Muonphilic Portals to Fermionic ADM

• The paper presents muonphilic portals with fermionic ADM coupling to muons via effective dimension-6 operators and UV completions.
• It details the methodology and constraints from direct detection, neutron-star heating, and collider bounds, emphasizing experimental signatures.
• The work outlines viable parameter space ensuring over 99% asymmetric relic density, contrasting EFT approaches with axial Lµ-Lτ models.

Muonphilic portals to fermionic asymmetric dark matter (ADM) form a well-motivated, minimal scenario in which dark matter couples primarily or exclusively to second-generation leptons—specifically muons—via effective operators or explicit new gauge interactions. These models are strongly motivated both by the ADM paradigm, which demands efficient annihilation for the symmetric thermal DM component, and by the search for dark matter candidates compatible with the observed baryon–dark matter relic density coincidence. The most robust implementations invoke either weak effective theory (WEFT) dimension-6 operators or ultraviolet (UV) completions based on gauged $L_\mu-L_\tau$. Viable muonphilic portals provide both novel phenomenology and distinctive experimental signatures, particularly relevant for future high-energy muon colliders (Roy et al., 29 Dec 2025).

1. Dimension-6 Muonphilic Operators in Weak EFT

Below a cutoff scale $\Lambda$, interactions between Dirac dark matter $\chi$ and the Standard Model (SM) muon $\mu$ are parameterized by the four-fermion Lagrangian: $$\mathcal{L}_{\rm WEFT} =\mathcal{L}_{\rm SM} +\sum_i\frac{C_i}{\Lambda^2}\,O_i +\sum_j\frac{C_j\,v_h}{\Lambda^3}\,O_j \quad(v_h=246\,\mathrm{GeV})$$ Here, $O_i$ are dimension-6 operators with dimensionless Wilson coefficients $C_i$ (set to 1 individually for phenomenological scans). The ten independent operators coupling only muons and $\chi$ are:

Operator	Structure
$O_{ss}$	$(\bar{\mu}\mu)(\bar{\chi}\chi)$
$O_{pp}$	$(\bar{\mu}\gamma^5\mu)(\bar{\chi}\gamma^5\chi)$
$O_{sp}$	$(\bar{\mu}\mu)(\bar{\chi}i\gamma^5\chi)$
$O_{ps}$	$(\bar{\mu}i\gamma^5\mu)(\bar{\chi}\chi)$
$O_{vv}$	$(\bar{\mu}\gamma^\mu\mu)(\bar{\chi}\gamma_\mu\chi)$
$O_{aa}$	$$(\bar{\mu}\gamma^\mu\gamma^5\mu)(\bar{\chi}\gamma_\mu\gamma^5\chi)$$
$O_{va}$	$$(\bar{\mu}\gamma^\mu\mu)(\bar{\chi}\gamma_\mu\gamma^5\chi)$$
$O_{av}$	$$(\bar{\mu}\gamma^\mu\gamma^5\mu)(\bar{\chi}\gamma_\mu\chi)$$
$O_{tt}$	$$(\bar{\mu}\sigma^{\mu\nu}\mu)(\bar{\chi}\sigma_{\mu\nu}\chi)$$
$O_{pt}$	$$(\bar{\mu}i\sigma^{\mu\nu}\mu)(\bar{\chi}\sigma_{\mu\nu}\gamma^5\chi)$$

Operators $O_{ss}$, $O_{ps}$, and $O_{va}$ yield $p$-wave suppressed annihilation rates; $O_{aa}$ is $s$-wave but its annihilation is helicity-suppressed by $m_\mu^2/m_\chi^2$. The rest are unsuppressed $s$-wave. Nuclear scattering for these operators is loop-induced by attaching photons to muon lines, rendering direct-detection rates negligible relative to tree-level quark-coupling models (Roy et al., 29 Dec 2025).

2. Gauged $L_\mu-L_\tau$ UV Models

Gauging $U(1)_{L_\mu-L_\tau}$ introduces a new $Z'$ mediator coupling only to $\mu$, $\tau$, and their corresponding neutrinos, along with the DM sector. Two UV-complete scenarios are relevant:

2.1 Vector-Coupled Dark Matter

The Lagrangian is: $$\mathcal{L}_V = \mathcal{L}_{\rm SM} -\frac14Z'_{\alpha\beta}Z'^{\alpha\beta} +\frac12m_{Z'}^2Z'_\alpha Z'^\alpha +\frac{\varepsilon}{2}Z'_{\alpha\beta}F^{\alpha\beta} +g'\,J^\mu_{L_\mu-L_\tau}Z'_\mu +g_\chi\bar{\chi}\gamma^\mu\chi\,Z'_\mu$$ with $g'$ the $Z'$–lepton coupling, $g_\chi=g'Q'(\chi)$ the vector DM coupling, and $m_{Z'}$ the $Z'$ mass. The $J^\mu_{L_\mu-L_\tau}$ current ensures the $Z'$ is muonphilic. DM annihilates as $\chi\bar{\chi}\to Z'Z'$, $\ell^+\ell^-$, and $\nu_\ell\bar{\nu}_\ell$ ($\ell=\mu,\tau$).

2.2 Axial-Coupled Dark Matter

Anomaly cancellation requires two singlets ($\chi$, $\psi$) and a complex scalar $S$ with chiral $U(1)'$ charges. After symmetry breaking, $S$ acquires a vev, mixing $\chi$ and $\psi$ into mass eigenstates $X_1$ and $X_2$. The lightest state $X_1$ couples axially: $$\mathcal{L} \supset g_{\rm ax}Z'_\mu\,\bar{X}_1\gamma^\mu\gamma^5X_1$$. Annihilation channels and kinematic suppression differ from the vector case due to the distinct chiral structure.

3. Relic Abundance and the Asymmetric Criterion

Fermionic ADM scenarios require that at least $99\%$ of the dark matter relic density survive in the asymmetric component. Using the comoving densities,

$$Y_\chi = \frac{n_\chi}{s}, \quad Y_{\bar\chi} = \frac{n_{\bar\chi}}{s}$$

and $Y_{\rm asy}=Y_\chi-Y_{\bar\chi}$, $Y_{\rm sym}=2Y_{\bar\chi}$, the symmetric relic after freeze-out is: $$Y_{\rm sym} = \frac{2\,Y_{\rm asy}}{\exp\left[Y_{\rm asy}\lambda\left(a/x_F+3b/x_F^2\right)\right]-1}$$ where $a$ and $b$ are the $s$- and $p$-wave coefficients of $\langle\sigma v\rangle$, $$\lambda=4\pi/(\sqrt{90}m_\chi M_{\rm Pl}\sqrt{g_*})$$, and $x_F$ the modified freeze-out parameter. The asymmetric-DM criterion imposes: $$Y_{\rm sym} \leq 0.01\,\frac{\Omega_{\rm DM}h^2}{2.76\times10^8}\,\frac{\rm GeV}{m_\chi}$$ For EFT operators, this sets an upper bound $\Lambda<\Lambda_1(m_\chi)$, and for UV completions, an upper limit of the form $m_{Z'}/\sqrt{g'g_\chi}<X(m_\chi)$ (Roy et al., 29 Dec 2025).

4. Experimental and Astrophysical Constraints

Muonphilic ADM models are constrained by several complementary probes:

• Direct Detection: DM scattering arises only at loop level (e.g., $O_{vv}$ yields $$\sigma_{vv}^{\rm DD}\simeq\frac{1}{9}\sigma_N^{(1)}L_\mu^2$$, with $$\sigma_N^{(1)}=\frac{\mu_N^2}{\pi}\left(\frac{\alpha Z}{\pi\Lambda^2}\right)^2$$ and $L_\mu^2=\ln^2(m_\mu^2/\Lambda^2)$). Current direct-detection experiments (LZ, PandaX-4T, PICO) exclude $O_{vv}$ and $O_{pt}$ for almost the entire allowed DM mass range.
• Vector $L_\mu-L_\tau$ Model: For $m_{Z'}^2\gg q^2$, $$\sigma_{\chi N} = \frac{1}{A^2}\frac{\mu_{\chi N}^2}{9\pi}\left(\frac{\alpha_{\mathrm{em}}g'g_\chi}{\pi m_{Z'}^2}\ln\frac{m_\tau^2}{m_\mu^2}\right)^2 Z^2$$.
• Axial $L_\mu-L_\tau$ Model: Loop-induced $Z'-\gamma$ mixing and velocity suppression: $$\sigma_{X_1N} = \frac{1}{A^2}\frac{\mu_{X_1N}^2}{\pi}\frac{v^2}{12}\left(\frac{\alpha g' g_{\rm ax}}{\pi m_{Z'}^2}\ln\frac{m_\tau^2}{m_\mu^2}\right)^2 Z^2$$.
• Neutron-Star Heating: The DM capture rate in neutron stars with muons present (BSk24-2 model) rules out $O_{va}$, $O_{aa}$, and $O_{tt}$ up to scales $\Lambda\sim10^2$–$10^3$ GeV for $m_\chi$ in the few-GeV–TeV range.
• Collider Bounds: CMS $pp\to Z\to4\mu$ at 13 TeV excludes large regions of the $g'$–$m_{Z'}$ parameter space in the pure vector model. The neutrino trident process ($\nu_\mu N\to\nu_\mu\mu^+\mu^-N$) requires $m_{Z'}/g'\gtrsim 500$ GeV (CCFR).
• Muon $g-2$: A loop-level $Z'$ contribution to $(g-2)_\mu$: $\Delta a_\mu=(g'^2/12\pi^2)(m_\mu^2/m_{Z'}^2)$, with the 2025 $2\sigma$ bound $\Delta a_\mu\leq1.7\times10^{-9}$ implying $$g' \lesssim m_{Z'}/(200\,\mathrm{GeV}) \sqrt{\Delta a_\mu^{\max}/2.36\times10^{-10}}$$.

5. Sensitivity at Future Muon Colliders

Prospective high-energy muon colliders (3–10 TeV, 1 ab$^{-1}$) enable distinctive probes via initial-state radiation (ISR) mono-photon searches: $\mu^+\mu^-\to\chi\bar{\chi}\,\gamma$

• EFT Operators: For $m_{Z'}\gg\sqrt{s}$, $\sigma\sim s/\Lambda^4$ (log-enhanced). Sensitivity at 3 TeV reaches $\Lambda \sim$ several TeV for $m_\chi\lesssim$ few 100 GeV; at 10 TeV, up to $\Lambda \sim 10$ TeV.
• Vector $L_\mu-L_\tau$ Model: Muon collider limits are weaker than existing direct detection, $g-2$, trident, and CMS bounds, except in a narrow $m_{Z'}\approx2m_\chi$ corridor.
• Axial $L_\mu-L_\tau$ Model: For $m_\chi\sim500$ GeV, a 3 TeV collider with moderate kinetic/mass mixing can probe beyond current astrophysical and collider exclusions (see (Roy et al., 29 Dec 2025), Fig. 13).

Event selection involves identifying a single isolated photon with transverse momentum, rapidity, and photon-energy-fraction cuts ($f_E\equiv E_\gamma/E_{\rm beam}$), optimizing for either EFT or on-shell $Z'$ regimes.

6. Viable Parameter Space and Portal Classification

Empirical constraints and collider projections yield the following summary of viable muonphilic ADM portals:

Portal Type	Viability
$O_{vv}, O_{pt}$	Ruled out by DD for most $m_\chi$
$O_{sp}, O_{ps}, O_{pp}$	Unconstrained by DD/trident/$g-2$, probed by NS heating up to $\Lambda\sim10^2$–$10^3$ GeV
$O_{va}, O_{av}, O_{aa}, O_{tt}$	Mostly NS-heating constrained; muon colliders probe at high $m_\chi$
Vector $L_\mu-L_\tau$	Excluded except ultra-narrow resonance $m_{Z'}\approx2m_\chi$
Axial $L_\mu-L_\tau$	Large viable space for $m_\chi\gtrsim50$ GeV; can be tested at future muon colliders

Thus, the viable muonphilic portals under current and projected constraints are (i) EFT operators with axial or pseudoscalar muon currents, and (ii) the gauged axial-vector $L_\mu-L_\tau$ UV completion. Upcoming muon colliders (3–10 TeV) will uniquely probe cutoff scales up to $\sim10$ TeV in the EFT, and extend sensitivity to the high-$m_\chi$, axial-$Z'$ parameter space beyond the reach of current astrophysical, collider, and precision observables (Roy et al., 29 Dec 2025).