R_{D^{(*)}}$ and $R_{K^{(*)}}$ fit results presented in Fig. 5 because of the
orthogonality between the $\lambda^{\prime}$ and $\lambda$ parameter space
mentioned before. As for the $(g-2)_{\mu}$-preferred region in the
$(m_{\widetilde{\nu}_{\tau}},\lambda_{233})$ parameter space, the main
difference with respect to Fig. 6 is that the allowed $\lambda$ couplings can
now be as low as 0.8, thus pushing the Landau pole to as high as $2.5\times
10^{16}$ GeV. The reason is that the relevant LHC constraint for
$\lambda_{233}$ comes from $\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ final state (in
contrast with the $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ final state in Fig. 6). We
did not find any 13 TeV LHC analysis in this channel, and using the old 8 TeV
analysis from Ref. [103], we obtain a lower bound of only 150 GeV on the
sneutrino mass. A dedicated 13 TeV analysis to update this bound is currently
underway.

##  V Collider Signals

Simple crossing symmetry arguments have been used to establish high-$p_{T}$
model-independent tests of the $R_{D^{(*)}}$ and $R_{K^{(*)}}$ anomalies in
the CMS and ATLAS experiments [50, 104, 105, 106, 83, 107, 16]. The basic idea
is that the underlying quark-level processes $b\to c\tau\nu$ for $R_{D^{(*)}}$
and $b\to s\ell^{+}\ell^{-}$ for $R_{K^{(*)}}$ necessarily imply, by crossing
symmetry, the existence of processes like $pp\to\tau\nu$,
$pp\to\ell^{+}\ell^{-}$, $pp\to b\tau\nu$ and $pp\to b\ell^{+}\ell^{-}$, which
can be searched for in the high-$p_{T}$ LHC experiments. In fact, a recent CMS
study has found a mild discrepancy in the ratio of differential
$\mu^{+}\mu^{-}$ to $e^{+}e^{-}$ pair-production cross sections [108], which
might turn out to be important for the $R_{K^{(*)}}$ anomaly. However, the
model-independent effective field theory treatments relating the low-energy
operators to the high-$p_{T}$ LHC signals might break down, if the new physics
cut-off scale is smaller than the LHC energies. Thus, it is important to
explore all possible high-$p_{T}$ LHC signals in the context of a given BSM
scenario in order to distinguish it from other BSM interpretations of the
flavor anomalies.

To this effect, we propose some striking LHC signals that could be used as an
independent probe of the allowed RPV3 parameter space shown in Figs. 5 and 6
at the high-$p_{T}$ LHC and future colliders. For the
$(m_{\widetilde{b}_{R}},\lambda^{\prime})$ parameter space in Fig. 5, we
propose the process $pp\to\overline{t}(t)\mu^{+}\mu^{-}$ mediated by an
sbottom; see Fig. 7.

Figure 7: Representative Feynman diagrams for the signal process
$pp\rightarrow\overline{t}\mu^{+}\mu^{-}$. There are similar diagrams for the
process $pp\rightarrow t\mu^{+}\mu^{-}$, however the SM background is larger
for top-quark final states, compared to the anti-top, so we only consider the
latter case for drawing the sensitivity contours in Fig. 5.

There is no $pp\to\overline{t}(t)\mu^{+}\mu^{-}$ final state in the SM, so the
dominant SM background comes from $pp\to\overline{t}(t)\mu^{+}\mu^{-}X$ where
$X$ can be either a light jet ($j$), $b$-jet, or a gauge boson decaying to
jets or leptons, which are somehow missed in the detector. We perform a
parton-level simulation for the signal and background processes using
MadGraph5_aMC@NLO v2.7.0 [109]. We assume that the (anti)top quark can be
identified from its decay products and use the minimal trigger cuts
$p_{T}^{t,\mu}>20$ GeV, $|\eta^{t,\mu}|<2.5$, $\Delta R^{\mu\mu}>0.4$ and
$\Delta R^{t\mu}>0.4$ for the $t(\overline{t})\mu^{+}\mu^{-}X$ final states.
To estimate the SM background, we further require that the $X$ containing
jets, leptons or missing transverse energy is soft enough to evade detection,
i.e. $p_{T}^{j,b,\ell}<20$ GeV and $\not{E}_{T}<20$ GeV.

For the RPV3 signal, we use the three benchmark points discussed in the
previous section. It is easy to see that the $pp\rightarrow t\mu^{+}\mu^{-}$
and $pp\rightarrow\overline{t}\mu^{+}\mu^{-}$ give similar cross-sections for
the signal, because in all three cases, $\lambda^{\prime}_{k13}=0$, and
therefore, the first-generation quarks do not participate in the initial
state. For our parameter choice, the main contribution comes from
$\overline{c}(c)g$ initial states as shown in Fig. 7. Since the $\overline{c}$
and $c$ contents in proton are very similar, the resulting cross-section
should also be the same. However, for the SM background, $t\mu^{+}\mu^{-}X$
final state has two times more background than $\overline{t}\mu^{+}\mu^{-}X$,
which mainly comes from the fact that the $u$ content in proton is much larger
than the $\overline{u}$ content. Therefore, we will only consider the
$pp\rightarrow\overline{t}\mu^{+}\mu^{-}$ final state to show our sensitivity
contours.

With the basic trigger cuts, we find that the total SM background for the
$pp\rightarrow\overline{t}\mu^{+}\mu^{-}(+X)$ final state at $\sqrt{s}=14$ TeV
is 0.4 fb, which is dominated by $X=j$. For comparison, the corresponding
signal cross section for the $*$ point in BP1 in Fig. 5 is only $1.5\times
10^{-3}$ fb. However, we can improve the signal-to-background substantially by
using their different kinematic features. First of all, the $\mu^{+}\mu^{-}$
in the SM background case mainly comes from $Z$ decay, so we expect the dimuon
invariant mass $M_{\mu^{+}\mu^{-}}$ to peak at the $Z$-mass and to drop
significantly at higher masses; see Fig. 8 (green). On the other hand, in our
RPV3 case, one of the muons in the final state comes from sbottom decay, so we
expect a longer tail in the $M_{\mu^{+}\mu^{-}}$ distribution, as confirmed in
Fig. 8 (red). Therefore, using an appropriate cut on $M_{\mu^{+}\mu^{-}}>400$
GeV, we can maximize the signal-to-background ratio. We find that the
corresponding signal at the $*$ point in BP1 and background cross sections
after the $M_{\mu^{+}\mu^{-}}$ cut are respectively $1.1\times 10^{-3}$ fb and
$4.2\times 10^{-4}$ fb. Further improvements in the signal-to-background can
in principle be achieved using the fact that for an on-shell sbottom decaying
to $\overline{t}\mu^{+}$, we expect a peak at the sbottom mass in the
invariant mass $M_{\overline{t}\mu^{+}}$ distribution for the signal, but not
for the background. However, since the final-state reconstruction involving
top quarks is somewhat involved, especially for the leptonic decay of the $W$
boson coming from the top, and also the sbottom mass is not known a priori (we
use it as a free parameter in Fig. 5), we refrain from using the
$M_{\overline{t}\mu^{+}}$ cut in our analysis.

Figure 8: Dimuon invariant mass distribution for the RPV3 signal at the $*$
point in BP1 (red) and SM background (green) in the
$pp\to\overline{t}\mu^{+}\mu^{-}$ channel at 14 TeV LHC.

Assuming an integrated luminosity of ${\cal L}=3000~{}{\rm fb}^{-1}$, we show
the $2\sigma$ signal significance in Fig. 5 by the green solid, dashed and
dot-dashed contours for $\sqrt{s}=14$, 27 and 100 TeV colliders, respectively.
We find that a portion of the overlap region explaining all flavor anomalies
can already be accessed at the HL-LHC, while the proposed future colliders
should be able to access the entire allowed parameter space.

Figure 9: Feynman diagram for the four-lepton signal from the sneutrino pair-
production in our RPV3 model.

Now for the $(m_{\widetilde{\nu}_{\tau}},\lambda)$ parameter space in Fig. 6
relevant only for the $(g-2)_{\mu}$ anomaly, we focus on the spectacular four-
muon final state [110] coming from the sneutrino pair-production, followed by
each sneutrino decaying into two muons via the $\lambda_{232}$ coupling; see
Fig. 9. Such multilepton channels are very clean even at the hadron colliders,
and in fact, the results of a recent ATLAS multilepton analysis [102] can
already be recast into a new bound on the sneutrino mass. Using the 95% CL
observed cross section limit of $0.044$ fb for the $4\ell$, off-$Z$ signal
region with $M_{4\mu}>400$ GeV999This also removes potential contributions
from heavy neutral Higgs to $ZZ$ final states. and the same selection criteria
as in Ref. [102], we obtain a lower bound of
$m_{\widetilde{\nu}_{\tau}}\gtrsim 670$ GeV, as shown by the purple-shaded
region in Fig. 6. This LHC constraint already rules out a big chunk of the
$(g-2)_{\mu}$-preferred region and pushes the $\lambda_{232}$ coupling toward
the perturbativity limit. The HL-LHC can completely cover the remaining
$(g-2)_{\mu}$-preferred region, as shown by the green curve in Fig. 6.

##  VI Discussion

In this section, we make a few remarks on our results before concluding our
discussion.

###  VI.1 Interplay between Anomalies

We find in Fig. 6 that only a narrow region in the
$(m_{\widetilde{\nu}},\lambda_{232})$ parameter space is allowed that could
explain the $(g-2)_{\mu}$ anomaly in our minimal RPV3 setup. It is worth
checking how does the $(g-2)_{\mu}$-preferred region gets affected on its own,
and more importantly, if dropping the $R_{D^{(*)}}$ and $R_{K^{(*)}}$
anomalies could open up more parameter space in Fig. 2 6. To this effect, we
find that allowing $\lambda^{\prime}_{213}\neq 0$ in Eq. (8), the
$(g-2)_{\mu}$ solution can be improved only slightly, compared to Fig. 6. With
$m_{\widetilde{b}_{R}}=1.23$ TeV (the minimum value allowed by LHC 13 TeV
data) and $|\lambda^{\prime}_{233}|=1.57$ (the maximum value allowed from the
$Z$-decay universality constraint $g^{\mu}_{A}/g^{e}_{A}$, where
$g^{\ell}_{A}$ is the axial-vector coupling of $Z$ to leptons), the lower
$3\sigma$ bound of $(g-2)_{\mu}$ starts at $(0.7\mathrm{~{}TeV},2.66)$ and
ends at $(0.93\mathrm{~{}TeV},\sqrt{4\pi})$ in the
$(m_{\widetilde{\nu}_{\tau}},\lambda_{232})$ parameter space. The lower
$2\sigma$ bound is still not reached for $\lambda_{232}<\sqrt{4\pi}$. Because
of the constraints $B\to K\nu\overline{\nu}$ and $K\to\pi\nu\overline{\nu}$,
$\lambda^{\prime}_{213},\lambda^{\prime}_{223}\approx 0$ and thus cannot
contribute much to the $(g-2)_{\mu}$ anomaly in this optimal $(g-2)_{\mu}$
case.

Nevertheless, it is important to mention that due to the orthogonality between
$(m_{\widetilde{\nu}_{\tau}},\lambda)$ and
$(m_{\widetilde{b}_{R}},\lambda^{\prime})$ parameter spaces in our RPV3
scenario, even if the four-muon signal completely rules out the
$(g-2)_{\mu}$-favored region in Fig. 6 (drawn for BP1, but similar for BP2 and
BP3 as we can see from Fig. 4(i)), the $R_{D^{(*)}}$ and $R_{K^{(*)}}$
anomalies can still be explained by the
$(m_{\widetilde{b}_{R}},\lambda^{\prime})$ parameter space shown in Fig. 5.
Similarly, suppose the $pp\to\overline{t}\mu^{+}\mu^{-}$ signal completely
rules out one of the overlap regions in Fig. 5, but it will not affect the
$(g-2)_{\mu}$ solution in Fig. 6.

###  VI.2 Leptoquark versus RPV3

Several BSM scenarios have been invoked to explain the flavor anomalies, but
very few have the ability to explain all the flavor anomalies simultaneously
in a minimal, theoretically well-motivated setup like the RPV3. Leptoquarks
(LQs) have been a popular choice, but a single scalar LQ solution has now been
disfavored by global fits [111]. A single vector LQ $U_{1}({\bf 3},{\bf
1},2/3)$ still remains a viable option [112, 113, 114], but must be embedded
in some ultraviolet completion like the Pati-Salam gauge group [115, 116,
117], thus necessarily requiring more particles to cancel gauge anomalies, and
hence, losing its minimality feature. Another alternative is to invoke more
than one scalar LQs [118, 119, 120, 121, 122, 123]. The right-sbottom with
$\lambda^{\prime}$ couplings in our RPV3 scenario behaves exactly like the
$SU(2)_{L}$-singlet LQ $S_{1}({\bf 3},{\bf 1},-1/3)$ originally invoked in
Ref. [124], which still gives an excellent fit to the $b\to c\tau\nu$ data,
including polarization observables and forward-backward asymmetry [125].
However, the same $S_{1}$ LQ cannot explain the $b\to s\mu^{+}\mu^{-}$ data
simultaneously [126], while being consistent with the low-energy constraints,
in particular from $B_{s}-\overline{B}_{s}$ mixing. This is a key difference
with RPV3, where a TeV-scale sbottom by itself can explain both $R_{D^{(*)}}$
and $R_{K^{(*)}}$, owing to a (partial) cancellation in the
$B_{s}-\overline{B}_{s}$ mixing [cf. Eq. (A.2)]. Another important difference
is the $\lambda$ coupling, which gives rise to the distinct four-lepton signal
in the RPV3 scenario and uniquely distinguishes our scenario from LQ models.

###  VI.3 Precision Tests

Apart from the collider tests proposed here, our RPV3 solution to the flavor
anomalies can also be probed via low-energy precision observables at LHCb and
Belle-II. For instance, the $*$ benchmark point in Fig. 5 predicts the ratio
[cf. Eq. (12)] $R_{B\to K\nu\overline{\nu}}=2.1$, which is just below the
Belle 95% CL upper limit of 3.2 [127, 128]. The future Belle-II sensitivity
can improve this limit by up to a factor of 5 [129], which should be able to
completely probe the overlap region. In particular, the red, yellow, and blue
overlap regions in Fig. 5 can be completely excluded for $R_{B\to
K\nu\overline{\nu}}<1.7$, 2.0 and 1.1 respectively. This is a distinct feature
of our RPV3 scenario.101010For instance, in the $U_{1}$ vector LQ case, there
is no tree-level contribution to $B\to K\nu\overline{\nu}$ and any prediction
involving loop-processes depends on the UV-completion details. Similarly,
future lattice improvements in the precision of the SM prediction for
$B_{s}-\overline{B}_{s}$ mixing could be fateful for the overlap region in
Fig. 5. In addition, hints of LFUV in other independent observables involving
the third generation, such as LFV $\tau$ and $B$ decays, $b\to c\mu\nu/b\to
ce\nu$ and baryonic decay modes like $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$,
would provide critical further tests of our proposal. Furthermore, there are
other lepton flavor violating $B$ and $\tau$ decays that could get enhanced
contributions from RPV3 within reach of Belle II sensitivity [16]. If the
flavor anomalies persist and grow in statistical significance, the precision
flavor observables mentioned above, in conjunction with the collider
observables discussed in Sec. V, might be able to uniquely distinguish our
RPV3 interpretation from other BSM interpretations.

###  VI.4 Caveats

In spite of all the above-mentioned attractive features of our RPV3 scenario,
there are a few weak points which we just lay out here for future
contemplation.

$\bullet$ Landau Pole: In the minimal RPV3 setup presented here, some of the
$\lambda^{\prime}$ and $\lambda$ couplings are required to be fairly large
$\gtrsim{\cal O}(1)$. Such large couplings would hit the Landau pole very
quickly, preventing the model from being valid all the way up to the gauge
coupling unification scale. For instance, the benchmark point shown in Fig. 6
hits the Landau pole at 12.4 TeV. There might be a way out in the general RPV-
MSSM with more parameters, but a detailed analysis of the full MSSM parameter
space is beyond the scope of this work.
$\bullet$ Neutrino Mass: The trilinear RPV couplings in Eqs. (1) and (2)
contribute to neutrino masses at one-loop level through the lepton-slepton and
quark-squark loops [130, 131, 54]. To ensure that the neutrino masses remain
small and satisfy the cosmological bound on the sum
$\sum_{i}m_{\nu_{i}}\lesssim 0.1$ eV [132], we require some degree of
cancellation between the soft trilinear $A$-terms and the $\mu\tan\beta$ term
[16], depending on the other SUSY parameters.
$\bullet$ Dark Matter: In RPV scenarios, the lightest supersymmetric particle
(LSP) is no longer stable, but decays to SM particles. Therefore, it cannot be
the dark matter of the Universe, unless it is sufficiently long-lived, which
requires extremely small values of the RPV couplings. In our RPV3 scenario
with ${\cal O}(1)$ RPV couplings, the neutralino LSP cannot be the dark
matter. However, a gravitino LSP with its naturally Planck-suppressed decays
can in principle have a lifetime much longer than the age of the universe, and
hence, be the dark matter [133].
$\bullet$ Hierarchy of RPV Couplings: For our numerical analysis, we have
treated the relevant RPV couplings as free parameters and find the best-fit
that explains the flavor anomalies. We find that some of the RPV couplings
need to be fairly large $\gtrsim{\cal O}(1)$, while some others need to be
hierarchically smaller, and yet others need to be extremely small or
vanishing. One could in principle invoke a flavor symmetry (similar to Ref.
[68] for instance) to explain such hierarchy between couplings; so this need
not be an insurmountable issue, although it would require further work.

##  VII Conclusion

The flavor anomalies might already be giving us the first glimpse of natural
supersymmetry with light third-generation sfermions and with $R$-parity
violating couplings. We have proposed a simple, testable RPV3 scenario that
simultaneously explains the $R_{D^{(*)}}$, $R_{K^{(*)}}$ and $(g-2)_{\mu}$
anomalies with TeV-scale sbottom and tau sneutrino which are easily accessible
at the HL-LHC. With experimental updates from LHCb, Belle-II and Fermilab muon
$(g-2)$ experiments in the next few years, as well as with better limits on
third-generation sfermion masses from the LHC, our knowledge of the anomalies
will surely evolve and the allowed RPV3 ranges shown in Figs. 5 and 6 may have
to be modified accordingly. But let us hope SUSY prevails in the end.

###### Acknowledgements.

We gratefully acknowledge many useful discussions with Wolfgang Altmannshofer
and Yicong Sui. BD would like to thank Marcela Carena for an important
suggestion that led to our Fig. 4. BD also thanks Lawrence Hall for suggesting
the $\lambda_{233}$ scenario for $(g-2)_{\mu}$. AS would like to thank Mattia
Bruno, Christoph Lehner, Kalman Szabo and Hartmut Wittig for helpful
discussions on the SM calculation of muon $g-2$ and related issues. AS also
thanks Yoav Afik for useful discussion on LHC searches relevant for RPV3. The
work of BD and FX is supported in part by the U.S. Department of Energy under
Grant No. DE-SC0017987 and by the MCSS funds. BD is also supported in part by
the Neutrino Theory Network Program and by a Fermilab Intensity Frontier
Fellowship. The work of AS was supported in part by the U.S. DOE contract #DE-
SC0012704.

##  Appendix A Low energy constraints

Despite the many free parameters our RPV3 scenario is remarkably well-
constrained by various low-energy flavor observables so much so that more
accurate measurements of $R_{D^{(*)}}$, $R_{K^{(*)}}$ and $(g-2)_{\mu}$
preserving the central values could have appreciable adverse consequences on
our RPV3 explanation of these anomalies. In this section, we summarize all
relevant constraints on our RPV3 scenario shown in Fig. 5. For more details
and additional constraints (which are weaker, and therefore not mentioned
here), see Ref. [16].

###  A.1 $B\to K^{(*)}\nu\overline{\nu}$

Figure 10: Contributions to $B\to K^{(*)}\nu\overline{\nu}$ via
$\lambda^{\prime}$ interactions in RPV3.

As shown in Fig. 10, $B\to K^{(*)}\nu\overline{\nu}$ receives a tree-level
RPV3 contribution via sbottom exchange. The branching ratio is given by

$\displaystyle R_{B\to K^{(*)}\nu\overline{\nu}}\ \equiv\ \frac{{\rm BR}(B\to
K^{(*)}\nu\overline{\nu})}{{\rm BR}(B\to
K^{(*)}\nu\overline{\nu})_{\text{SM}}}$ $\displaystyle\ =\
\frac{1}{3}\left|\delta_{ii^{\prime}}-\frac{v^{2}\pi
s_{w}^{2}}{2\alpha_{\text{em}}}\frac{\lambda^{\prime}_{i33}}{V_{tb}V_{ts}^{*}}\left(\frac{\lambda^{\prime}_{i^{\prime}23}}{m_{\widetilde{b}_{R}}^{2}}+\frac{\lambda^{\prime}_{i^{\prime}32}}{m_{\widetilde{b}_{L}}^{2}}\right)\frac{1}{X_{t}}\right|^{2}.$
(12)

with the top loop function $X_{t}=1.469\pm 0.017$ [134] and $s_{w}$ being the
weak mixing angle. We consider both $\widetilde{b}_{L}$ and
$\widetilde{b}_{R}$ exchanges assuming that
$m_{\widetilde{b}_{L}}=m_{\widetilde{b}_{R}}$ for numerical purposes. An
experimental upper bound for this ratio exists: $R_{B\to
K^{(*)}\nu\overline{\nu}}<5.2$ at 95% CL [135, 136], which was adopted for our
original parameter setting and indicated in Fig. 5 as the solid brown line.
However, stronger upper bounds of $R_{B\to K\nu\overline{\nu}}<3.9$ and
$R_{B\to K^{*}\nu\overline{\nu}}<2.7$ have been quoted by Belle but at 90% CL
[127]. In order to make a fair comparison with the other low-energy and
collider bounds which are all given at 95% CL, we have derived an approximate
$95\%$ CL equivalent bound using the Belle data provided in Ref. [127]. We get
$R_{B\to K^{*}\nu\overline{\nu}}\lesssim 3.2$, where we have used the
theoretical uncertainty from Ref. [128] and have also taken into account the
propagation of uncertainty. This $95\%$ CL upper limit is shown in Fig. 5 by
the dashed brown line.

###  A.2 $B_{s}-\overline{B}_{s}$ Mixing

Figure 11: Relevant contributions to $B_{s}-\overline{B}_{s}$ mixing via
$\lambda^{\prime}$ couplings in RPV3.

Experimentally, the mass difference $\Delta M_{B_{s}}$ in neutral $B_{s}$
meson mixing is measured with excellent precision, $\Delta
M_{B_{s}}=(17.757\pm 0.021)$ ps-1 [13], dominated by LHCb and still
statistically limited. On the other hand, the SM prediction $\Delta
M_{B_{s}}^{\text{SM}}=(19.3\pm 1.7)$ ps-1 [137, 16] has sizable uncertainties
stemming mainly from the hadronic matrix elements and the CKM matrix element
$V_{cb}$.

In RPV3, additional contributions can arise at the tree level from sneutrino
exchange, or at the one-loop level from box diagrams with sbottoms,
sneutrinos, or stops (see Fig. 11). For the mass difference, we obtain

$\displaystyle\Delta M_{B_{s}}^{\rm RPV}\ =\
\frac{2}{3}m_{B_{s}}f_{B_{s}}^{2}\left|P^{VLL}_{1}\frac{\lambda^{\prime}_{i23}\lambda^{\prime}_{j33}\lambda^{\prime}_{j23}\lambda^{\prime}_{i33}}{128\pi^{2}m_{\widetilde{b}_{R}}^{2}}\right.$
$\displaystyle\quad\left.+P^{LR}_{1}\frac{\lambda^{\prime}_{i23}\lambda^{\prime}_{j33}\lambda^{\prime}_{i32}\lambda^{\prime}_{j33}}{128\pi^{2}m_{\widetilde{b}_{R}}m_{\widetilde{b}_{L}}}+P^{LR}_{2}\frac{\lambda^{\prime}_{332}\lambda^{\prime}_{323}}{2m_{\widetilde{\nu}}^{2}}\right|\,,$
(13)

where

$P^{VLL}_{1}\ =\ 0.80\,,\,P^{LR}_{1}\ =\ -2.52\,\,{\rm and}\,\,P^{LR}_{2}\ =\
3.08~{},$ (14)

are the updated hadronic $P$ factors from Ref. [138] with the latest lattice
input from Ref. [139] (see also Refs. [140, 141]), $f_{B_{s}}=(274\pm 8)$ MeV
is the $B_{s}$ decay constant, and $i,j$ are neutrino-flavor indices in the
box graphs. Combining our SM prediction with the experimental result we obtain
the following bound at 95% C.L. on

$0.78\ <\ \left|\frac{\Delta M_{B_{s}}}{\Delta M_{B_{s}}^{\text{SM}}}\right|\
<\ 1.12~{},$ (15)

which constrains the RPV3 contribution in Eq. (A.2). This bound is indicated
as the magenta-shaded region in Fig. 5.

###  A.3 $D^{0}\to\mu^{+}\mu^{-}$

Figure 12: Contribution to $D^{0}\to\mu^{+}\mu^{-}$ from $\lambda^{\prime}$ in
RPV3.

As shown in Fig. 12, there is a tree-level contribution from sbottom exchange
to this rare $D^{0}$ decay width which can be expressed as

$\displaystyle\Gamma(D^{0}\to\mu^{+}\mu^{-})$ $\displaystyle\ =\
\frac{1}{128\pi}\left|\frac{\lambda^{\prime}_{2j3}\lambda^{\prime}_{2j^{\prime}3}V_{uj^{\prime}}V_{cj}}{m_{\widetilde{b}_{R}}^{2}}\right|^{2}f_{D}^{2}$
$\displaystyle\times m_{D}m_{\mu}^{2}\sqrt{1-4m_{\mu}^{2}/m_{D}^{2}}\,,$ (16)

where $f_{D}=(212\pm 1)$ MeV is the $D^{0}$ decay constant. Using the
experimental upper bound on this branching ratio [142] of $7.6\times 10^{-9}$
at 95% CL, we calculate the corresponding bound on the RPV3 parameter space,
as shown by the purple-shaded region in Fig. 5.

###  A.4 $Z\to\ell\overline{\ell}^{\prime}$

Figure 13: Contribution to $Z\to\ell\overline{\ell}^{\prime}$ from
$\lambda^{\prime}$ in RPV3.

This process gets modified by top-sbottom loops, as shown in Fig. 13. A change
in the $Z$ decay from the SM prediction will affect the ratios of the vector
and axial-vector couplings of the $Z$ boson with different lepton flavors.
Experimental measurements on these couplings are [91]

$\displaystyle\left(\frac{g_{V}^{\tau}}{g_{V}^{e}}\right)_{\rm exp}$
$\displaystyle\ =\ 0.9588\pm 0.02997\,,$ (17)
$\displaystyle\left(\frac{g_{A}^{\tau}}{g_{A}^{e}}\right)_{\rm exp}$
$\displaystyle\ =\ 1.0019\pm 0.00145\,.$ (18)

The contributions to these ratios from RPV3 are given by

$\displaystyle\left(\frac{g_{V}^{\tau}}{g_{V}^{e}}\right)_{\rm SM+RPV}$
$\displaystyle\ =\ 1-\frac{2\,\delta g_{\ell_{3}\ell_{3}}}{1-4\,s_{w}^{2}}\,,$
$\displaystyle\left(\frac{g_{A}^{\tau}}{g_{A}^{e}}\right)_{\rm SM+RPV}$
$\displaystyle\ =\ 1-2\,\delta g_{\ell_{3}\ell_{3}}\,,$

where

$\displaystyle\delta g_{\ell_{i}\ell_{j}}$ $\displaystyle\ \simeq\
\frac{3y_{t}^{2}}{32\sqrt{2}G_{F}\pi^{2}}\frac{\lambda^{\prime}_{i33}\lambda^{\prime}_{j33}}{m_{\widetilde{b}_{R}}^{2}}\left[\log\left(\frac{m_{\widetilde{b}_{R}}}{m_{Z}}\right)-0.612\right]\,.$
(19)

Taking $i$, $j$ both equal to 3 and using Eqs. (17) and (18), we derive a
bound on the RPV3 parameter space, as shown by the violet-shaded region in
Fig. 5.

###  A.5 $b\to s\gamma$

Figure 14: Contribution to $b\to s\gamma$ from $\lambda^{\prime}$ couplings in
RPV3.

The branching ratio of $b\to s\gamma$ has been measured [13] as:

${\rm BR}(b\to s\gamma)_{\rm exp}\ =\ (3.43\pm 0.21\pm 0.07)\times 10^{-4}\,,$
(20)

which is consistent with SM [143]:

$\displaystyle{\rm BR}(b\to s\gamma)_{\rm SM}\ =\ (3.36\pm 0.23)\times
10^{-4}\,.$ (21)

In RPV3, there are one-loop contributions involving both left- and right-
handed sbottoms (see Fig. 14). Comparing this to the difference between the
experimental and SM results, we obtain the following bound at 95% CL:

$\displaystyle|\lambda^{\prime}_{223}\lambda^{\prime}_{233}|$ $\displaystyle\
\lesssim\ 0.025\left(\frac{100\rm{\ GeV}}{m_{\widetilde{b}_{R}}}\right)^{-2}$
(22) $\displaystyle|\lambda^{\prime}_{232}\lambda^{\prime}_{233}|$
$\displaystyle\ \lesssim\ 0.01\left(\frac{100\rm{\
GeV}}{m_{\widetilde{b}_{L}}}\right)^{-2}$ (23)

This is shown by the grey-shaded region in Fig. 5.

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# The Smoke of Zaanen

Philip W. Phillips  Department of Physics and Anthony J. Leggett Institute of
Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana,
IL 61801, USA

Theoretical physics suffered a major loss with the death of my dear friend Jan
Zaanen on January 18. This note is my remembrance of him.

Figure 1: Jan and Christa: July 18, 2023, Leiden

Johannes (Jan) Zaanen, a resonant, much-needed, deftly insightful voice in our
field of strongly correlated quantum matter, died Jan. 18 2024 in Leiden, the
place of his birth. He was 66. I received the news from Erik van Heumen, a
colleague of his at U. Amsterdam who was handling most of Jan’s scientific and
public matters once he took ill. In fact, they had just co-written a paper,
which they submitted to Science, on the breakdown of spectral sum rules in the
underdoped regime of the copper-oxide high-temperature superconductors. This
was Jan’s favorite problem. He threw the kitchen sink at it always looking for
something new. His methodology ranged from more traditional, early Hartree-
Fock[1] studies on stripes in the cuprates ( though it was highly influential,
he never recounted it fondly), the local density approximation augmented with
the Hubbard $U$[2] (LDA+U) to his more exotic mining[3] of the gauge gravity
duality to gain traction on the strange metal T-linear resistivity problem in
the cuprates. As we had both become gauge-gravity duality converts, we would
attend joint meetings with string theorists and discuss how to really push
this technology to realistic finite-density matter. In fact, in our last Zoom
call on Dec. 30, 2023, (a meeting for which I was an hour late because of a
time-zone mixup, Jan entreating me “We had a date, i kept hands free and now i
am waiting in vain for you, not great”) , the old problem of Sawatzky’s[4]
dynamical spectral weight transfer came up as a possible route to strange
metallicity. To the end, the strange metal remained his burning light. We had
agreed to write a joint paper on it putting all the details of Mottness into
the mix. Along with physics, Jan laid plain his frailty. Talking to me from
his hospice bed, he stated what I could see all too clearly: “I am basically
paralyzed you know and the mind is the last to go.” He talked about empathy
and “humankind being pretty nice.” He continued, “You know Philip, I am back
to being human. Physics keeps me going but now I’m human.”

Obituaries are unwieldy because of the obvious weight that goes with writing
them. What is difficult to communicate is a snapshot into the person’s actual
life, how it was like for them, not just a global summary. It is for this
reason that I have decided to bring Jan to the fore with a story. This story,
‘The smoke of Zaanen,’ is a love story. Not a typical one though. Most love
stories are private, lots of intimate exchanges, long looks and commitments
that last. But this one took place out in the open, typically in the front of
buildings. We all saw it happen, except the beginning. I had heard the story
before but never thought much of it. Jan always tells stories but listening to
him on March 21, 2023 in the hospital as fluid was being drained from his
lungs made his words more paramount. Human frailty focuses the empathic mind.
Jan was suffering from complications from two surgeries to relieve esophageal
cancer.

When I saw Jan in July (2022) in Stockholm, there was no hint of
hospitalization in his future. At that meeting, he chose the Swedish means of
transportation, the stand-up two-wheel slightly motorized scooter, to get
around Stockholm. He seemed like a child again, each morning regaling me and
others within earshot of the joy of 2-wheel standing mobility around
Stockholm. I settled for the bus, the lack of European cell service being the
bridge too far, or more aptly the expense I was not willing to incur. I saw
him take off one morning with others from the workshop at Nordita and indeed
he seemed a natural at it, no reticence there. I suspected he must have ridden
a rudimentary version of the modernized Segway transport when he was young. It
turns out Jan and I spent part of our childhoods on the most southern islands
in the Caribbean, Jan in Curucao and Tobago for me. Jan was conceived on a
family vacation to Costa Rica. After a brief stint back in Curucao, he was
shuttled to Leiden where he entered the world in the very hospital in which I
was then visiting him, Leiden University Medical Center. Immediately after,
they returned to Curucao for the next three and a half years giving shape to
his Caribbean roots. Perhaps this is part of our mutual bond, he intonated.

Back to March 2023. In between nurses’ visits , we continued our conversation.
I noticed the level of the fluid in the drain container was rising. Jan
insisted it wasn’t and continued, doing things his way, another of Jan’s
traits. As a goodbye to the Caribbean, Jan’s parents took a trip to Mexico.
Jan’s father, a mechanical engineer for Shell, was on kid duty that day in the
open-air market in Mexico City as his mother was taking a break. Wandering
through the market with his sister, Mieke, born in Curucao, he noticed a quite
irresistible lump on the ground which he proceeded to eat. Much to his
father’s surprise, it was horse shit. Yes, Jan ate horse shit. The cure for
the dysentery that ensued was a strict diet of ripe bananas. How they managed
to smuggle enough bananas on their plane carrying them back to the Netherlands
remains a mystery, a story shrouded in Dutch customs lore: the Dutch family
with a year’s supply of bananas and an extremely hungry kid.

Like all of us who grew up in the Caribbean, winter is harsh. Hearing a knock
on the door one day, he opened it and was greeted with what seemed like a
fierce animal. It was the chill of winter, his first. Nothing in Curucao
readied him to be pierced so unceremoniously with just the opening of a door.
There was no warning, just an enduring cold. Throughout his life, Jan hated
winter, thoughts back to Curucao his only escape.

Around age 13, Jan discovered another escape, and here lies the love story.
One day his school had a class outing to the countryside in Drenthe just
outside Groningen. Two of his class mates, Marianna and Maartje took him on a
hike away from the crowd. Jan was a good-looking boy but shy and nerdy so
there was no resistance to being led by two precocious girls, also 13. They
tired and soon spotted a park bench. I know what you’re thinking but that’s
not it. This is a PG story about a nerdy school boy who would become a world-
renowned physicist, coiner of Planckian dissipation[5] and the classifier[6]
of band gaps in transition metal compounds which led to his 1985 thesis with
the man who “showed him physics,” George Sawatzky.

There is of course continuity in how we become who we are. Or perhaps not? But
we’ll see. The girls, placing Jan firmly on the bench, said, “we have
something for you.” Jan liked Maartje but was too shy to say so. Marianna sat
on his right knee and Maartje on his left. Jan heard some undecipherable
rustling but he kept his eyes closed as instructed. Was this the moment in
this nerd’s life? In a flash Jan envisioned his future as a rock and roller
guitarist (something he did try before grad school)–perfect training at hand.
Ever so carefully, Marianna took out wrapping paper and tobacco and rolled a
cigarette and placed it in between Jan’s lips. Maartje took out a lighter and
lit the cigarette. He then opened his eyes as Maartje clasped his hand and led
it to meet the cigarette. What an introduction to smoking. “You know Philip,
the first 4 cigarettes are pretty bad, but then the fifth one, that’s an
experience,” Jan said. They returned to the group, Jan now in love. He
discovered smoking and it would forever be linked to Marianna and Maartje,
each puff taking him back to that day in Drenthe where his world changed to
the Jan Zaanen we knew.

I first met Jan in 1997/98 in Trieste and of course it was on one of his
notorious smoking breaks, complete with the hypnotic rolling of tobacco. He
stopped me and said he liked my talk (on the Kravchenko problem as I recall)
and proceeded to say some things I barely followed. Jan’s accent took a while
to decipher but his love for smoking was as clear as a crystal day. When I saw
him smoking in Trieste, I remember thinking why doesn’t this man just buy the
damn cigarettes in a store. With the back story revealed, I now see the
continuity. I now look back on that first meeting in Trieste differently.
There was a certain intensity and earnestness in Jan’s eyes when he smoked. It
was as if something were being revealed to him. I now have an idea what. From
the first cigarette Marianna rolled for him ever so deliberately, smoking
would always take him back to that park bench in Drenthe when he was 13. He
remained purely platonic with Maartje, and of course Marianna, but their mark
on him was indelible.

Jan was not a man of few words. He was voluble and wide ranging in his views
on physics. I hung out with him because we had similar likes and dislikes. He
liked children, my son no exception as we spent much time together in Kyoto.
We both shared a dislike for conventional approaches to correlated electron
problems. So strong was Jan’s drive to do something new that he turned down a
tenured job at Stuttgart for a post-doc at Bell Labs in 1987. For him, Bell
Labs “was playing on Mt Olympus.” So why not. He scaled those heights and was
rewarded with the position he filled throughout the remainder of his life,
Professor of physics at Lorentz’ institution, University of Leiden. He was a
Spinoza Award recipient, the highest honor in the Netherlands in science as
well as a member of the Dutch Royal Academy. In July 2023, many of us convened
in Leiden for a celebration of Jan’s remarkable life in physics. Of course the
meeting was organized by Erik van Heumen. We all paid our debt to Jan for
inspiring us to think differently. Undoubtedly, his non-traditional background
as a chemist as an undergraduate prepared him to think outside the box in
theoretical physics. This commonality in our backgrounds cemented the bond
between us.

The day after the meeting, I rode a bike out to see Jan and Christa at their
home. The picture I include here is from that visit. Jan and I finally talked
about what was on my mind in March, 2023 when I visited him in Leiden
Hospital, quantum gravity. It turns out we were thinking about the same
problem: Penrose’s demonstration that the gravitational self energy places
restrictions on the time scale for a quantum superposition of two spatially
separated objects. One of his colleagues at Leiden, Tjerk Oosterkamp[7], is
performing experiments to test Penrose’s ideas. The synergy with Jan never
ceased to surprise me. Our conversations were always endless. It turns out Jan
was writing a book about this very problem while I was trying to link Penrose
to a recent proposal by Cotler/Strominger[8] in which unitarity is abandoned.
Jan’s book, to be published soon by Oxford Press, is entitled ironically “On
Time.” That’s exactly what he ran out of. Time. His mark on us all is
indelible and his life way too short. We will all miss Jan but his loss is
harshest for his family, Christa Buschmann, his widow, his son, Filip, his
sister, Mieke, and his favorite niece, Amber.

## References

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