# Anomalies in Particle Physics

Andreas Crivellin  Physik-Institut, Universität Zürich, Winterthurerstrasse
190, CH–8057 Zürich, Switzerland  Paul Scherrer Institut, CH–5232 Villigen
PSI, Switzerland  e-mail<EMAIL_ADDRESS>Bruce Mellado  School of
Physics and Institute for Collider Particle Physics, University of the
Witwatersrand, Johannesburg, Wits 2050, South Africa  iThemba LABS, National
Research Foundation, PO Box 722, Somerset West 7129, South Africa
<EMAIL_ADDRESS>
###### Abstract

The currently accepted mathematical description of the fundamental
constituents and interactions of matter is the Standard Model of particle
physics. Its last missing particle, the famous Higgs boson, was observed at
the Large Hadron Collider at CERN in 2012. However, it is clear that the
Standard Model cannot be the ultimate theory of Nature, and e.g. cannot
account for Dark Matter or non-vanishing neutrino masses (and does not include
gravity). In fact, searches for physics beyond the SM have been intensified
since the Higgs boson discovery. In this article, we review the hints for new
physics, called “anomalies”, obtained in particle physics experiments within
the last years. We consider both direct high-energy searches for new
resonances at the LHC and indirect low-energy precision experiments. These
anomalies range from the nuclear scale (approximately the mass of the proton)
to the electroweak scale (i.e. the mass of the Higgs boson) to the TeV scale
(the highest scale directly accessible at the LHC), therefore spanning over
four orders of magnitude. After discussing the experimental and theoretical
status of the anomalies, we summarize possible explanations in terms of new
particles and new interactions. In particular, new Higgs bosons and
leptoquarks are promising candidates. Discovery prospects and implications for
future colliders are discussed.

Key points:

  * •

The Standard Model of Particle Physics is the currently accepted mathematical
theory describing the fundamental constituents of matter as well as their
interactions, and its final completion was the discovery of the Higgs particle
at the Large Hadron Collider at CERN in 2012.

  * •

The Standard Model cannot account for the existence of Dark Matter or non-
vanishing neutrino masses and must therefore be extended, however, a plethora
of viable options exist.

  * •

In recent years, several interesting deviations from the Standard Model
predictions, called “anomalies”, were found, both in high-energy searches at
the LHC and in low-energy precision observables.

  * •

These anomalies range from precision measurements of properties of the muon to
hints for new scalar bosons to the existence of heavy resonances at the TeV
scale.

  * •

They can be explained by supplementing the Standard Model with new particles
and new interactions, in particular, additional Higgs bosons, new fermions and
new strongly interacting particles.

  * •

While the data of the third run of the Large Hadron Collider will already be
able to establish the existence of such new particles, future colliders, like
FCC, ILC or CEPC, as well as new precision experiments are needed for a
comprehensive and precise study of their properties.

Website summary:

The Standard Model of particle physics is the currently accepted theory of the
fundamental constituents of matter and their interactions. We review the
status of hints for new physics, which, if confirmed, would require the
extension of the Standard Model by new particles and new interactions.

##  1 Introduction

The Standard Model (SM) of particle physics is the currently accepted
mathematical description of the fundamental constituents of matter and their
interactions (excluding gravity). More specifically, matter, i.e. quarks and
leptons, are fermions (spin 1/2 particles whose wave functions are invariant
under rotation of $4\pi$). A proton contains two up-quarks (with electric
charge $+2/3$) and one down-quark (with charge $-1/3$), while a neutron
consists of one up-quark and two down-quarks. Electrons constitute the hull of
atoms. Together with the nearly mass-less and very weakly interacting
neutrinos they form the class called leptons. All fermions appear in three
copies, called generations or flavours, that only differ in mass. The electron
is accompanied by its heavy cousins the muon and the tau. The more massive
versions of the up-quark, are called charm and top while strange and bottom
(sometimes also called “beauty”) are the more massive versions of the down-
quark. Only first-generation fermions are stable, while the heavier
generations are short-lived and decay to the lighter flavours.

The forces between the fermions are mediated by gauge interactions. The
corresponding gauge group is $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. These
are “local” symmetries, meaning that they hold independently at any point in
space-time. Due to quantisation, these interactions result in force particles,
the gauge bosons. The electromagnetic force is mediated by the photon, the
weak force (corresponding to the $SU(2)_{L}$ factor) by the $W$ and $Z$ gauge
bosons and the strong force ($SU(3)_{c}$) by eight gluons. While quarks are
charged under the strong force and thus interact with gluons, neutrinos only
feel the weak force and the charged leptons (electron, muon and tau) as well
have electromagnetic interactions. Importantly, all flavour violation in the
SM is induced by the couplings of the $W$ boson to up and down quarks via the
Cabibbo-Kobayashi-Maskawa (CKM) matrix.

Finally, we have the famous Higgs particle [1, 2], which was discovered at the
Large Hadron Collider at CERN in 2012 [3, 4]. The field from which the Higgs
boson originates spontaneously breaks the gauge symmetry of the weak force and
$U(1)_{Y}$ to the electromagnetic gauge group ($U(1)_{\rm EM}$) and gives
masses to the $W$ and $Z$ bosons as well as to all (fundamental) fermions and
the Higgs boson itself. The particle content of the SM is summarized in Fig.
1.

Figure 1: Particle content of the Standard Model: Fermions consisting of
quarks (gray) and leptons (green) as well as the gauge bosons (red) and the
Higgs (blue).

Therefore, the SM is now complete. It has been extensively tested and verified
by precision experiments within the last decades and no discovery of any
particle beyond the SM ones has been announced [5]. However, it is clear that
the SM cannot be the ultimate fundamental theory of Nature: In addition to
many theoretical arguments for the existence of beyond the SM (BSM) physics,
the SM e.g. cannot account for the observations of Dark Matter (DM)
established at cosmological scales (since it does not contain a weakly
interacting particle with the right relic abundance), nor for the non-
vanishing neutrino masses required by neutrino oscillations, because in the SM
neutrinos are necessarily massless due to the absence of a right-handed
partner.

Unfortunately, no right-handed neutrinos have been observed and the Dark
Matter direct detection experiments did not see any signal [6]. Thus, we know
that the SM cannot be the ultimate theory of Nature. However, there are many
options, spanning a very large mass range (from several keV to the scale of
Grand Unification at around $10^{15}$GeV), how one can account for Dark Matter
and neutrino masses. Therefore, more experimental information on physics
beyond the SM, preferably deviations from its predictions (in best case in the
form of new resonances) is imperative to make progress towards a theory
superseding the SM.

The search for BSM physics was further intensified in the last decade, both in
direct high-energy searches (mainly at the LHC) and in precision experiments
testing the SM via quantum fluctuations of new particles. In fact, an
increasing number of hints for new physics, called “anomalies”, have been
reported. They span over a huge energy range, from precision measurements of
muon properties (the anomalous magnetic moment of the muon), over semi-
leptonic $B$ meson (quark bound states containing a $b$ quark) decays, the
measurement of the $W$ boson mass, to direct LHC searches and even non-
resonant searches for particle too heavy to be produced directly at the LHC.

While probability theory and statistics tell us that one cannot expect that
all these anomalies will be confirmed in the future, it is also unlikely that
all of them are just statistical flukes. Therefore, it is important to assess
the strengths and weaknesses of these anomalies and to see to which extensions
of the SM they point to predict other signals for future verification or
falsification. In this article, we will review the status of these anomalies,
give an overview of how they can be explained by BSM physics and give an
outlook on their future implications.

##  2 Anomalies

Let us now review the status of these anomalies. We will present them in
increasing order of the corresponding energy scale.

###  2.1 Anomalous magnetic moment of the muon ($a_{\mu}$)

While the Dirac equation predicts that the $g$ factor of any fundamental
fermion is exactly 2, the famous prediction of Quantum Electro Dynamics (the
quantum field theory of electromagnetism) by Schwinger [7] was a positive
shift of $a_{\ell}=(g-2)_{\ell}/2=\alpha/(2\pi)$ (see left diagram in Fig. 2
a)). Nowadays, the accuracy has dramatically increased. The combined value of
the Brookhaven E821 result [8] and the $g-2$ experiment at Fermilab [9, 10]
deviate from the SM prediction of the $g-2$ theory initiative [11] by
$5.0\,\sigma$. However, this SM prediction is based on measurements of
$e^{+}e^{-}\to$hadrons [12, 13, 14] and does not include newer results for
hadronic vacuum polarization from lattice simulations of Quantum Chromo
Dynamics (the quantum field theory describing the strong interactions) [15]
nor the latest measurement of $e^{+}e^{-}\to$hadrons111Hadrons are bound
states of quarks which form due to the confining nature of QCD (i.e. because
the coupling increases at low energies). by the CMD 3 collaboration [16] which
would render the SM prediction closer to the measurement. Since these tensions
between the different SM predictions are not understood yet, one can only say
that a positive shift in $a_{\mu}$ of the order of $10^{-9}$ is preferred but
a reliable estimate of the significance is not possible at the moment.

###  2.2 Cabibbo Angle Anomaly (${\rm CAA}$)

The CKM matrix $V$ is by construction unitary [17] (since it must conserve
probability) and we know from experiments that it has a hierarchical
structure; while the size of the diagonal elements is close to unity, the off-
diagonal elements are small. One can test the SM prediction
$\sum_{j}V_{ij}V^{*}_{jk}=\delta_{ik}$ experimentally. In this context, the
Cabibbo angle [18], which parametrizes the mixing between the first two quark
generations, is particularly interesting as it dominates the first and second
row and column relations. In fact, a deficit in first-row and first-column CKM
unitarity, which can be traced back to the fact that $V_{ud}$ extracted from
beta decays [19, 20] (see left diagram in Fig. 2 b)) does not agree with
$V_{us}$ ($V_{cd}$) determined from kaon222Pions are bound states containing
only up and down quarks while kaons are are made of a light quark (up or down)
and a strange quark. The $D$ ($B$) meson contains a charm (bottom) quark and a
light quark, while $B_{s}$ is made of a (anti-)bottom and a strange quark. A
$\phi$ is a strange anti-strange bound state and a “$*$” refers to an excited
bound state. and tau decays ($D$ decays), when comparing them via CKM
unitarity. Furthermore, there is also a disagreement between the
determinations of $V_{us}$ from $K\to\mu\nu$ [21] and $K\to\pi\ell\nu$ [22]
decays. The significance of these deviations crucially depends on the
radiative (quantum) corrections to beta decays [23, 24, 20] and on the
treatment of the tensions between kaon [25, 26, 27] and tau decays [28]. In
summary, both tensions are slightly below the $3\sigma$ level.

###  2.3 Lepton flavour universality violation in tau decays
($\tau\to\mu\nu\bar{\nu}$)

Because the interactions of the $W$ boson with leptons and neutrinos are
flavour universal, the (leading terms of the) amplitudes for
$\ell\to\ell^{\prime}\nu\nu$ are predicted to be the same for all lepton
generations. However, combining the ratios of branching ratios ${\rm
Br}(\tau\to\mu(e)\nu\bar{\nu})/{\rm Br}(\mu\to e\nu\bar{\nu})$ and ${\rm
Br}(\tau\to\mu\nu\bar{\nu})/{\rm Br}(\tau\to e\nu\bar{\nu})$ [29, 28], leads
to an $\approx 2\sigma$ preference for constructive new physics (NP) at the
per-mille level in $\tau\to\mu\nu\bar{\nu}$ [30]. While here the SM prediction
is quite clean, the significance is limited by statistics and the experimental
problem of reconstructing the event due to missing energy.

###  2.4 Charged current tauonic $B$ decays ($b\to c\tau\nu$)

These charged current transitions, mediated at tree-level by a $W$ boson in
the SM (see left diagram in Fig. 2 c)), have significant branching ratios (up
to ${\mathcal{O}}(10^{-2})$). With light leptons, they are used to extract the
CKM element $V_{cb}$ and the result is consistent with the global CKM fit [31,
32]. However, the ratios (of branching ratios) $R({{D^{\left(*\right)}}})$ =
${\rm Br}({B\to{D^{\left(*\right)}}\tau\nu})$/ ${\rm
Br}({B\to{D^{\left(*\right)}}\ell\nu)})$, are measured to be bigger than the
SM predictions by approximately 20%, resulting in a $\gtrapprox 3\sigma$
significance [28] for NP related to tau leptons.

###  2.5 Flavour changing neutral current semi-leptonic $B$ decays ($b\to
s\ell^{+}\ell^{-}$)

Like all flavour changing neutral current processes, $b\to s\ell^{+}\ell^{-}$
transitions are loop suppressed (i.e. only induced via quantum fluctuations)
within the SM (see left diagram in Fig. 2 d) for an example) since only the
couplings of the charged $W$ can violate quark flavour. This results in small
branching ratios, up to a few times $10^{-6}$. While the previous hints [33]
for lepton flavour universality violation in the ratios
$R({K^{\left(*\right)}})={\rm Br}({B\to
K^{\left(*\right)}\mu^{+}\mu^{-}})/{\rm Br}({B\to
K^{\left(*\right)}e^{+}e^{-}})$ were not confirmed [34] and
$B_{s}\to\mu^{+}\mu^{-}$ [35, 36] now agrees quite well with the SM prediction
[37, 38], there are several $b\to s\mu^{+}\mu^{-}$ observables that
significantly deviate from the SM predictions. This includes the angular
observable $P_{5}^{\prime}$ [39, 40], the total branching ratio ${\rm
Br}({B\to K\mu^{+}\mu^{-}})$[41, 42], ${\rm Br}({B_{s}\to\phi\mu^{+}\mu^{-}})$
[43, 44] and also semi-inclusive observables [45]. As a result, global fits
find a preference for NP at the $5\sigma$ level [46, 47, 48]. Recently, the
Belle-II collaboration reported an excess in the closely related $B\to
K^{*}\nu\nu$ decay [49].

###  2.6 Asymmetries in $Z$ decays ($Z\to b\bar{b}$)

Since the $Z$ boson couplings to fermions are chiral (see left diagram in Fig.
2 e)), i.e. are different for left-handed and right-handed ones, one can
measure asymmetries in its decays (with respect to the beam direction). The
measurement by the LEP collaboration of the forward-backward asymmetry in
$Z\to b\bar{b}$ [50] deviates from its SM prediction by $\approx 2\sigma$.
Similarly, there is an $\approx 2\sigma$ tension in the lepton asymmetry
parameter $A_{\ell}$ [51], mainly due to the electron channel. The
corresponding suggested deviations from the SM coupling at the sub-permille
level.

###  2.7 $W$ boson mass ($m_{W}$)

In general, three parameters are sufficient to parameterise completely (at
tree level) the electro-weak sector of the SM. They are usually taken to be
the Fermi constant $G_{F}$, the fine-structure constant $\alpha$ and the $Z$
boson mass since these are measured most precisely. In this input scheme, the
$W$ mass is not a free parameter but can be calculated as a function of
$G_{F}$, $\alpha$ and $m_{Z}$ (and the Higgs and the top mass which enter at
the loop-level). The CDF II result [52] shows a very strong $7\sigma$ tension
with the SM prediction. However, LHC [53, 54, 55, 56] and LEP results [57] are
closer to the SM and thus in tension with CDF II value. Therefore, employing a
conservative error estimate that increases the uncertainty finds a tension of
$3.7\,\sigma$ [51].333This average does not include the latest ATLAS result
[58] superseding Ref. [53], which however has a small impact on the fit.

SM NP SM NP

Figure 2: Feynman diagrams showing some of the processes where anomalies are
observed. The left diagrams depict the SM process, while the right-handed ones
show a possible NP explanation. $\left.a\right)$ Schwinger term contribution
to $a_{\mu}$ and LQ explanation $\left.b\right)$ beta decay in the SM and
modification via a vector-like quark $\left.c\right)$ $W$ contribution to
$R(D^{(*)})$ and LQ effect $\left.d\right)$ $W$ box contribution to $b\to
s\ell^{+}\ell^{-}$ in the SM and $Z^{\prime}$ effect $\left.e\right)$ $Z\to
b\bar{b}$ and its modification via vector-like quarks $\left.g\right)$ top
pair production and decay in the SM and new Higgses “polluting” the
measurement $\left.g\right)$ di-di-jet production in the SM and NP
contribution via DQs h$)$ $pp\to e^{+}e^{-}$ in the SM and LQ contribution.

###  2.8 LHC Multi-Lepton Anomalies ($e\mu(+b)$)

The “multi-lepton anomalies”, are LHC processes with two or more leptons in
the final state (see Ref. [59] for a review), with and without
$b$-jets444Since quarks and gluons are confined at low energies, they do not
appear as free particles in a detector but rather hadronize and give
signatures called jets ($j$). A $b$-jet is such an experimental signature
containing a bottom quark., where statistically large disagreements with the
SM predictions have been observed [60, 61, 62, 63, 64, 65]. The excesses can
be summarized as follows

Final state | Characteristics | SM backgrounds | Significance
---|---|---|---
$\ell^{+}\ell^{-}$+($b$-jets) [62, 66, 65] |  $m_{\ell\ell}<100$ GeV, $(1b,2b)$ | $t\overline{t},Wt$ | $>5\sigma$
$\ell^{+}\ell^{-}$+(no jet) [61, 67] |  $m_{\ell\ell}<100$ GeV | $W^{+}W^{-}$ | $\approx 3\sigma$
$\ell^{\pm}\ell^{\pm},3\ell$ \+ ($b$-jets) [64, 68, 69] | Moderate $H_{T}$ | $t\overline{t}W^{\pm},t\overline{t}t\overline{t}$ | $>3\sigma$
$\ell^{\pm}\ell^{\pm},3\ell,($no$\;b$-jet) [63, 70, 71] | In association with $h$ | $W^{\pm}h(125),WWW$ | $\gtrapprox 4\sigma$
$Z(\rightarrow\ell\ell)\ell,($no$\;b$-jet) [62, 72] |  $p_{\rm T}^{Z}<100$ GeV | $ZW^{\pm}$ | $>3\sigma$

The fact that the leptons in these channels are non-resonant, i.e. no peak in
the invariant mass spectrum is observed, shows that, at least within the SM,
they are related to leptonic $W$ decays. A statistically particularly
significant disagreement is observed in differential lepton distributions in
$t\bar{t}$ measurement [62, 65] (see left diagram in Fig. 2 f)). For all SM
simulations used, ATLAS finds such a high $\chi^{2}$ value that they conclude
[66]: “No model (SM simulation) can describe all measured distributions within
their uncertainties.” Because excesses also appear in $WW$ signatures without
jets (where SM $t\overline{t}$ production is strongly suppressed) and in
$Wh/3W$, $t\bar{t}W$, $t\overline{t}t\overline{t}$ and $ZW$ production with
low $Z$ boson transverse momentum ($p^{Z}_{T}$), this indicates that the
excess may not be due to the mismodelling of $t\overline{t}$ production and
decay. In addition, there is a hint for a resonant $t\bar{t}$ excess at around
$400\,$GeV [73].

###  2.9 Higgs-like signals ($yy=\gamma\gamma,\tau\tau,WW,ZZ$)

New particles that are directly produced at colliders show up as bumps in the
otherwise continuous invariant mass spectrum of the corresponding decay
products. For scalar bosons, di-photon distributions are very sensitive: Even
though they have in general small rates because they are loop-suppressed, the
experimental signature is very clear. In fact, there are several hints for di-
photon resonances at $95\,$GeV [74, 75], $\approx 152$ GeV [76]555The mass of
this excess is consistent with invariant mass of di-leptons in the multi-
lepton anomalies. and also $\approx 680\,$GeV [77, 78]. The hint at $95\,$GeV
is supported by a di-taus excess reported by CMS [79], a $ZH$ signal (with
$H\to b\bar{b}$) by LEP [80] as well as the $WW$ channel [61, 81]. The
$\gamma\gamma$ (plus missing energy) hint at 152 GeV is supported by several
signals in associated production [82, 76], including $WW+$missing energy [81].
Combining all channels, global significances of $3.8\sigma$ and $4.9\sigma$
are found for 95 GeV and 152 GeV, respectively, if for the latter, a
simplified model with $pp\to H\to SS^{*}$ is assumed [83].

###  2.10 (di-)di-jet resonances ($jj(-jj)$)

A particle decaying into two quarks (or two gluons) results in a di-jet event
at the LHC. ATLAS [84] observed a weaker limit than expected in resonant di-
jet searches slightly below $1\,$TeV. Furthermore, CMS [85] found hints for
the (non-resonant) pair production of di-jet resonances with a mass of
$\approx 950$ GeV with a local (global) significance of 3.6$\sigma$
(2.5$\sigma$). This compatibility suggests that both excesses might be due to
the same new particle $X$, once directly (resonantly) produced in proton-
proton collisions ($pp\\!\to\\!X\\!\to\\!jj$), once pair produced via a new
state $Y$ ($pp\\!\to\\!Y^{(*)}\\!\to\\!XX\\!\to\\!(jj)(jj)$). In fact, Ref.
[86] finds a global $3.2\sigma$ significance at $m_{Y}\approx 3.6\,$TeV. In
the latest analysis, ATLAS finds a di-di-jet excesses [87] at $\approx
3.3\,$TeV with a di-jet mass of $850\,$GeV which could be compatible with the
CMS one once the quite poor jet energy resolution is taken into account.

###  2.11 Non-resonant di-electrons ($q\bar{q}\to e^{+}e^{-}$)

If the mass of a particle exceeds the energy reach of a collider, its impact
can still be seen by looking at the high-energetic end of the spectrum of a
distribution where such effects are most relevant because they possess a
relative enhancement w.r.t. the SM (see left-diagram in Fig. 2 h)). In such a
non-resonant search for high-energetic oppositely charged leptons, CMS and
ATLAS observe more electrons than expected in the SM [88, 89]. Because the
number of observed muons is compatible with the SM prediction, this is a sign
of lepton flavour universality violation and the ratio of muons over electrons
provided by CMS has the advantage of reduced theoretical uncertainties [90].
Performing a model-independent fit, one finds the NP at a scale of $10\,$TeV
with order one couplings can improve over the SM hypothesis by $\approx
3\sigma$ [91].

###  2.12 Summary

The anomalies observed in particle physics are summarized in Fig. 3, together
with their corresponding energy scale, showing that they range over at least
five orders of magnitude. While one cannot expect that all anomalies will be
confirmed, it is also statistically unlikely that all will turn out as flukes.
Therefore, it is important to investigate their implications for NP in order
to assess possible correlations among them and identify signatures for future
verification (or falsification).

Figure 3: Compilation of various anomalies ordered according to the
corresponding energy scale.

##  3 New Physics Explanations

For a consistent renormalizable extension of the SM, only scalars bosons (spin
0), fermions (spin 1/2) and vectors bosons (spin 1) are at our disposal,
provided that in the latter case a Higgs-like mechanism of spontaneous
symmetry-breaking exists to give them masses. Here we will focus on the
following extensions of the SM:

  * •

Leptoquarks (LQs): Scalar or vector bosons that carry color and couple quarks
directly to leptons [92, 93]. These particles were first proposed in the
context of quark-lepton unification at high energies, namely the Pati-Salam
model [94] and Grand Unified Theories (GUTs) [95]. Furthermore, in the
R-parity violating MSSM (see e.g. Ref. [96] for a review) the supersymmetric
partners of quarks can have the properties of LQs.

  * •

Diquarks (DQs): Scalar bosons which are either triplets or sextets of
$SU(3)_{c}$ and couple to a quark and an anti-quark. They are predicted by
GUTs based on the $E_{6}$ symmetry group [97] and appear in the R-parity
violating MSSM.

  * •

$Z^{\prime}$ bosons: Neutral heavy vector bosons. They can be singlets under
$SU(2)_{L}$ but also the neutral component of an $SU(2)_{L}$ multiplet. These
particles can be resonances of the SM $Z$, e.g. Kaluza-Klein excitations of
the SM $W$ in composite [98] or extra-dimensional models [99], or originate
from an abelian symmetry (like $B-L$ [94]) or gauged flavour symmetries [100].

  * •

$W^{\prime}$ bosons: Electrically charged but QCD neutral vector particles.
They can have similar origins as $Z^{\prime}$ bosons but also come for a left-
right symmetry [101].

  * •

Vector-like Quarks (VLQs): For vector-like fermions in general left-handed and
right-handed fields have the same quantum numbers under the SM gauge group
(unlike SM fermions) and can thus have masses independently of EW symmetry
breaking, meaning that they can be arbitrarily heavy. They appear in GUTs
[102], as resonances of SM fermions in composite or extra-dimensional models
[103] and as the supersymmetric partners of SM vectors and scalars [104].

  * •

Vector-like Leptons (VLLs): These particles can have similar origins as VLQs.
In addition, they are involved in the type I [105, 106] and type III [107]
seesaw mechanisms used for giving masses to the light active neutrinos as
required by neutrino oscillations.

  * •

New scalars ($S$): Scalars could be supersymmetric partners of SM fermions
[104], but also scalar fields of different representations under the SM gauge
group can be added. Most commonly, a copy of the SM Higgs, an $SU(2)_{L}$
doublet with hypercharge $0$, leading to a two-Higgs doublet model [108, 109].
Note that we do not include coloured scalars with the properties of DQs or LQs
here.

###  3.1 Anomalous magnetic moment of the muon ($a_{\mu}$)

Since the anomalous magnetic moment of the electron is measured and predicted
much more precisely than $a_{\mu}$, the resulting bounds on NP are stringent
[110, 111, 112]. Therefore, the effect in $a_{\ell}$, unlike the famous
Schwinger term, must violate lepton flavour universality [113]. Furthermore,
because the deviation from the SM prediction is as large as its EW
contribution, new physics must be either quite light, e.g. a light
$Z^{\prime}$ boson only coupling to muons and tau lepton [114], or if it is
heavy (at the TeV scale) must possess an enhancement factor (see Ref. [115]
for a recent overview on NP in $a_{\mu}$). This can be provided via the
mechanism of chiral enhancement, meaning that the chirality flip does not
originate from the small muon Yukawa coupling but from a larger coupling of
other particles to the SM Higgs. In the MSSM, this factor is $\tan\beta$, the
ratio of the two vacuum expectation values of the two Higgs fields [116, 117]
but also models with generic new scalars and fermions can explain $a_{\mu}$
[118, 119, 120, 121]. Furthermore, and there are two scalar LQs ($S_{1}$ and
$S_{2}$) that address $a_{\mu}$ via a $m_{t}/m_{\mu}$ enhancement [122, 123,
113] (see Fig. 2 a)). This leads to interesting predictions for
$h\to\mu^{+}\mu^{-}$ [124] and $Z\to\mu^{+}\mu^{-}$ [125] that can be measured
at future colliders.

###  3.2 Cabibbo Angle Anomaly (CAA)

A sub-permille effect suffices to explain the CAA. The disagreement between
the two determinations of $V_{us}$ can only be explained via a right-handed
quark current pointing towards vector-like quarks (see Fig. 2 b)). The deficit
in first-row and first-column CKM unitarity can be explained via left-handed
(i.e. SM-like) NP in beta decays. Both an effect in beta decays or in the
Fermi constant (determined from muon decay, i.e. $\mu\to e\nu\nu$), which is
needed to extract $V_{ud}$, is possible. Therefore, we have four options
[126]: 1) a direct (tree-level) modification of beta decays 2) a direct (tree-
level) modification of muon decay 3) a modified $W$-$\mu$-$\nu$ coupling
entering muon decay 4) a modified $W$-$u$-$d$ coupling entering beta decays
(the effect of a modified $W$-$w$-$\nu$ drops out). Option 1) could in
principle be realized by a $W^{\prime}$ [127] or a LQ [128], however, in the
latter case, stringent bounds from other flavour observables arise.
Possibility 2) can be achieved by adding a singly charged $SU(2)_{L}$ singlet
scalar [129], a $W^{\prime}$ [127] or $Z^{\prime}$ boson with flavour
violating couplings [130]. Option 3) and 4) can be achieved by vector-like
leptons [131, 132] and vector-like quarks [133, 134, 135, 136], respectively.
However, without a compensating effect [137], explaining the CAA via a
modification of $G_{F}$ increases the tension in the $W$ mass.

###  3.3 Lepton flavour universality violation in tau decays
($\tau\to\mu\nu\nu$)

Explanations of $\tau\to\mu\nu\nu$ are very similar to the ones of the CAA via
a modified Fermi constant (with $\tau\to\mu\nu\nu$ taking the role of $\mu\to
e\nu\nu$). In addition to the options discussed above, a $Z^{\prime}$ boson
coupling to muons and tau leptons can generate the desired effect via box
diagrams [138].

###  3.4 Charged current tauonic $B$ decays ($b\to c\ell\nu$)

Because this transition occurs at tree-level in the SM, also a tree-level NP
effect is necessary to obtain the needed effect of $O(10)$% w.r.t. the SM
(assuming heavy NP with perturbative couplings). Therefore, charged Higgses
[139, 140, 141], $W\textquoteright$ bosons [142] (with or without right-handed
neutrinos) or LQs [143, 123, 144, 145] are candidates. While there is a small
region in parameter space left that can account for $R(D^{(*)})$ with charged
Higgses [146, 147], LHC searches constrain $W^{\prime}$ solutions [142, 148],
leaving LQ as the probably best solutions (see Fig. 2 c)). However, also for
LQs constraints from $B_{s}-\bar{B}_{s}$ mixing, $B\to K^{(*)}\nu\nu$ and LHC
searches must be respected such that the $SU(2)_{L}$ singlet vector LQ [149,
150, 151, 152, 153, 154, 155] or the singlet-triplet model [156, 157, 158] are
particularly interesting.

###  3.5 Flavour changing neutral current semi-leptonic $B$ decays ($b\to
s\ell^{+}\ell^{-}$)

Because $R({K^{\left(*\right)}})$ requires dominantly lepton flavour universal
NP and $B_{s}\to\mu\mu$ constraints axial couplings to leptons. Such a NP
effect at the required level of $O(20\%)$ (w.r.t. the SM) can be most
naturally obtained via [159]: 1) A $Z\textquoteright$ boson with lepton
flavour universal but flavour-violating couplings to bottom and strange quarks
[160, 161] (see Fig. 2 d)). However, due to the bounds from
$B_{s}-\bar{B}_{s}$ mixing [162], the LHC (see e.g. [163]), and LEP [164] a
full explanation requires some tuning in $B_{s}-\bar{B}_{s}$ mixing by a
right-handed $sb$ coupling [165] or a cancellation with Higgs contributions
[166]. Furthermore, $K^{0}-\bar{K}^{0}$ and $D^{0}-\bar{D}^{0}$ mixing require
an approximate global $U(2)$ flavour symmetry [167]. 2) $\tau$ or charm loop
effect via an off-shell photon penguin [168]. The LQ representations which can
give such a tau loop are $S_{2}$ LQ [169], the $U_{1}$ LQ [170] or the
combination of $S_{1}+S_{3}$ [157]. The 2HDM with generic flavour structure
[171] can generate the desired effect via a charm loop $C_{9}^{U}$ [172, 173].
Alternatively, a DQ solution is possible [174].

###  3.6 Asymmetries in $Z$ decays ($Z\to b\bar{b}$)

An explanation of $Z\to b\bar{b}$ requires vector-like quarks mixing with the
SM ones at tree-level because the necessary NP effect is sizable [175, 176,
177] (see Fig. 2 e)). $A_{e}$ could be explained by vector-like leptons or
$Z-Z^{\prime}$ mixing.

###  3.7 $W$ boson mass ($m_{W}$)

The tension in the $W$ mass is most easily explained by a tree-level effect,
e.g. an $SU(2)_{L}$ scalar triplet that acquires a vacuum expectation value
[178] or via $Z-Z^{\prime}$ mixing in case $Z^{\prime}$ is an $SU(2)_{L}$
singlet [179]. However, loop effects of new particles with masses below or at
the TeV scale are possible as well [180], such as VLQs [181] or LQs [182].

Figure 4: Summary of the anomalies together with the implications for
extending the SM with new particles: Leptoquarks (LQ), vector-like leptons
(VLL), vector-like quarks (VLL) electrically neutral scalars ($S$), neural
gauge bosons ($Z^{\prime}$), charged gauge bosons ($W^{\prime}$) and diquarks
(DQ).

###  3.8 Multi-lepton anomalies ($e\mu(+b)$)

The multi-lepton anomalies can be explained by the associated production of
new scalars (i.e. via the decay of a heavier scalar into two lighter ones). In
particular, the deviations from the SM predictions in the differential
distributions of leptons in the measurements of $t\bar{t}$ decays can be
resolved by the production of a new neutral Higgs $H$ that decays into two
lighter ones $S$ and $S^{\prime}$ which subsequently decay to $W$ bosons and
$b$ quarks, respectively [65] (see Fig. 2 f)). This setup is preferred over
the SM hypothesis by more than $5\sigma$ and points towards $m_{S}\approx
150\,$GeV. Similarly, the excess in $h\to WW$ can be described by a new Higgs
boson decaying to $WW$ [61, 81] and the same sign lepton signals with $b$ jets
by the associated production of $H$ with top quarks where again $H\to
SS^{\prime}\to WWbb$ [62].

###  3.9 Higgs-like signals ($yy=\gamma\gamma,\tau\tau,WW,ZZ$)

These hints for resonances point towards the extension of the Higgs sector of
the SM because only scalars can decay to photons. For the $95\,$GeV excess, at
least an $SU(2)_{L}$ doublet [183], triplet [184] or even a more complex
scalar sector is needed [185]. If one aims at also addressing the di-Higgs
excess (i.e. $650\,{\rm GeV}\to b\bar{b}(90\,{\rm GeV})+\gamma\gamma(125\,{\rm
GeV})$), the resonant pair production of the SM Higgs and a new scalar is
required [186]. To account for the $152\,$GeV excess an even larger scalar
sector is necessary since the bulk of the signal is in associated production.
In fact, not only the most significant excesses are related to missing energy,
the $WW$ signal can also be explained for $m_{S}\approx 150\,$GeV, i.e. the
decay chain $pp\to H\to(S\to\gamma\gamma,WW)+(S^{\prime}\to$invisible)
describes data well.

###  3.10 (di-)di-jet resonances ($jj(-jj)$)

Here two options come to mind [86]: two scalar DQ (see Fig. 2 g)) or new
massive gluons seem to be the most plausible candidates [86]. Concerning the
latter, a specific example is based on an $SU(3)_{1}\times SU(3)_{2}\times
SU(3)_{3}$ gauge group, broken down to $SU(3)$ colour via two bi-triplets.

###  3.11 Non-resonant di-electrons ($q\bar{q}\\!\to\\!e^{+}e^{-}$)

As this analysis involves non-resonant electrons that do not originate from
the on-shell production of a new particle, NP must be heavier than the energy
scale of the LHC (or enter in the $t$-channel). This can be achieved with NP
at the 10 TeV scale with order one coupling to first-generation quarks and
electrons [136]. Therefore, $Z^{\prime}$ bosons [187] or LQs [128] (see Fig. 2
h)) have the potential to explain the CMS measurement.

###  3.12 Summary and connections

The anomalies discussed above, together with the extensions of the SM to which
they point, are shown in Fig. 4. One can see that many extensions point
towards new Higgs-like scalars. In particular, the agreement between the mass
of the scalar suggested by the multi-lepton anomalies and the $\gamma\gamma$
excess around $152\,$GeV is striking. LQ are also interesting candidates and,
in particular, allow for a combined and correlated explanation of $b\to
c\tau\nu$ and $b\to s\ell^{+}\ell^{-}$ via the tau-loop [188, 189]. Finally,
$Z\to b\bar{b}$, $m_{W}$ and the CAA could be explained by VLQ. Of course, in
a UV complete model, many more possible connections exist, as can be seen from
Fig. 4, offering interesting open research directions.

##  4 Comparison, conclusions and outlook

Let us now compare the anomalies concerning their experimental and theoretical
features:666Please note that even though we try to be objective here, the
impact of personal opinion is unavoidable.

  * •

Anomalous magnetic moment of the muon ($a_{\mu}$):
+ Precise and confirmed direct measurements
– Standard Model prediction plagued by hadronic uncertainties
– Tensions within hadronic light-by-light measurements and with lattice QCD
– Quite large NP effect needed; model building is challenging

  * •

Cabibbo Angle Anomaly (CAA):
+ Natural place to search for NP since only sub-permille effect w.r.t. the SM is needed
– Only one (competitive) measurement of $K\to\mu\nu$ available
– Beta decays need hadronic theory input to extract $V_{ud}$

  * •

Lepton flavour universality violating in tau decays ($\tau\to\mu\nu\nu$)
+ Precise SM prediction
+ Sensitive to small NP effects
– Challenging measurement
– Small statistical significance

  * •

Flavour changing neutral current $B$ decays ($b\to s\ell^{+}\ell^{-}$)
+ Many different observables measured
+ Consistent picture
+ Large significance
+ Possible connection to $b\to c\tau\nu$
– Sensitive to form factors and other hadronic input

  * •

Charged current tauonic $B$ decays ($b\to c\tau\nu$)
+ Measurements from different collaborations
+ Small theory uncertainty
+ Possible connection to $b\to s\ell^{+}\ell^{-}$
– Difficult measurement
– Limited significance
– Large effect needed; challenging model building

  * •

Asymmetries in $Z$ decays ($Z\to b\bar{b}$):
+ Precise SM prediction
– Limited significance

  * •

$W$ mass ($m_{W}$)
+ Theoretically clean
+ Statistically significant
+ Very sensitive to NP; many natural NP explanations
– Tensions among the measurements

  * •

Multi-lepton anomalies ($e\mu(+b)$):
+ Statistically very significant
+ Large multiplicity of signatures
+ Coherent picture
+ Consistent with the Higgs-like signals
– SM prediction often difficult
– Complex SM extension needed

  * •

Higgs-like signal ($yy$)
+ Statistically significant
+ Many different channels
+ Motivated by the multi-lepton anomalies
– Possible look-elsewhere effect

  * •

(di-)di-jet ($jj(-jj)$)
+ Agreement between different measurements
– Poor mass resolution
– Challenging theory explanation

  * •

Non-resonant di-electrons ($q\bar{q}\to e^{+}e^{-}$):
+ Agreement between ATLAS and CMS
+ Ratio theoretically clean
– Limited statistics
– Electrons are difficult LHC signatures

The anomalies are also compared in Table 1 w.r.t. several criteria which try
to answer the following questions:

  * •

Experimental signature: Is the experimental environment clean? Is the signal
well separated from background?

  * •

Experimental consistency: Do multiple independent measurements exist? Are they
in agreement with each other?

  * •

Standard Model prediction: How accurate and reliable is the SM prediction? Are
the conflicting results?

  * •

Statistical significance: How sizable are the deviations from the SM
predictions?

  * •

NP explanation: Are there models that can naturally account for the excess?
Are they in conflict with other observables?

  * •

Consistent connection: Are the connections to other anomalies via the same new
particle or model? How direct is this connection?

Here, + reflects a positive assessment, – a negative one and 0 means neutral,
i.e. positive and negative aspects compensate to a good approximation.

| $\begin{array}[]{c}{\rm{Exp}}{\rm{.}}\\\ {\rm{signature}}\end{array}$ | $\begin{array}[]{c}{\rm{Exp}}{\rm{.}}\\\ {\rm{consistency}}\end{array}$ | $\begin{array}[]{c}{\rm{SM}}\\\ {\rm{prediction}}\end{array}$ | $\begin{array}[]{c}{\rm{statistical}}\\\ {\rm{significance}}\end{array}$ | $\begin{array}[]{c}{\rm{NP}}\\\ {\rm{explanation}}\end{array}$ | $\begin{array}[]{c}{\rm{consistent}}\\\ {\rm{connection}}\end{array}$
---|---|---|---|---|---|---
$a_{\mu}$ | + | – | – | + | – | –
$\rm{CAA}$ | + | 0 | 0 | – | + | +
$\tau\to\mu\nu\nu$ | – | 0 | + | – | + | +
$b\to s\ell\ell$ | + | + | 0 | + | 0 | +
$b\to c\tau\nu$ | – | + | + | 0 | – | +
$Z\to b\bar{b}$ | + | 0 | + | – | 0 | 0
$m_{W}$ | 0 | – | + | + | + | +
$e\mu\left({+b}\right)$ | 0 | + | 0 | + | 0 | +
$yy$ | + | + | + | 0 | + | +
$jj({jj})$ | 0 | + | + | 0 | 0 | –
$pp\to ee$ | 0 | + | + | – | 0 | –
Table 1: Comparison of the different anomalies in particle physics in terms of
various features. See main text for details.

Finally, let us discuss the future implications of these anomalies.

  * •

$a_{\mu}$: While the experimental situation concerning the direct measurement
seems settled, there will be updates on HVP e.g. by Belle-II [190] and MUonE
[191] aims at an independent determination with a completely different method.
Furthermore, lattice QCD simulations will deliver improved results within the
next years.

  * •

CAA: Here improved measurements and theory calculations of beta decays will be
available within the next years [192]. Furthermore, NA62 could measure
$(K\to\mu\nu)/(K\to\pi\mu\nu)$ to asses the possibility of right-handed
currents [27] and the PIONEER experiment will measure pion beta decay [193] to
determine $V_{us}$, which is theoretically accurately predicted.

  * •

$\tau\to\mu\nu\nu$: Because a clear experimental environment is necessary,
this measurement can be done at the Belle-II [190] or at the FCC-ee [194] or
CEPC [195] using tau leptons from $Z$ decays.

  * •

$b\to c\ell\nu$: $R(D^{(*)})$ and related ratios can be measured at Belle-II,
by LHCb with run 3 data and the parked $B$ data from CMS [196].

  * •

$b\to s\ell^{+}\ell^{-}$: The best hope to solve the bottleneck concerning the
SM predictions are lattice calculations in combinations with other non-
perturbative methods like dispersion relations [197].

  * •

$m_{W}$: Since with increasing luminosity (like at the high-luminosity LHC)
the measurement of the $W$ mass at a hadron collider becomes even more
difficult, very precise results would be possible with a future electron-
positron collider like ILC [198], CLIC [199], FCC-ee or CEPC. However, also
LHCb could help to solve the puzzle since it does not use the full LHC
luminosity.

  * •

$e\mu(+b)$: Full next-to-next-to-leading order calculation [200] of all
processes including off-shell effects [201] will help to determine the SM
background. On the experimental side, already run 3 of the LHC should provide
enough events such that statistics is not the limiting factor anymore.

  * •

$yy$: Given the current strength of the excesses, LHC run 3, but at the latest
the high-luminosity LHC [202], should suffice to verify or falsify them.
However, to fully explore their properties, a Higgs factory would be
desirable.

  * •

$jj(-jj)$ & $q\bar{q}\\!\to\\!e^{+}e^{-}$: LHC run 3 should suffice to
determine the validity of these excesses.

Clearly, particle physics is currently a very exciting area of research. While
the SM has been consolidated over the last five decades, hints of new
particles and new interactions are emerging. Despite originating from very
different experiments and ranging over five orders of magnitude in energy, the
task is to find combined explanations to verify or falsify their predictions
in the future. However, one has to take into account that most likely not all
anomalies will be confirmed by ongoing and forthcoming experimental efforts.
However, already establishing one of these hints beyond a reasonable doubt
would lead particle physics into a new era, the BSM age.

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## Acknowledgements

The work of A.C. is supported by a professorship grant from the Swiss National
Science Foundation (No. PP00P2_211002). B.M. gratefully acknowledges the South
African Department of Science and Innovation through the SA-CERN program, the
National Research Foundation, and the Research Office of the University of the
Witwatersrand for various forms of support.

## Author contributions

The main part of the writing was done by A.C. while B.M. contributed to the
part concerning the multi-lepton anomalies, the di-di-jet excesses and the
Higgs-like signals.


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