# Topological Portal to the Dark Sector

Joe Davighi<EMAIL_ADDRESS>Theoretical Physics Department, CERN, 1211
Geneva 23, Switzerland  Admir Greljo<EMAIL_ADDRESS>Department of
Physics, University of Basel, Klingelbergstrasse 82, CH 4056 Basel,
Switzerland  Nudžeim Selimović<EMAIL_ADDRESS>Istituto
Nazionale di Fisica Nucleare, Sezione di Padova, Via Francesco Marzolo 8,
35131 PD, Italy

###### Abstract

We propose a topological portal between quantum chromodynamics (QCD) and a
dark QCD-like sector, present only for a unique coset structure after
confinement, which connects three QCD to two dark pions. When gauged, it
serves as the leading portal between the two sectors, providing an elegant
self-consistent scenario of light thermal inelastic dark matter. The inherent
antisymmetrization leads to diminished annihilations at later times and
suppressed direct detection. However, novel collider signatures offer
tremendous prospects for discovery at Belle II.

##  I Introduction

The quest to unravel the mysteries of dark matter (DM) stands as a pivotal
challenge in contemporary physics. That DM is composed of a hidden particle
physics sector is a well-motivated hypothesis ParticleDataGroup:2022pth .
While DM has primarily revealed its presence through gravitational
interactions, the existence of other portals may not only be feasible but
perhaps essential in elucidating aspects of cosmological evolution and
accounting for the observed relic abundance Gondolo:1990dk .

In this letter, we propose a novel portal, generating a link between quantum
chromodynamics (QCD) and a dark QCD-like theory. Our starting point is to seek
a (hitherto overlooked) non-trivial topological interaction between the two
theories in their confined phases. After a comprehensive and rigorous
topological analysis of the possible global symmetries characterizing QCD-like
dark sectors, we discover that there is a unique coset structure featuring a
non-trivial Wess–Zumino–Witten (WZW) term Wess:1971yu ; Witten:1983tw  which
connects three QCD pions to two dark pions.

Having established the existence and uniqueness of this topological portal
operator, we delve into the resulting phenomenology. The topological portal
enables dark number conservation through a $\mathbb{Z}_{2}$ symmetry, ensuring
the stability of the lightest dark pion. Furthermore, after quantum
electrodynamics (QED) is properly included, the portal operator yields 4-point
interactions involving one photon, one neutral QCD pion ($\pi^{0}$ or $\eta$),
and two dark pions. The chiral Lagrangian power counting identifies these
(appropriately gauged) WZW terms as the leading portal between the two
sectors, which thus defines the dominant phenomenology. Prompted by this, we
investigate whether such an operator can establish the correct relic
abundance. Intriguingly, we uncover a self-consistent scenario whereby DM
achieves thermal freeze-out at temperatures below the QCD phase transition.
This phenomenon is primarily governed by late-stage interactions, captured
correctly by the topological operator, hinting at the presence of DM in the
GeV range Zurek:2024qfm .

The topological operator, being a differential form and therefore completely
antisymmetric under exchanging pairs of fields, necessarily couples two
different dark pion species. After freeze-out governed by $\chi_{1}\chi_{2}$
coannihilations, the heavier dark pion rapidly decays, leaving the lighter as
the residual DM. This elegantly leads to diminished DM annihilation into SM
particles at later times for the topological portal, skirting otherwise
stringent annihilation constraints from CMB anisotropies on elastic $s$-wave
scattering Galli:2011rz ; Planck:2018vyg . In addition, direct detection
experiments lack sensitivity due to the kinematic suppression of inelastic
scatterings. Thus, we stumble upon an elegant and natural realization of the
light thermal inelastic DM scenario Tucker-Smith:2001myb ; Izaguirre:2015zva .

The key to probing this new portal lies in collider experiments. Novel and
previously unexplored collider signatures are anticipated in current
$e^{+}e^{-}$ flavor factories, calling for the design of new search
strategies. Depending on the mass splitting, the heavy dark pion exhibits
decays that are either displaced or detector-stable. The future at Belle II
looks incredibly bright, offering the potential to explore much of the
intriguing parameter space defined by the relic abundance.

##  II QCD-Dark topological portal EFT

We consider a dark QCD-like sector characterized by a global symmetry-breaking
transition $K\to H$, that we suppose arises from chiral symmetry breaking in
an underlying dark gauge theory, delivering dark pions as the DM. The relevant
degrees of freedom are the QCD pions plus dark pions, which are collectively
described by a 4-d non-linear sigma model (NLSM) on a product coset

$X=\frac{SU(3)_{L}\times SU(3)_{R}\times K}{SU(3)_{L+R}\times H}\cong
SU(3)\times\frac{K}{H}\,,$ (1)

with global symmetry $G=SU(3)_{L}\times SU(3)_{R}\times K$. In addition to the
usual terms appearing in the EFT construction of Callan, Coleman, Wess, and
Zumino Callan1969 , which require a $G$-invariant metric on $X$, the action
also admits non-trivial $G$-invariant topological interactions that can be
constructed without a metric. In this paper, we explore EFTs in which such a
topological interaction provides a portal through which DM interacts with the
SM.

Recall that in pure QCD there already appears a topological interaction, the
WZW term Wess:1971yu ; Witten:1983tw . This term originates from the existence
of a bi-invariant closed 5-form
$\omega_{5}=\frac{-i}{480\pi^{3}}N_{c}\mathrm{Tr}\left(g^{-1}dg\right)^{5}$ on
$SU(3)$, where $g$ is the $SU(3)$-valued pion field. The action, evaluated on
4-d spacetime $\Sigma$, can be defined by extending $g(x)$ to a 5-d bulk
manifold $X$ whose boundary is $\Sigma$ and integrating $\omega_{5}$ thereon.
Requiring the path integral phase $e^{2\pi i\int_{X}\omega_{5}}$ be
independent of the choice of the bulk $X$ forces the coefficient $N_{c}$ be an
integer Witten1983 . Famously, $N_{c}$ is fixed to be the number of colors in
QCD by anomaly matching.

In addition to the invariant closed 5-form, QCD also features an invariant
closed 3-form, $\omega_{3}\sim\mathrm{Tr}(g^{-1}dg)^{3}$. In fact, there are
no other bi-invariant forms on $SU(3)$ with which one could construct
topological terms involving QCD pions. Simply by virtue of its degree,
$\omega_{3}$ does not appear in the pure QCD action.111It does, however,
appear in the topologically-conserved current in QCD, which can be identified
with baryon number Witten1983 . But by coupling to a dark sector coset, this
3-form can be used to construct a second topological term.222There can also be
‘theta-like’ topological terms corresponding to closed 4-forms with $U(1)$
coefficients Davighi:2018inx . But these are total derivatives (locally) that
do not give new Feynman rules, and so we neglect them in our pursuit of a
topological portal interaction. Such a mixed topological term requires there
be a $K$-invariant closed 2-form $\Omega_{2}$ on the dark coset $K/H$, which
we can ‘wedge’ with $\omega_{3}$ to form a $G$-invariant333 Because $G$ is
semi-simple, $G$-invariance of the closed 5-form is enough to guarantee
$G$-invariance of the WZW action Davighi:2018inx ; Davighi:2020vcm , giving
similar invariance criteria to DHoker1994 . Because of the product structure
of $X$, one can show that closedness, integrality, and invariance properties
can be separately required of $\omega_{3}$ and $\Omega_{2}$.  closed 5-form.

Which QCD-like dark sectors $K/H$ feature such an invariant 2-form? Let us
accept the standard lore that the viable coset patterns take the form $SU(N)$,
$SU(N)/SO(N)$, or $SU(2N)/Sp(2N)$, depending on whether the dark quarks that
form the chiral condensate transform in a complex, real, or pseudo-real
representation respectively. Because these $K/H$ are not just homogeneous
spaces but, moreover, symmetric spaces, one can show that invariant forms are
here in 1-to-1 with de Rham cohomology classes.444When $K/H$ is a symmetric
space, it follows that all $K$-invariant forms are closed schwarz1994 . This
means there are no invariant exact forms and that invariant forms are in
1-to-1 with classes in invariant cohomology Chevalley1948 . Furthermore, every
closed form on a homogeneous space is cohomologous to an invariant one
schwarz1994 .  Therefore, a dark coset $K/H$ features a topological portal
interaction iff $H^{2}_{\mathrm{dR}}(K/H)\neq 0$. From Tab. 1, we find there
is a unique choice of dark coset that features such a topological portal term,
which is

$K/H=SU(2)/SO(2)\cong S^{2}\,.$ (2)

The closed, $K$-invariant 2-form in question is simply the unit-normalized
volume form $\Omega_{2}$ on $S^{2}$.

|  | Portal |  |  | SIMP
---|---|---|---|---|---
$p$ | $~{}~{}1~{}~{}$ | $2$ | $~{}~{}3~{}~{}$ | $~{}~{}4~{}~{}$ | $5$
$H^{p}(SU(2))$ | $0$ | $0$ | $\mathbb{R}$ | $-$ | $-$
$H^{p}(SU(n)),\,n\geq 3$ | $0$ | $0$ | $\mathbb{R}$ | $0$ | $\mathbb{R}$
$H^{p}(SU(2)/SO(2))$ | $0$ | $\mathbb{R}$ | $-$ | $-$ | $-$
$H^{p}(SU(3)/SO(3))$ | $0$ | $0$ | $0$ | $0$ | $\mathbb{R}$
$H^{p}(SU(4)/SO(4))$ | $0$ | $0$ | $0$ | $\mathbb{R}$ | $\mathbb{R}$
$H^{p}(SU(n)/SO(n)),\,n\geq 5$ | $0$ | $0$ | $0$ | $0$ | $\mathbb{R}$
$H^{p}(SU(2n)/Sp(2n)),\,n\geq 2$ | $0$ | $0$ | $0$ | $0$ | $\mathbb{R}$
Table 1: de Rham cohomology in low degrees for all homogeneous spaces that are
expected to arise as pNGB manifolds in QCD-like theories
cartan1959demonstration . Entries shaded blue (of which there is only one)
feature a topological portal interaction with the pions of QCD; entries shaded
orange feature a 5-point dark pion WZW term, as employed in the SIMP mechanism
Hochberg:2014dra ; Hochberg:2014kqa . No coset features both portal and SIMP
interactions.

The topological term is then constructed from the $G$-invariant closed
(integral555The normalisation of the QCD part is subtle – see Witten1983  for
the original derivation, and Davighi:2018inx ; Lee:2020ojw ; Freed:2006mx  for
more modern treatments.) 5-form

$\omega_{\text{portal}}=\frac{n}{24\pi^{2}}\mathrm{Tr}\left(g^{-1}dg\right)^{3}\wedge\Omega_{2}\,,\quad
n\in\mathbb{Z}\,.$ (3)

Like the WZW term, this interaction has an integer-quantized coefficient, with
$n$ determined by the details of the UV completion (which we do not specify).
We now expand this 5-form in terms of the pion and dark pion fields. First,
consider the QCD part. Expanding $g=\exp(2i\pi_{a}(x)t_{a}/f_{\pi})$ where
$t_{a}=\frac{1}{2}\lambda_{a}$ with $\lambda_{a}$ being the Gell-Mann
matrices,666We use a convention in which the QCD pion kinetic term is
$f_{\pi}^{2}\mathrm{Tr}(\partial_{\mu}g\,\partial^{\mu}g^{-1})/4$, with
$f_{\pi}=92$ MeV.
$g^{-1}dg=\frac{2i}{f_{\pi}}t_{a}d\pi_{a}+\mathcal{O}(\pi^{2})$ implies

$\displaystyle\mathrm{Tr}(g^{-1}dg)^{3}$
$\displaystyle=(2/f_{\pi}^{3})f_{abc}\,d\pi_{a}\wedge d\pi_{b}\wedge
d\pi_{c}+\mathcal{O}(\pi^{4})\,,$ (4)

where $f_{abc}$ are the $SU(3)$ structure constants defined such that
$[t_{a},t_{b}]=if_{abc}t_{c}$, and where we use antisymmetry of the wedge
product to write
$\mathrm{Tr}(t_{a}t_{b}t_{c})\mapsto\mathrm{Tr}(t_{a}[t_{b},t_{c}])/2$, which
is proportional to $f_{abc}$.

Now consider the ‘dark part’. Let $\chi_{1}/f_{D}$ and $\chi_{2}/f_{D}$ be the
usual polar coordinates on $S^{2}$, with ranges $\chi_{1}/f_{D}\in[0,\pi]$,
$\chi_{2}/f_{D}\in[0,2\pi)$, and where $f_{D}$ is the dark pion decay
constant. Then $\Omega_{2}=\epsilon_{ij}d\chi_{i}\wedge d\chi_{j}/(8\pi
f_{D}^{2})$. Combining with the QCD part, eq. (3) gives

$\omega_{\text{portal}}=\frac{n}{96\pi^{3}f_{\pi}^{3}f_{D}^{2}}f_{abc}\epsilon_{ij}d\pi_{a}d\pi_{b}d\pi_{c}d\chi_{i}d\chi_{j}\,,$
(5)

now suppressing the wedges. Finally, by invoking the Poincaré lemma (and
multiplying by $2\pi i$) we convert this to a topological Lagrangian, which is
a locally-defined 4-form that we can integrate directly over spacetime:

$\mathcal{L}_{\text{portal}}^{e=0}=\frac{in\epsilon^{\mu\nu\rho\sigma}}{48\pi^{2}f_{\pi}^{3}f_{D}^{2}}f_{abc}\epsilon_{ij}\pi_{a}\partial_{\mu}\pi_{b}\partial_{\nu}\pi_{c}\partial_{\rho}\chi_{i}\partial_{\sigma}\chi_{j}\,.$
(6)

Just as is the case for the familiar WZW term of QCD, it is crucial to gauge
the electromagnetic subgroup $U(1)_{Q}\subset SU(3)_{L+R}$, generated by
$Q=t_{3}+t_{8}/\sqrt{3}$, in order to derive the leading phenomenological
consequences of this Lagrangian. While the gauging of WZW terms is in general
complicated Witten1983 ; Hull1989 ; HULL1991379 ; Garcia-Compean:2010xgv ;
Brauner:2018zwr , here we wish to gauge only an abelian subgroup in an
effective 2d WZW term — the QCD part. Applying a recent formula of Yonekura
Yonekura:2020upo , the procedure here simplifies to shifting $\omega_{3}$ by
$\frac{-e}{4\pi^{2}}F\wedge\mathrm{Tr}\left(Qg^{-1}dg\right)$ where $F=dA$ is
the QED field strength. Evaluating the trace picks up contributions only from
the $\pi^{0}$ and $\eta$. Wedging with $\Omega_{2}$ to form the ‘gauged portal
interaction’ 5-form, we find the extra term that comes from gauging QED is

$\tilde{\mathcal{L}}_{\text{portal}}-\mathcal{L}_{\text{portal}}^{e=0}=\frac{ne\,\epsilon^{\mu\nu\rho\sigma}}{16\pi^{2}f_{\pi}f_{D}^{2}}\left(\pi^{0}+\frac{\eta}{\sqrt{3}}\right)F_{\mu\nu}\partial_{\rho}\chi_{1}\partial_{\sigma}\chi_{2}\,,$
(7)

having again used the Poincaré lemma and multiplied by a factor $2\pi i$. The
full interaction is $\tilde{\mathcal{L}}_{\text{portal}}$.

The gauged part of this operator, which couples $\gamma\pi^{0}(\eta)$ to the
pair of dark pions, is an effective dimension-7 operator in our EFT. In
addition to providing a completely new portal to the dark sector, we also
expect this operator to be the leading EFT interaction that couples dark to
light states here, at least in terms of EFT power counting (ignoring symmetry-
breaking spurion insertions). Indeed, the first non-topological interaction
that involves both QCD and dark pions is the dimension-8 interaction
$\frac{1}{f_{\pi}^{2}f_{D}^{2}}(D_{\mu}\pi_{a}D^{\mu}\pi^{a})(\partial_{\nu}\chi_{i}\partial^{\nu}\chi^{i})$,
where $D_{\mu}=\partial_{\mu}+ieA_{\mu}$ is the covariant derivative. This
dominance of the topological portal means it ought to determine the relic
abundance of the dark pions to a first approximation.

##  III Relic abundance from the topological portal

We are now ready to investigate the phenomenology of the topological portal in
cosmology and at colliders. For simplicity, we study the Boltzmann equations
Gondolo:1990dk  in the approximation of negligible mass splitting,777We
comment on the mass splitting at the end of this section. describing the
evolution of $\chi_{1}+\chi_{2}$ yield, $Y_{\chi}$, defined as the ratio of
the combined dark pion number density $n_{\chi}$ to the entropy density $s$,

$\frac{dY_{\chi}}{dx}=-\sqrt{\frac{\pi g_{*}}{45}}\frac{M_{\rm
P}m_{\chi}}{x^{2}}\langle\sigma v\rangle\left(Y^{2}_{\chi}-Y_{\rm
eq}^{2}\right)\,.$ (8)

The evolution parameter is $x=m_{\chi}/T$ where $m_{\chi}$ is the dark pion
mass, $M_{\rm P}$ is the Planck mass, and $g_{*}$ is the effective number of
relativistic degrees of freedom. The thermally averaged cross section times
the Møller velocity,
$v=\sqrt{|\mathbf{v}_{1}-\mathbf{v}_{2}|^{2}-|\mathbf{v}_{1}\times\mathbf{v}_{2}|^{2}}$,
can be expressed as a single integral Gondolo:1990dk

$\langle\sigma
v\rangle=\frac{\int_{4m_{\chi}^{2}}^{\infty}\sigma\sqrt{s}(s-4m_{\chi}^{2})\,{\rm{K}}_{1}(\sqrt{s}/T)\,{\rm{d}}s}{8m_{\chi}^{4}T\,{\rm
K}_{2}^{2}(m_{\chi}/T)}\,,$ (9)

with $\sigma$ being the unpolarised cross section for the process
$\chi_{1}\chi_{2}\to\pi^{0}\gamma$, $s=(p_{\chi_{1}}+p_{\chi_{2}})^{2}$, and
${\rm K}_{i}$ are the modified Bessel functions of the $i$-th order. Finally,
the equilibrium yield $Y_{\rm eq}$ is a function of the evolution parameter
$x$, namely $Y_{\rm eq}={45x^{2}{\rm K}_{2}(x)}/(4\pi^{4}g_{*})$.

For our topological portal interaction, we find the following unpolarised
cross section

$\displaystyle\sigma=\frac{n^{2}}{1536\,\pi^{4}}\,\frac{\alpha_{\rm
Q}}{f_{\pi}^{2}f_{D}^{4}}\,s^{3/2}\sqrt{s-4m_{\chi}^{2}}\,,$ (10)

where $\alpha_{\rm Q}=e^{2}/(4\pi)$ is the QED coupling constant, and we
summed over the photon polarisations. After the integration in Eq. (9), we
obtain

$\langle\sigma v\rangle=\frac{n^{2}}{64\,\pi^{7/2}}\,\frac{\alpha_{\rm
Q}x_{f}^{4}}{f_{\pi}^{2}}\,\frac{x\,{\rm
G}^{3,0}_{1,3}\left(\genfrac{}{}{0.0pt}{}{-2}{-\frac{9}{2}\,-\frac{1}{2}\,\frac{1}{2}}|\,x^{2}\right)}{{\rm
K}_{2}^{2}(x)}\,,$ (11)

where $x_{f}=m_{\chi}/f_{D}$, and ${\rm G}(x^{2})$ is the Meijer G-function
ADAMCHIK1995283 .

Figure 1: The dark pion yield as a function of $x=m_{\chi}/T$. The different
lines show qualitatively different possibilities for setting the relic
abundance, depending on the cosmological history of dark pions. See the main
text for details.

Depending on the cosmological history of the dark pions, Eq. (8) can describe
different scenarios setting the dark pion relic abundance. On the one hand, if
the dark pions establish thermal equilibrium with the bath before the
topological operator turns on at the QCD phase transition, then standard DM
freeze-out occurs via $\chi\chi\to\gamma\pi^{0}$ at $x\sim 23$, after the
topological portal is activated. This is shown by the blue line in Fig. 1, for
which the topological operator turns on at $x_{\rm QCD}:=m_{\chi}/T_{\rm
QCD}=12$. On the other hand, if the initial yield of the dark pions is
negligible, and they are produced only after the QCD phase transition, they
proceed with a quick thermalization followed by the freeze-out, and the
correct relic abundance can still be attained. This is shown by an orange line
in Fig. 1, for which the QCD phase transition occurs at $x=x_{\rm QCD}=17$.
Consequently, the correct yield today ($Y_{\chi}$ for $x\to\infty$) can be
achieved irrespective of the dark pion cosmological history. The topological
portal thus robustly sets the correct relic abundance,

$\Omega_{\chi}h^{2}\approx\frac{2\,x_{\rm QCD}\,T_{\rm
QCD}\,Y_{\infty}}{3.6\cdot 10^{-9}\,{\rm GeV}}\approx 0.12\,,$ (12)

through an interplay between the QCD phase transition onset relative to the
dark pion mass, determined by $x_{\rm QCD}$, and the $x_{f}=m_{\chi}/f_{D}$
ratio. In both scenarios, the QCD phase transition should happen no later than
$x=x_{\rm max}=23$; after this point, the dark pion yield drops below the
value necessary to account for the DM relic abundance. This limiting case is
shown by a red line in Fig. 1. The upper bound on $x_{\rm QCD}$ translates to
an upper bound on $m_{\chi}$. Specifically, to explain the relic abundance via
the topological portal points towards light thermal DM with $m_{\chi}\lesssim
3.7$ GeV.
In the computation, we used the QCD phase transition temperature, $T_{\rm
QCD}\sim 160\,\,{\rm MeV}$, and the number of effective degrees of freedom,
$g_{*}\sim 18$, assuming the topological operator turns on only after the QCD
phase transition ParticleDataGroup:2022pth . The fact that the correct relic
abundance demands $f_{D}\sim\mathcal{O}(5.5-7)\times\sqrt{n}\,m_{\chi}$
ensures that the dark pions are indeed the lightest dark states and forbids
their decay to e.g. dark glueballs.

$\Delta m_{\chi}$ | $\lesssim 1.7m_{\pi^{0}}$ | $\,\,\gtrsim 1.7m_{\pi^{0}}\,\,\,$
---|---|---
Signature | $\pi^{0}+\not{E}_{T}$ | $\pi^{0}+\not{E}_{T}+\text{DV}(\pi^{0}\gamma\not{E}_{T})$
Table 2: Collider signatures. Here ‘DV’ indicates a displaced secondary
vertex. The values of $\Delta m_{\chi}$ for which $\chi_{2}$ lifetime is
approximately $10^{-7}$ sec depend on the value of $m_{\chi_{1}}$, and vary
from $[1.3-2.1]\,m_{\pi^{0}}$ in the mass region
$m_{\chi_{1}}\in[1-3.5]\,$GeV.

The non-zero mass splitting, $\Delta m_{\chi}=m_{\chi_{2}}-m_{\chi_{1}}$,
leads to a suppression of the co-annihilation cross section. The dominant
effect can be captured by introducing an exponential suppression factor
$e^{-x\Delta}$ in Eq. (11), where $\Delta:=\frac{\Delta
m_{\chi}}{m_{\chi_{1}}}$ Griest:1990kh ; DAgnolo:2018wcn . The resulting
$f_{D}$ which fits the relic abundance is, accounting for the mass splitting,

$f_{D}(\Delta)\approx f_{D}(0)\,e^{-\frac{x_{\rm max}\Delta}{4}}~{},$ (13)

where the factor of $1/4$ appearing in the exponent comes from the dependence
$\langle\sigma v\rangle\propto x_{f}^{4}$ in Eq. (11). This is a numerically
large suppression effect that we should account for in understanding the
viable parameter space; for example, for
$\Delta=5\times\frac{m_{\pi^{0}}}{3.5\,\text{GeV}}$ one finds
$f_{D}(\Delta)/f_{D}(0)\approx 1/3$.

##  IV Novel collider phenomenology

With a non-zero mass splitting, $\chi_{2}$ decays to $\chi_{1}$ shortly after
freeze-out, leaving $\chi_{1}$ as the sole dark matter component. As a result,
DM annihilations at later times are suppressed, avoiding otherwise stringent
indirect detection constraints from CMB anisotropies Galli:2011rz ;
Planck:2018vyg . Likewise, direct detection experiments are ineffective for
the inelastic $\chi_{1}\to\chi_{2}$ up-scattering.

How, then, can we test this scenario? Interestingly, collider experiments
offer a promising avenue through
$e^{+}e^{-}\to\gamma^{*}\to\chi_{1}\chi_{2}\pi^{0}$ production. In particular,
the required collider energy, high luminosity, and hermetic environment of
Belle II result in exceptional sensitivity, potentially covering the full
parameter space set by the DM relic abundance. The dark pion mass splitting
$\Delta m_{\chi}$ suggests different search strategies, as summarised in Tab.
2. For a small mass splitting, $\chi_{2}$ is detector stable,888When $\Delta
m_{\chi}<m_{\pi^{0}}$, the decay $\chi_{2}\to\chi_{1}\gamma\gamma\gamma$
through an off-shell $\pi^{0}$ gives a lifetime $\gtrsim\mathcal{O}(1)$ sec
where the cosmological bounds kick in. Thus, the interesting mass range for
colliders is $\Delta m_{\chi}>m_{\pi^{0}}$. We note that the decay
$\chi_{2}\to\chi_{1}\gamma$, naïvely induced at 1-loop, is zero.  otherwise,
$\chi_{2}$ decays at a displaced vertex.999Prompt decays, which would require
a large mass splitting, are less motivated given Eq. (13). Neither of these
signatures have been explored in dedicated experimental analyses so far.

Figure 2: Limits by the B-factories on the parameter space of dark pions with
zero mass splitting. The signature corresponds to $\pi^{0}+\not{E}_{T}$, where
$\pi^{0}$ is identified as a photon such that mono-$\gamma$ searches are used.
The red lines delineate the parameter space points giving the correct DM relic
abundance (solid: $\Delta m_{\chi}=0$, dashed: $\Delta m_{\chi}=m_{\pi^{0}}$,
dotted: $\Delta m_{\chi}=2m_{\pi^{0}}$). Both Belle II exclusion lines
correspond to the projections Belle-II:2018jsg , while BaBar already performed
a mono-$\gamma$ search BaBar:2014zli .

To illustrate the potential of Belle II, we focus on the small $\Delta
m_{\chi}$ scenario (detector-stable $\chi_{2}$). We leave the other case for
future work. The energy of $\pi^{0}$ originating from
$e^{+}e^{-}\to\gamma^{*}\to\pi^{0}\chi_{1}\chi_{2}$ is such that the two
photons from $\pi^{0}\to\gamma\gamma$ cannot be sufficiently separated and are
detected as a single photon Belle-II:2018jsg . Therefore, we recast the
existing (proposed) dark photon search at BaBar BaBar:2014zli  (Belle II
Belle-II:2018jsg ) in the mono-$\gamma$ channel, keeping in mind that a
dedicated analysis would outperform our study, see also Duerr:2019dmv . We
simulate the signal events using MadGraph5_aMC@NLO Alwall:2014hca  and apply
the same event selection as reported in Belle-II:2018jsg . The overall
analysis efficiency is slightly less compared to the dark photon search,
resulting in the limits shown in Fig 2.

The red lines in the figure produce the correct DM relic abundance for three
values of the mass splitting, and all terminate at $m_{\chi_{1}}\lesssim 3.7$
GeV as discussed in Sec. III. Already with $20\,\rm{fb}^{-1}$, Belle II can
test a significant portion of the parameter space, slightly more than the
existing BaBar exclusion based on $56\,\rm{fb}^{-1}$. This is due to the
improved single photon trigger efficiency at Belle II. The projections for the
full luminosity of $50\,\rm{ab}^{-1}$ (dashed orange) are extremely promising,
covering most of the parameter space of interest. Finally, should Belle II
discover such a signal in $\pi^{0}\chi_{1}\chi_{2}$, then observing the
production of $\eta\chi_{1}\chi_{2}$ would serve as the next target (and
arguably a ‘smoking gun’ for this topological portal), since it is predicted
to occur at a fixed rate relative to the pion channel – see Eq. (7).

##  V Outlook

In this letter, we postulate a unique topological portal operator between QCD
and a dark QCD-like sector, which successfully realizes the light thermal
inelastic DM scenario while offering exciting signatures at Belle II.

Future explorations will be on two fronts. Firstly, developing an ultraviolet
completion for our topological operator is crucial. We anticipate this
involving new chiral fermions near (below) the $f_{D}$ scale that match
anomalies. Those fermions are likely charged under both dark and visible
sectors, which would imply striking model-dependent signatures, such as semi-
visible jets, observable at the LHC and future high-energy colliders.
Secondly, whether Nature has selected this unique and subtle portal for
primary communication between the visible and the dark sector is a question
that Belle II could resolve, a course of action we highly recommend for the
collaboration.

## Acknowledgement

We are grateful to Tim Cohen, Javier M. Lizana, and especially Nakarin
Lohitsiri for the discussions. We further thank Nakarin Lohitsiri for
providing helpful comments on the manuscript. NS has received funding from the
INFN Iniziative Specifica APINE. The work of JD and NS was partially carried
out at the University of Zürich, supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement 833280 (FLAY) and by the Swiss National
Science Foundation (SNF) under contract 200020-204428. AG acknowledges the SNF
for funding through the Eccellenza Professorial Fellowship “Flavor Physics at
the High Energy Frontier” project number 186866.

## References

  * (1) Particle Data Group collaboration, R. L. Workman et al., _Review of Particle Physics_ , _PTEP_ 2022 (2022) 083C01.
  * (2) P. Gondolo and G. Gelmini, _Cosmic abundances of stable particles: Improved analysis_ , _Nucl. Phys. B_ 360 (1991) 145–179.
  * (3) J. Wess and B. Zumino, _Consequences of anomalous Ward identities_ , _Phys. Lett._ 37B (1971) 95–97.
  * (4) E. Witten, _Global Aspects of Current Algebra_ , _Nucl. Phys. B_ 223 (1983) 422–432.
  * (5) K. M. Zurek, _Dark Matter Candidates of a Very Low Mass_ , 2401.03025.
  * (6) S. Galli, F. Iocco, G. Bertone and A. Melchiorri, _Updated CMB constraints on Dark Matter annihilation cross-sections_ , _Phys. Rev. D_ 84 (2011) 027302, [1106.1528].
  * (7) Planck collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6, [1807.06209].
  * (8) D. Tucker-Smith and N. Weiner, _Inelastic dark matter_ , _Phys. Rev. D_ 64 (2001) 043502, [hep-ph/0101138].
  * (9) E. Izaguirre, G. Krnjaic and B. Shuve, _Discovering Inelastic Thermal-Relic Dark Matter at Colliders_ , _Phys. Rev. D_ 93 (2016) 063523, [1508.03050].
  * (10) J. C. G. Callan, S. R. Coleman, J. Wess and B. Zumino, _Structure of phenomenological Lagrangians. 2._ , _Phys. Rev._ 177 (1969) 2247–2250.
  * (11) E. Witten, _Current Algebra, Baryons, and Quark Confinement_ , _Nucl. Phys._ B223 (1983) 433–444.
  * (12) J. Davighi and B. Gripaios, _Homological classification of topological terms in sigma models on homogeneous spaces_ , _JHEP_ 09 (2018) 155, [1803.07585].
  * (13) J. Davighi, B. Gripaios and O. Randal-Williams, _Differential cohomology and topological actions in physics_ , 2011.05768.
  * (14) E. D’Hoker and S. Weinberg, _General effective actions_ , _Phys. Rev._ D50 (1994) R6050–R6053, [hep-ph/9409402].
  * (15) A. Schwarz, _Topology for physicists_.  Grundlehren der mathematischen Wissenschaften. Springer New York, 1994\.
  * (16) C. Chevalley and S. Eilenberg, _Cohomology theory of Lie groups and Lie algebras_ , _Trans. Am. Math. Soc._ 63 (1948) 85–124.
  * (17) H. Cartan, _Démonstration homologique des théoremes de périodicité de bott, ii. homologie et cohomologie des groupes classiques et de leurs espaces homogenes_ , _Séminaire Henri Cartan_ 12 (1959) 1–32.
  * (18) Y. Hochberg, E. Kuflik, T. Volansky and J. G. Wacker, _Mechanism for Thermal Relic Dark Matter of Strongly Interacting Massive Particles_ , _Phys. Rev. Lett._ 113 (2014) 171301, [1402.5143].
  * (19) Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky and J. G. Wacker, _Model for Thermal Relic Dark Matter of Strongly Interacting Massive Particles_ , _Phys. Rev. Lett._ 115 (2015) 021301, [1411.3727].
  * (20) Y. Lee, K. Ohmori and Y. Tachikawa, _Revisiting Wess-Zumino-Witten terms_ , _SciPost Phys._ 10 (2021) 061, [2009.00033].
  * (21) D. S. Freed, _Pions and Generalized Cohomology_ , _J. Diff. Geom._ 80 (2008) 45–77, [hep-th/0607134].
  * (22) C. M. Hull and B. J. Spence, _The gauged nonlinear $\sigma$ model With Wess-Zumino term_, _Phys. Lett._ B232 (1989) 204–210.
  * (23) C. M. Hull and B. J. Spence, _The geometry of the gauged sigma-model with Wess-Zumino term_ , _Nuclear Physics B_ 353 (1991) 379 – 426.
  * (24) H. Garcia-Compean and P. Paniagua, _Gauged WZW Models Via Equivariant Cohomology_ , _Mod. Phys. Lett. A_ 26 (2011) 1343–1352, [1008.0206].
  * (25) T. Brauner and H. Kolešová, _Gauged Wess-Zumino terms for a general coset space_ , _Nucl. Phys. B_ 945 (2019) 114676, [1809.05310].
  * (26) K. Yonekura, _General anomaly matching by Goldstone bosons_ , _JHEP_ 03 (2021) 057, [2009.04692].
  * (27) V. Adamchik, _The evaluation of integrals of bessel functions via g-function identities_ , _Journal of Computational and Applied Mathematics_ 64 (1995) 283–290.
  * (28) K. Griest and D. Seckel, _Three exceptions in the calculation of relic abundances_ , _Phys. Rev. D_ 43 (1991) 3191–3203.
  * (29) R. T. D’Agnolo, C. Mondino, J. T. Ruderman and P.-J. Wang, _Exponentially Light Dark Matter from Coannihilation_ , _JHEP_ 08 (2018) 079, [1803.02901].
  * (30) Belle-II collaboration, W. Altmannshofer et al., _The Belle II Physics Book_ , _PTEP_ 2019 (2019) 123C01, [1808.10567].
  * (31) BaBar collaboration, J. P. Lees et al., _Search for a Dark Photon in $e^{+}e^{-}$ Collisions at BaBar_, _Phys. Rev. Lett._ 113 (2014) 201801, [1406.2980].
  * (32) M. Duerr, T. Ferber, C. Hearty, F. Kahlhoefer, K. Schmidt-Hoberg and P. Tunney, _Invisible and displaced dark matter signatures at Belle II_ , _JHEP_ 02 (2020) 039, [1911.03176].
  * (33) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079, [1405.0301].