Beyond the Standard Model: Hunting for Dark Matter in Pion Decay

Author: Denis Avetisyan

A new analysis details how upcoming experiments leveraging pion decays could reveal the existence of sterile neutrinos and other particles composing the universe’s hidden dark sector.

The PIENU experiment predicts the expected detection rate of <span class="katex-eq" data-katex-display="false">\pi^{+}\to\mu^{+}\nu_{\mu}X</span> decays as a function of muon kinetic energy, demonstrating how varying the mass of the undetected particle <i>X</i> impacts signal prediction, with analyses at both low and high energies-corresponding to branching ratios of <span class="katex-eq" data-katex-display="false">3\times 10^{-5}</span> and <span class="katex-eq" data-katex-display="false">6\times 10^{-5}</span>-yielding results consistent with previously extracted residuals. — The PIENU experiment predicts the expected detection rate of $\pi^{+}\to\mu^{+}\nu_{\mu}X$ decays as a function of muon kinetic energy, demonstrating how varying the mass of the undetected particle X impacts signal prediction, with analyses at both low and high energies-corresponding to branching ratios of $3\times 10^{-5}$ and $6\times 10^{-5}$ -yielding results consistent with previously extracted residuals.

This review examines the potential of exotic pion decays, particularly those studied by the PIONEER experiment, to constrain models of light dark matter and new physics beyond the Standard Model.

Despite compelling evidence for dark matter, the fundamental nature of dark sector particles remains elusive, motivating searches beyond standard model extensions. This paper, ‘Exploring Invisible New Physics with Exotic Pion Decays’, presents a detailed analysis of sensitivity to light, invisible dark sector particles through exotic pion decay channels, including sterile neutrinos and scalar/vector bosons. We demonstrate that the planned PIONEER experiment has the potential to improve current limits on exotic pion branching ratios by at least one order of magnitude, complementing constraints from other searches. Could precise measurements of these rare decays unlock new insights into the composition and interactions of the hidden dark universe?

The Emerging Gaps: When Standard Models Fail

Despite its extraordinary predictive power and consistent validation through decades of experimentation, the Standard Model of particle physics remains incomplete. Observations of gravitational effects suggest the existence of dark matter, a substance comprising roughly 85% of the universe’s mass, yet no Standard Model particle can fully account for its properties. Furthermore, neutrino oscillation experiments demonstrate that neutrinos possess mass – a feature not initially included in the Standard Model, necessitating modifications to accommodate this discovery. These discrepancies, alongside other unresolved puzzles, strongly indicate that the Standard Model represents only an effective theory, a piece of a larger, more fundamental framework waiting to be unveiled. The search for physics beyond this established model is therefore not a quest to disprove it, but rather to complete it, extending its reach to encompass the entirety of observed phenomena and unlock a deeper understanding of the universe.

The relentless pursuit of physics beyond the Standard Model increasingly relies on exquisitely precise measurements and the search for exceedingly rare particle decays. These investigations don’t seek direct collisions creating new, massive particles – instead, they meticulously examine known processes for subtle deviations from theoretical predictions. Minute discrepancies in a particle’s magnetic moment, or an unexpected rate for a normally suppressed decay, could signal the influence of virtual particles mediating new forces, or even extra dimensions. Experiments like those at the Large Hadron Collider and dedicated facilities are pushing the boundaries of measurement accuracy, while specialized detectors are designed to sift through vast datasets, hunting for events that are statistically improbable under the Standard Model, effectively illuminating the shadows where new physics might reside. This approach complements direct searches, offering a powerful, indirect pathway to uncover the fundamental building blocks and forces governing the universe.

Recent investigations into the behavior of leptons – fundamental particles including electrons and muons – have revealed intriguing deviations from predictions made by the Standard Model of particle physics. These anomalies, observed in experiments involving particle decays and collisions, suggest that leptons may be interacting with previously unknown forces or particles. Simultaneously, numerous experiments designed to meticulously track energy and momentum have detected instances of “missing energy” – energy that cannot be accounted for by known particles. This missing energy strongly implies the existence of weakly interacting, invisible particles, such as those posited in dark matter theories, escaping detection. The convergence of these unusual lepton behaviors and missing energy signatures provides compelling, albeit indirect, evidence for physics beyond the Standard Model, driving ongoing searches for new particles and interactions at facilities like the Large Hadron Collider and in dedicated precision experiments.

Current experimental constraints, including those from anomalous magnetic moments, mono-photon searches, beam dumps, pion decay (PIENU, red), and potential PIONEER sensitivity (blue), limit the couplings of dark sector particles to electrons as a function of particle mass for various models including scalars, axion-like particles, and spin-1 bosons with vector or axial-vector couplings.

Whispers of New Particles: Candidates Emerge

Sterile neutrinos are hypothesized neutral leptons that do not interact via any of the forces described in the Standard Model, beyond gravity. Their existence is motivated by the See-Saw Mechanism, which provides a natural explanation for the observed small masses of active neutrinos – electron, muon, and tau neutrinos. This mechanism postulates that the smallness arises from the mixing of active neutrinos with much heavier sterile neutrino states. The mass of the sterile neutrino is inversely proportional to the smallness of the active neutrino mass, potentially placing sterile neutrinos in the mass range of keV to GeV. Crucially, sterile neutrinos within this mass range are considered viable warm dark matter candidates, potentially resolving some discrepancies between the cold dark matter model and observations of structure formation in the universe. Searches for sterile neutrinos focus on detecting their decay products or observing their effects on neutrino oscillation experiments.

Axion-like particles (ALPs) and dark vector bosons represent well-motivated candidates for dark matter, differing from weakly interacting massive particles (WIMPs) in their predicted interaction strengths and masses. ALPs are pseudo-scalar bosons that arise in extensions of the Standard Model addressing the strong CP problem, and their interactions with Standard Model particles are typically suppressed by an energy scale $f_a$ . Dark vector bosons, conversely, are associated with a new $U(1)$ gauge symmetry and interact via a kinetic mixing portal with the Standard Model photon and Z boson. Critically, both ALP and dark vector boson models predict the existence of a “hidden sector” – a set of particles that interact primarily with each other and weakly with the Standard Model, potentially explaining phenomena beyond dark matter, such as dark radiation or anomalies in muon $g-2$ .

Scalar particles, encompassing hypothetical bosons with integer spin-0, continue to be considered extensions to the Standard Model due to their relatively weak experimental constraints compared to other candidate particles. Unlike vector bosons or fermions, the absence of strong theoretical limitations on their mass and coupling strengths allows for a broad parameter space, potentially mediating interactions beyond those currently known. These particles do not necessarily need to couple to Standard Model particles, opening the possibility of a ‘hidden sector’ of interactions, or they may couple weakly, leading to subtle effects difficult to detect with current instrumentation. While direct detection remains a challenge, ongoing searches at colliders and through precision measurements aim to identify potential signals arising from scalar particle production or their influence on established processes.

Current experimental constraints, including those from the anomalous magnetic moment of the muon (gray), pion decay (red), and anticipated PIONEER sensitivity (blue), limit the coupling strength of dark sector particles to muons as a function of particle mass for scalar, axion-like, and spin-1 mediators with vector or axial-vector couplings.

Probing the Invisible: Experimental Signatures

The PIENU experiment at the Paul Scherrer Institute has established constraints on the properties of sterile neutrinos through the observation of pion decays. By searching for rare and exotic decay modes of charged pions, specifically $\pi^+ \rightarrow e^+ \nu_e X$ and $\pi^+ \rightarrow \mu^+ \nu_\mu X$ , where X represents undetected particles, PIENU has set upper limits on the branching ratios for these processes. These limits are then used to constrain the mass and coupling strength of potential sterile neutrinos, providing valuable input for models beyond the Standard Model. The experiment’s sensitivity is determined by the collected pion decay statistics and the efficiency of identifying signal events amidst background processes.

The PIONEER experiment is designed to substantially improve current searches for rare pion decays, with the potential to enhance existing upper limits on branching ratios by one to two orders of magnitude. This improvement in sensitivity will be achieved through increased data collection and refined analysis techniques, enabling the exploration of previously unconstrained parameter space for physics beyond the Standard Model. Specifically, PIONEER aims to achieve an upper limit on $BR(\pi^+ \rightarrow e^+ \nu_e X)$ of 1×10^-9 and $BR(\pi^+ \rightarrow \mu^+ \nu_\mu X)$ in the few x 10^-8 range, representing a significant advancement over existing PIENU results. The experiment’s target event statistics of 10¹⁰ events for the $\pi^+ \rightarrow \mu^+ \nu_\mu$ channel will be crucial to achieving this enhanced sensitivity.

The PIONEER experiment is designed to improve current limits on the branching ratio $BR(\pi^+ \rightarrow e^+ \nu_e X)$ to $1 \times 10^{-9}$ . This represents approximately one order of magnitude improvement over existing measurements obtained by the PIENU experiment. This enhanced sensitivity is specifically targeted at searches for exotic decays involving masses around 100 MeV, allowing for exploration of previously inaccessible parameter space and providing more stringent constraints on potential new physics beyond the Standard Model.

Beam dump experiments and monophoton searches represent complementary approaches to detecting physics beyond the Standard Model, each optimized for different hypothetical particles. Beam dump searches, characterized by directing a high-energy particle beam into a shielding block, are particularly sensitive to weakly interacting particles that can be produced in the beam but do not penetrate the shielding, such as dark vector bosons. These particles are identified by their decay products emerging from the shielding. Conversely, monophoton searches focus on detecting single photons resulting from the decay of axion-like particles (ALPs), which interact very weakly with ordinary matter and are expected to produce photons as a primary decay signature. The differing detection strategies and target particles make these experiments crucial for exploring a broad range of potential dark sector candidates.

The PIONEER experiment is designed to collect $10^{10}$ π⁺→μ⁺ν_μ decay events. This substantial increase in event statistics, compared to previous experiments like PIENU, directly translates to enhanced sensitivity in searches for rare or previously unobserved decay modes. A larger dataset minimizes statistical uncertainties, allowing for more precise measurements and the potential to detect exceedingly rare processes with branching fractions below current experimental limits. The increased statistics are crucial for probing beyond the Standard Model, specifically in searches for new physics contributing to muon-neutrino final states.

The PIONEER experiment is projected to achieve an upper limit on the branching ratio $BR(\pi^+ \rightarrow \mu^+ \nu_\mu X)$ of a few x 10^-8. This represents a significant improvement over current limits established by the PIENU experiment, which are approximately one to two orders of magnitude higher. The increased sensitivity is expected through the collection of 10¹⁰ target events in the $\pi^+ \rightarrow \mu^+ \nu_\mu$ decay channel, enabling a more precise search for rare or previously unobserved decay modes and constraining the properties of potential new particles contributing to these decays.

Constraints on sterile neutrino mixing angle <span class="katex-eq" data-katex-display="false">|U_{eN}|^2</span> as a function of mass <span class="katex-eq" data-katex-display="false">m_N</span> exclude regions shaded in red and gray, with PIONEER’s sensitivity estimates (blue) potentially reaching a see-saw target (yellow) for neutrino masses between 64 and 130 MeV, as compared to existing bounds from PIENU and NA62. — Constraints on sterile neutrino mixing angle $|U_{eN}|^2$ as a function of mass $m_N$ exclude regions shaded in red and gray, with PIONEER’s sensitivity estimates (blue) potentially reaching a see-saw target (yellow) for neutrino masses between 64 and 130 MeV, as compared to existing bounds from PIENU and NA62.

Refining the Framework: A Path Forward

The precise measurement of pion decay branching ratios – the probability of a pion decaying into specific particles – serves as a critical cornerstone for testing the Standard Model of particle physics and searching for deviations that could signal new phenomena. These ratios are not merely descriptive; they are highly sensitive probes of fundamental interactions and provide stringent constraints on theoretical models. Any discrepancy between experimental measurements and Standard Model predictions would necessitate revisions to existing theories or the introduction of new particles and forces. For instance, subtle variations in the branching ratios involving muons and electrons could indicate the existence of undiscovered heavy leptons or leptoquarks, particles that mediate interactions between leptons and quarks. Consequently, experiments dedicated to meticulously determining these branching ratios, often achieving parts-per-million precision, are paramount in the ongoing quest to unravel the mysteries beyond our current understanding of the universe.

Effective Field Theory (EFT) offers a uniquely versatile approach to particle physics by allowing researchers to explore potential “new physics” without needing a complete understanding of the underlying, yet-undiscovered, particles and interactions. Instead of directly modeling these unknown entities, EFT parameterizes their effects using a series of operators built from the known Standard Model fields. These operators, characterized by coefficients representing the strength of the new interactions, effectively capture how deviations from Standard Model predictions might manifest in experimental data. $\mathcal{L}_{EFT} = \mathcal{L}_{SM} + \sum_{i} c_i O_i$ , where $c_i$ are the Wilson coefficients and $O_i$ are the higher-dimensional operators. This strategy allows physicists to systematically analyze experimental results-such as those from pion decay-and constrain the possible values of these coefficients, providing crucial clues about the nature of physics beyond the Standard Model, even if the ultimate source remains hidden.

The pursuit of physics beyond the Standard Model necessitates a synergistic approach, demanding the careful integration of experimental observations with rigorous theoretical predictions. No single experiment possesses the definitive power to unveil new phenomena; instead, a compelling signal emerges from the convergence of data collected by diverse facilities, each probing complementary aspects of particle interactions. Simultaneously, theoretical frameworks – such as Effective Field Theory – provide the necessary tools to interpret these complex datasets, allowing physicists to disentangle established physics from potential deviations hinting at undiscovered particles or forces. This combined analysis doesn’t merely confirm or refute hypotheses; it refines the search itself, narrowing the parameter space and guiding future experiments towards the most promising avenues for exploration, ultimately striving to resolve the fundamental mysteries of the universe.

The expected number of <span class="katex-eq" data-katex-display="false">\pi^{+}\to e^{+}\nu_{e}X</span> decays at the PIENU experiment, modeled after Batell:2017cmf with a branching ratio of <span class="katex-eq" data-katex-display="false">3\times10^{-7}</span>, aligns with residuals extracted from PIENU:2021clt as a function of positron energy for varying values of the X mass. — The expected number of $\pi^{+}\to e^{+}\nu_{e}X$ decays at the PIENU experiment, modeled after Batell:2017cmf with a branching ratio of $3\times10^{-7}$ , aligns with residuals extracted from PIENU:2021clt as a function of positron energy for varying values of the X mass.

The pursuit of understanding exotic pion decays, as detailed in the analysis, reveals a fascinating landscape where established models meet the potential for new physics. This research doesn’t impose order, but rather seeks to observe the patterns that emerge from these complex interactions. As Confucius stated, “Study the past if you would define the future.” The PIONEER experiment, by meticulously examining branching ratios and decay signatures, operates on this principle-leveraging detailed observation to illuminate previously unseen phenomena. It acknowledges that control over these subatomic events is illusory; instead, the experiment aims to influence understanding through careful measurement and analysis, recognizing self-organization within the dark sector as the true governing principle.

The Horizon Beckons

The search for physics beyond the Standard Model often feels like mapping a forest by observing the fall of leaves. This analysis of pion decays, and the anticipated capabilities of PIONEER, highlights a particularly intriguing pattern in those fallen leaves – a potential signal of interactions with a hidden sector. The limits placed on sterile neutrinos, and the sensitivity to anomalous decays, are not endpoints, but rather the sharpening of tools. The universe does not intend to reveal its secrets; it merely allows them to be discovered, if one observes closely enough.

Current explorations, while rigorous, remain tethered to assumptions about the nature of dark sector particles. The focus on sterile neutrinos, for instance, is a natural starting point, but the forest may contain trees of entirely different species. Future progress will likely require a broadening of the search – a willingness to consider decay modes and particle properties not currently favored by prevailing models. The most interesting discoveries often lie at the edges of what is considered plausible.

Ultimately, the value of experiments like PIONEER resides not in confirming or refuting specific hypotheses, but in charting the unknown. The universe evolves without a forester, yet follows rules of light and water. The experiment will not find new physics, but will reveal it, should it exist, as a natural consequence of local interactions. The true horizon is not a destination, but the ever-receding line where ignorance meets observation.

Original article: https://arxiv.org/pdf/2601.06254.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Emerging Gaps: When Standard Models Fail

Whispers of New Particles: Candidates Emerge

Probing the Invisible: Experimental Signatures

Refining the Framework: A Path Forward

The Horizon Beckons

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