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refname refsname disposition

# Topportunities at the LHC:
Rare Top Decays with Light Singlets

Henning<EMAIL_ADDRESS>Seth<EMAIL_ADDRESS>Lian-Tao
<EMAIL_ADDRESS>
###### Abstract

The discovery of the top quark, the most massive elementary particle yet
known, has given us a distinct window into investigating the physics of the
Standard Model and Beyond. With a plethora of top quarks to be produced in the
High Luminosity era of the LHC, the exploration of its rare decays holds great
promise in revealing potential new physics phenomena. We consider higher-
dimensional operators contributing to top decays in the SMEFT and its
extension by a light singlet species of spin 0, 1/2, or 1, and exhibit that
the HL-LHC may observe many exotic top decays in a variety of channels. Light
singlets which primarily talk to the SM through such a top interaction may
also lead to distinctive long-lived particle signals. Searching for such long-
lived particles in top-quark decays has the additional advantage that the SM
decay of the other top quark in the same event provides a natural trigger.

###### Contents

  1. 1 Introduction
  2. 2 Operators of interest
  3. 3 Top-quark decay channels
  4. 4 Light BSM singlet particles
    1. 4.1 Signals from the decays of BSM singlet particles
    2. 4.2 Complementary constraints
  5. 5 Conclusion

##  1 Introduction

As we look forward to the continued success of the Large Hadron Collider
project and its transformation into the High Luminosity era, we must adapt to
the evolving strengths of the program. With the enormous number of proton
collisions to be observed, we can search for new physics hiding in extremely
rare processes—so long as their signatures are unique enough. In this work we
focus on the possibilities for discovering new physics in rare top quark
decays, as we anticipate the projected $\sim 10^{10}$ top quarks to be
produced through Standard Model processes at the HL-LHC.

Assuming new physics particles mediating the top rare decay are heavy, we
parameterize the interactions relevant for these processes by higher-
dimensional operators. We consider the operators with only SM particles as
well as those extended by a single new, light singlet field of various spins,
and systematically examine which higher-dimensional operators lead to top
decays at observable rates. Having new physics appear in association with the
top quark is a feature of familiar approaches to the hierarchy problem, as new
degrees of freedom are introduced to cancel the top quark’s contribution to
the Higgs mass, and there is no small Yukawa to suppress their interactions
with the SM fields. On the other hand, light BSM singlets feature in many of
the most well-studied extensions of the SM, from axion-like particles to
sterile neutrinos to dark photons. Then on general grounds we are motivated to
search for all the possible ways by which light singlets may appear in
association with top quarks, for which an effective field theory construction
of operators is well-suited. We consider especially those cases in which the
new light species may be long-lived, which can lead to distinct signals, and
we will examine the extent to which this possibility is consistent with their
coupling to the SM in top quark decay operators. Whenever one adds new light
states to the SM there are other constraints to be confronted, and we also
discuss qualitatively the complementary constraints from low-energy probes of
flavor physics.

Existing experimental searches for rare top-quark decays concentrate on
flavor-changing-neutral-current (FCNC) decays to SM final states. These
include searches for top-quark decays into a light quark and a $Z$ boson [1,
2, 3, 4, 5, 6], a photon [7], a Higgs boson [8, 9, 10, 11, 12, 13, 14, 15], or
two leptons [16]. For other work on the prospects of rare top-quark decays to
SM states, we refer to e.g. [17, 18, 19, 20, 21, 22]. In addition, also a few
searches for top-quark decays into a light BSM particle have been performed.
This includes searches for top quarks decaying into a charged Higgs boson (see
e.g. the recent ATLAS:2018gfm,CMS:2019bfg,CMS:2022jqc,ATLAS:2023bzb) as well
as a neutral scalar [27]. We mention also some recent phenomenological
discussions of top decays which include light BSM (pseudo)scalars (e.g
Carmona:2022jid,Banerjee:2018fsx,Bhattacharyya:2022umc), fermions (e.g. [31]),
or vectors (e.g. [32, 33]). Notably, the direct constraints on the width of
the top quark [34] are uncertain to $\mathcal{O}(1)$ GeV. Hence, the search
for exclusive final states remains the most sensitive probe for the top-quark
rare decays.

We consider especially those cases in which the new light species may be long-
lived, which can lead to distinct signals, and we will examine the extent to
which this possibility is consistent with their coupling to the SM in top-
quark decay operators. Whenever one adds new light states to the SM there are
other constraints to be confronted, and we also discuss qualitatively the
complementary constraints from low-energy probes of flavor physics. The
prospect of long-lived BSM particles has garnered increased attention in
recent years [35] due to their distinctive experimental signatures and
resulting low backgrounds [36, 37, 38], and will be an important component of
collider searches moving forward [39, 40, 41, 42, 43, 44]. In comparison with
some of the commonly studied long-lived particle (LLP) production benchmarks,
such as through the Higgs portal, the top quark to LLP processes studied here
have a distinct advantage that the SM decay of the other top quark in the same
event provides a natural trigger.

This article is organized as follows. In Section 2, we organize the pertinent
top quark decay operators in the SMEFT extended by light singlets. In Section
3 we discuss the top quark decay channels induced by these operators and give
the rates. In Section 4 we consider the possible signatures of the outgoing
light singlets as well as the constraints from complementary flavor physics
searches. We conclude in Section 5.

##  2 Operators of interest

To evince the worthwhile phenomenological opportunities present in studying
top decays at HL-LHC, we study the different types of operators through which
the top quark might decay. It is our interest to consider a representative
array of operators leading to different signatures. Therefore, we include
representative operators for every operator class but leave out operators for
which we expect a similar collider phenomenology (for example in the class of
four-fermion operators, we only include the scalar current operators but leave
out the vector current operators). Then it is natural to orient ourselves
within the SM effective field theory (SMEFT) (see e.g. Grzadkowski:2010es) and
organize our thinking around an expansion in powers of
$\Lambda_{\text{NP}}^{-1}$, the scale of new physics. However, our EFT of the
light fields will contain not only the SM but additional possible singlet
fields of spin zero, half, and one. These are a new light scalar $S$, a right-
handed neutrino $N$, or a vector $Z^{\prime\mu}$. Recently, Song:2023jqm has
cataloged operators involving these new fields, so provides a useful check on
our considerations. Another possibility to introduce light BSM particles,
which we do not consider here, is to add a second doublet to the SM Higgs
sector (i.e., in the framework of Two-Higgs doublet models).

SM dim 6
---
$\mathcal{O}^{6}_{quqd}$ | $\left(\bar{Q}_{Li}u_{Rj}\right)\left(\bar{Q}_{Lk}d_{Rl}\right)$
$\mathcal{O}^{6}_{qule}$ | $\left(\bar{Q}_{Li}u_{Rj}\right)\left(\bar{L}_{Lk}e_{Rl}\right)$
$\mathcal{O}^{6}_{\Phi\Phi qu\Phi}$ | $(\Phi^{\dagger}\Phi)(\bar{Q}_{Li}u_{Rj}\tilde{\Phi})$
$\mathcal{O}^{6}_{\Phi D\Phi qq}$ | $(\Phi^{\dagger}i\overset{{}_{\leftrightarrow}}{D}_{\mu}\Phi)(\bar{Q}_{Li}\gamma^{\mu}Q_{Lj})$
$\mathcal{O}^{6}_{qu\Phi G}$ | $(\bar{Q}_{Li}\sigma^{\mu\nu}\tau^{A}u_{Rj})\widetilde{\Phi}G_{\mu\nu}^{A}$
$\mathcal{O}^{6}_{qu\Phi W}$ | $(\bar{Q}_{Li}\sigma^{\mu\nu}\tau^{I}u_{Rj})\widetilde{\Phi}W_{\mu\nu}^{I}$
$\mathcal{O}^{6}_{qu\Phi B}$ | $(\bar{Q}_{Li}\sigma^{\mu\nu}u_{Rj})\widetilde{\Phi}B_{\mu\nu}$

BSM dim 5
---
$\mathcal{O}^{5}_{SqDq}$ | $S(\bar{Q}_{Li}\not{D}Q_{Lj})$
$\mathcal{O}^{5}_{SQu\Phi}$ | $S(\bar{Q}_{Li}u_{Rj}\widetilde{\Phi})$
BSM dim 6
$\mathcal{O}^{6}_{qdlN}$ | $\left(\bar{Q}_{Li}d_{Rj}\right)\left(\bar{L}_{Lk}N\right)$
$\mathcal{O}^{6}_{qulN}$ | $\left(\bar{Q}_{Li}u_{Rj}\right)\left(\bar{N}L_{Lk}\right)$
$\mathcal{O}^{6}_{dueN}$ | $\left(\bar{e}_{Rj}^{c}u_{Rj}\right)\left(\bar{d}_{Rk}N\right)$
$\mathcal{O}^{6}_{qqNN}$ | $\left(\bar{Q}_{Li}\gamma_{\mu}Q_{Lj}\right)\left(\bar{N}\gamma^{\mu}N\right)$
$\mathcal{O}^{6}_{SSqDq}$ | $S^{2}(\bar{Q}_{Li}\not{D}Q_{Lj})$
$\mathcal{O}^{6}_{SSqu\Phi}$ | $S^{2}(\bar{Q}_{Li}u_{Rj}\widetilde{\Phi})$
$\mathcal{O}^{6}_{qu\Phi Z^{\prime}}$ | $(\bar{Q}_{Li}\sigma^{\mu\nu}u_{Rj})\widetilde{\Phi}F^{\prime}_{\mu\nu}$

Table 1: Overview of operators inducing top-quark decays into SM particles
(left) and BSM particles (right). $Q_{L}$ is a Dirac spinor containing the
left-handed quark doublet. The right-handed quarks are denoted by $u_{R}$ and
$d_{R}$, respectively. $L_{L}$ and $e_{R}$ are the left- and right-handed
lepton doublets/singlets, respectively. $i$, $j$, $k$, and $l$ are the quark
and lepton generation indices. Other indices are suppressed. The Higgs doublet
is called $\Phi$. $D$ denotes the SM covariant derivative, and
$\not{D}\equiv\gamma^{\mu}D_{\mu}$. The SM vector-boson field strengths are
called $G$, $W$, and $B$. The BSM singlets are called $S$ (scalar) and $N$
(Dirac fermion). The $Z^{\prime}$ field strength is denoted by $F^{\prime}$.
The superscript “$c$” denotes charge conjugation.

We will concern ourselves with solely the lowest-dimensional operators up to
dimension 6, which will suffice for us to find top quark decay channels
involving either solely SM states or additionally any of our new light
singlets. We list these operators in Table 1, separating out those which
involve solely SM particles and those which include a new light state. With
only the SM, new operators through which the top quark might decay begin in
SMEFT at dimension six. These operators can be divided into different
categories: four-fermion operators, Yukawa-like operators, current
interactions, and dipole operators. Our aim is to study a representative
sample, rather than a comprehensive one, so we choose operators exhibiting an
array of final state signatures. But there are many choices for possible
tensor structures in the four-fermion operators especially, as well as another
possibility for the Higgs-quark current-current interaction
$\mathcal{O}^{6}_{\Phi D\Phi qq}$ (see e.g. Grzadkowski:2010es). We leave a
detailed study comparing these possibilities to future work and here content
ourselves to study examples of the possible final states.

In the case where we add a BSM singlet scalar $S$, we can write down new
operators at dimension five, which consist of a SM marginal operator with a
scalar added on. There are multiple such operators containing the top quark,
but they are related by the quark equations of motion. We non-redundantly
choose one Yukawa-like and one kinetic-like operator in which the top decay
amplitudes arise directly. This will allow us to easily exhibit the different
effects these operators may have in the phenomenology of the $S$ itself in
Section 4. Similar operators appear at dimension six involving two scalar
singlets, such that these operators do not destabilize $S$. The fermion
singlet $N$ can enter into four-fermi operators like those in the SMEFT, and
we consider examples involving one or two fermion singlets, but again do not
comprehensively study all the possibilities. The BSM vector boson $Z^{\prime}$
can be used to construct an additional dimension-six dipole operator.111We do
not consider the case where the top quark is directly charged under a flavored
$Z^{\prime}$ gauge symmetry, which would allow new marginal operators
contributing to exotic top decays e.g. $\bar{u}_{R3}\not{Z^{\prime}}u_{R2}$.
Since such gauge symmetries are not respected by the SM Yukawas, they must be
broken far above the electroweak scale in order to generate the Yukawas, and
then the $Z^{\prime}$ will not be accessible in top decays. It would be
interesting to understand precisely whether there is any room for observable
$\bar{t}\to Z^{\prime}\bar{c}$ at small gauge coupling where one could
conceivably avoid the flavor constraints, but this requires more detailed
flavor analysis than we pursue.

We will find branching ratios which result in thousands to millions of exotic
top decays at the HL-LHC. So it is conceivable that even further suppressed
decays could still result in many rare top decays with potentially visible
signatures. Some amplitudes of top decays to SM final states through operators
up to dimension 13 have recently been studied [47]. An interesting line of
further pursuit would be to consider models introducing new light SM singlets
where the leading effects on top quarks are in even higher-dimensional
operators—perhaps the case of an additional discrete symmetry requiring some
number $n$ of light singlets as decay products. We leave such efforts on
maximizing the rare top decay discovery potential to future work.

##  3 Top-quark decay channels

Based on the operators presented in Section 2, we now discuss the various top-
quark decay channels facilitated by these operators. We calculate the
corresponding decay widths using MadGraph5_aMC@NLO (version 3.4.2) [48, 49]
with the necessary UFO model file [50, 51] generated using FeynRules [52, 53].
For calculating the expected number of events at the HL-LHC, we employ the SM
prediction for the $t\bar{t}$ cross section at $13.6\;\text{TeV}$ of
$923.6\;\text{pb}$222This value has been calculated at the next-to-next-to-
leading order in perturbative QCD, including soft-gluon resummation at the
next-to-next-to-leading logarithmic order, and using a top-quark mass of
$172.5\;\text{GeV}$ (see Czakon:2011xx and references therein). and a
luminosity value of $3000\;\text{fb}^{-1}$.

Figure 1: Branching ratios (lower axis) and expected number of events at HL-
LHC (upper axis) for the various top-quark decay channels induced by operators
listed on the left-hand side. The results for SM final states are shown in
green colors at the upper side of the plot; the results for BSM final states,
in red colors at the bottom. The numerical results have been derived setting
$\Lambda_{\text{NP}}=1\;\text{TeV}$ and all Wilson coefficients equal to one.
Moreover, all masses of BSM particles have been set to $10\;\text{GeV}$. The
black dotted lines indicate existing collider constraints.

An overview of the branching ratios for the various decay channels is
presented in Fig. 1. For this Figure, we set
$\Lambda_{\text{NP}}=1\;\text{TeV}$. All Wilson coefficients are fixed to one
($c=1$). Regarding the flavour structure of the operators, we choose to set
the coupling involving the top and the first-generation quarks to a non-zero
value, while the coupling involving the second-generation quarks is fixed to
zero. Consequently, the quark in the final state is a first-generation quark
(similar results are expected for second-generation quarks in the final
state). For final state photons, we place a lower transverse momentum cut of
$20\;\text{GeV}$ to avoid infrared divergences.

In the upper part of Fig. 1, the branching ratios for the top-quark decays
into SM final states are shown. The four-fermion operators induce decays
either into three jets or into two leptons and a jet. The corresponding
branching ratios are of ${\cal O}(10^{-5})$ corresponding to ${\cal
O}(10^{4})$ events expected at HL-LHC. The various decay channels induced by
dipole operators ($t\to Wj,Zj,\gamma j,jj$) generally yield higher branching
ratios of ${\cal O}(10^{-2})$ corresponding to ${\cal O}(10^{7}-10^{8})$
events expected at HL-LHC. Most of these decay channels have already been
probed experimentally. The currently strongest experimental 95% confidence
level upper limits are

  * •

$\text{BR}(t\to Zu)<1.7\cdot 10^{-4}$ and $\text{BR}(t\to Zc)<2.4\cdot
10^{-4}$ [6],

  * •

$\text{BR}(t\to\gamma u)<1.2\cdot 10^{-5}$ and $\text{BR}(t\to\gamma
c)<4.5\cdot 10^{-5}$ [7],

  * •

$\text{BR}(t\to Hu)<6.9\cdot 10^{-4}$ and $\text{BR}(t\to Hc)<9.4\cdot
10^{-4}$ [15],

  * •

$\text{BR}(t\to ue\mu)\lesssim 10^{-7}$ and $\text{BR}(t\to ce\mu)<10^{-6}$
[16].

These limits are indicated in Fig. 1 in the form of dotted black lines. There
may additionally be indirect, low-energy constraints on these SMEFT operators
from flavor physics observables. We refrain from placing these in the above
figure, as these complementary data do not diminish the usefulness of direct
searches, but we refer to recent works with which such limits may be derived
e.g.
DeBlas:2019ehy,Aebischer:2018iyb,Straub:2018kue,GAMBITFlavourWorkgroup:2017dbx,EOSAuthors:2021xpv.

In the lower part of Fig. 1, the branching ratios for the top-quark decays
into one or more BSM particles (plus accompanying SM particle(s)) are shown
setting the BSM masses to $10\;\text{GeV}$. As is visible, large branching
ratios of ${\cal O}(10^{-2}-10^{-1})$ can be obtained for two-body final
states containing either $S$ or $Z^{\prime}$. The decays into a light scalar
and a jet are already constrained by a recent experimental search which
constrains $\text{BR}(t\to uS,cS)\cdot\text{BR}(S\to b\bar{b})\lesssim
10^{-4}-10^{-3}$ (for $20\;\text{GeV}<m_{S}<160\;\text{GeV}$) [27]. For
depicting this limit in the Fig. 1, we assume a $\text{BR}(S\to b\bar{b})=1$
and extrapolate the limit from $m_{S}=20\;\text{GeV}$ to $10\;\text{GeV}$.
This search can, however, be completely evaded if $S$ decays less-frequently
to bottom quarks. The operators $\mathcal{O}^{5}_{SqDq}$ and
$\mathcal{O}^{5}_{SQu\Phi}$ also induce further three-body decays with an
additional gauge boson in the final state. Their branching ratios are
significantly smaller making them phenomenologically less relevant (with the
potential exception of the $t\to S\gamma j$ channel, where the additional
photon could help with background separation). One additional final state with
one BSM particle is the $t\to N\ell j$ decay. This channel has a comparably
low branching ratio of ${\cal O}(10^{-6})$, which would, however, still result
in a sizeable number of events at the HL-LHC. The branching ratios with two
BSM particles in the final state are of similar size. Apart from the decay
channels shown in Fig. 1, a few other possibilities like $t\to Z^{\prime}Hj$
or $t\to SSZj$ exist which are, however, highly suppressed.

We also want to stress here again that the provided numbers are only a rough
indication of the potential size of these decay modes. The results are derived
for a fixed UV scale $\Lambda_{\text{NP}}$ (but can easily be rescaled). This
scale (and also the Wilson coefficients) could very well be lower or higher.

Figure 2: Decay widths for top-quark decays including BSM final states as a
function of the mass of the BSM particle. The results are shown setting
$\Lambda_{\text{NP}}=1\;\text{TeV}$ and all Wilson coefficients equal to one.

We study the mass dependence of the top quark decays into one or two BSM
particles in Fig. 2. The two-body decay widths are shown in the upper left
panel. As expected, the decay widths shrink if the mass of the BSM particle in
the final state is raised (by up to an order of magnitude in the considered
mass window). The dependence on the BSM mass is stronger for higher masses,
where the phase space is rapidly shrinking.

The same behaviour is visible for the three-body final states with one BSM
particle in the final state (see upper right and lower left panels of Fig. 2).
For final states with two BSM particles the drop of decay widths is even more
rapid (see lower right panel of Fig. 2).

As mentioned above, an additional possibility for rare top-quark decays can be
introduced by adding an additional scalar doublet to the SM Higgs sector. Such
a Two-Higgs-Doublet model can lead to top-quark decays into a bottom quark as
well as a charged Higgs boson $H^{\pm}$. Since this possibility has been
widely studied in the literature — discussing cases in which the charged Higgs
boson decays directly to SM particles or first into other BSM states (see e.g.
Akeroyd:2016ymd,Bahl:2021str,Cheung:2022ndq,Hu:2022gwd,Krab:2022lih,Bhatia:2022ugu,Kim:2023lxc,Mondal:2023wib,Fu:2023sng,Sanyal:2023pfs,Li:2023btx
for a review and recent works) — as well as probed experimentally (see e.g.
the recent ATLAS:2018gfm,CMS:2019bfg,CMS:2022jqc,ATLAS:2023bzb), we do not
consider it in this work.

##  4 Light BSM singlet particles

In this Section we discuss the phenomenology of the light singlets produced in
rare top decays.

###  4.1 Signals from the decays of BSM singlet particles

The phenomenology of light singlet particles produced at colliders will depend
sensitively on their lifetimes $\tau$. There are three different
possibilities:

  * •

Prompt decaying BSM particles — Prompt decays into SM particles, with a decay
length $c\tau\lesssim 1\,\text{mm}$, can be induced by introducing additional
operators with a large enough Wilson coefficient. While the inclusion of
multiple operators involving the new light singlet fields is in some sense
‘non-minimal’, one realistically expects that many operators are generated in
a UV completion barring some additional structure forbidding them. For
example, the $N$ may interact via the relevant ‘neutrino portal’ $y_{N}\Phi
L_{L}N$ leading to mixing with the SM neutrinos. The $S$ and $Z^{\prime}$
singlets could e.g. decay into bottom quarks, light quarks, leptons, or two
photons. This case has been studied in detail in
Banerjee:2018fsx,Bhattacharyya:2022ciw,Bhattacharyya:2022umc.

An additional, irreducible possibility for prompt decays into SM particles is
a decay involving the operator(s) responsible for the top-quark decay into the
light BSM particle. This case is studied in detail below.

  * •

Stable BSM particles — For the top-quark decay operators involving two BSM
particles, the respective BSM particle is stable if no additional operators
involving only one instance of the BSM field are introduced. From a UV
perspective, such a situation can be generated e.g. by imposing a
$\mathbb{Z}_{2}$ symmetry. In this case, the BSM particles would manifest
themselves in the form of a missing transverse energy signal.

  * •

Long-lived BSM particles — The BSM particles can also be long-lived
($1\,\text{mm}\lesssim c\tau$). The most striking signatures possible appear
when the decays are displaced from the interaction point and result in a rich
phenomenology. Such signatures typically have no SM backgrounds, and so the
threshold for discovery may be just a few events at the HL-LHC, making
extremely rare such processes nonetheless visible. One such signature has been
studied in Carmona:2022jid, investigating the decay of top quarks into long-
lived axion-like particles.

While this is the most interesting possibility, we are not necessarily free to
dictate it, as there may be an upper bound on the lifetime of the new state we
are considering induced by the top-quark decay operator through which it was
produced. We will calculate such upper bounds below. Additional decay channels
can of course always be opened by introducing additional operators with a
small enough Wilson coefficient (or a large enough scale suppression).

Although detector sizes are typically $\lesssim 10\;\text{m}$, for larger
lifetimes the rate of decay in the detector falls only linearly in $c\tau$ and
it is proportional to the volume of the detector component which is sensitive
to the signal, allowing ample detection prospects on the tail of the decay
distribution. The search for LLPs in rare top-quark decays has a distinct
advantage in comparison to some of the other production mechanisms. For
example, a well-studied benchmark is the production of LLPs from rare Higgs-
boson decay through the Higgs portal coupling. Typically, additional hard
radiation is needed to trigger on this kind of signal. However, in the case of
rare top-quark decay, the SM decay of the other top in the same event provides
a natural trigger without further suppressing the signal rate. Eventually,
these modes become too rare, and the signature in our colliders is simply
missing transverse energy mimicking the situation in which they are stable.
This limits the information one may uncover about the new states even if such
exotic top decays are observed.

Figure 3: Feynman diagrams for tree-level decays of the scalar singlet $S$
when the new physics interaction involves the $b$-quarks directly (left panel)
or when an off-shell $W$ appears in the intermediate diagram (middle panel).
Analogous tree-level decays can occur for $Z^{\prime}$. In the right panel, we
show a five-body tree-level decay of $N$ through an off-shell $W$. The red
vertex denotes the dimension five or six operators responsible for the
respective top-quark decay into a light BSM particle.

Figure 4: Feynman diagrams for loop-induced decays of the scalar singlet $S$
(left panel), the gauge boson $Z^{\prime}$ (middle panel), and the sterile
neutrino $N$ (right panel). The red vertex denotes the dimension five or six
operator responsible for the respective top-quark decay into a light BSM
particle.

We proceed with a calculation of the lower bounds on the decay width of the
BSM particles originating from diagrams involving the top-quark decay
operators. We consider three different possibilities:

  * •

The decay is induced at the tree level involving no off-shell particles. Such
a decay occurs for example for the $\mathcal{O}^{5}_{SqDq}$ operator, which
allows for $S\to bd$ or $S\to bs$ as the operator is constructed from the
left-handed quark doublets (see left diagram of Fig. 3).

  * •

The decay is induced at the loop level with an internal $W$ boson to exchange
the top with a lighter quark, as in Fig. 4.

  * •

The decay is induced at the tree level involving an off-shell top-quark and a
(potentially off-shell) $W$ boson, which further decays either hadronically or
leptonically (see middle and right diagrams of Fig. 3).

To calculate these decay rates, we employ MadGraph5_aMC@NLO for the tree-level
decays and the packages FeynArts [72, 73], FormCalc [74], and LoopTools [74]
for the loop-induced decays—having generated the necessary model file using
FeynRules.

Note that the appearance of a $W$ boson in the later two possibilities to
exchange the heavy top for the lighter bottom means that these decay widths
can be sensitive to the interplay of flavor with chirality in the operators.
For example, for an operator containing $\bar{Q}_{L1}u_{R3}$, the right-handed
top quark can be easily switched to left-handed by its large Yukawa, such that
a $W$ boson can easily be coupled to the top quark resulting in sizeable loop-
induced decay rates. But in the opposite case $\bar{Q}_{L3}u_{R1}$ it is very
costly to change the up-quark chirality, and the leading decay channel may
become the four-body tree-level decay with an off-shell $W$. These
considerations made little difference for the top-quark decay rates but can
have a large impact on the phenomenology of the light singlets.

Figure 5: Upper left: Mass dependence of the decay width of the scalar singlet
$S$ for operators coupling the top quark to first-generation quarks. Upper
right: Mass dependence of the various decay modes of the scalar singlet $S$
induced by the $\mathcal{O}^{5}_{Sq_{1}Dq_{3}}$ operator. Bottom: Mass
dependence of the decay width of the scalar singlet $S$ for operators coupling
the top quark to second-generation quarks. The results are shown setting
$\Lambda_{\text{NP}}=1\;\text{TeV}$ and all Wilson coefficients equal to one.

The numerical results for the decay width of the scalar singlet $S$ (and the
corresponding decay length) induced by operators coupling the top quark to
first-generation quarks are shown in the upper left panel of Fig. 5. The decay
width of $S$ strongly depends on the nature of the operator inducing the
decay(s) as well as on the mass of $S$.

Starting with the $\mathcal{O}^{5}_{Sq_{1}Dq_{3}}$ operator, the $S$ is
decaying promptly for $m_{S}>m_{b}$ and long-lived for $m_{S}<m_{b}$. The
contribution of the various decay channels is broken down in the upper right
panel of Fig. 5. The tree-level decay to $bd$ dominates in the region of
$m_{S}>m_{B}$. For $m_{S}<m_{B}$, the decay width is orders of magnitude
smaller and originates from loop-induced decays where the top quark is traded
for a first- or second-generation quark via the $W$ boson in the loop
(receiving a suppression by the off-diagonal entries of the CKM matrix). The
off-shell four-body decay into three light and one $b$ jet is only relevant
for high $m_{S}$, for which at least the intermediary $W$ boson can be on-
shell.

For the $\mathcal{O}^{5}_{Sq_{3}Dq_{1}}$ operator, the tree-level decays are
proportional to the down quark mass, which strongly suppresses it.333The
derivative acting on $q_{1}$ can be rewritten in terms of the mass of the
quark on which it is acting using the Dirac equation. For the same reason,
also the loop-induced and the four-body tree-level decays are strongly
suppressed.

For the Yukawa-type operators ($\mathcal{O}^{5}_{Sq_{1}u_{3}\Phi}$ and
$\mathcal{O}^{5}_{Sq_{3}u_{1}\Phi}$), there are no tree-level two-body decays
and the tree-level four-body decays give a sizeable rate only for large
$m_{S}$. For the $\mathcal{O}^{5}_{Sq_{3}u_{1}\Phi}$ operator, also the loop-
induced decay channels are suppressed due to the chirality structure of the
operator resulting in $S$ being long-lived for large parts of the considered
parameter space. For the $\mathcal{O}^{5}_{Sq_{1}u_{3}\Phi}$ operator, on the
other hand, the loop-induced decay channels are sizeable resulting in $S$
being long-lived only for $m_{S}\lesssim m_{b}$.

In the lower panel of Fig. 5, we look at the changes in the $S$ decay rate if
it couples the top quark to a second-generation quark. In comparison to first-
generation operators in the upper panels of Fig. 5, the main changes are
visible for the $\mathcal{O}^{5}_{Sq_{3}Dq_{2}}$ and
$\mathcal{O}^{5}_{Sq_{3}u_{2}\Phi}$ operators. For the
$\mathcal{O}^{5}_{Sq_{3}Dq_{2}}$ operator, both the tree-level and one-loop
two-body decays are significantly enhanced due to the strange-quark mass being
substantially larger than the down-quark mass. For the same reason, the loop-
level two-body decay induced by the $\mathcal{O}^{5}_{Sq_{3}u_{2}\Phi}$
operator is significantly enhanced. In general, the possibility of $S$ being
long-lived is more constrained if it couples the top quark to second-
generation quarks instead of first-generation quarks. This conclusion also
holds for the fermionic and vector singlets $N$ and $S$. Therefore, we in the
following only discuss the case in which they couple the top quark to first-
generation quarks.

Figure 6: Mass dependence of the decay widths of the gauge boson $Z^{\prime}$
(left panel), and the sterile neutrino $N$ (right panel) into SM particles.
The results are shown setting $\Lambda_{\text{NP}}=1\;\text{TeV}$ and all
Wilson coefficients equal to one.

In the case of the singlet vector $Z^{\prime}$ (see left panel of Fig. 6), the
nature of its decay depends strongly on the flavour structure of the dipole
operator. In the case of $\mathcal{O}^{6}_{q_{1}u_{3}Z^{\prime}}$, the two-
body loop-induced decay dominates for lower $Z^{\prime}$ masses whereas for
higher masses the tree-level four-body decay gives a sizeable contribution.
These decays are prompt whenever the $Z^{\prime}$ mass is sufficiently above
the bottom-quark mass. However, for small enough $Z^{\prime}$ masses such that
the only available channels are to light quarks, these decays have additional
CKM suppression which can make the $Z^{\prime}$ long-lived. In fact in the
benchmark point of Fig. 6 it transitions quickly to being so long-lived as to
be missing energy, but in detail, this shape will depend on the size of the
coupling. In the case of $\mathcal{O}^{6}_{q_{3}u_{1}Z^{\prime}}$, the loop-
induced decay is strongly suppressed due to the chirality structure of the
operator making $Z^{\prime}$ long-lived if $m_{Z^{\prime}}\lesssim
80\;\text{GeV}$. For higher masses, the tree-level four-body decay gives a
substantial contribution.

For the right-handed neutrino $N$, its decays are further suppressed by the
three-body phase space required with only the minimal four-fermion operators
we have added. This makes a fermion singlet a natural candidate for being
long-lived. The size of the $N$ decay width, however, depends sensitively on
the structure of the operator inducing the top-quark decay (see right panel of
Fig. 6). The $\mathcal{O}^{6}_{qdlN}$ and $\mathcal{O}^{6}_{q_{3}u_{1}lN}$
operators induce tree-level three-body decays resulting in prompt decays once
$m_{N}\gtrsim 10\;\text{GeV}$. In the case of the
$\mathcal{O}^{6}_{q_{1}u_{3}lN}$ and $\mathcal{O}^{6}_{dueN}$ operators no
tree-level three-body decays exist. Moreover, also the loop-induced three-body
decays are strongly suppressed due to the chirality structure of the
operators. This makes the five-body tree-level decays the leading decay
channels resulting in $N$ being long-lived throughout almost the complete
considered parameter region.

Figure 7: Lifetimes of the singlets $S$ (blue), $N$ (orange), and $Z^{\prime}$
(green) interacting through the indicated operators over a range of singlet
masses. The longest lifetimes are achieved for $m_{\text{BSM}}=1\;\text{GeV}$
at the upper end (in some cases they exceed the upper limit of the plot); the
hatched regions denote masses larger than $100\;\text{GeV}$. Lifetimes well
below the dashed line at $1\text{mm}$ produce prompt decays rather than
displaced ones.

This discussion clearly shows that the minimal decay width of the light
singlets depends strongly on their mass. In summary, the following collider
signatures arise in the absence of other non-top operators:

  * •

$m_{\text{BSM}}\lesssim m_{b}$ — If the singlet mass is below the bottom-quark
mass, the singlet is likely long-lived. For $S$ and $Z^{\prime}$, the
resulting collider signature is a pair of jets originating from a displaced
vertex. Similarly, for $N$, three jets originate from a displaced vertex.

  * •

$m_{\text{BSM}}>m_{b}$ — If the singlet mass is above the bottom-quark mass,
the singlet can be long-lived or decay promptly depending mainly on the
chirality structure of the involved operator. For $S$ and $Z^{\prime}$, the
collider signature is either a di-jet or a di-jet plus $W$ boson final state
originating potentially from a displaced vertex. For $N$, one additional jet
is present in the final state.

To provide an overview, we show in Fig. 7 the range of lifetimes for the
singlets $S$, $N$, and $Z^{\prime}$ for the various operators (leaving out
those operators which induce tree-level $B$ meson decays, see Section 4.2). As
discussed, the longest lifetimes are expected for the singlet fermion $N$.

While we have been focusing on the minimal cases, the signal for the LLP can
in principle be much richer, as the singlet could have other decay modes in
addition to those dictated by the operators involved in the rare top-quark
decays. This is particularly interesting in cases where the minimal decay
width of the new species is very small, as adding new decay channels with
comparable partial widths does not dramatically change the decay lifetime. For
example, light right-handed neutrinos $N$ minimally coupled to the top quark
may be so long-lived that they appear as missing energy at the LHC. But of
course, the $N$ may interact via the relevant ‘neutrino portal’ $y_{N}\Phi
L_{L}N$ or additional higher-dimensional operators
$(Q_{L1}\bar{d}_{R1})(L_{L}\bar{N})/\Lambda^{2}$ which reduce its lifetime and
provide distinctive long-lived $N\rightarrow e^{+}+\pi^{-}$ decays in the
tracker. For the scalar, a small coupling $SF^{\mu\nu}F_{\mu\nu}$ (or the
$\mathcal{CP}$-odd version) can lead to displaced photon signatures for which
search strategies have been developed [75, 76].

###  4.2 Complementary constraints

Our goal in this Section is to outline the relevant constraints arising from
low-energy probes of flavor physics. Our perspective is that the operators we
study for rare top decays will be one contribution to these flavor signatures,
but there may well be others, and it is possible in a full model there may be
cancellations in contributions to any particular observable. While we will
discuss the signatures of various operators in flavor observables, and indeed
it is possible for them to be quite constraining, none of this dampens the
usefulness of direct searches for the complementary rare top decay signatures.
Since we are here concerned with these effective operators in particular
rather than in constraining complete models of UV physics which include them,
we will only perform a qualitative analysis since a precise evaluation is
beyond the scope of the current work and would likely also depend on the
concrete UV scenario.

First, let us briefly address the preference of this new physics for top
quarks: From the point of view of the gauge quantum numbers of the Standard
Model particles, there is no reason for some new singlet to couple to some
particular flavors, or for that matter to prefer quarks to leptons. One might
then worry from this perspective whether our consideration of such ‘top-
philic’ operators comprises some severe theoretical fine-tuning. At the same
time, the top quark is special in the flavor structure of the Standard Model.
It is also expected to play a crucial role in the dynamics of electroweak
symmetry breaking. If the related new physics to address these questions is
also responsible for rare decays into singlets, a preference for the top quark
can then be understood. While we do not embed our effective operators in full
ultraviolet models, we can have in the back of our minds simple scenarios
where such a structure does come about. For example, at higher energies these
singlets may be charged under a flavored gauge symmetry that has been
spontaneously broken at energies above $\Lambda_{\text{NP}}$, such as gauged
horizontal baryon number $U(1)_{B_{1}-B_{3}}$ or third-generation baryon minus
lepton number $U(1)_{(B-L)_{3}}$. For further such considerations we refer to,
for example,
Alonso:2017uky,Babu:2017olk,Alonso:2017bff,Bordone:2017bld,Greljo:2018tuh,Fox:2018ldq,Elahi:2019drj,
among many such works.

Figure 8: Sample Feynman diagrams for (upper left) loop-induced decays of $B$
mesons to missing energy; (upper right) loop-induced mixing of $D$ mesons;
(lower left) tree-level mixing of $B_{s}$ mesons; (lower right) loop-level
decays of $Z$ bosons.

Given that our new physics is manifestly flavour-violating, there can be
strong constraints from the effects of flavor-changing neutral currents. This
is most pertinent for the scalar $S$ which has flavor-changing interactions at
dimension-five, so we discuss this possibility first. The possible effects
naturally divide into four cases depending upon the mass and couplings:

  * •

Sub-mesonic scalar — When the new singlet species is light enough (e.g., below
the $B$-meson mass scale $m_{B}$), it may be produced in meson decays [84].
Contributions to meson decays e.g. $B^{+}\rightarrow\pi^{+}+S$ can appear as
diagrammed in the upper left of Fig. 8. With $\mathcal{O}(1)$ coupling, the
decay width can be far larger than in the SM where it proceeds through an off-
shell $W$ into a pair of fermions and is suppressed by phase space and an off-
diagonal CKM. In fact, such exotic decays are probed by searches for decays
such as $B^{\pm}\rightarrow\pi^{\pm}\nu\bar{\nu}$, which are constrained now
down to a branching ratio below $\sim 10^{-5}$, and similarly for decays to
kaons [85]. Resultingly, extremely light singlet bosons with flavor-changing
dimension-5 interactions must have quite suppressed couplings, though in
detail the constraint will depend on which operator is turned on along the
lines of the $S$ lifetime discussion. If the bottom coupling is present
already at tree-level, as with $(c/\Lambda_{NP})S(\bar{Q}_{Li}\not{D}Q_{Lj})$,
this gives a strong constraint which is roughly
$c^{2}(1\;\text{TeV}/\Lambda_{NP})^{2}\lesssim 10^{-11}$. If instead there is
a coupling to light right-handed quarks, as for
$S(\bar{Q}_{L3}u_{R1}\tilde{\Phi})$, a small Yukawa coupling is needed to flip
the chirality and allow a $W$ loop. In which case the dominant decay may
instead be at tree-level through an off-shell $W$, for example,
$B^{0}\rightarrow S+\pi^{-}+e^{+}+\nu_{e}$, which would appear as extra
contributions to the SM $\text{Br}(B^{0}\rightarrow\pi^{-}+e^{+}+\nu_{e})$
which has been measured at $\sim 10^{-4}$ [86]. 444We note that the
application of these constraints does assume the $S$ is invisible, and so one
could imagine adding further operators allowing even a light $S$ to decay to
mesons or charged leptons and hide amongst the uncertainties in those decay
modes.

  * •

Flavor-anarchic scalar — If the new light species couples to all three
generations (that is, if both $t$–$u$–$S$ and $t$–$c$–$S$ couplings exist),
then a loop of top quarks and the $S$ can induce oscillations among flavored
neutral mesons, such as in the upper right diagram of Fig. 8. Such effects in
the SM appear at one loop and are suppressed by a small CKM angle
$V_{\text{small}}$ and the electroweak scale. These mixing phenomena have been
measured (e.g. LHCb:2021ykz) and agree with the SM predictions, so at the very
least our new contributions must be subdominant. The effect appears here with
the same diagram topology, so the scale of our operator needs only be mildly
higher to overcome to the CKM suppression, very roughly $\Lambda_{NP}^{4}\gg
v_{EW}^{4}/V_{\text{small}}^{2}$. Since the top quark branching ratio scales
as $\Lambda_{\text{NP}}^{2}$, this is not too significant a constraint on
exotic decays into $S$.

  * •

Non-chiral scalar — If the scalar couples both to up-type and down-type
quarks, as with $\mathcal{O}^{5}_{SqDq}$, then tree-level exchanges can lead
to $B^{0}$ or $B_{s}^{0}$ oscillations, as in the lower left diagram of Fig.
8. The SM predictions for these phenomena are in good agreement with the data
(see e.g. LHCb:2016gsk). Here the new effect appears at tree-level, so is
significantly enhanced, and the requirement of being smaller than the SM rates
is roughly $\Lambda_{\text{NP}}^{2}\gg
v_{\text{EW}}^{4}/(m_{B}^{2}V_{\text{small}}^{2})$, which is far more severe a
constraint on the rate of exotic top quark decays. These effects do not appear
if we consider instead the operator $\mathcal{O}^{5}_{SQu\Phi}$ without the
analogous down-quark Yukawa-like operator, as can be enforced by $S$ being
charged under a Peccei-Quinn symmetry.

  * •

Flavorful, up-philic scalar — If the new physics couples the third generation
to only one of the light generations, and only to the up-type quarks, the
prior effects are absent. Then the most relevant effects arise from closing
the top-quark loop with a $Z$ boson coupling, and the resulting constraint is
from affecting electroweak precision. For example, at one loop there is a new
contribution to the width of the $Z$ boson shown in the lower right diagram of
Fig. 8. While the $Z$ boson width has been measured quite precisely (see e.g.
ALEPH:2005ab), this contribution is easily reduced to below the level of the
many SM 1-loop diagrams.

In the case where $S$ is endowed with a $\mathbb{Z}_{2}$ symmetry such that it
couples only in pairs, the same diagrams appear but with an extra scalar leg.
This leads to additional $\Lambda_{\text{NP}}^{-2}$ suppression from having a
higher-dimensional operator, as well as additional loop suppression from
closing the extra line. The constraints in this case are thus much weaker.

Similarly for a singlet vector $Z^{\prime}$, the same diagrams are present as
in the case of a single $S$, so approximately similar considerations apply.
Since the leading operator is at dimension six, there is additional
suppression from each BSM vertex of $m_{t}/\Lambda_{\text{NP}}$ or
$m_{Z^{\prime}}/\Lambda_{\text{NP}}$, which softens the constraints relative
to the scalar case. Again the least stringently constrained option is for a
massive $Z^{\prime}$ which couples solely to up-type quarks and only to one of
the light generations.

Figure 9: Sample Feynman diagrams for complementary effects of top decay
operators with a singlet fermion $N$, including (upper left) leptonic decays
of $B$ mesons; (upper right) decays of $D$ mesons; (lower left) decays of
pions; (lower right) semi-visible decays of $Z$ bosons.

The singlet fermion $N$ can lead to a similar variety of flavor-changing
phenomena, though here the primary effects are in leptonic decays rather than
in oscillations. The lowest operators are again at dimension six, making them
relatively safe compared to the impact of a light scalar singlet. For light-
enough $m_{N}\ll m_{b}$, one may have $W$-loop decays of $B$-mesons to $e+N$
or fully invisible $\nu+N$, as in the upper left diagram of Fig. 9. Such modes
have been searched for and are constrained at the level of
$\mathcal{O}(10^{-6})$ [90] in the branching ratio. Since the SM width is
suppressed by phase space and CKM, this implies a tight constraint on such
light fermion couplings. Similarly to the scalar considerations, if the
operator is the fully right-handed $\mathcal{O}^{6}_{dueN}$, then instead the
dominant induced decay mode may be at tree-level with an off-shell $W$, as
$B^{0}\rightarrow\pi^{-}+e^{+}+N$. As mentioned above, the SM
$\text{Br}(B^{0}\rightarrow\pi^{-}+e^{+}+\nu_{e})$ has been measured at $\sim
10^{-4}$. Moving to heavier species, for fully flavored couplings, a loop of
$t$ and $N$ leads to new leptonic $D$-meson decays, as in the upper right
diagram of Fig. 9). These are tightly constrained to a branching ratio of
$\lesssim 10^{-7}$ [91], though the new physics effect here scaling as
$\Lambda_{\text{NP}}^{-8}$ makes this constraint not so severe. Even with only
one flavor of coupling there are still decays of quarkonia to
$\ell^{+}\ell^{-}$ (see lower left diagram of Fig. 9). Especially dangerous
would be couplings to the first generation which give extra leptonic decay of
pions, but even $J/\psi\rightarrow e^{+}e^{-}$ is measured quite precisely
[92]. Again the faster scaling of this process with $\Lambda_{\text{NP}}$
compared to the top decays makes this less severe, but it is still an
important consideration. The additional contribution to the $Z$ boson width is
in this case further suppressed either by phase space (as in the lower right
diagram of Fig. 9) or another loop.

##  5 Conclusion

The Large Hadron Collider, having achieved in its infancy huge success as a
discovery machine, is maturing into a precision machine for its next decades
of cutting-edge exploration into the structure of matter. Rare decays of top
quarks offer an exciting window into potential deviations from Standard Model
physics, either with flavor-changing effects or in the observation of new
light singlets.

To set up the framework of our study, we parameterized the interaction
relevant to the rare decay of the top quark by a set of EFT operators,
including all pertinent ones up to dimension six. We then described the decay
final states, including the ones with only SM particles, and those including
an additional singlet. In both of these cases, there are channels with a large
number of signal events at the HL-LHC, leading to great discovery potential.

We then focused on the case of light singlet scalars, fermions, or vector
bosons. To get an overview of the possible experimental signatures, we
distinguished three different cases: promptly decaying singlets, long-lived
singlets, and stable singlets. We discussed under which conditions each of
these possibilities can be realized. Depending on the concrete operator
inducing the exotic top-quark decay, the light singlets can be long-lived
resulting in interesting experimental signatures. We also qualitatively
discussed complementary constraints from flavor physics and electroweak
precision measurements.

In this initial work, our intent has been to point to these opportunities in
general, and there are many avenues for future work. As highlighted above, the
large number of rare top decays possible with dimension-six operators raises
the prospect of even higher-dimensional operators still being usefully probed.
It would be useful to understand more precisely all the possible final state
signatures, and to ensure that experimental searches exist for all the useful
cases. Finally, it would also be interesting to make further contact with
particular ultraviolet models where one could understand in more detail the
effects of complementary constraints from flavor physics and other possible
correlated signatures.

## Acknowledgments

HB acknowledges support from the Alexander von Humboldt foundation. SK is
supported by an Oehme Postdoctoral Fellowship from the Enrico Fermi Institute.
LTW is supported by the DOE grant DE-SC0013642. The Feynman diagrams shown in
this article have been generated using FeynGame [93].

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