Constraining axion-like particles from rare pion decays

Ultraviolet completions for axion-like particles (ALPs) lighter than the neutral pion generically induce ALP-neutral pion mixing, and are therefore sensitive to direct constraints on the mixing angle. For ALPs below the pion mass, we demonstrate that strong and novel bounds on the ALP-pion mixing angle can be extracted from existing rare pion decay data, measured by the PIENU and PIBETA experiments.


INTRODUCTION
Searches for axion-like particles (ALPs) are a powerful means to probe various extensions of the Standard Model (SM), including models of dark matter, baryogenesis, and solutions to the strong CP problem (see e.g. [1][2][3][4]). In the MeV to GeV mass range, the strongest known constraints on such particles arise mainly through their coupling to photons. At large couplings, LEP bounds are applicable from diphoton searches. For a range of somewhat smaller couplings, bounds from beam-dump and fixed-target experiments, such as Charm/Nu-Cal, E137, and E141 apply. Future or current experiments, such as NA62, SeaQuest, and Belle II will probe some of the yet unexplored mass-coupling parameter space.
The phenomenology of an ALP, a, in this mass range may, however, be re-parametrized in terms of its mixing with light unflavored hadrons. The generic nature of this mixing makes it an attractive phenomenological quantity to explore. In particular, UV completions for such ALPs will typically generate ALP-pion mixing, and are therefore sensitive to mixing constraints. In this paper, we demonstrate how strong additional bounds on ALPpion mixing can be extracted from existing rare pion decay data. We derive these bounds for the mass range 10 MeV m a < m π , extendible down to the massless limit with improved form factor treatments.
Charged pion decays are a promising avenue for ALP searches: The high precision measurements of the chirally-suppressed π + → eν ("π e2 ") and of the phasespace suppressed π + → π 0 eν ("π β ") branching ratios can place strong constraints on Br[π + → aeν], because the latter has neither of these suppressions. Because any process producing a π 0 in the final state will also produce an ALP in the final state via mixing, these constraints can in turn be transformed into bounds on ALP-pion mixing by estimating the π + → aeν amplitude from its mixing with π + → π 0 eν (cf. Refs. [5,6]).
The PIENU experiment [7,8] currently provides the highest precision measurement of the π + → eν branching ratio from decays of stopped charged pions: The world average Br[π + → eν] = (1.230±0.004)×10 −4 [9]. A multicomponent background plus signal fit to the measured positron energy spectra has been used in previous stud-ies to tightly constrain contributions from heavy sterile neutrinos decays, i.e. π + → eN [7,10] (see also [11]), and a similar approach has been used for constraining Majoron-neutrino couplings [12]. In this paper, we derive ALP-pion mixing constraints from the PIENU spectra via a similar analysis, leveraging the observation that the irreducible background from Br[π + → π 0 eν] is much smaller than the experimental precision. (This is not the case for e.g. K → πeν versus K → eν, for which reason we do not study bounds from semileptonic kaon decays.) Further, the PIBETA experiment [13] provides the highest precision measurement of the rare π + → π 0 eν decay, Br[π + → π 0 eν] = (1.036 ± 0.006) × 10 −8 , including a measurement of the opening angle spectrum of the daughter π 0 → γγ process. This spectrum has a kinematic edge that is highly sensitive to the γγ invariant mass. We show that it generates even tighter constraints on Br[π + → (a → γγ)eν] for a small m a range.
Previous analyses have considered bounds on ALPpion mixing using constraints on K + → π + + invisible and estimating the K + → π + a amplitude from mixing with K + → π + π 0 (see e.g. Ref. [14]). While powerful, these bounds implicitly require suppression of the ALPtop quark coupling, which can otherwise generate large short-distance s → d penguin contributions. E.g. in the case of universal ALP-quark coupling, the penguins are enhanced by compared to the mixing amplitudes [6,[15][16][17], and naively dominate the K + → π + a amplitudes. By contrast, the semileptonic processes we consider arise from tree-level chargedcurrent amplitudes. Short-distance contributions are expected to enter only at higher loop and electroweak order, far smaller than the hadron mixing contributions we probe. In the context of UV completions, the bounds we derive are therefore independent from kaon bounds.

ALP-PION MIXING
The framework we are considering consists of an ALP, a, coupled to SM quarks or gauge bosons, with mass below the pion mass, m a < m π . We assume no tree-level ALP-lepton couplings, and consider only the case that the branching ratio to diphotons is dominant. The low energy effective field theory of ALP-SM interactions may be matched onto the chiral Lagrangian of the light hadrons, such that the ALP-SM interactions involve either mixings with SM hadrons or higherdimension derivatively-coupled interactions to hadrons or gauge bosons (see e.g. Ref. [18]). In the regime that the ALP-SM effective couplings are perturbative, the physical ALP state a = (cos ϑ + . . .) a 0 + sin ϑ π 0 + . . . (1) in which the angle ϑ encodes the mixing of the ALP and QCD neutral pion eigenstates, and the ellipsis indicates mixings with other hadrons as allowed by parity and angular momentum conservation. An amplitude involving a π 0 generates a contribution to an associated ALP amplitude, via mixing with an off-shell π 0 , viz.
in which other contributions may involve mixing with other hadrons, or other UV operators. The ALP-π 0 mixing arises in the chiral Lagrangian via the mixed kinetic term ε ∂ µ a ∂ µ π 0 . In UV-complete models, ε can be generated either through the ALP-gluon coupling, or the ALP coupling to light quarks (for the expression of ε, see e.g. Ref. [18]). It is straightforward to show that, in the limit ε 1, sin ϑ m 2 a ε/(m 2 π − m 2 a ). Other sources of isospin-breaking may generate additional mass mixing terms, that further modify sin ϑ. Hereafter, we shall treat sin ϑ as a purely phenomenological mixing parameter -keeping in mind that it may be re-expressed in terms of UV quantities in a model-dependent way -and seek to develop direct sin ϑ constraints.

ALP LIFETIME
The amplitude for the diphoton mode a → γγ -the dominant decay mode for m a < m π -presents a simple manifestation of Eq. (2): It always receives a contribution from ALP-pion mixing, such that γγ|a = γγ|π * 0 π * 0 |a + . . . = γγ|π * 0 sin ϑ + . . ., in which additional model-dependent UV contributions from a direct coupling to photons, g aγ aF µνF µν (F µν = ε µνρσ F ρσ ), may also be present. The diphoton width is then (choosing f π = 130 MeV) with an effective ALP-photon coupling, g eff aγ , and where the coupling of the pion to photons is The limits we explore from PIENU and PIBETA data are sensitive to the ALP lifetime, as it determines whether the ALP is e.g. prompt or invisible with respect to the detector. Since the lifetime (3) is in general an independent parameter from sin ϑ, for the purposes of setting sin ϑ limits we shall explore three lifetime regimes: i) The prompt regime, i.e. g eff aγ is sufficiently large for the ALP to decay within the timing/displacement resolution of the detector, possibly via a large g aγ , ii) The invisible regime, i.e. g eff aγ is sufficiently small for meta-stable ALPs, possibly via tuning of g aγ against the mixing contribution, iii) The mixing regime, i.e g eff aγ sin ϑ g πγ .

PION SEMILEPTONIC DECAYS TO ALPS
At tree-level, the π + → aeν parton-level amplitude (we remind the reader that we assume no leading-order ALP-lepton interactions), (4) The second term is electroweak suppressed compared to the first, and can be neglected. The first term contains the ALP-hadron matrix element of the form in which M 0 and M + are each complete sets of (multi)hadronic states, with appropriate quantum numbers. The axial vector matrix element M 0 |dγ µ γ 5 u|π + vanishes by parity and angular momentum conservation. The leading chirally-unsuppressed contribution to the second term of Eq. (5) arises from a virtual ρ * exchange, and is therefore suppressed by m 2 a /m 2 ρ . The overall dominant contribution to the matrix element is then generated via the off-shell π * 0 'insertion' in the first term, as in Eq. (2), so that the π + → a amplitude The π + → π 0 eν decay is conventionally computed by applying the so-called conserved vector current (CVC) hypothesis and by mapping to the µ → eνν process, as done in e.g. Ref. [19]. Our estimates for π + → π * 0 eν will instead be informed by the similar K + → π 0 eν process, using the language of form factors. This is a similar, but more general, approach to that of Refs. [5,6], that studied π + → (a → ee)eν in the context of the longdefunct 1.8 MeV axion anomaly [20].
The hadronic π + → π * 0 matrix element may be represented by form factors, defined (for SM currents) via in which q = p + − p 0 , the difference of the charged and neutral pseudoscalar momenta, with masses m + and m 0 , respectively. We have defined dimensionless form factors , such that f 0 couples only to the lepton mass. Here c π = 2 × 1/ √ 2 is a coupling combinatoric factor multiplied by a Clebsch-Gordan coefficient.
, and where we have neglected small electroweak corrections [19]. In the regime m + − m 0 m e , the electron mass terms may be neglected.
Following from the Ademollo-Gatto theorem [21][22][23], one expects that f + (q 2 = 0) 1 up to corrections that are expected to scale as ∼ (m 2 The matrix element may be expressed as an analytic function of a conformal expansion parameter z = ( [24], so that provided |z| 1, the form factor should be approximately linear in w or q 2 . In the analogous K + → π 0 eν system, f K + (q 2 ) is wellapproximated by a linear function from f (q 2 = 0) 1 to f (q 2 = q 2 max ) ∼ 1.2, and |z| max 0.098. Thus requiring a sufficiently small z, say |z| max 0.3 -equivalent to r 0.1 or m 0 10 MeV -and approximating should provide a lower bound on f + , yielding an O(1) level conservative estimate for the π + → aeν rate. (In the massless positron limit, applying the approximation (8) to Eq. (7) yields a partial width in agreement with e.g. Eq.
(1) of Ref. [13] or Eq. (7.12) of Ref. [19] Combining Eqs. (6)- (8) with the π + → ν partial width, one then obtains the ratio of branching ratios Using Eq. (9), we proceed to set bounds on sin ϑ from rare pion decay data. These bounds rely, in part, on fits to the positron energy spectrum in the parent rest frame. At truth level, the positron energy is bounded by 0 ≤ E e ≤ m + (1 − r 2 )/2, and Overlaid are typical event topologies for the prompt (decaying to photons, blue) and invisible (green) ALP scenarios. Right: Schematic cross-section of the PIBETA detector configuration including the target, tracking and calorimeter elements (gray). Overlaid are typical event topologies for the minimum truth-level opening angle configuration of a π 0 (red) and lighter prompt ALP (orange) diphoton decay.

PIENU RESIDUALS BOUND
The PIENU experiment [8] measures the π + → eν branching ratio from a sample of stopped pions, by determining the positron yield in the electromagnetic (EM) inclusive decay π + → eν(γ) compared to the cascade π + → (µ → eνν)ν(γ). The main experimental components comprise a target, silicon strips and wire chambers for high precision tracking, a positron calorimeter to reconstruct the positron energy, and a semi-hermetic calorimeter array to capture EM showers. The combined calorimeter energy is given by the sum of positron energy and EM showers, E cal = E e + E EM . This necessarily also includes contributions from π + → eνγ. A sketch of the PIENU detector configuration is shown on the left in Fig. 1.
The relevant backgrounds include not only the π + → µ → e cascade, but also contributions from pion decaysin-flight, stopped muon decays, and radiative µ decays to energetic photons. Their branching ratios overwhelmingly dominate the signal mode. Timing cuts are used to suppress these large backgrounds compared to the prompt π → eν(γ) modes. A simultaneous fit of the timing distributions for both signal and backgrounds then permits measurement of the ratio, R e/µ = Γ[π → eν(γ)]/Γ[π → µν(γ)], at the 10 −3 level, from which the π + → eν branching ratio is inferred.
The E cal distribution for the π + → eν mode is sharply peaked at (m 2 + + m 2 e )/2m + 69.8 MeV, with an additional low-energy tail arising from EM shower losses. The (timing-cut-suppressed) backgrounds from the π + → µ → e cascade or muon decays-in-flight are, by contrast, smoothly distributed in the low-energy region E cal < E 0 52 MeV, the endpoint.
Refs. [7,10] perform a precision fit of the measured E cal distribution in the low-energy region to the com- bination of the (simulated) π + → eν(γ) low-energy tail and the background distributions. The bin residuals of this fit can be used to place strong constraints on additional prompt contributions from exotic π + → eX, where X has sufficient invariant mass to push the signal E cal distribution into the low-energy fit region. Refs. [7,10] consider the case that X = N , a heavy sterile neutrino.
In Fig. 2 we show the residuals of Ref. [10] used for such an analysis, normalized against the π + → eν branching ratio. In this work we consider instead X = aν, making use of Eq. (9) to convert the bound on branching ratios to a bound on sin ϑ. (More precise limits will require a dedicated analysis fitting the X = aν signal template simultaneously with the background components.) In the regime that the ALP is long-lived enough to be invisible to the detector, the E cal distribution receives no additional contributions from a → γγ. Overlaid on Fig. 2 we show the corresponding binned positron energy spectra (thick lines) from π + → aeν decays for m a = 40 and 80 MeV, including quoted acceptance corrections [10].
In the prompt ALP regime, however, daughter photons of the ALP may contribute to the measured E cal in the event. For m a ∼ m π , the ALP is slow enough that one photon may hit the PIENU positron calorimeter within the ∼ 20% positron acceptance [10], as sketched in Fig. 1 (deposition in the outer calorimeters is required to be < 2 MeV [10], thereby excluding hard photon contributions in those). But for m a m π , the ALP momentum may back-react against the lepton system, such that the daughter photons, which decay in a narrow cone around the ALP momentum, miss the acceptance. In Fig. 2 we show the corresponding binned positron spectra (thin lines) for the same two mass benchmarks. The heavier 80 MeV benchmark is slightly altered by a longer tail. While Ref. [10] does not quote the bin residual correlations, one may reproduce quoted π + → eN bounds assuming nearby bins are uncorrelated. We therefore extend this assumption to treat all bins as uncorrelated over the measured energy range. Under this assumption, in the left panel of Fig. 3 we show the corresponding 95% CL exclusion regions in the sin 2 ϑ-m a parameter space, for both the invisible (green) and prompt (blue) regimes, which are very similar. The excluded regions in sin 2 ϑ for the prompt and invisible cases differ at most by O(1) and extend down to sin 2 ϑ 10 −5 . This corresponds to branching ratios as small as O(10 −8 ).
A full study of regimes outside the prompt or invisible limits requires simulation of the PIENU response when the EM shower is somewhat spatially or time-displaced from the prompt decays, but still within the detector acceptance. As a proxy for such a study, we characterize whether the ALP is prompt or invisible by considering whether the mean characteristic ALP displacement from decay-in-flight, βγ cτ , is inside the target or outside the calorimeter radius, respectively: We treat the PIENU target size as ∼ 1 cm and the calorimeter size as ∼ 1 m.
Independent of the relationship between sin ϑ and the lifetime (cf. Eq. (3)), requiring a prompt ALP -βγ cτ < 1 cm -directly implies a lower bound on g eff aγ . Over the ALP mass ranges consider in this paper, we have checked that this bound is far smaller than, and is therefore not saturated by, the direct g eff aγ bounds from LEP tri-photon searches [28,29]. Electron fixed-target experiments such as NA64 [30] and LDMX [31], as well as Belle(II) [26] and BaBar [32], also have invisible ALP searches. However, for these experiments, the ALP production and lifetime is controlled by g eff aγ , independent of sin ϑ. Hence these constraints do not appear in the left panel of Fig. 3.
The pure mixing regime (g aγ = 0 in Eq. (3)) fixes the relationship between the π + → aeν branching ratio and the ALP lifetime, and may therefore interpolate between the prompt and invisible regimes in different parts of the sin 2 ϑ-m a space. For the pure mixing regime, in Fig. 3 the regions βγ cτ < 1 cm (> 1 m) are above (below) the red dot-dashed contours. Above the 1 cm contour, the prompt exclusion should be a good proxy for the mixing regime one. Similarly, below the 1 m contour the invisible regime should well-represent the mixing regime exclusion.
In the mixing regime, one may make use of the relation g eff aγ sin ϑ g πγ to recast beam-dump, collider, and fixed-target experiment bounds on g eff aγ onto the sin 2 ϑm a space. For m a < m π , the relevant bounds are set by the CHARM/Nu-Cal [33][34][35][36], E137 [37], E141 [38] and LEP [28,29] experiments, corresponding in the right panel of Fig. 3 to the regions shaded in gray. As shown in the figure, above the gray regions, we may use the prompt regime exclusion as a proxy for the mixing regime. We see that the PIENU data places powerful new constraints on ALPs in the mixing regime for m a 25 MeV.
These constraints will be complemented in the future In this regime the mean characteristic decay length regions βγ cτ < 1 cm (dot-dashed red line and above) and βγ cτ > 1 m (dot-dashed red line and below) approximately delineate where the prompt and invisible regime exclusions apply, respectively. Also shown are exclusions from CHARM/Nu-Cal, E137, E141 and LEP (grey), and projected reaches for SeaQuest (yellow line and below) [25], Belle II (green line and below) [26], PrimEx (red line and above) and GlueX (orange line and above) [27].
by proton fixed-target beam-dump experiments, such as SeaQuest [25] searching for 3γ signatures or Belle II monophoton searches [26]. In the right panel of Fig. 3, we show the SeaQuest (yellow, 10 20 protons on target) and Belle II (green) reaches as representatives of experiments capable of setting upper bound limits in the sin 2 ϑ-m a parameter space in the pure mixing regime. Part of the sin 2 ϑ-m a space may also be tested by NA62 running in beam-dump mode [39], and FASER [40]. A slightly larger region of parameter space will be probed by the SHiP experiment [41]. In a complementary manner, sizable values of the mixing angle sin ϑ can be also tested [27] in the future by the PrimEx and GlueX experiments (region above the red and orange lines, respectively).

PIBETA DIPHOTON BOUND
The PIBETA experiment [13,42] directly measures the rare π + → (π 0 → γγ)eν branching ratio, by triggering on the prompt π 0 → γγ decay in coincidence with a positron track from a sample of stopped pions. The main detector elements relevant to this discussion are a near-spherical electromagnetic calorimeter, as well as cylindrical multiwire proportional tracking chambers surrounded by plastic scintillator. A schematic of the experiment is shown on the right in Fig. 1.
The photon showers are required to each have energy E γ > m µ /2, beyond the kinematic endpoint of stopped µ → eνν background decays. The overall normalization of the π + → π 0 eν rate is obtained via comparison with a large prescaled sample of non-prompt single positron track events, including both π + → eν and inflight µ → eνν backgrounds. This comparison entails a simultaneous fit of signal and background kinematic and timing distributions.
Reconstruction of the diphoton pair includes measurement of the diphoton opening angle in the lab frame. At truth level, the diphoton opening angle is bounded via The maximum (minimum) cosine corresponds to diphoton emission perpendicular (parallel) to the π + direction of flight in the π 0 rest frame, generating a sharp kinematic edge (smooth kinematic endpoint) in the θ γγ spectrum. Because the upper bound increases as m a decreases, the prompt diphoton decay of an ALP in π + → (a → γγ)eν with m a < m π may then produce diphoton showers with truth-level opening angles beyond the π 0 edge at ∼ 176 • . In Fig. 1 we show schematically the maximum truth-level cos θ γγ configuration for a π 0 compared to a lighter ALP. In practice, the finite detector-level angular resolution smears out the reconstructed θ γγ distribution and thus the θ γγ edge. Including this effect, for an angular smearing σ θγγ 2.25 • and requiring both photons' energy E γ > 53 MeV [42], we show in Fig. 4 the expected θ γγ distributions for several m a benchmarks, as well as for π 0 , compared to the measured θ γγ spectrum in the range 160 • ≤ θ γγ ≤ 180 • [13]. We see that the π 0 spectrum (grey) agrees well with the data. For m a 110 MeV, the photon energy cut significantly suppresses the θ γγ spectrum in the 160-180 • range. FIG. 4. PIBETA reconstructed diphoton opening angle distribution (black) for π + → (π 0 → γγ)eν, normalized to unity. Also shown are π + → aeν binned spectra for the prompt regime, with ma = 110, 120, 130 MeV and mπ. The spectra are normalized such that Γ[π + → aeν]/Γ[π + → π 0 eν] = 1.
The PIBETA experiment does not provide residuals for the fit of the simulated π + → π 0 eν opening angle spectrum to the data. Instead, we can extract an approximate, estimated bound on sin 2 ϑ, by conservatively requiring that the integrated contribution to the θ γγ spectrum in the 160-180 • range from π + → aeν not exceed the quoted 0.6% uncertainty for the π + → π 0 eν branching ratio. In Fig. 3 we show the corresponding exclusion (see purple region on the left panel). This exclusion will likely be much stronger if the full differential information shown in Fig. 4 can be incorporated. This approximate bound from PIBETA data sets the most stringent bound on the mixing angle sin ϑ for prompt regime ALPs with masses above ∼ 100 MeV. A future analysis of data for θ γγ < 160 • could lead to stringent constraints for a broader range of ALP masses below 100 MeV.

CONCLUSIONS AND OUTLOOK
Models for axion-like-particles (ALPs) generically predict a mixing between the ALP and the SM neutral pion. In this paper we have derived strong new constraints on ALP-pion mixing, by extracting direct constraints on the π + → aeν branching ratio from the rare pion decay data measured by the PIENU and PIBETA experiments.
In the pure mixing regime, these constraints complement existing exclusions as well as the reaches of future planned experiments, leading to near complete coverage of the sin 2 ϑ-m a space over many decades of the mixing angle for 10MeV m a m π . Beyond this regime, the constraints provide exclusions for the very wide range of UV ALP models that generate ALP-pion mixing. Because they arise from charged current tree-level processes, these exclusions can probe UV models that are charac-teristically different from those probed by similar bounds extracted from K + → π + + invisible decays.
Our approximate treatments of the detector responses can be improved by dedicated ALP analyses in future π + → eν or π + → π 0 eν measurements, that account for e.g. bin correlations, effects of displaced ALP decays, and/or make use of other differential information. Our results also rely on theoretical approximations, expected to introduce no more than O(10%) uncertainties, that may be improved with more detailed treatments of the π + → π * 0 form factors. This in turn would permit extension of these bounds to even lower ALP masses, below ∼ 10 MeV.