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# Dilaton Forbidden Dark Matter

Thomas Appelquist  Department of Physics, Sloane Laboratory, Yale University,
New Haven, Connecticut 06520, USA  James Ingoldby  Institute for Particle
Physics Phenomenology, Durham University, Durham, UK  Maurizio Piai
Department of Physics, College of Science, Swansea University, Singleton Park,
Swansea, Wales, UK

###### Abstract

Dilaton effective field theory (dEFT) describes the long distance behavior of
certain confining, near-conformal gauge theories that have been studied via
lattice computation. Pseudo-Nambu- Goldstone bosons (pNGBs), emerging from the
breaking of approximate, continuous, internal symmetries, are coupled to an
additional scalar particle, the dilaton, arising from the spontaneous breaking
of approximate scale invariance. This effective theory has been employed to
study possible extensions of the standard model. In this paper, we propose a
complementary role for dEFT, as a description of the dark matter of the
universe, with the pNGBs identified as the dark-matter particles. We show that
this theory provides a natural implementation of the “forbidden” dark matter
mechanism, and we identify regions of parameter space for which the thermal
history of dEFT yields the measured dark matter relic density.

††preprint: IPPP/24/XX††preprint: IPPP/24/17

##  I Introduction

The nature of the dark matter of the universe is one of the most important
mysteries in fundamental science [1]. If dark matter takes the form of (non
relativistic) new particles, its non-gravitational interactions with the
standard model (SM) must be very weak. Yet, current constraints on cold dark
matter (CDM) are compatible with a vast range of masses and interaction
strengths (see Ref. [2], and references therein).

Global fits of cosmological data within the $\Lambda_{\rm CDM}$ paradigm (the
standard cosmological model) determine the present cold dark matter energy
density to be $\Omega_{\rm CDM}h^{2}=0.120\pm 0.001$ [3],111$h\simeq 0.674(5)$
is Hubble’s constant, $H_{0}$, in units of $100\,{\rm km}\,{\rm s}^{-1}{\rm
Mpc}^{-1}$, and $\Omega_{\rm CDM}$ is the CDM energy density normalized to
criticality, $\rho_{c}\equiv\frac{3H_{0}^{2}}{8\pi
G_{N}}\,=1.878\,\times\,10^{-26}h^{2}{\rm kg}^{2}m^{-3}$ [4]. CDM providing
about 26-27% of the energy content of the universe.

A natural mechanism for populating the universe with dark matter and
explaining the present abundance is freezeout; in the early universe, the dark
matter was in thermal equilibrium with the SM plasma, decoupling as the
universe expanded, and decreasing its interaction rate with the SM. When this
mechanism is realized, the relic density, $\Omega_{\rm CDM}h^{2}$, is
determined by the annihilation and replenishment cross sections of dark matter
particles.

The dark matter particles could be composite, emerging in a new strongly
coupled dark sector. As discussed in Ref. [5], and the review [6], this
possibility may be relevant to anomalies in small-scale structure [7, 8].
Furthermore, if the new strong dynamics leads to a first-order phase
transition in the early universe, this might yield a relic stochastic
gravitational-wave background [9, 10, 11], detectable in experiments such as
LISA [12] or ET [13].

One intriguing extension of the freezeout idea is that, within the dark
sector, CDM particles annihilate into heavier states, through a process that
would be forbidden at zero temperature, but is allowed at finite temperature.
This mechanism is known as forbidden dark matter (FDM). See Refs. [14, 15],
and [16, 17] for applications. The thermal suppression in the FDM mechanism
allows for realistically large relic densities at freezeout, but with smaller
CDM masses and a broad range of self-interaction strengths.

Here, we propose a natural realization of FDM within dEFT. This effective
theory extends the conventional chiral Lagrangian with its pNGB fields, $\pi$,
to include a dilaton field, $\chi$. It has been extensively studied, in our
Refs. [18, 19, 20, 21, 22, 23] and in Refs. [24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. See also the precursors in Refs.
[43, 44] and Ref. [45]. dEFT emerges as the low-energy description of certain
underlying gauge theories amenable to lattice studies. One notable example is
the $SU(3)$ gauge theory with $N_{f}=8$ fundamental Dirac fermions [46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. Lattice studies of this theory
have reported evidence for the presence of a light singlet, scalar particle,
along with $N^{2}_{f}-1$ pNGBs, in the accessible range of fermion masses.

Although the $\mathrm{SU}(3)$ gauge theory provides a UV-complete environment
in which to implement the forbidden dark matter dynamics, we restrict
attention to only the lightest spin-$0$ states, and describe the dark matter
within dEFT, ignoring all other, heavier, composite states. Our approach
therefore can apply to other gauge theories that admit a dEFT low-energy
description [59, 60, 61, 30, 37, 34].

We take the dilaton mass, $M_{d}$, and the pNGB multiplet mass, $M_{\pi}$, to
be free parameters, with $M_{d}>M_{\pi}$. In the $\mathrm{SU}(3)$ gauge
theory, throughout the fermion-mass range explored on the lattice, the dilaton
is somewhat heavier than the pNGBs, which are the dark matter particles. This
hierarchy is expected to persist down to the chiral limit, since explicit
breaking of conformal symmetry remains even there. Finally, we discuss the
coupling of the dark sector to the standard model, which must be present to
achieve thermal equilibrium between the sectors.

The description of a dark sector in terms of pNGB fields (but no dilaton) has
appeared before in the literature on strongly interacting massive particles
(SIMPs) [62, 63, 64, 65]. There, the depletion takes place through a
$3\rightarrow 2$ process, with the relevant $5$-point interaction arising from
a Wess-Zumino-Witten term. Our dEFT contains $3\rightarrow 2$ processes even
at leading order, but, in the parameter range of interest, they are less
important than the dominant $2\rightarrow 2$ (forbidden) processes.

Models in which pNGB dark matter is coupled to a dilaton, but which are not of
the FDM type, have been identified in the literature [66, 67, 68]. We also
note that an FDM model including a dilaton has appeared recently [69], though
in this case the dark matter is not a pNGB. All these approaches are rather
different from strongly coupled models in which the dark matter candidate is a
baryon [70].

In Section II, we summarize dEFT for describing the dark matter pNGBs, $\pi$,
and the dilaton, $\chi$. The dEFT interactions yield the annihilation process
$\pi\pi\rightarrow\chi\chi$, its inverse, and also the self-interaction among
the $\pi$s. In Section III, we compute the relic density, mapping out the
space of allowed masses and coupling strengths. The interaction with the SM is
discussed in Section IV. We summarize and discuss open questions in Section V.

##  II Dilaton Effective Field Theory

Our framework is defined by the three-term Lagrangian

${\cal L}={\cal L}_{\text{SM}}+{\cal L}_{D}+{\cal L}_{\text{int}}\,,$ (1)

where ${\cal L}_{\text{SM}}$ describes the standard model and
$\mathcal{L}_{D}$ describes a new composite dark sector with a confinement
scale that can be below the electroweak scale. The (highly suppressed)
interaction between the two sectors is described by
$\mathcal{L}_{\text{int}}$.

The dark-sector Lagrangian, $\mathcal{L}_{D}$, is taken to be that of dEFT.
Its form, reviewed in Ref. [23], is given by

${\cal
L}_{D}=\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\,+\frac{F_{\pi}^{2}}{4}\left(\frac{\chi}{F_{d}}\right)^{2}\,\mbox{Tr}\left[\partial_{\mu}\Sigma(\partial^{\mu}\Sigma)^{\dagger}\right]\\\
+\frac{M_{\pi}^{2}F_{\pi}^{2}}{4}\left(\frac{\chi}{F_{d}}\right)^{y}\,\mbox{Tr}\left[\Sigma+\Sigma^{\dagger}\right]\,-\,V(\chi)\,,$
(2)

where $M_{\pi}$ and $F_{\pi}$ are the the mass and decay constant of the
composite pNGBs, $\Sigma$ includes the multiplet of pNGB fields with
$\Sigma^{\dagger}\Sigma=\mathbf{1}$, and $\chi$ is the dilaton field. The
potential, $V(\chi)$, includes a term that explicitly breaks conformal
symmetry, and the exponent $y$ is a constant. The dilaton field has a full
potential arising from $V(\chi)$ and the third term of Eq. (2):

$\displaystyle W(\chi)\equiv
V(\chi)-\frac{M^{2}_{\pi}F^{2}_{\pi}N_{f}}{2}\left(\frac{\chi}{F_{d}}\right)^{y}\,,$
(3)

and acquires a nonzero vacuum expectation value (VEV)
$\langle\chi\rangle=F_{d}$ at the potential minimum, breaking approximate
scale invariance spontaneously. Expanding about the minimum via the
redefinition $\chi=F_{d}+\bar{\chi}$ determines the dilaton mass and self-
interactions:

$\displaystyle
W(\bar{\chi})=\text{constant}+\frac{M^{2}_{d}}{2}\bar{\chi}^{2}+\frac{\gamma}{3!}\frac{M^{2}_{d}}{F_{d}}\bar{\chi}^{3}+\dots\,,$
(4)

where $\gamma$ is a constant greater than 2 [45], which depends on $y$ and the
form of the term in $V(\chi)$ that explicitly breaks scale invariance.

To compute the dark-matter relic density, we need to know the cross section
for the forbidden process, $\pi\pi\rightarrow\chi\chi$, and its inverse,
$\chi\chi\rightarrow\pi\pi$. These depend on the parameters of dEFT: the
masses, $M_{d}^{2}$ and $M_{\pi}^{2}$, the decay constants, $F_{d}^{2}$ and
$F_{\pi}^{2}$, the pNGB count, $N_{\pi}=N_{f}^{2}-1$, and the dimensionless
parameters $y$ and $\gamma$.

Lattice studies of specific gauge theories, which provide ultraviolet
completions for dEFT, yield (model-dependent) information about some of the
dEFT parameters. For example, in the $\mathrm{SU}(3)$ gauge theory with
$N_{f}=8$ fundamental fermions, fits of dEFT to lattice data constrain the
parameter $y$ to be close to 2 [37, 38, 39], a value also supported by
theoretical arguments [71]. For simplicity, we set $y=2$ in the following,
although recovering the more general expressions is straightforward. The
parameter $\gamma$ is less well determined, and we keep it general.

The lattice studies of the $N_{f}=8$ theory explore only a particular range of
$M_{\pi}$ values (in lattice units). Throughout this range, $M_{d}$ is larger
than $M_{\pi}$, but of the same order. We take this to be the case in our
description of the dark sector. In these data sets, fits to dEFT predictions
indicate that the ratio $F_{\pi}^{2}/F_{d}^{2}\simeq 0.1$ [37, 38, 39],
compatible with the expectation that $F_{\pi}^{2}/F^{2}_{d}\sim N_{f}^{-1}$.

The effective expansion parameter of dEFT, at momentum scales of order
$M_{\pi}$ or smaller, is of order $M_{\pi}^{2}N_{\pi}/(4\pi F_{d})^{2}$. For
the range of fermion masses explored in the most recent $N_{f}=8$ lattice
study [58], $M_{\pi}/F_{\pi}\approx 4$, leading to an expansion parameter of
order $0.5$. By reducing fermion masses in the underlying theory,
$M_{\pi}/F_{\pi}$ and the effective expansion parameter can be made smaller.
We assume $M_{\pi}/F_{\pi}$ is small enough for us to reliably use dEFT to
compute the relevant cross sections at freezeout, where the temperatures at
play are small compared to $M_{\pi}$. After specifying the parameters of dEFT,
there remains only the question of the overall mass scale of the dark sector.
We consider a range of possibilities below the electroweak scale.

The dark sector must couple to the standard model, with enough strength to
maintain thermal equilibrium between the sectors during freezeout, and yet
without enough strength to overwhelm the forbidden mechanism within the dark
sector. We discuss this interaction in Section IV, noting that a sizable range
of very weak couplings satisfy these constraints.

##  III Relic Density

In the FDM mechanism, the population of the dark matter pNGBs is depleted
through a $2\rightarrow 2$ scattering process to dilatons. These then decay
into SM particles through weak contact interactions to be discussed in Section
IV. The relic density is determined by the Boltzmann equation

$\displaystyle\frac{\partial n_{\pi}}{\partial t}+3Hn_{\pi}=$
$\displaystyle-\langle\sigma_{2\pi\rightarrow 2\chi}v\rangle
n^{2}_{\pi}+\langle\sigma_{2\chi\rightarrow
2\pi}v\rangle\left(n_{\chi}^{\text{eq}}\right)^{2}\,,$ (5)

where $H$ is the Hubble scale and $n_{\pi}$ is the total number density of
pNGBs (summed over all flavors). We assume that due to the contact interaction
the dilaton remains in thermal equilibrium with the SM during freezeout. For
the relevant temperatures, $T<M_{\pi},\,M_{d}$, we represent the pNGB and
dilaton number densities using the non-relativistic equilibrium expression

$\displaystyle
n^{\text{eq}}_{i}=w_{i}\left(\frac{M_{i}T}{2\pi}\right)^{3/2}e^{-M_{i}/T}\,,$
(6)

where $i=\pi,\,d$, is the pNGB or dilaton, and $w_{i}$ counts the number of
species ($w_{d}=1$ and $w_{\pi}=N_{\pi}$). The quantities
$\langle\sigma_{2\pi\rightarrow 2\chi}v\rangle$ and
$\langle\sigma_{2\chi\rightarrow 2\pi}v\rangle$ are the thermally averaged
cross section and inverse cross section.

The cross section $\langle\sigma_{2\pi\rightarrow 2\chi}v\rangle$, vanishing
at $T=0$ since $M_{\pi}<M_{d}$, is nonzero at finite $T$, and given in terms
of $\langle\sigma_{2\chi\rightarrow 2\pi}v\rangle$ by

$\displaystyle\langle\sigma_{2\pi\rightarrow
2\chi}v\rangle=\frac{(1+\Delta)^{3}}{N^{2}_{\pi}}e^{-2\Delta
x}\langle\sigma_{2\chi\rightarrow 2\pi}v\rangle\,,$ (7)

where $x=M_{\pi}/T$ and $\Delta=(M_{d}-M_{\pi})/M_{\pi}$. The forbidden rate
is exponentially suppressed when $T\ll M_{d}-M_{\pi}$. Equation (7) can be
derived by integrating the two cross sections against the Maxwell-Boltzmann
velocity distribution, or by using the principle of detailed balance [15].

Employing Eq. (7) for the thermally averaged $\pi\pi\rightarrow\chi\chi$ cross
section in the Boltzmann equation yields

$\frac{\partial n_{\pi}}{\partial
t}+3Hn_{\pi}=-\langle\sigma_{2\chi\rightarrow 2\pi}v\rangle\\\
\times\left(\frac{n^{2}_{\pi}}{N_{\pi}^{2}}(1+\Delta)^{3}e^{-2\Delta x}-\
\left(n_{\chi}^{\text{eq}}\right)^{2}\right)\,.$ (8)

The thermally averaged cross section, $\langle\sigma_{2\chi\rightarrow
2\pi}v\rangle$, can be computed at tree level in dEFT. For the case $y=2$,
this cross section is given approximately by

$\displaystyle\langle\sigma_{2\chi\rightarrow
2\pi}v\rangle=\frac{M^{2}_{\pi}N_{\pi}}{36\pi
F^{4}_{d}}\sqrt{\Delta(2+\Delta)}(1+\Delta)(5+\gamma)^{2}\,,$ (9)

where we have neglected corrections of order $1/x=T/M_{\pi}$. We note that
$\gamma$ cannot be too large or else the cross section, Eq. (9), would exceed
unitarity bounds, signaling that the dEFT is outside its domain of validity.

To solve the Boltzmann equation (Eq. (8)), we assume a radiation dominated
universe so that the Hubble scale, $H$, is given by the expression

$\displaystyle H=\pi\sqrt{\frac{g(T)}{90}}\frac{T^{2}}{M_{\text{pl}}}\,,$ (10)

where $M_{\text{pl}}=1/\sqrt{8\pi G}$, with $G$ Newton’s gravitational
constant, and $g(T)$ is a continuous function that counts the effective number
of relativistic degrees of freedom.

It is then convenient to recast Eq. (8) in terms of the co-moving number
density, $Y_{\pi}=n_{\pi}/s$, where $s$ is the entropy density given by the
expression

$\displaystyle s(T)=\frac{2\pi^{2}}{45}h(T)T^{3}\,,$ (11)

and $h(T)$ is a different function that counts the effective number of degrees
of freedom. Free particles with masses $m_{i}\ll T$ contribute equally to the
functions $h(T)$ and $g(T)$. However, these functions differ in general due to
mass thresholds and interactions.

We construct $h(T)$ and $g(T)$ using tabulated values taken from micrOMEGAs6.0
[72]. Those values are derived from determinations of the functions presented
in Ref. [73], themselves obtained using lattice QCD calculations of the
equation of state [74].

In terms of $x=M_{\pi}/T$, the Boltzmann equation for $Y_{\pi}$ takes the form

$\displaystyle\frac{dY_{\pi}}{dx}=-\frac{\xi(M_{\pi}/x)}{x^{2}}e^{-2\Delta
x}\left(1+\Delta\right)^{3}\left[Y_{\pi}^{2}-\left(Y^{\text{eq}}_{\pi}\right)^{2}\right]\,,$
(12)

where $Y_{\pi}^{eq}=n_{\pi}^{eq}(T)/s(T)$, with $n_{\pi}^{\text{eq}}(T)$ given
by Eq. (6), while

$\displaystyle\xi(T)\equiv\frac{2\pi\sqrt{10}M_{\pi}M_{\text{pl}}}{15N_{\pi}^{2}}\langle\sigma
v\rangle\frac{h(T)}{\sqrt{g(T)}}\left[1+\frac{1}{3}\frac{d\ln h}{d\ln
T}\right]\,.$ (13)

We solve Eq. (12) numerically to obtain the co-moving number density at late
times $Y_{\pi}(\infty)$, and from it the relic density of CDM today. A
boundary condition must be provided. We take it to be $Y_{\pi}$ at a higher
temperature, before freezeout but with $T<M_{\pi}$ ($x>1$) so that dEFT is
applicable and the pNGBs are moving at a nonrelativistic typical speed. At
which point we set $Y_{\pi}$ to its non-relativistic, thermal-equilibrium
value $Y_{\pi}^{eq}$ (the right-hand side of Eq. (12) causes $Y_{\pi}$ to
approach $Y_{\pi}^{eq}$ in the range $x>1$, independently of its behavior at
higher temperatures). The requisite late time value $Y_{\pi}(\infty)$ is then
insensitive to the behavior of $Y_{\pi}$ at smaller $x$, earlier in
cosmological history.

Figure 1: In red, the numerical solution to Eq. (12) for the yield function
$Y_{\pi}$ is shown for the following choice of model parameters:
$M_{\pi}=1\,\text{GeV}$, $M_{\pi}/F_{\pi}=4$, $F^{2}_{\pi}/F^{2}_{d}=0.1$,
$\Delta=0.3$, $\gamma=3$ and $y=2$. The equilibrium yield
$Y_{\pi}^{\text{eq}}$ is plotted as the solid black line for comparison. The
freezeout temperature, $x_{f}$, is represented as the vertical dashed line.
The yield at late times (large $x$) is plotted as the dashed horizontal line.

We plot $Y_{\pi}(x)$ in Fig. 1, for an illustrative choice of dEFT parameters.
Its qualitative behavior is similar for a wide range of parameter choices. The
late-time value, $Y_{\pi}(\infty)$, which depends on dEFT parameters,
determines the relic density of CDM today through the expression

$\displaystyle\Omega_{\rm
CDM}h^{2}=\frac{M_{\pi}s_{0}Y_{\pi}(\infty)}{\rho_{c}/h^{2}}\,,$ (14)

where $s_{0}=2970\text{ cm}^{-3}$ is the entropy density today. Setting the
relic density to its observed value $\Omega_{CDM}h^{2}=0.120\pm 0.001$ [3]
allows us to derive a constraint on the allowed parameter space of dEFT.

The form of this constraint depends on the parameter sensitivity of
$Y_{\pi}(\infty)$, determined through the numerical solution to Eq. (12). In
Fig. 2, we plot the allowed values of $M_{\pi}$ versus $\Delta$ for three
fixed values of the ratio $M_{\pi}/F_{\pi}$. These correspond to three fixed
values of the effective expansion parameter of dEFT.

Fig. 2 can be understood qualitatively by inspection of Eqs. (12) and (13),
which determine $Y_{\pi}(\infty)$. It can be argued that this quantity grows
approximately linearly with $M_{\pi}$ and exponentially with $\Delta$. This
behavior can be exhibited explicitly if Eq. (12) is solved analytically, which
is possible if the evolution takes place at temperatures $T$ where $g(T)$ and
$h(T)$ are approximately constant. These functions however vary rapidly for
temperatures around the QCD confinement scale, giving rise to the kinks
visible in Fig. 2 for $M_{\pi}\sim 3$ GeV.

Figure 2: The bands indicate the parts of the parameter space for which the
dark-matter relic density is within 10% of its observed value. The red, gray
and blue colors correspond to three values of the quantity $M_{\pi}/F_{\pi}$,
which determines the dark matter self interaction coupling. In the bottom
right corner, there are portions of the bands shaded in paler colors, for
which the dark-matter mass falls below the lower bound shown in Eq. (17),
setting $\sigma_{\text{max}}/M_{\pi}=2\,\text{cm}^{2}/g$. For reference, we
have also set $F^{2}_{\pi}/F^{2}_{d}=0.1$, $\gamma=3$ and $y=2$.

As indicated by the shading in Figure 2, the range of $M_{\pi}$ values is
further constrained, from below, by the strength of elastic dark matter
scattering. The behavior of merging galaxy clusters limits the size of this
cross section to lie below the bound [75, 76]

$\displaystyle\frac{\sigma}{M_{\pi}}\leq\frac{\sigma_{\text{max}}}{M_{\pi}}\approx
2\,\text{cm}^{2}/\text{g}\,.$ (15)

The elastic cross section has the form

$\sigma=\frac{M^{2}_{\pi}}{128\pi
F^{4}_{\pi}N^{2}_{\pi}}\bigg{[}\frac{a^{2}}{4}-\frac{b^{2}M^{2}_{\pi}F^{2}_{\pi}}{(4M^{2}_{\pi}-M^{2}_{d})F^{2}_{d}}\\\
+\frac{c^{2}M^{4}_{\pi}F^{4}_{\pi}}{(4M^{2}_{\pi}-M^{2}_{d})^{2}F^{4}_{d}}\bigg{]}\,,$
(16)

where the parameters $a^{2}$, $b^{2}$, and $c^{2}$ are listed in Table 1. In
the regime $F_{\pi}^{2}\ll F_{d}^{2}$, the contribution of the dilaton to Eq.
(16) is negligible and the terms proportional to $b^{2}$ and $c^{2}$ can be
dropped. Our thermal cross section then coincides with the result of Ref.
[62]. For the $N_{f}=8$ theory, the lower bound on $M_{\pi}$ becomes

$\displaystyle M_{\pi}\geq 11.8\text{ MeV
}\left(\frac{M_{\pi}}{F_{\pi}}\right)^{4/3}\left(\frac{2\,\text{cm}^{2}/\text{g}}{\sigma_{\text{max}}/M_{\pi}}\right)^{1/3}\,.$
(17)

$a^{2}N^{2}_{f}$ | $8(N^{2}_{f}-1)(3N^{4}_{f}-2N^{2}_{f}+6)$
---|---
$b^{2}N_{f}$ | $64(N^{2}_{f}-1)(2N^{2}_{f}-1)$
$c^{2}$ | $256(N^{2}_{f}-1)^{2}$
Table 1: Coefficients that appear in the $2\pi\rightarrow 2\pi$ scattering
cross section in Eq. (16).

Using the numerical solution of Eq. (12), we can also determine the freezeout
temperature, $T_{f}$. We take this to be the temperature below which the
comoving number density of pNGBs, $Y_{\pi}(x)$, begins to rise significantly
above the value it would have if it were in thermal equilibrium with the rest
of the SM $Y_{\pi}^{\text{eq}}(x)$, indicating that the pNGBs have decoupled
from the SM bath. Following the convention of [77], we take
$Y_{\pi}(x_{f})=2.5Y^{\text{eq}}_{\pi}(x_{f})$, where $x_{f}=M_{\pi}/T_{f}$.
See Figs. 1 and 3.

Figure 3: Plot of $x_{f}=M_{\pi}/T_{f}$ as a function of the dark-matter mass,
$M_{\pi}$, for different choices of $M_{\pi}/F_{\pi}$. The value of $\Delta$
has been adjusted to ensure that the dark matter relic density is equal to its
observed value. For reference, we have also set $F^{2}_{\pi}/F^{2}_{d}=0.1$,
$\gamma=3$ and $y=2$.

###  III.1 Subdominant $3\rightarrow 2$ Processes

In addition to the $2\rightarrow 2$ annihilations of dark-matter pNGBs to
dilatons, the population of pNGBs can also be depleted through $3\rightarrow
2$ processes, as in the composite SIMP models of Refs. [62, 63, 64, 65].
There, with the dark matter described only by pNGBs (no dilaton), the relevant
five-point interaction arises from the Wess-Zumino-Witten term. In our
framework, we must also account for the $3\rightarrow 2$ process,
$\pi\pi\pi\rightarrow\chi\pi$, with an outgoing dilaton, which arises at
leading order in dEFT. It is unsuppressed thermally, as long as
$M_{d}<2M_{\pi}$.

The relative importance of this process during freezeout is given by the ratio

$\displaystyle
R=\left.\frac{n^{\text{eq}}_{\pi}\langle\sigma_{3\pi\rightarrow\chi\pi}v^{2}\rangle}{\langle\sigma_{2\pi\rightarrow
2\chi}v\rangle}\right|_{x=x_{f}}\,.$ (18)

The denominator $2\rightarrow 2$ cross section is given by Eqs. (7) and (9),
which at freezeout, with $\Delta$ and $\gamma$ in the relevant range, is of
order

$\displaystyle\langle\sigma_{2\pi\rightarrow
2\chi}v\rangle\approx\frac{M^{2}_{\pi}e^{-2\Delta
x_{f}}}{F^{4}_{d}N_{\pi}}\,.$ (19)

The numerator $3\rightarrow 2$ cross section, with mass dimension $-5$, arises
from a tree level amplitude of order $M^{2}_{\pi}/(F_{\pi}^{2}F_{d})$. The
phase-space integral is similar to that of the $2\rightarrow 2$ cross section.
Thus we expect the cross section to be roughly of order

$\displaystyle\langle\sigma_{3\pi\rightarrow\chi\pi}v^{2}\rangle\approx\frac{M_{\pi}}{F^{4}_{\pi}F^{2}_{d}N_{\pi}}\,.$
(20)

Using Eq. (6) for the equilibrium number density, we have

$\displaystyle R\approx
N_{\pi}\frac{M^{2}_{\pi}F^{2}_{d}}{F^{4}_{\pi}}x_{f}^{-3/2}e^{-(1-2\Delta)x_{f}}\,.$
(21)

Thus $R$ is exponentially suppressed due to the factor $n_{\pi}^{eq}$
associated with the $3\rightarrow 2$ process, provided that $\Delta<0.5$. This
restriction, implemented in Fig. 2, is satisfied by the LSD lattice data, and
for a range of $M_{\pi}$ values closer to the chiral limit. We note that the
relative importance of $2\rightarrow 2$ and $3\rightarrow 2$ processes has
also been discussed in a different set of models [78, 79, 80, 81, 82].

##  IV Interaction with the SM

The FDM mechanism and the form of Eq. (5) incorporate thermal equilibrium
between the standard model and the dark-sector dilaton. To provide the
necessary coupling between the dilaton and the standard model, we introduce
weak contact interactions of the form

$\displaystyle\mathcal{L}_{\text{int}}=\epsilon
F_{d}^{4-d_{\text{SM}}}\left(\frac{\bar{\chi}}{F_{d}}\right)\mathcal{O}_{\text{SM}}\,,$
(22)

where $\bar{\chi}\equiv\chi-F_{d}$, and $\mathcal{O}_{SM}$ is an SM-singlet
operator with engineering dimension $d_{\text{SM}}$. We envision that this
interaction arises at high scales, well above those of the dEFT and the SM,
leading to a small value for the dimensionless parameter $\epsilon$. We do not
describe the operators $\mathcal{O}_{SM}$ in detail here, noting only two key
features. Firstly, they are built from SM fields that are sufficiently light
after electroweak symmetry breaking to contribute to the decay of the dilaton.
Secondly, since the dilaton, $\chi$, emerges within only the dark sector, they
are not directly associated with the possibility of spontaneous breaking of
approximate scale symmetry in the SM. Thus the operators $\mathcal{O}_{SM}$
are not constrained to be related to the trace of the energy momentum tensor
of the SM.

We denote the inclusive decay rate of the dilaton into SM particles as
$\Gamma_{\chi\rightarrow\text{SM}}$. It must be large enough to ensure that
the dark sector and the SM reach thermal equilibrium well before the freezeout
process begins. We take this to be at a common temperature of order $M_{\pi}$,
just as the dark matter is becoming non-relativistic. The lower limit is taken
to be

$\Gamma_{\chi\rightarrow\text{SM}}\gtrsim H_{T=M_{\pi}}\,.$ (23)

It is also important that the direct annihilation of pNGBs to SM particles
through a virtual dilaton does not dominate the forbidden annihilations. This
leads to the inequality

$\langle\sigma_{2\pi\rightarrow\text{SM}}v\rangle\lesssim\langle\sigma_{2\pi\rightarrow
2\chi}v\rangle\,,$ (24)

where the strongest bound comes when the right hand side is taken to be the
cross section at roughly the freezeout temperature. There,
$\langle\sigma_{2\pi\rightarrow 2\chi}v\rangle\approx
H_{T=T_{f}}/n_{\pi}^{\text{eq}}(T_{f})$.

Equation (24) leads to an upper bound on $\Gamma_{\chi\rightarrow SM}$ since
the dominant contribution to the left-hand side comes from an s-channel
dilaton exchange. The general “resonance” expression for this cross section is

$\displaystyle\langle\sigma_{2\pi\rightarrow\text{SM}}v\rangle=\frac{M^{5}_{d}\Gamma_{\chi\rightarrow\text{SM}}}{4N_{\pi}M^{2}_{\pi}F^{2}_{d}\left(4M^{2}_{\pi}-M^{2}_{d}\right)^{2}}\,,$
(25)

where we have assumed a narrow width and taken $y=2$, as before. Since
$2M_{\pi}>M_{d}$ throughout the parameter space we consider, the dilaton is
unable to decay to two pNGBs, and is well off shell. However, the resonance
formula may still be applied in this case [83].

We combine Eqs. (23) - (25), employing Eq. (7) for
$\langle\sigma_{2\pi\rightarrow 2\chi}v\rangle$. This leads to the order-of-
magnitude double-sided bound

$\displaystyle H_{T=M_{\pi}}\lesssim\Gamma_{\chi\rightarrow\text{SM}}\lesssim
H_{T=T_{f}}\frac{M_{\pi}N_{\pi}F^{2}_{d}}{n_{\pi}^{\text{eq}}(T_{f})}\,,$ (26)

where we have disregarded numerical factors in the upper bound, anticipating
that its relative largeness is determined dominantly by the exponential
suppression of $n_{\pi}^{\text{eq}}$ in the denominator. The two bounds insure
a sizable allowed range for the decay rate $\Gamma_{\chi\rightarrow SM}$, but
restricted to very small values relative to $M_{d}\approx M_{\pi}$.

If the decay rate $\Gamma_{\chi\rightarrow SM}$ arises from a single
interaction of the form Eq. (22), we can roughly bound the parameter
$\epsilon$. For the case $d_{\text{SM}}=3$, and taking the decay to be into
two light SM states, we have
$\Gamma_{\chi\rightarrow\text{SM}}\approx\epsilon^{2}M_{d}/16\pi$. For the
choice $M_{\pi}/F_{\pi}=1$ for example, and taking $M_{\pi}\approx M_{d}=1$
GeV, the forbidden mechanism is viable providing
$10^{-9}\lesssim\epsilon\lesssim 10^{-4}$. Both the lower and upper bounds
grow with $M_{\pi}$, but remain very small for the range of allowed mass
values shown in Fig. 2.

The magnitude of the weak dilaton-SM interactions in Eq. (22), the structure
of the operators $\mathcal{O}_{\text{SM}}$, and the mass scale $M_{\pi}$ will
determine possible experimental signatures of the dark matter. These could
include dilaton production and decay in collider searches as well as signals
in direct and indirect detection experiments. It will be valuable to develop
predictions for these phenomena, which will involve further modeling of the
dilaton-SM interactions.

##  V Summary and Discussion

We have proposed a description of composite dark matter based on dilaton
effective field theory (dEFT). The dark-matter particles are pseudo-Nambu-
Goldstone bosons (pNGBs) arising from an underlying, near conformal gauge
theory. Lattice studies of near conformal gauge theories have led also to a
relatively light scalar particle, identified as an approximate dilaton in
dEFT. In the fermion-mass range of the lattice studies, the dilaton mass
$M_{d}$ is of order the pNGB mass $M_{\pi}$, but somewhat larger.

Thus the dEFT provides a natural implementation of the forbidden dark-matter
framework [14, 15], in which freezeout of the dark-matter population is
described by $2\rightarrow 2$ scattering to somewhat heavier particles. The
process takes two pNGBs to two dilatons (which then transition rapidly to SM
particles). Forbidden at zero temperature by the dilaton-pNGB mass gap, this
process becomes allowed but suppressed at finite temperatures small compared
to $M_{\pi}$.

The dark-sector dEFT we have employed derives from an underlying
$\mathrm{SU}(3)$ gauge theory with $N_{f}=8$ Dirac fermions. After describing
the dEFT, along with an effective coupling to the SM, we have discussed the
computation of the relic dark-matter density in terms of the late-time, co-
moving number density $Y_{\pi}(\infty)$ (Eq. (14)). Setting the relic density
to its observed value has then allowed us to derive a constraint on the
allowed parameter space of the dEFT.

We have adopted the value $F_{\pi}^{2}/F_{d}^{2}\sim 0.1$ for the ratio of
pNGB and dilaton decay constants, as suggested by the lattice computations. We
have then considered several values for the ratio $M_{\pi}/F_{\pi}$, which
determines the interaction strength of the dEFT. One, $M_{\pi}/F_{\pi}=4$, is
typical of the current range of lattice data. It leads to an effective dEFT
coupling strength of order $0.5$ (at the edge of weak coupling). The other,
smaller values for $M_{\pi}/F_{\pi}$ will emerge from lattice studies closer
to the chiral limit, and place the dEFT further inside the weak coupling
range.

With either of these choices for $M_{\pi}/F_{\pi}$, a correlated set of values
for $M_{\pi}$ and $\Delta\equiv(M_{d}-M_{\pi})/M_{\pi}$ yields the measured
relic dark-matter density, as shown in Fig. 2. For the case
$M_{\pi}/F_{\pi}=1$, for example, $M_{\pi}$ can range up to roughly $10$ GeV
without the need to fine tune $\Delta$ close to $0$. It can range down to
roughly $10$ MeV, keeping $\Delta$ below $0.4$, ensuring that the forbidden
process dominates the SIMP ($3\rightarrow 2$) process. Freezeout of the dark
matter takes place at temperatures of order $M_{\pi}/25$, where the dEFT is
reliable and the dark matter is non-relativistic.

We have briefly discussed the effective interaction that must be present
between the dEFT dark sector and the particles of the SM, observing that a
range of very weak couplings are allowed, indicating that the interactions
arise from new physics at a very high scale. The detailed form of these
interactions will dictate possible experimental signatures of the dark matter.

The particular dEFT we have employed to describe the dark sector, linked
reassuringly to a specific underlying gauge theory studied on the lattice,
should be regarded as only one possibility. Other underlying gauge theories
that yield a light scalar could lead to other dEFTs describing all the
emergent light bound states.

###### Acknowledgements.

We gratefully acknowledge Suchita Kulkarni for providing comments on earlier
drafts of our manuscript.
Open Access Statement – For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising. Research Data Access Statement – No new data were
generated for this manuscript.

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