QCD axion dark matter in the dark dimension

Here we briefly discuss axions that originate from higher-dimensional gauge fields. For a more comprehensive understanding, please refer to recent refs. Reece:2024wrn ; Choi:2024ome .

Firstly, we demonstrate that the theory of an higher-dimensional $p$-form gauge field can generate massless four-dimensional axions. Assume that spacetime consists of $d=(4+n)$ dimensions, which is manifested as a warped product compactification

where $X$ represents the four-dimensional spacetime, and $Y$ denotes the $n$-dimensional space. The metric is given by

where $w(y)\geq 0$ is the warping. If there exists a $p$-form gauge field on $M$, then for every independent non-torsion $p$-cycle in $Y$, there corresponds a distinct massless, periodic four-dimensional axion field. Considering a $p$-form U(1) gauge field

where $\wedge$ is the wedge product, it includes the local gauge transformations

and the large gauge transformations

where $b_{p}(Y)$ is the Betti number, and $\omega_{p}^{(i)}$ represents the corresponding cohomology class. In $d=(4+n)$ dimensions, the standard kinetic term for the $p$-form gauge field can be described by

where $e_{p}$ represents the $p$-form gauge coupling, $\Phi(x,y)$ denotes the scalar modulus field, and $\star$ is the Hodge star operation, which transforms a $p$-form into a $(d-p)$-form in $d$-dimensional spacetime. The equation of motion of $A_{p}$ is given by

Assuming a background solution where $\Phi=\Phi(y)$ is independent of $x$, an ansatz for a perturbed four-dimensional axion field around this background is expressed as

where $\omega_{p}$ is a $p$-form on $Y$. In general, considering a collection of such axions, the ansatz generalizes to

where $\widehat{\omega}_{p}^{(i)}$ represents the warped harmonic representative. Plugging this into eq. (9), the four-dimensional kinetic term is given by

with the matrix

The diagonal entries of the kinetic matrix $k_{ij}$ can be interpreted as the squares of the axion decay constants, denoted by

In the context of large gauge transformations, the redundancy manifests as a precise periodicity constraint on the four-dimensional axions, expressed as

Above is the depiction of massless four-dimensional axions originating from higher-dimensional $p$-form gauge fields. Additionally, there are effects that contribute to a small mass for these four-dimensional axions, such as couplings to objects carrying a $A_{p}$ gauge charge, and Chern-Simons interactions involving $A_{p}$ with itself or other gauge fields. The existence of such axions represents a topological characteristic of the theory. Specifically, there is no requirement for supplementary frameworks such as supersymmetry to elucidate why an axion in an extra dimension is light; it is inherently exponentially suppressed in mass, as long as it does not acquire a tree-level mass contribution from the topological terms. Consequently, they alleviate the axion quality problem significantly Reece:2024wrn .

In this subsection, we briefly review the QCD axion in the dark dimension.

The dark dimension scenario Montero:2022prj predicts one extra mesoscopic dimension in the micron range, which is achieved by applying various Swampland principles to the dark energy. The dark energy in Planck units $\Lambda$ is a hierarchically small parameter. Since the emergence of any small parameters within the framework of quantum gravity necessitates the presence of towers of light states, which can arise due to either some dimensions expanding to a large size or a critical string becoming light Lee:2019wij , the sole experimentally allowed model featuring large extra dimensions in this context is one that involves a single large extra dimension with the length

corresponding to the five-dimensional Planck mass

where $M_{p}$ is the Planck mass in our four-dimensional spacetime. In this scenario, the SM fields should be localized in the fifth dimension, as any deviation would result in the production of a tower of light particles for each SM field — a phenomenon inconsistent with experimental observations.

The QCD axion within the context of the dark dimension was recently investigated in ref. Gendler:2024gdo . That work delves into two scenarios: one where the QCD axion is localized on the SM brane, and another where the axion propagates freely within the bulk of the fifth dimension. They find that when the QCD axion is localized on the SM brane, the axion decay constant is subject to an upper bound, as dictated by the WGC:

The WGC Arkani-Hamed:2006emk asserts that, for compatibility with quantum gravity, there must exist a U(1) charged particle whose gravitational interaction is weaker than its U(1) gauge interaction. In Planck units, this translates to the existence of a “superextremal” particle that satisfies the following inequality

where $Q=qg$, with $q$ being the quantized charge and $g$ the gauge coupling, $\mathcal{Q}$ and $M$ represent the charge and mass of an extremal black hole, respectively. The rationale behind this conjecture is that, in the absence of such a particle adhering to the WGC bound, an extremal charged black hole would be unable to discharge through the Hawking evaporation and would instead decay into a charged remnant, which poses issues with the Bekenstein entropy bound Bekenstein:1973ur . Notice also that in the context of multiple U(1) gauge interactions, there exists a strong version of the WGC Cheung:2014vva . For particle species $i$, with charge vector $\vec{q}_{i}$, mass $m_{i}$, and defining the vector $\vec{z}_{i}\equiv\vec{q}_{i}/m_{i}$ (in units where an extremal black hole has $|\vec{z}_{\rm BH}|=1$); the mild WGC is equivalent to the fact that the convex hull of the vectors $\vec{z}_{i}$ contains the unit ball. For recent reviews on this topic, see $\rm e.g.$ refs. Harlow:2022ich ; vanBeest:2021lhn ; Reece:2023czb ; Rudelius:2024mhq .

The WGC extends its applicability beyond mere particles, encompassing any charged state associated with a $p$-form gauge field. Specifically, for any given $p$-form gauge field, there ought to exist a ($p-1$)-dimensional state, characterized by a charge-to-mass ratio that surpasses or equals that of an extremal ($p-1$) black brane. When applied to the $0$-form gauge field, namely the axion, we have the inequality

where $S_{\rm inst}$ represents the instanton action, and $f_{a}$ is the axion decay constant. It’s worth noting that, due to the absence of an extremal solution for the instanton, this inequality encompasses an undefined factor of order one. Given that the dilute instanton gas approximation necessitates $S_{\rm inst}\gtrsim 1$ to maintain control over the instanton expansion, we can deduce the axion WGC bound

Now, let’s consider a four-dimensional brane localized in the fifth dimension and apply the WGC to particles localized on this brane. This yields

which can be rewritten by

In the most extreme scenario, where $L_{5}\sim M_{5}^{-1}$, the inequality simplifies to

In this case, we obtain the axion WGC bound Gendler:2024gdo

Alternatively, this inequality can be derived by considering axion propagation throughout the entire five-dimensional bulk. Combining this with the observational lower bound on the classical QCD axion window, $f_{a}\gtrsim 10^{8}-10^{9}\,{\rm GeV}$, the QCD axion localized on the SM brane falls within a narrow range for the axion decay constant

In this context, it’s reasonable to consider the QCD axion as the DM candidate, which is the focus of this work. However, as discussed in Gendler:2024gdo , the QCD axion within this range can only contribute a small fraction to the overall DM density.

Here we discuss the QCD axion DM in the dark dimension scenario through the misalignment mechanism Preskill:1982cy ; Abbott:1982af ; Dine:1982ah .

Since the extra-dimensional axion is not associated with the cosmological PQ phase transition, as discussed in ref. Reece:2023czb , we can consider it within the canonical pre-inflationary scenario where the PQ symmetry is spontaneously broken during inflation. The low-energy effective Lagrangian of the QCD axion, stemming from the QCD non-perturbative effects, can be described by

where $\theta=\phi/f_{a}$ is the QCD axion angle, $\phi$ represents the QCD axion field, $m_{a}$ and $f_{a}$ denote the axion mass and decay constant, respectively. Notice that the QCD axion mass is temperature-dependent, which is given by

where $T_{\rm QCD}\simeq 150\,\rm MeV$ is the critical temperature of the QCD phase transition, and $b\simeq 4.08$ is an index derived from the dilute instanton gas approximation Borsanyi:2016ksw . The mass at temperatures below $T_{\rm QCD}$ is referred to as the zero-temperature QCD axion mass, which is given by

where $m_{\pi}$ and $f_{\pi}$ represent the mass and decay constant of the pion, respectively, $m_{u}$ and $m_{d}$ are the up and down quark masses.

At high cosmic temperatures, the QCD axion field remains frozen at its initial misalignment angle $\theta_{i}$. As the temperature decreases, the axion field begins to oscillate when its mass becomes comparable to the Hubble parameter

where $T_{i,a}$ is the oscillation temperature

The axion initial energy density at $T_{i,a}$ is given by

where $m_{a,i}$ is the axion mass at $T_{i,a}$, $f(\theta_{i})$ is the anharmonicity factor Lyth:1991ub , and $\chi\simeq 1.44$ is a numerical factor Turner:1985si . For temperatures $T_{0}<T<T_{i,a}$, the QCD axion energy density is adiabatic invariant with the comoving number

where $T_{0}$ is the current cosmic microwave background (CMB) temperature, and $a$ is the scale factor. The QCD axion energy density at $T_{0}$ is given by

where $a_{x}$ denotes the scale factor at the temperature $T_{x}$. Then the current QCD axion DM abundance can be described by

where $h\simeq 0.68$ is the reduced Hubble constant, $\rho_{\rm crit}$ is the critical energy density, $g_{*}(T)$ and $g_{*s}(T)$ are the numbers of effective degrees of freedom for the energy density and the entropy density, respectively. In order to explain the observed DM abundance, $\Omega_{\rm DM}h^{2}\simeq 0.12$ Planck:2018vyg , we require an $\sim\mathcal{O}(1)$ initial misalignment angle

Now considering the QCD axion DM in the dark dimension scenario with a narrow axion decay constant $f_{a}$ ranging from $10^{9}$ to $10^{10}\,\rm GeV$, we can obtain a small fraction of the overall DM density Gendler:2024gdo

where the initial misalignment angle $\theta_{i}$ is assumed to be of order one.

Then, the situation begins to take on a captivating turn. This is because, in general, the issue we face is the overproduction of the QCD axion DM abundance. Specifically, when the decay constant of the QCD axion exceeds $10^{12}\,\rm GeV$ and we only consider an initial misalignment angle of order unity, the axion DM abundance will be overproduced. Therefore, in this scenario, fine-tuning the initial misalignment angle emerges as a viable resolution. However, as analyzed above, when considering the QCD axion in the context of dark dimension, the values of the axion decay constant within the range of $\sim 10^{8}-10^{9}\,{\rm GeV}$ are inadequate to explain the observed DM abundance. Therefore, our subsequent endeavor will be to explore how to enhance the QCD axion DM abundance for relatively smaller decay constants $f_{a}\sim 10^{8}-10^{9}\,{\rm GeV}$. Specifically, we will consider the effect of two-axion mass mixing to address this issue Cyncynates:2023esj ; Li:2024okl .

In this subsection, we investigate the enhancement of the QCD axion DM abundance within the context of a simple two-axion mixing scenario. In particular, we consider the mixing between one QCD axion and one ALP.

The low-energy effective Lagrangian that describes this two-axion mixing can be formulated as follows

with the general mixing potential

where $\Theta=\varphi/f_{A}$ is the ALP angle, $\varphi$ represents the ALP field, $m_{A}$ and $f_{A}$ denote the ALP mass and decay constant, respectively, and $n_{ij}$ are the domain wall numbers. Notice that the ALP field is considered as the simplest case with a constant mass $m_{A}$. For our purpose, here the domain wall numbers should be taken as¹¹1The mixing potential for this particular selection was first explored in ref. Cyncynates:2023esj , where it was used to study the mixing between the QCD axion and sterile axion, ultimately resulting in the heavy QCD axion as DM. Additionally, for a different selections of domain wall numbers, the axion mixing potential can lead to a suppression of the QCD axion abundance, see $\rm e.g.$ refs. Kitajima:2014xla ; Daido:2015cba ; Ho:2018qur ; Li:2023xkn ; Li:2023uvt ; Li:2024okl ; Murai:2024nsp .

Given that the axion oscillation amplitudes are significantly smaller compared to their respective decay constants, the mass mixing matrix can be described by

By diagonalizing the mass mixing matrix, we can derive the heavy ($a_{h}$) and light ($a_{l}$) mass eigenstates

which represent the two distinct axion states arising from the mixing process. These eigenstates are associated with the mass eigenvalues $m_{h}(T)$ and $m_{l}(T)$, respectively.

Next we discuss the resonant conversion within the context of axion mixing. For our analysis, we consider the general prerequisites for axion resonant conversion, which are expressed as follows

The resonant conversion temperature $T_{R}$ is given by solving $d\left(m_{h}^{2}(T)-m_{l}^{2}(T)\right)/dT=0$, yielding the precise expression

In the scenario where $f_{A}\gg f_{a}$, it is possible to approximate the temperature $T_{R}$ such that the value of $m_{a}(T_{R})$ is approximately equal to $m_{A}$, leading to

See figure 1 for the distribution of $T_{R}$ as a function of the axion mass ratio $m_{a,0}/m_{A}$ with the QCD axion decay constant $f_{a}=1.0\times 10^{10}\,\rm GeV$. This figure highlights the dramatic changes in $T_{R}$ as the ALP decay constant $f_{A}$ increases. We can observe that $T_{R}$ demonstrates an asymptotic behavior when $f_{A}$ is slightly greater than $f_{a}$, and it is not necessary for $f_{A}$ to be significantly larger than $f_{a}$ for this distribution to manifest.

At temperatures above $T_{R}$, the heavy mass eigenstate $a_{h}$ primarily consists of the ALP, while the light mass eigenstate $a_{l}$ is dominated by the QCD axion. As the cosmic temperature decreases, the mass eigenvalues $m_{h}(T)$ and $m_{l}(T)$ converge at $T=T_{R}$ and then diverge. Below the temperature $T_{R}$, $a_{h}$ becomes the QCD axion, and $a_{l}$ transforms into the ALP. Here, the axion energy transition at $T_{R}$ is presumed to be adiabatic, which is approximately valid when

where $T_{i,a}$ represents the QCD axion oscillation temperature. In this case, the ALP begins oscillating prior to $T_{i,a}$. Subsequently, it is crucial to examine the range of $m_{A}$, which can be characterized by

where $m_{a,i}\equiv m_{a}(T_{i,a})$. Therefore, for distinct values of the QCD axion decay constant $f_{a}=10^{9}$ and $10^{10}\,\rm GeV$, respectively, we obtain the following ALP mass ranges

In order to determine the abundance of QCD axion DM in the context of axion mixing, we should begin with the ALP field. At high temperatures, the ALP field is frozen at its initial misalignment angle, $\Theta_{i}$. It begins to oscillate at the temperature $T_{i,A}$ with the initial energy density

where $T_{i,A}$ is given by $3H(T_{i,A})=m_{A}$. For temperatures $T_{R}<T<T_{i,A}$, the ALP energy density is adiabatic invariant with the comoving number $N_{A}\equiv\rho_{A}a^{3}/m_{A}$ and it at $T_{R}$ is expressed as

where $a_{i,A}$ and $a_{R}$ are the scale factors at $T_{i,A}$ and $T_{R}$, respectively. At $T=T_{R}$, the energy density of the ALP, $\rho_{A,R}$, is transferred to the QCD axion, $\rho_{a,R}$. For temperatures below $T_{R}$, the QCD axion energy density is adiabatic invariant and it at $T_{0}$ is given by

where $T_{0}$ is the present CMB temperature, and $a_{0}$ is the scale factor at $T_{0}$. Comparing this with the case without mixing, we find that the QCD axion energy density can be modified by a factor

Now, given the initial misalignment angles ($\theta_{i}$ and $\Theta_{i}$) of $\sim\mathcal{O}(1)$, and in order to account for the total DM density with $f_{a}\sim 10^{9}-10^{10}\,\rm GeV$, we simply require that

In this case, the QCD axion DM abundance can be approximated as follows

where we neglect some trivial factors. By assuming a typical ALP mass of $m_{A}=10^{-5}\,\rm eV$, and considering the observed cold DM abundance $\Omega_{\rm DM}h^{2}\simeq 0.12$ with the QCD axion in the dark dimension, where $f_{a}\sim 10^{9}-10^{10}\,\rm GeV$, we find that the ALP decay constant falls within the range

Utilizing eqs. (59) and (65), we also show in figure 2 the allowed ALP parameter space in the $\{m_{A},1/f_{A}\}$ plane. The red and blue solid lines depict the scenarios where $f_{a}=10^{9}\,\rm GeV$ and $10^{10}\,\rm GeV$, respectively. The shaded gray quadrilateral area bounded by these two lines represents the allowed ALP parameter space within our scenario, encompassing the range that accounts for the overall DM abundance via resonant conversion. We also find that a significant portion of ALPs falling within this specified range have the potential to be detected by future experiments.

Additionally, in the context of considering a scenario with multiple QCD axions mass mixing Li:2024okl , the abundance of QCD axion DM can also be effectively enhanced, depending on the relationship among the QCD axion decay constants. That scenario should also be capable of explaining the issue of QCD axion DM abundance in the dark dimension, requiring further investigation.