sign in sign up
<<
Home , Axion, X17 particle

D 103, 055018 (2021)

Signals of the QCD with of 17 MeV=c2: Nuclear transitions and light decays

Daniele S. M. Alves * Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 3 November 2020; accepted 11 February 2021; published 23 March 2021; corrected 30 March 2021)

The QCD axion remains experimentally viable in the mass range of Oð10 MeVÞ if (i) it couples predominantly to the first generation of SM ; (ii) it decays to eþe− with a short lifetime −14 τa ≲ 4 × 10 s; and (iii) it has suppressed isovector couplings, i.e., if it is piophobic. Remarkably, these are precisely the properties required to explain recently observed anomalies in nuclear deexcitations, to wit: the eþe− emission spectra of isoscalar magnetic transitions of 8Be and 4He nuclei showed a “bumplike”

feature peaked at meþe− ∼ 17 MeV. In this paper, we argue that on-shell emission of the QCD axion (with the aforementioned properties) provides an extremely well-motivated, compatible explanation for the observed excesses in these nuclear deexcitations. The absence of anomalous features in other measured transitions is also naturally explained: piophobic axion emission is strongly suppressed in isovector magnetic transitions and forbidden in electric transitions. This QCD axion hypothesis is further corroborated by an independent observation: a ∼2–3σ deviation in the measurement of Γðπ0 → eþe−Þ from the theoretical expectation. This paper also includes detailed estimations of various axionic signatures in rare light meson decays, which take into account contributions from low-lying QCD resonance exchange, and, in the case of rare decays, the possible effective implementations of ΔS ¼ 1 octet enhancement in chiral perturbation theory. These inherent uncertainties of the effective description of the strong interactions at low energies result in large variations in the predictions for hadronic signals of the QCD axion; in spite of this, the estimated ranges for rare meson decay rates obtained here can be probed in the near future in η=η0 and kaon factories.

DOI: 10.1103/PhysRevD.103.055018

I. INTRODUCTION of “intensity frontier” experiments initiated in the 1970s. This earlier period, however, was driven partly by studies The past decade has seen a resurgence of interest in the of hadronic and physics, and partly by searches phenomenology of new light with feeble inter- actions with the Standard Model (SM) [1–3]. Motivations for the Higgs and the QCD axion. Indeed, in its have been varied, spurred from the growing belief that original incarnation, the QCD axion was part of the dark might be part of a more complex dark sector electroweak Higgs sector and had its mass spanning Oð100 –1 Þ – with additional matter and force carriers [4–8],butalso the range of keV MeV [14 17]. With increas- because light dark sectors could be parasitically explored ing constraints and no discoveries, laboratory searches for in the broader U.S. and worldwide neutrino program the QCD axion withered away in the early 1990s. By then, [9–12]. Experimental signatures of dark sectors are being the consensus was that the Peccei-Quinn (PQ) mechanism searched for by a diverse suite of experiments ranging had to take place at much higher energy scales, resulting 1 “ ” – from beam dumps/fixed targets to meson factories. This in the invisible axion [18 20]. The tradeoff for fore- effort drew on the legacy of an earlier, very active period going PQ symmetry breaking at the electroweak scale was an ultralight axion with the correct cosmological relic abundance to explain [21–23],whichhas *[email protected] 1See, e.g., talks at the kickoff meeting of the RF6 SNOW- been the focus of several ongoing and proposed experi- MASS Working Group, “Dark Sectors at High Intensities,” ments [24–35]. August 12–13, 2020, [13]. Nonetheless, motivations for the scale of PQ symmetry breaking are a matter of theoretical prejudice. In the Published by the American Physical Society under the terms of original PQWW (Peccei-Quinn-Weinberg-Wilczek) axion the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to model, a single mechanism to break PQ and electroweak the author(s) and the published article’s title, journal citation, symmetries tackled two major puzzles at once—the and DOI. Funded by SCOAP3. absence of CP violation in the strong interactions and

2470-0010=2021=103(5)=055018(25) 055018-1 Published by the American Physical Society DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

2 the generation of for SM particles. Conversely, Despite these large uncertainties stemming from χPT, it ≳ 109 axion models with high PQ breaking scales fPQ GeV is still possible to parametrize the dependence of a variety could simultaneously address the strong CP problem of hadronic signatures of the axion in terms of its isovector and the origin of dark matter. In this paper, we focus on and isoscalar mixing angles, while remaining agnostic yet another possibility, whereby the PQ mechanism is about their magnitudes. The usefulness of such paramet- realized by new dynamics close to the QCD scale. rization is manifest when confronting experimental data, Considering that the solution to the strong CP problem not only in constraining the axion’s hadronic mixing provided by the PQ mechanism is intimately connected angles, but also in interpreting experimental anomalies with the nonperturbative dynamics of QCD, it is not as potential signals of the QCD axion. This will be the farfetched to suppose that their scales should not be underlying philosophy of this study.3 separated by over 10 orders of magnitude. Indeed, such Complementary, the underlying motivation for this study wide separation of scales makes the delicate cancellation is a combination of the long-standing puzzle posed by the mechanism of the strong CP phase vulnerable to spoiling strong CP problem, and three independent experimental effects, such as nonperturbative quantum effects, anomalies. The first two refer to bumplike excesses observed which, based on general arguments, are expected to violate in specific magnetic transitions of 8Be and 4He nuclei via global symmetries [37–41] (see also [42] for a shared point eþe− emission, with (naïve) significances of 6.8σ [43] and of view). Furthermore, existing anomalies in nuclear tran- 7.2σ [44], respectively. The third is related to the 0 þ − sitions [43,44] and in the π decay width to e e [45],if persistently high central value observed for the width confirmed as beyond the Standard Model (BSM) phenom- Γðπ0 → eþe−Þ, whose most recent and precise measure- ena, would strongly support the possibility of a low PQ scale ment, performed by the KTeV Collaboration in 2007 [45], axion, as we shall discuss. showed a discrepancy from the theoretical expectation in the A light BSM sector realizing the PQ mechanism at a SM at the level of ∼2–3.2σ [48–51]. In combination, these scale of OðGeVÞ cannot be completely generic, however. anomalies point to a common BSM origin: a new short-lived Any new degrees of freedom must either have weak or boson with mass of ∼16–17 MeV, coupled to light nongeneric couplings to avoid existing experimental con- and , and decaying predominantly to eþe− (see also straints (which is the case of the electrophilic, muophobic, [52] for connections with other anomalies). As an ad hoc and piophobic QCD axion studied in [46]), or they must explanation, there are only two possibilities for the and “ ” have predominantly hadronic couplings and blend in of this hypothetical new boson: it can either be a with the QCD resonances in the spectral range of pseudoscalar (JP ¼ 0−), or an axial vector (JP ¼ 1þ), in ∼400 –2 MeV GeV. The phenomenology of the latter is order to simultaneously account for these three excesses.4 quite challenging to predict and to probe experimentally. Further constraints push these two possibilities into peculiar On the other hand, the inevitable pseudo-Goldstone degree regions of parameter space, which may require contrived and/ of freedom, manifested as the QCD axion, is much more or baroque UV completions.5 At face value, neither of them is amenable to phenomenological studies using Chiral particularly compelling, leading many to believe that these Perturbation Theory (χPT). Indeed, a robust prediction of χPT is that the mass of the QCD axion should lie in the ∼ 1–20 3 range ma MeV when its decay constant is This same philosophy was adopted by the authors of [47] in f ∼ Oð1–10Þ GeV. For generic models in this range, the study of hadronically coupled axionlike particles (ALPs). a − the axion mixing angle with the neutral is quite large, 4In particular, the 1 protophobic proposed by θ ∼ Oð Þ ∼ Oð0 01–0 1Þ Feng et al. in [53,54] as an explanation of the 8Be anomaly aπ fπ=fa . . , and strongly excluded by − þ 4 −4 cannot be emitted in the 0 → 0 transition of He, nor does it bounds on rare pion decays, which require θ π ≲ Oð10 Þ. 0 þ − a contribute non-negligibly to Γðπ → e e Þ.In[55], Feng et al. However, as shown in [46], axion-pion mixing can be proposed an alternative explanation of the 4He anomaly, whereby suppressed well below its generic magnitude if the axion the eþe− excess stems from the deexcitation of the overlapping couples exclusively with light quarks, u and d, with PQ- 0þ nuclear state. Recently, [56] argued that the protophobic 8 charge assignments qu ¼ 2qd . In this special region of vector boson hypothesis is excluded as an explanation of the Be PQ PQ anomaly. parameter space, the phenomenology of the QCD axion is 5For instance, in the axial-vector case, the model building no longer dominated by its isovector couplings; instead, it required to circumvent stringent bounds from -neutrino is largely determined by its isoscalar mixings with the η and scattering restricts the axial-vector couplings of the 1þ state to 0 A ¼ −2 A η . As such, it inherits the same strong dependence light quarks to satisfy gu gd [57,58]; axial-vector models of the η and η0 on higher order terms in the chiral expansion, also typically require many ad hoc degrees of freedom to cancel gauge anomalies in the UV. In the axion case, in order to suppress and its hadronic couplings suffer from Oð1Þ uncertainties. 0 a − π mixing, the PQ charges of the up and down quarks must u ¼ 2 d satisfy qPQ qPQ, with (nearly) vanishing PQ charges for the 2Amusingly, the original axion was allegedly nicknamed other quarks. Such flavor alignment, combined with the fact that ∼ Oð Þ higglet by and . Higglet was also fPQ GeV , requires nontrivial UV completion at the weak the terminology used by Bill Bardeen and Henry Tye in [36]. scale; see [46].

055018-2 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

0 anomalies are either the result of experimental systematics (iii) The a − π mixing must be suppressed, θaπ ≲ and/or poorly understood SM effects. In our opinion, this Oð10−4Þ, in order to respect upper bounds þ þ þ − illustrates the paradoxical predicament of the light dark sector on Brðπ → e νeða → e e ÞÞ. intensity frontier program: the generic models it seeks to A simple phenomenological IR model realizing the discover or rule out are not strongly motivated, and, at least requirements above can be easily incorporated in the historically, it has been the case that experimental excesses post-electroweak symmetry breaking SM Lagrangian by without theoretically compelling interpretations tend to be ascribing axionic phases to the masses of the up-, received with strong skepticism. down-quark, and electron, Fortunately, this predicament might not be warranted γ5 u here. Nuclear transitions via axion emission and (modified) i q a=fa mu → mue PQ ; rare meson decays are smoking gun signatures of the QCD 5 d iγ q a=fa axion which have been predicted over three decades ago md → mde PQ ; – 5 [59 66]. The fact that some of these signatures have iγ qe a=f → PQ a ð Þ appeared in 8Be, 4He, and π0 decays, and can be consis- me mee ; 1 tently explained by a QCD axion variant which remains f ¼ e ∼ Oð1Þ experimentally viable (albeit with peculiar properties of where qPQ (f u, d, e) are PQ charges, with qPQ u ¼ 2 d electrophilia, muophobia, and piophobia), should be taken and qPQ qPQ. Importantly, no additional operators with cautious optimism. should be present in this specific basis, such as derivative After a brief overview of the most relevant properties of couplings of the axion to quark axial currents, or the usual the piophobic QCD axion in Sec. II, we obtain the linear coupling of the axion to the dual field strength parameter space of axion isoscalar couplings favored by operator. the 8Be and 4He anomalies, and, taking into account nuclear In this IR model, requirement (ii) mentioned above has and hadronic uncertainties, show that they significantly been imposed by fiat. Requirement (i) follows from the þ − overlap, favoring the QCD axion emission hypothesis as a axion’s coupling to e e , which dominates its decay width, single explanation of both anomalies (Sec. III). We then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi turn to axion signals in rare meson decays. In Sec. IVA,we m qe m 2 4m2 0 Γð → þ −Þ¼ a PQ e 1 − e ð Þ obtain the parametric dependence of η=η dielectronic a e e 2 ; 2 8π f m decays on the axion’s isoscalar mixing angles. In a a Sec. IV B, we calculate the rate for axio-hadronic decays −15 0 4 × 10 s of the η and η mesons in the framework of Resonance ⇒ τ ≈ : ð3Þ a ð e Þ2 Chiral Theory, an effective “UV completion” of χPT that qPQ incorporates low-lying QCD resonances and extends the principle of Dominance. Finally, in Sec. V, For ma ∼ 16–17 MeV, existing bounds on the electron’s we investigate various axionic decays of charged and PQ charge are very mild, limiting its range to 1 3 ≲ j e j ≲ 2 ’ neutral , considering distinct possible implementa- = qPQ . The upper bound is set by KLOE s 2015 tions of octet enhancement in χPT and their effect on search for visibly decaying dark [67], whereas the axionic kaon decay rates. We conclude in Sec. VI. lower bound is set by the 2019 results from CERN’s SPS NA64 fixed target experiment [68,69], constraining the τ ≲ 0 4 10−13 II. BRIEF OVERVIEW OF THE axion lifetime to a . × s. The sensitivities of PIOPHOBIC QCD AXION future experiments to the axion’s electronic couplings (such as fixed targets and eþe− colliders) have been explored Generic models of the QCD axion with mass of ∼16–17 in [46]. MeV are largely excluded. However, as investi- From (1) and standard χPT at leading order, the axion gated in [46], all experimental constraints to date can be mass is given by avoided in this mass range if the axion satisfies a few pffiffiffiffiffiffiffiffiffiffiffiffi specific requirements which are as follows: j u þ d j −13 qPQ qPQ mumd mπfπ (i) It must be short lived (τa ≲ 0.4 × 10 s) and decay m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð4Þ þ − a 1 þ ϵ ð þ Þ predominantly to e e in order to avoid limits s mu md fa from beam dump and fixed target experiments, as well as constraints from charged kaon decays such with þ → πþð → γγ Þ as K a ; invisible . 2 2 m m mπ m (ii) The PQ charges of second and third generation SM ϵ ≈ u d 1 þ 6 K ≃ 0 04: ð Þ s ð þ Þ2 2 2 . 5 fermions must vanish or be suppressed, in order to mu md mK mη0 avoid limits from the anomalous magnetic ð − 2Þ u 2 ¼ d ¼ 1 ¼ dipole moment, g μ, and from upper bounds on It follows then that for qPQ= qPQ and ma þ − radiative decays: J=Ψ;ϒ→γða→e e Þ. 16.7 MeV, the axion decay constant is fa ≃ 1030 MeV.

055018-3 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

u d We will benchmark ma, fa, q ,andq to these values for magnitude and instead simply parametrize the physical PQ PQ 6 the remainder of this paper. axion current as For generic parameter space of QCD axion models, the aphys ≡ ∂ quark mass hierarchy mu;d ≪ ms typically induces a Jμ fa μaphys hierarchy of axion-meson mixing angles, θ π ≫ θ η; θ η0 , a a a fa ð3Þ ðudÞ ðsÞ ≡ ðfπ∂μa þ θ πJ þ θ η J þ θ η J Þ; ð9Þ resulting in the isovector couplings of the axion dominating a 5μ a ud 5μ a s 5μ fπ its experimental signatures. This is not the case for the 0 piophobic axion we are considering. Here, the a − π where mixing angle, to leading order in χPT, is given by ¯ ð3Þ u¯γμγ5u − dγμγ5d ðm qu − m qd Þ J5μ ≡ ≡ fπ∂μπ3; ð10aÞ θ j ¼ − fπ u PQ d PQ 2 aπ χPT LO fa mu þ md ¯γ γ þ ¯γ γ ðqu − qd Þ 1 ðudÞ u μ 5u d μ 5d PQ PQ J5μ ≡ ≡ fπ∂μηud; ð10bÞ þ ϵs ; ð6Þ 2 2 1 þ ϵs ð Þ s¯γμγ5s u d s which, after taking q =2 ¼ q ¼ 1 and m =m ¼ J ≡ pffiffiffi ≡ fπ∂μη : ð10cÞ PQ PQ u d 5μ 2 s 0.485 0.027 from [70], results in

−3 The axionic field a and the neutral meson degrees of θaπjχ ¼ð−0.02 3Þ × 10 : ð7Þ PT LO freedom π3, ηud, and ηs in (10) mix among themselves to yield the physical degrees of freedom (i.e., the mass ’ It is clear from (6) and (7) that the axion s piophobia eigenstates) a , π0, η, and η0. In particular, the implica- is the result of an accidental cancellation in χPT’s phys tion of (9) is that any strong or weak process involving the leading order contribution to θ π. This cancellation stems a currents in (10) will have a corresponding axion signature from the near numerical coincidence between m =m u d for which one of the neutral mesons in the amplitude gets qd =qu ¼ 1=2 and PQ PQ . replaced by a properly weighted by the appropriate χ ’ phys Unfortunately, PT s prediction (7) alone is not precise mixing angle. enough to be useful. We instead have resort to observation With the parametrization in (9), it is straightforward to θ to determine the allowed range for aπ with better precision. obtain the axion’s couplings to photons and . 3 2σ This can be achieved by requiring that the . excess in Specifically, below the QCD confinement scale, the electro- ’ Γðπ0 → þ −Þ KTeV s measurement of e e [45] be the result magnetic anomaly of the physical axion current (9) leads to of π0 − a mixing, which yields [46] pffiffiffi α 5 2 ˜ μν ð−0.6 0.2Þ L ⊃ θ π þ θ η þ θ η aFμνF ; ð11Þ θ j ¼ 10−4 ð Þ a 4π a 3 a ud 3 a s aπ KTeV e × : 8 fπ qPQ which, combined with (2), yields the axion decay width and θ Given the suppressed value (8) for aπ, this model features branching ratio to two photons, an atypical hierarchy of mixing angles, θ π ≪ θ η; θ η0 , a a a pffiffiffi which results in the isoscalar couplings of the axion 2 2 3 5 2 α ma dominating its experimental signatures. This aggravates Γða → γγÞ¼ θaπ þ θaη þ θaη ; 3 ud 3 s 4πfπ 4π the loss of χPT’s usual predictive power in axion phenomenology—given its state of the art, χPT cannot ð12Þ θ θ 0 numerically pin down the isoscalar mixing angles aη, aη pffiffi 2 with good accuracy. As argued in [46], θ η, θ η0 receive θ þ 5 θ þ 2 θ a a −7 1 aπ 3 aη 3 aη Oð1Þ Oð 4Þ ⇒ Brða → γγÞ≈ 10 × ud s : contributions from operators at p in the chiral ðqe Þ2 10−3 expansion, many of which have poorly determined Wilson PQ coefficients. ð13Þ Any substantive theoretical progress in better determin- ’ ð − 2Þ ing the axion’s hadronic couplings is unlikely to be The axion s contribution to g e stemming from its accomplished anytime soon. Indeed, such efforts might couplings to electrons and photons has been worked out be superseded by future experimental results which will be in [46]. able to either exclude or narrow down the preferred ranges ’ 6 for the axion s isoscalar couplings. With this in mind, in We omit the dependence of (9) on e¯γμγ5e, which has no this study we choose to remain agnostic about their bearing on the axion-meson mixing angles.

055018-4 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

Finally, expressing the axion nuclear couplings generi- Axion emission rates for the magnetic dipole transitions cally as of 8Be have already been worked out in [46]; we briefly review the main results here to make this section self- L ¼ ¯ γ5ð ð0Þ þ ð1Þ τ3Þ ð Þ aNN aNi gaNN gaNN N; 14 contained. We then estimate the expected axion emission rate for the transition of 4He inves- the parametrization in (9) yields the following isovector and tigated by Krasznahorkay et al. and show that the reported isoscalar axion- couplings, respectively: excess rates for both nuclei favor the same range of axion isoscalar mixing angles. ð1Þ ¼ θ ¼ θ ðΔ − Δ Þ mN ð Þ gaNN aπgπNN aπ u d ; 15a fπ A. Evidence for the QCD axion in 8Be transitions pffiffiffi ð0Þ m In [43], the MTA Atomki experiment selectively popu- ¼ðθ ðΔ þ Δ Þþ 2θ Δ Þ N ð Þ 8 gaNN aη u d aη s : 15b ud s fπ lated specific excited states of the Be nucleus by impinging a beam of with finely tuned energy on a 7Li target. Above, N is the nucleon isospin doublet, m is the nucleon They then measured the energy and angular correlation of N þ − mass, and Δq quantifies the matrix elements of quark axial- e e pairs emitted in deexcitations of these states to the 8 currents in the nucleon via 2sμΔq ¼hNjq¯γμγ5qjNi, with sμ ground state of Be. From these measurements, they were the nucleon spin vector. The combination in (15a) is well able to reconstruct final state kinematic variables, such as þ − determined from β decay, the of the e e pair, meþe− . The nuclear levels of interest deexcited to the ground state via magnetic

Δu − Δd ¼ gA ≃ 1.27: ð16Þ dipole (M1) transitions,

8 8 þ − On the other hand, estimations for Δu þ Δd and Δs based Be ð17.64Þ → Beð0Þþe e ; on data from semileptonic decays, deep ΔE ¼ 17.64 MeV; ΔI ≈ 1; ð18aÞ inelastic scattering, and lattice calculations vary widely – [71 83], ranging from 8Beð18.15Þ → 8Beð0Þþeþe−; 0.09 ≲ Δu þ Δd ≲ 0.62 and − 0.35 ≲ Δs ≲ 0: ð17Þ ΔE ¼ 18.15 MeV; ΔI ≈ 0: ð18bÞ

8 þ In the following, we will use (15) to fit the recent 8Be and Above, Beð0Þ is the JP ¼ 0 isospin-singlet ground state 8 8 8 4He anomalies, and (9) to obtain various rare meson decays. of the Be nucleus, and Be ð17.64Þ and Be ð18.15Þ are JP ¼ 1þ excited states, whose isospin quantum numbers ¼ 1 ¼ 0 III. NUCLEAR TRANSITIONS are predominantly I and I , respectively, but are nonetheless isospin mixed, One of the smoking gun signatures of in the Oð – Þ 8 mass range keV MeV are magnetic nuclear deexci- j Be ð17.64Þi ¼ sin θ1þ jI ¼ 0iþcos θ1þ jI ¼ 1i; ð19aÞ tations via axion emission [59–61]. Indeed, such signals 8 have been extensively searched for during the 1980s j Be ð18.15Þi ¼ cos θ1þ jI ¼ 0i − sin θ1þ jI ¼ 1i: ð19bÞ [84–93]. However, since the energy of typical nuclear transitions ranges from a few keV to a few MeV, past Their level of isospin mixing, quantified by θ1þ,was searches did not place meaningful bounds on axions estimated by ab initio quantum Monte Carlo techniques ≳ 2 heavier than ma MeV. [54,94] and by χEFT many-body methods [58] to fall in the Recently, the MTA Atomki Collaboration led by A. approximate range 0.18 ≲ sin θ1þ ≲ 0.43. Following Krasznahorkay reported on the observation of bumplike [54,58], we will consider a narrower range for sin θ1þ þ − excesses in the invariant mass distribution of e e pairs which more accurately describes the width of the electro- 8 4 emitted in the deexcitation of specific states of Be and He magnetic transition 8Beð18.15Þ → 8Beð0Þþγ, nuclei [43,44]. The energy difference ΔE between the nuclear levels involved in these particular transitions is 0.30 ≤ sin θ1þ ≤ 0.35: ð20Þ atypically high, a priori allowing on-shell emission of particles as heavy as ∼17–18 MeV. Furthermore, consis- In the MTA Atomki experiment [43], a bumplike feature tent with the allowed values of angular momentum and in the meþe− distribution of the ΔI ≈ 0 transition (18b) was parity carried away by the axion (JP ¼ 0−; 1þ; 2−; 3þ; …), observed on top of the monotonically falling spectrum these excesses appeared in magnetic (but not electric) expected from SM internal pair conversion (IPC) [43].A transitions. Also, consistent with the emission of a pio- statistical significance of 6.8σ was reported for this deviation phobic axion, these excesses were observed only in relative to the IPC expectation. Additionally, it was claimed predominantly isoscalar (but not isovector) transitions. in [43] that the excess events were consistent with the

055018-5 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021) emission of an on-shell resonance, generically labeled “X,” ¼ð16 70 35 0 5 Þ with mass of mX . . stat . syst MeV, promptly decaying to eþe−. This excess was later corroborated by the same collaboration with a modified experimental setup [95], with a combined fit yielding a relative branching ratio of

Γ X ≈ ð6 1Þ 10−6 ð Þ Γ × ; 21 γ 8Beð18.15Þ with respect to the radiative γ width of this 8 8 transition, Be ð18.15Þ → Beð0Þþγ,ofΓγð18.15Þ≈ ð1.90.4ÞeV [96]. As for the meþe− spectrum of the ΔI ≈ 1 transition (18a), no statistically significant deviation from the IPC expect- ation was observed. References [53,58] inferred a naïve upper bound of

Γ X ≲ Oð10−6ÞðÞ Γ 22 γ 8Beð17.64Þ for the deexcitation rate of 8Beð17.64Þ via on-shell emission of this hypothetical “Xð17Þ” resonance. If it is confirmed that the observed excess originates from new, beyond the SM phenomena, as opposed to effects or experimental systematics, it could indeed be explained by the piophobic QCD axion. The prediction FIG. 1. Fits, constraints, and sensitivity projections in the for axion emission rates from magnetic dipole nuclear parameter space of the axion isoscalar couplings. The upper transitions was first worked out by Treiman and Wilczek (lower) plot assumes the same (opposite) relative sign between θ θ aηud and aηs . The orange and yellow bands enclose the range of [59] and independently by Donnelly et al. [60] back in the 8 4 late 1970s. For the two transitions in (18), the axion-to- isoscalar mixing angles that can explain the Be and He anomalies, respectively, benchmarking Δu þ Δd and Δs to the emission rate is (see also [61,63,90]) values shown; cf. (25), (33). The shaded gray regions are P ð þ → πþð → ðIÞ 8 2 excluded by the conservative upper bound Br K a Γ 1 g hIj Be i 2 3=2 þ − −5 a ¼ P I¼0;1 aNN 1 − ma e e ÞÞ ≲ 10 (under different scenarios for octet enhancement ð Þ ð Þ 8 2 ; ð0Þ þ − Γ 8 2πα ðμ I − η I Þh j i Δ χ η → γ Be I¼0;1 I Be E in PT) and by current bounds on e e , assuming qe ¼ 1=2; cf. (34a) and (35a). The dashed gray (red) lines ð23Þ PQ show the expected reach from measurements of (or bounds on) η0ðηÞ → þ − j8 i e e , assuming that future experiments will have where Be denotes one of the states in (18), and its sensitivity to the branching ratios predicted in the SM, (34b) overlap with the isospin eigenstates jI ¼ 0i and jI ¼ 1i and (35b),withOð1Þ precision. ð0Þ follows from (19). The quantities μ ¼ μp þ μn ¼ 0.88 ð1Þ and μ ¼ μp − μn ¼ 4.71 are, respectively, the isoscalar 1σ and isovector nuclear magnetic moments, and ηð0Þ, ηð1Þ within the range favored by the KTeV anomaly fit, (8), e 1 2 ≤ e ≤ 2 parametrize ratios of nuclear matrix elements of convection while also varying qPQ in the range = qPQ , and the ð1Þ and magnetization currents [89]. In particular, ηð0Þ ¼ 1=2 nuclear structure parameters θ1þ and η in the ranges (20) due to total angular momentum conservation. The nuclear and (24), respectively. We obtain ηð1Þ structure dependent parameter , to the best of our pffiffiffi 8 knowledge, has not been calculated for Be; we therefore − ðθ ðΔ þ Δ Þþ 2θ Δ Þj8 aηud u d aηs s Be ð18.15Þ conservatively vary ηð1Þ in the range ≈ ð1.1–6.3Þ × 10−4: ð25Þ ð1Þ −1 ≤ η j8 ≤ 1 ð Þ Be : 24 θ θ Figure 1 displays the parameter space in aηud versus aηs Combining (23) with (15), (16), (18b), and (19b), we can favored by the 8Be anomaly (orange bands) under the infer the axion isoscalar mixing angles that yield the assumptions of Δu þ Δd ¼ 0.52, Δs ¼ −0.022 [97], and observed excess rate (21). For concreteness, we vary θaπ equal (upper plot) or opposite (lower plot) relative sign

055018-6 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

θ θ ¼ð16 84 0 16 0 20 Þ between aηud and aηs . These bands shift non-negligibly as mass of ma . . stat . syst MeV, and Δu þ Δd and Δs are varied within the ranges in (17). deexcitation width,7 Finally, we conclude this discussion by using (23) and −5 Γj4 4 ≈ 3 9 10 ð Þ (25) to predict the axion emission rate for transition (18a) He ð21.01Þ→ Heð0Þþa . × eV: 30

Γ a ≈ ð0 008–1Þ 10−6: ð Þ It is encouraging that not only the same resonance mass Γ . × 26 4 γ 8Beð17.64Þ (within error bars) is favored by fits to both the He and 8Be excesses, but also that they appear in magnetic and Indeed, this rate can be down by as much as 2 orders of (dominantly) isoscalar transitions, compatible with the magnitude below the sensitivity of published results to date, interpretation of piophobic axion emission. To further but could potentially be detectable if sufficient statistics is support this hypothesis, we must obtain the range of accumulated in this channel. axion isoscalar mixing angles compatible with the observed rate. According to Donnelly et al. [60], the width 0− → 0þ B. Evidence for the QCD axion in 4He transitions of axionic emission in nuclear transitions is estimated to be8 More recently, the same collaboration led by A. Krasznahorkay investigated transitions of a different 2 j⃗ j5 4 Γ j ≈ pa j ð0Þ ð0Þ þ ð1Þ ð1Þ j2 ð Þ nucleus, He [44]. With a 900 keV proton beam bom- a M0 2 2 aM0 gaNN aM0 gaNN ; 31 3 ð2I þ 1Þ barding a H fixed target, this experiment populated the first N mN Q two excited states of 4He, j⃗ j ≈ pwhereffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIN is the isospin of the excited nuclear state, pa 4 P þ 2 2 He ð20.49Þ;J¼0 ;I¼0; Γ¼0.50MeV; ð27aÞ ΔE − ma is the magnitude of the axion’s spatial- momentum in the rest frame of the decaying nucleus, Q 4Heð21.01Þ;JP ¼0−;I¼0; Γ¼0.84MeV; ð27bÞ is a typical nuclear momentum transfer (of order the nucleus Fermi momentum, Q ≈ kF ≈ 250 MeV), and þ − ð0Þ ð1Þ and similarly measured the emission of e e pairs a 0, a 0 involve nuclear matrix elements of magnetization þ M M from deexcitations of these states to the I ¼ 0, JP ¼ 0 currents, and are of Oð1Þ, unless forbidden by isospin 4 ground state, denoted here by Heð0Þ. Such transitions are conservation. For an isoscalar transition such as (28b), (31) allowed via reduces to

4 4 þ − ðE0Þ He ð20.49Þ → Heð0Þþðγ → e e Þ; 2ðΔ 2 − 2Þ5=2 Γ ≈ j ð0Þ j2 E ma j ð0Þ j2 ð Þ a aM0 2 2 gaNN : 32 ΔE ¼ 20.49 MeV; ΔI ¼ 0; ð28aÞ M0;ΔI¼0 mNQ

4 4 þ − ð0Þ ðM0Þ He ð21.01Þ → Heð0Þþða → e e Þ; Using (32) and (15b), and varying aM0 in the range 1 3 ≤ j ð0Þ j ≤ 3 ΔE ¼ 21.01 MeV; ΔI ¼ 0; ð28bÞ = aM0 , we find that the axionic deexcitation width of the M0 transition (28b) in 4He yields the observed but forbidden to occur via the following processes: rate (30) if

4 4 þ − ðE0Þ He ð20.49Þ ↛ Heð0Þþða → e e Þ; ð29aÞ 7No error bars were provided for (30) in [44]. 8In [55], the calculated rate for pseudoscalar emission in this − þ ðM0Þ 4Heð21.01Þ ↛ 4Heð0Þþðγ → eþe−Þ: ð29bÞ 0 → 0 transition assumed a nonderivatively coupled pseudo- scalar “X” (see the effective operator in Eq. (39) of [55]), resulting in an amplitude with no momentum dependence Above, E0 and M0 refer, respectively, to the electric Γ ∝ jp⃗ j þ − (Eq. (49) of [55]) and an emission rate scaling as X X . monopole (JP ¼ 0 ) and magnetic monopole (JP ¼ 0 ) Under this assumption, the authors of [55] concluded that the rate multipolarities of these transitions. of pseudoscalar emission in this transition would be 6 orders of After cuts, background subtraction, and accounting for magnitude larger than the experimentally favored rate. We point out that their conclusion hinged on their assumption that the contributions to the meþe− spectrum from (28a) and from leading effective operator at the nuclear level mediating this external pair conversion originating from the radiative transition was a relevant operator of dimension 3. In the case of proton capture reaction 3Hðp; γÞ4He, a suggestive bumplike the QCD axion, this assumption is not valid, since the axion only couples derivatively to nuclear axial currents. For the QCD axion, excess was observed in the final meþe− distribution, with a 7 2σ the leading effective nuclear operator is dimension 5, resulting in statistical significance of . . Under the assumption that ⃗ 2 an amplitude scaling as ∝ jpaj , and therefore an emission rate this excess originated from on-shell emission of a narrow ⃗ 5 scaling as Γa ∝ jpaj . Note that the axionic amplitude is still resonance from the M0 transition (28b), the fit to the isotropic, as it should be for a monopole transition, despite its data performed in [44] yielded a favored resonance nontrivial momentum dependence. For details, see [60,61,98].

055018-7 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

pffiffiffi 0 − ðθ ðΔ þ Δ Þþ 2θ Δ Þj4 A. Dielectronic η and η decays aηud u d aηs s He ð21.01Þ π0 → þ − ≈ ð0.58–5.3Þ × 10−4; ð33Þ Just as the precise KTeV measurement of e e offered the best determination of θaπ, future observations of η → þ − η0 → þ − which is compatible with the range of axion isoscalar e e and e e could narrow down the ranges 8 for the axion isoscalar mixing angles θ η and θ η . Present mixing angles favored by the Be excess, (25). In Fig. 1,we a ud a s θ θ bounds on these dileptonic branching ratios [105,106] are likewise display the parameter space in aηud versus aηs 4 still 2 orders of magnitude away from sensitivity to the favored by the He anomaly (yellow bands) under the predicted SM rate [107–109], same assumptions for Δu þ Δd, Δs and relative sign between θ η and θ η used in the computation of the a ud a s Brðη → eþe−Þ < 7 × 10−7; ð34aÞ 8Be orange bands. The 4He yellow bands also shift non- exp negligibly as Δu þ Δd and Δs are varied within the Brðη → eþe−Þ ≈ ð4.6–5.4Þ × 10−9; ð34bÞ ranges in (17). SM It is remarkable that the piophobic QCD axion is able to and simultaneously explain the reported rate of anomalous excesses in deexcitations of two very different nuclei, 0 þ − −8 8 4 Brðη → e e Þ < 0.56 × 10 ; ð35aÞ Be and He, as shown by the overlap between the favored exp ranges for the axion isoscalar mixing angles (25) and (33), ðη0 → þ −Þ ≈ ð1–2Þ 10−10 ð Þ or, equivalently, by the overlap between the yellow and Br e e SM × : 35b orange bands in Fig. 1. This weakens the case for a nuclear Indeed, the highly suppressed SM contribution to these physics origin of the observed features in the meþe− spectra of these transitions [99]. And the fact that “unexplained” dileptonic channels makes them potentially sensitive to a variety of interesting new physics scenarios. features are absent in the meþe− spectrum of several other measured transitions—(18a) being one example—also In anticipation of a future discovery of these decay makes it less straightforward to “explain away” the modes, we obtain the axionic contribution to the dileptonic ηð0Þ → þ − − ηð0Þ observed excesses as poorly understood experimental decays e e due to a mixing. Assuming that systematics. We therefore reiterate our point, stated in this effect dominates these rates (i.e., that interference with the Introduction (Sec. I), that the anomalies in 8Be and 4He the SM amplitudes can be neglected), we have transitions, and their quantitative compatibility with pre- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 2 dicted signals from the QCD axion, should not be quickly m ð0Þ q m 4 Γðηð0Þ → þ −Þ ≈ η PQ e θ 1 − me ð Þ dismissed. A cautiously optimistic attitude and support for e e 8π aηð0Þ 2 : 36 fa m ð0Þ an independent verification of these measurements are η certainly warranted, as well as further exploration of other θ θ 0 isoscalar magnetic nuclear transitions with ΔE ≳ 17 MeV, The mixing angles aη and aη can be reexpressed in þ terms of θ η and θ η using the parametrization [110] and radiative π =p=n capture reactions with significant a ud a s magnetic components [100].

jηi¼cos ϕudjηudi − sin ϕsjηsi; ð37aÞ IV. η AND η0 DECAYS jη0i¼sin ϕ jη iþcos ϕ jη i; ð37bÞ In light of the anomalies in nuclear deexcitations dis- ud ud s s cussed in the previous section, a natural next step is to from which it follows that (34a) and (35a) translate into investigate other systems where the hadronic couplings of relatively weak bounds on the axion isoscalar mixing the piophobic QCD axion could be more precisely deter- angles, mined or more stringently constrained. In this section, we consider rare decays of η and η0 mesons, which, with the 0 0 014 η=η . prospect of future factories, could become powerful jθ ηj¼jcos ϕ θ η − sin ϕ θ η j ≲ ; ð38aÞ a ud a ud s a s j e j future probes of axions and hadronically coupled ALPs qPQ more generally [101]. These include the second phase of the JLab Eta Factory (JEF) program [102], expected to 0.01 improve existing bounds on rare η decays by two orders of jθ η0 j¼jsin ϕ θ η þ cos ϕ θ η j ≲ : ð38bÞ a ud a ud s a s j e j magnitude, and the REDTOP experiment [103,104],a qPQ planned η=η0 factory projected to deliver as many as 13 11 0 ϕ ¼ 39 8 ϕ ¼ 41 2 10 η mesons and 10 η mesons. These will offer an Taking ud . ° and s . ° from [110] and con- 0 j e j¼1 2 unprecedented opportunity to study rare η=η decays and servatively assuming qPQ = for concreteness, we probe BSM physics. display the bounds (38) in Fig. 1.

055018-8 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

θ θ χ In Fig. 1, we also show contours of aηud and aηs (dashed QCD description and the low-energy PT framework, lines) for which the axionic contribution to ηð0Þ → eþe− by encoding the most prominent features of nonpertur- becomes comparable to that of the SM. These contours bative strong dynamics [127,128]. There does not θ θ appear to be consensus in the literature, however, on can be interpreted as the sensitivity to aηud and aηs in the hypothetical scenario of a future observation of which low-lying resonances should be included as these processes showing an Oð1Þ deviation from the degrees of freedom in the RχT Lagrangian, and which branching ratios predicted in the SM, (34b) and (35b). resonances should be regarded as dynamically gener- It goes without saying that the actual experimental sensi- ated poles due to strong S-wave interactions [129,130]. 0 “ ” tivity of future η=η -factories to θ η and θ η could be Examples of such ambiguous poles include the a ud a s σð500Þ κð700Þ substantially better if Brðηð0Þ → eþe−Þ could be measured and the . Oð10%Þ In this subsection, we estimate the rates for η → ππa with better than precision, and if the uncertainties η0 → ππ χ in the SM theoretical predictions could be reduced to the and a using R T. In both cases, we find that χ percent level. the leading order PT predictions for these decay rates are significantly modified by inclusion of resonance exchange η → ππ B. Axio-hadronic η and η0 decays amplitudes. In particular, for a, there is substantial destructive interference between the leading order ampli- η η0 Hadronic decay channels of and mesons could tude from the Oðp2Þ quartic term and the amplitudes in principle be hiding promising signals of the QCD generated by tree-level resonance exchange. This is axion and/or other hadronically coupled ALPs. Among corroborated by performing the same calculation in the most obvious modes are the three-body final states 4 ordinary χPT at Oðp Þ, where one finds that the LECs, ηð0Þ → π0π0a; πþπ−a, which have only recently been in particular L4, L5,andL6, provide Oð1Þ contributions explored in the literature [47,111]. Indeed, the amplitudes 2 that destructively interfere with the Oðp Þ amplitude. For for these processes receive a direct contribution from the 0 η → ππa, the contributions from resonance exchange leading order potential term in the chiral Lagrangian9 and χ Oðp4Þ could in principle result in considerably large branching (alternatively, from PT interactions at )arethe ratios. The difficulty with studying hadronic η and η0 dominant effect in a significant portion of the parameter space and may enhance this decay rate by an order decays lies in reliably predicting their rates. One of the χ earliest examples where this difficulty was encountered of magnitude over the leading order PT prediction. χ was in the calculation of η → 3π, which was significantly The justification for favoring the R T framework over Oð 4Þ χ underestimated by χPT at leading order [112–115]. Indeed, p - PT for this calculation is that the former is it has long been understood that contributions from chiral expected to better capture the Dalitz phase space of the logarithms and strong final state rescattering could not final state, which is relevant when extracting the event be neglected in the computation of η → 3π [116–122]. acceptance due to momentum cuts in experimental analy- Similarly, neither the total width nor the Dalitz phase ses (in particular due to the e selection criteria). Indeed, space of η0 → ηππ is properly described by χPT at we will find that there is strong variation of the ampli- Oðp2Þ [123,124]. tude’s momentum dependence as we vary the assumptions Previous studies [125,126] have shown that such inter- and parameters of the RχT description, which implies a mediate energy processes can be satisfactorily described strong variation in the estimated sensitivity of existing by extending χPT to include low-lying meson resonances and future experimental analyses. Unfortunately, the carrying nonlinear realizations of SUð3Þχ—such as vectors variations in these assumptions cannot be narrowed down (ρ, ω, K , ϕ, …), axial vectors (a1, f1, K1, …), scalars (a0, without further input from experiment. Our main con- 0 f0, σ, κ, …), and pseudoscalars (η , πð1300Þ, …)—and clusion, therefore, is that one cannot reliably predict assuming the principle of “resonance dominance” (an neither the total branching ratio, nor the Dalitz phase extension of vector meson dominance), whereby the space, of the decays ηð0Þ → ππa. Under reasonable low-energy constants (LECs) of the Oðp4Þ chiral assumptions, our RχT-based estimates vary over 2 orders Lagrangian are saturated by mesonic resonance exchange. of magnitude in branching ratio, Brðηð0Þ → ππaÞ ∼ This framework, dubbed Resonance Chiral Theory (RχT), Oð10−4–10−2Þ. Nonetheless, this motivates dedicated has been quite successful phenomenologically as an reanalyses of existing data in final states of ηð0Þ → interpolating effective theory between the short-distance ππeþe−, as well as dedicated searches for eþe− reso- nances in these final states in future η=η0 factories. 9More explicitly, these leading order quartic couplings do In order to motivate our use of RχT, and also to justify not contain derivatives of the axion field, nor do they “descend” our later approximation of retaining only low-lying scalar from ordinary mesonic quartic terms via axion-meson mixing. resonances, we begin by obtaining the main contributions Nonetheless, they are still consistent with the axionP’s pseudo- ð0Þ 0 0 ð −1Þ−1 to the amplitude Aðη → π π aÞ in ordinary χPT at Goldstone because they are proportional to q mq and therefore vanish in the limit of a massless quark. Oðp4Þ. The Lagrangian is [131]

055018-9 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

2 2 fπ † μ fπ L j 4 ¼ ½ þ ½2 ð Þ þ χPT Oðp Þ 4 Tr DμU D U 4 Tr B0Mq a U H:c: 1 − 2η2 þ ½ † μ 2 þ ½ † ½ μ † ν 2 M0 0 L1Tr DμU D U L2Tr DμU DνU Tr D U D U † μ † ν † μ þ L3Tr½DμU D UDνU D UþL4Tr½DμU D UTr½2B0MqðaÞU þ H:c: † μ 2 þ L5Tr½DμU D Uð2B0MqðaÞU þ H:c:Þ þ L6Tr½2B0MqðaÞU þ H:c: 2 þ L7Tr½2B0MqðaÞU − H:c: þ L8Tr½ð2B0MqðaÞUÞð2B0MqðaÞUÞþH:c: − ½ μν † þ μν † þ ½ † μν ð Þ iL9Tr FR DμUDνU FL DμU DνU L10Tr U FR UFLμν : 39

ij i j 2 Above, fπ ¼ 92 MeV; δ B0 ¼ −hq q¯ i=fπ; M0 parametrizes the OðGeVÞ contribution to the mass of the chiral singlet η0 4 from the strong axial anomaly; Li (i ¼ 1; …; 10) are the Oðp Þ χPT LECs [see, e.g., [132] for a review of χPT and definitions of all the terms in (39)]; MqðaÞ is the axion-dependent quark mass matrix, transforming as an octet spurion of SUð3Þχ, 0 1 iqu a=f m e PQ a B u C d ð Þ ≡ B iq a=fa C ð Þ Mq a @ mde PQ A; 40

ms and U is the nonlinear representation of the pseudo-Goldstone chiral nonet, pffiffiffi 2 U ¼ Exp i φaλa fπ 0 0 η η 1 pπ ffiffi þ p8ffiffi þ p0ffiffi πþ Kþ B 2 6 3 C B 0 η η C φaλa ≡ B π− − pπ ffiffi þ p8ffiffi þ p0ffiffi K0 C ð Þ with @ 2 6 3 A: 41 η η K− K¯ 0 − pffiffiffiffiffiffi8 þ p0ffiffi 3=2 3 Collecting the terms in (39) that provide the dominant contributions to Aðηð0Þ → π0π0aÞ, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ˆ 1 þ L4=2 m m L ⊃ 2 2 ð u þ d Þ u d bη bπ2b χPT mπfπ qPQ qPQ 2 ud a 4 ˆ ð þ Þ Oðp Þ 1 þ L6 mu md  ˆ ð ˆ þ 2 ˆ Þ 2 − L5 ∂b bη bπ ∂bμbπ b − L5 L4 bη ∂b bπ ∂bμbπ b þ O mπ ð Þ ˆ μ ud a ˆ ud μ a 2 ; 42 2ð1 þ L4=2Þ 4ð1 þ L4=2Þ mK where we have defined the dimensionless fields depending on assumptions and chosen observables, it is ! well established from fits to fK=fπ that L5 is positive and rffiffiffi −3 π0 1 η 2 η relatively large, L5 ∼ ð1–3Þ × 10 . It is then easy to see bπ ≡ bη ≡ pffiffiffi 8 þ 0 b≡ a ð Þ ; ud ; a ; 43 from (42) and (44) that the contributions to Aðη → π0π0aÞ fπ 3f8 3f0 fa from the first and second terms in (42) are comparable in magnitude and destructively interfere with each other. This the dimensionless LECs, leads to a suppressed rate for η → π0π0a relative to the 2 2 naïve Oðp Þ estimation in χPT, which, however, is quite 32mη ˆ ≡ ∼ Oð103Þ ð Þ 2 Li 2 Li Li; 44 sensitive to the value of L5. On the other hand, the Oðp Þ fπ contribution to Aðη0 → π0π0aÞ from the first term in (42) b may be subdominant to that of the second term, which is and the dimensionless derivative ∂μ ≡∂μ=mη. The kinetic Oð ˆ 02 2Þ terms, omitted in (42), have been canonically normalized. parametrically larger by a factor of L5mη =mη . This While there is large variation in the literature of the inferred may lead to an order-of-magnitude enhancement of the rate 0 0 0 values for L4 and L6 from fits to experimental data, for η → π π a.

055018-10 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

FIG. 2. Contributions to the amplitude Aðηð0Þ → ππaÞ in the framework of RχT. Left graph: leading order quartic term. Middle and right graphs: exchange of low-lying scalar resonances. 0 1 pa0ffiffi þ pf0ffiffi þ While it is now straightforward to extract χPT’s pre- 2 6 a0 ðηð0Þ → π0π0 Þ B C diction for Br a using (42), we will instead B − 0 0 C ¼ B a − paffiffi þ pfffiffi C ð Þ pivot to RχT, from which ordinary χPT can be recovered by S @ 0 2 6 A: 46 integrating out the low-lying meson resonances. Under the − pfffiffiffiffiffiffi0 assumption of resonance dominance, RχT predicts that the 3=2 ð0Þ relevant LECs contributing to η → ππa (L4, L5, and L6) ð980Þ ð980Þ are saturated by the exchange of scalar resonances. We will Above, a0 and f0 are shorthand for a0 and f0 , therefore omit the low-lying pseudoscalar, vector, and respectively [133], and we have not explicitly identified the scalar mesons with nonzero strangeness, since they do not axial-vector resonances from our discussion. Following ð0Þ the notation in [127],wehave contribute to η → ππa. Accounting for the tadpole-induced nonzero vacuum 2 1 L ⊃ fπ ½ † μ − 2η2 expectation value of f0, RχT Tr DμU D U M0 0 4 2 pffiffiffi 2 4 2 ð 2 − 2 2Þ fπ mK mπ= þ Tr½2B0M ðaÞU þ H:c: hf0i¼− pffiffiffi c ; ð47Þ 4 q 3 m 2 mf0 † μ þ cdTr½SDμU D U and canonically normalizing the kinetic terms, we can þ c Tr½B0ðSM ðaÞþM ðaÞSÞU þ H:c:; ð45Þ m q q extract from (45) the RχT interactions contributing where S is the low-lying JPC ¼ 0þþ meson octet,10 to ηð0Þ → π0π0a,

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b hb i 1 þ cpd ffiffiffiffiffiffif0 3=2 m m L ⊃ 2 2ð u þ d Þ u d RχT mπfπ qPQ qPQ 2 b hb i ð þ Þ 1 þ cpm ffiffiffiffiffiffif0 mu md 3=2 2 3 pffiffiffi pffiffiffi 6 2 2 2 2 7 6b 2b rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b b pffiffiffi b b b b7 × 4ηudbπ a − cma0bπ a − cm f0 ηud a5 b hb i 3 1 þ cpd ffiffiffiffiffiffif0 3=2 2 3 2 2 pffiffiffi mηfπ 6 1 7 þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 6 2 b ∂b bπ ∂bμbη þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ∂b bπ ∂bμbπ7 ð Þ cd4 a0 μ ud pffiffiffi f0 μ 5; 48 b hb i b hb i 1 þ cpd ffiffiffiffiffiffif0 6 1 þ cpd ffiffiffiffiffiffif0 3=2 3=2

where, following the notation for the dimensionless fields and derivatives in (42), we have additionally introduced

a0ð980Þ b f0ð980Þ 10Unlike some studies in the literature, we do not ba0 ≡ ; f0 ≡ ð49Þ þþ fπ fπ assume the large-Nc limit and do not include a 0 chiral singlet resonance in our analysis. Furthermore, following þþ and the dimensionless couplings, Refs. [129,130], we consider the broad 0 state f0ð500Þ [aka σð500Þ] a dynamically generated pole due to strong b ≡ cd b ≡ cm ð Þ S-wave interactions, and therefore do not include it as a degree cd ð 2Þ ; cm ð 2Þ : 50 of freedom in (45) and (46). fπ= fπ=

055018-11 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

It is straightforward to recover (42) from (48) by [134] (such as sum rules between two-point correlators integrating out the scalar resonances and making the of two scalar vs two pseudoscalar currents [135], and following identifications: vanishing of scalar form factors at q2 → ∞ [136,137]) have been used to estimate the scalar octet couplings ≈ ≈ ð Þ ma0 mf0 mS; 51 to be jcdj ∼ jcmj ∼ fπ=2, with cdcm > 0. Here, we con- 2 servatively vary these values by 20% in our calculations cdcm cdcm cm L4 ¼ − ;L5 ¼ ;L6 ¼ − : ð Þ ð0Þ 3 2 2 6 2 52 of Γðη → ππaÞ, but retain, for simplicity, the assumption mS mS mS of jcdj¼jcmj. Early fits to a0ð980Þ → πη [127], along with large-Nc With the relevant interactions in (48), we can then obtain assumptions and imposition of short-distance constraints the tree-level amplitude11 Aðηð0Þ → ππaÞ (see Fig. 2),

ð0Þ 0 0 ð0Þ þ − Aηð0Þ→ππ ≡ Aðη → π π aÞ¼Aðη → π π aÞ a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ hˆ i 1 þ cpd ffiffiffiffiffiffif0 2 3 2 fπ mπ m m = ¼ 2 ð u þ d Þ u d Cηð0Þ q q 2 2 ˆ PQ PQ cˆ hf0i fa fπ ðmu þ mdÞ 1 þ pm ffiffiffiffiffiffi 3=2 2 6 2cˆ cˆ 1 pπ :pπ 1 þ d m 1 2 × 4 ˆ 2 2 cˆd hf0i 3 − ð þ Þ − Γ 1 þ pffiffiffiffiffiffi mf0 pπ1 pπ2 i f0 mf0 3=2 3 p ð0Þ :pπ p ð0Þ :pπ 7 − η 1 − η 2 ð Þ 2 2 2 2 5; 53 − ð ð0Þ − Þ − Γ − ð ð0Þ − Þ − Γ ma0 pη pπ1 i a0 ma0 ma0 pη pπ2 i a0 ma0

2 ¼ð þ Þ2 where we have neglected subdominant contributions of integration over invariant masses mπ1π2 pπ1 pπ2 2 2 2 2 2 Oðmπ=m ð0Þ Þ. Above, the equality between amplitudes with ¼ð þ Þ ¼ð ð0Þ − Þ η and mπ2a pπ2 pa pη pπ1 should be per- neutral versus charged is due to isospin symmetry; formed over the Dalitz phase space (see, e.g., the PDG ð0Þ pη , pπ1 , and pπ2 are relativistic 4-momenta; and review on Kinematics [133] for explicit expressions for the Dalitz plot boundaries). θ θ In Fig. 3, we show the branching ratios for ηð0Þ → π0π0a, ≡ fπ cospffiffiffi 8 − fπ psinffiffiffiffiffiffiffiffi0 ð Þ Cη ; 54a ηð0Þ → πþπ− f8 3 f0 3=2 a [computed by integrating the differential decay rate in (55) over the final state phase space] as a fπ sin θ8 fπ cos θ0 function of the RχT couplings bcd, bcm of the low-lying scalar Cη0 ≡ pffiffiffi þ pffiffiffiffiffiffiffiffi : ð54bÞ f8 3 f0 3=2 octet to the pseudo-Goldstone mesons; see Eqs. (45), (48), and (50). As mentioned previously, we assume, for sim- 0 b b For the η=η mixing angles and decay constants above, plicity, that cd ¼ cm, and vary their magnitudes by 20% jb j¼jb j¼1 we will adopt the values from the unconstrained fit around their expected values of cd cm in the large- in [110], namely, θ8 ¼ −24°, θ0 ¼ −2.5°, f8 ¼ 1.51fπ, Nc limit. The range covered by the bands are due to the and f0 ¼ 1.29fπ. uncertainties in the masses and widths of the scalar Finally, we can obtain the differential decay rate from (53), resonances a0ð980Þ and f0ð980Þ—following the PDG [133], we varied these parameters independently within ð0Þ ¼ð960–1000Þ dΓðη → ππaÞ the following ranges: ma0 ;mf0 MeV, Γ ¼ð40–100Þ Γ ¼ð10–200Þ 4 a0 MeV, f0 MeV. 1 ð2πÞ 2 ¼ jA ð0Þ j Φ ð ð0Þ ; Þ The lack of predictive power of our treatment, with an η →ππa d 3 pη pa;pπ1 ;pπ2 2 ð0Þ Sπ1π2 mη estimated range of branching ratios spanning 2 orders of 1 1 1 2 2 2 ¼ jA ð0Þ j dmπ π dmπ ; ð55Þ ð2πÞ3 32 3 η →ππa 1 2 2a 11 Aðηð0Þ → ππ Þ ππ Sπ1π2 mηð0Þ We ignore corrections to a from final state rescattering, based on the conclusions from [116,118] that these effects correct the η → 3π amplitude by modest amounts of Sπ π Sπþπ− ¼ 1 where 1 2 is the standard combinatorial factor ( , Oð10%Þ, and on Ref. [117], which finds somewhat larger Sπ0π0 ¼ 2!), dΦ3 is the three-body phase-space differential rescattering corrections, of ∼70%, which are still subdominant element, and, when obtaining the total decay rate, the relative to other sources of uncertainties in our estimations.

055018-12 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

0.05 0.012 B3 0.010 0.01 normalized B1 QED 0 0.005 0.008 e 0

e

' p B2 d

' 0.006

' Br 0.001 e Br e 5 10 4 2 0.004 1

0.002 ma 16.7 MeV 1 10 4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 d 0.000 cd cm 0 50 100 150 200 pe e MeV ηð0Þ → ππa FIG. 3. Estimated branching ratios for as a η → πþπ− function of the scalar octet couplings to the light pseudoscalar FIG. 4. The differential rate for a as a function j⃗þ − j ≡ j⃗þ þ ⃗− j¼⃗ mesons, cf. (45), (48), and (50). The bands result from varying of pe e pe pe pa, for three benchmark choices of RχT parameters specified in Table I. For comparison, we also the masses and widths of the scalar resonances, a0 and f0, η → πþπ− þ − within their experimental uncertainties. For the dark narrow show the differential rate of the SM process e e , ¼ ¼ 980 labeled “QED.” bands, their masses are fixed to ma0 mf0 MeV, and their Γ ¼ð40–100Þ Γ ¼ widths are varied within the ranges a0 MeV, f0 þ − ð10–200Þ MeV. The broader bands result from additionally vary- reconstructed e e vertex was within a 2.5 cm distance ¼ð960–1000Þ from the beampipe. This cut could have significantly ing their masses within the ranges ma0 ;mf0 MeV. impacted the acceptance of the axion signal, depending ð0Þ −4 −2 magnitude, Brðη → ππaÞ ∼ Oð10 –10 Þ, is due to the on the meþe− experimental resolution. Other event χPT and RχT parameters falling on a special range of selection requirements on the momenta of the π and values that, within uncertainties, can lead to substantial e charged tracks could in principle have rejected a large destructive inference between the LO amplitude and the fraction of the axion signal as well. þ − þ − amplitudes originating from exchange of low-lying scalar An earlier measurement of Brðη → π π ðγ → e e ÞÞ resonances. This is perhaps unsurprising, considering that by the CELSIUS/WASA Collaboration observed, in hind- even the SM hadronic decays of the η and η0 could not be sight, an upward fluctuation of the expected signal correctly predicted, but only “postdicted,” and their exper- [139,140]. Indeed, considering KLOE’s more precise imentally determined branching ratios and Dalitz plot measurement of this branching ratio, the CELSIUS/ parameters have been used to verify the validity of various WASA analysis should have expected 10 SM signal treatments and assumptions, such as RχT, QCD sum rules, events. It observed 24 events in the signal region, of large-Nc limit, dispersive methods, etc. [116–126]. which it determined that 7.7 were from background, and In particular, the upper range of our estimations, 16.3 were from the SM signal. Assuming, conservatively, Brðηð0Þ → ππaÞ ∼ Oð10−2Þ, is probably excluded or in that the 14 “excess events” were instead due to η → tension with observations, though no dedicated searches πþπ−a decays, and taking into account the relative signal for an eþe− resonance in ηð0Þ →ππeþe− final states have ever acceptance due to the minimum transverse momentum 12 been performed, to the best of our knowledge. However, the requirement of jp⃗Tj > 20 MeV for charged particles, we þ − lower range Brðηð0Þ → ππaÞ ∼ Oð10−4–10−3Þ likely remains estimate that branching ratios as large as Brðη → π π aÞ∼ −3 experimentally allowed, and within the sensitivity of ð1 − 3Þ × 10 could be compatible with the CELSIUS/ upcoming η=η0 factories, such as the JLab Eta Factory WASA measurement, although, without access to non- (JEF) and the REDTOP experiment. public information on details of the experimental analysis, The most recent and precise measurement of the this estimation is at the level of an educated guess. SM decay η → πþπ−ðγ → eþe−Þ, which shares the Finally, the two existing measurements of Brðη0 → same final state of η → πþπ−a, was performed by πþπ−ðγ → eþe−ÞÞ, performed independently by the the KLOE Collaboration at the Frascati ϕ-factory Φ DA NE [138]. While their measurement yielded 12 ðη → πþπ− þ −Þ¼ð2 68 0 09 0 07 Þ 10−4 We performed a simple MC event simulation to estimate the Br e e . . stat . syst × , geometric acceptance resulting from the event selection require- j⃗e j 20 it is nontrivial to infer any bounds from this analysis ment of pT > MeV, properly taking the momentum on Brðη → πþπ−aÞ. This is because, without proper dependence of the amplitudes into account. This was done for Monte Carlo simulations, one cannot determine how both the SM signal using the amplitude in [139], as well as for the axion signal, assuming a few RχT benchmark parameters (see the background rejection requirements would have þ − discussion below). We neglected contributions to the signal affected the η → π π a signal efficiency. In particular, efficiency from other event selection requirements and worked this search rejected events with meþe− < 15 MeV whose in the approximation of η mesons decaying at rest.

055018-13 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

TABLE I. Benchmarked RχT parameters for the examples in Fig. 4 and the resulting prediction for the total decay rate of η → πþπ−a. Γ Γ jb j¼jb j ðη → πþπ− Þ ma0 (MeV) a0 (MeV) mf0 (MeV) f0 (MeV) cd cm Br a B1 980 40 980 200 1.125 0.96 × 10−3 B2 980 50 980 100 1.125 1.1 × 10−3 B3 1000 50 1000 100 1.125 0.49 × 10−3

CLEO [141] and BESIII [142] Collaborations, were com- experiments—such as NA62 at CERN [143] and KOTO at bined by the PDG [133] to give Brðη0 → πþπ−eþe−Þ¼ J-PARC [144]—has been on K → πνν¯, there is an under þ1.3 −3 ð2.4−0 9 Þ × 10 . However, both experimental analyses explored opportunity to search for BSM resonances in . þ − reported large external photon-conversion backgrounds e e final states with low meþe−, motivated not only by the in the signal region, peaked in the range meþe− ¼ piophobic QCD axion, but also by visibly decaying ALPs ð8–25Þ MeV (CLEO; see Fig. 2(d) of [141]) and meþe− ¼ and dark photons more generally [42]. Furthermore, the ð10–20Þ MeV (BESIII; see Fig. 2 of [142]). Events falling highly suppressed a → γγ decay mode might be a com- γγ within these meþe− windows were excluded from the analy- petitive final state in kaon decay searches for which ses’ inference of the SM branching ratio. Since events from backgrounds in the mγγ ∼ 17 MeV signal region are tamer η0 → πþπ−a would have fallen precisely in this region where than the eþe− backgrounds. In such cases, final states with the photon conversion background peaked, it is difficult to K → ða → γγÞþSM can be obtained by combining the estimate how strong a potential axion signal could have been. relevant branching ratios BrðK → a þ SMÞ estimated in Simply requiring that the axion signal strength does not this section with Brða → γγÞ in (13). overpredict the number of events attributed to photon- The appearance of the axion in kaon decay final states conversion yields a conservative limit of Brðη0 → πþπ−aÞ≲ occurs via mixing with the neutral octet mesons, π0, η,andη0. few × 10−2, which is not particularly useful. Therefore, the axionic amplitudes can be obtained from We end this section by remarking that an additional ordinarySMamplitudes properly reweighted by axion-meson challenge with estimating the sensitivity of current and mixing angles. While this prescription is straightforward future experiments to ηð0Þ → ππa is the uncertainty in the for estimating “axio-leptonic” kaon decays such as Kþ → þ final state Dalitz phase space, which affects the signal μ νμa (as wewill show in Sec. VA),itisambiguousfor “axio- acceptance resulting from event selection cuts. Consider, ” þ → πþ 0 → π0 þ − hadronic kaon decays such as K a, KS;L a,and for instance, the differential decay rate dΓðη → π π aÞ= 0 → ππ KL a. Firstly, the two-body hadronic width of the CP- djp⃗j as a function of the axion’s 3-momentum jp⃗j. The 0 0 0 þ − −5 a a even neutral kaon, ΓðK → π π ; π π Þ ≈ 0.73 × 10 eV, dependence of this rate on jp⃗ j¼jp⃗þ þ p⃗− j ≡ jp⃗þ − j S a e e e e is enhanced by roughly 3 orders of magnitude relative varies dramatically depending of the numerical values to the two-body hadronic width of the charged kaon, chosen for the masses, widths, and couplings of the scalar ΓðKþ → πþπ0Þ ≈ 1.1 × 10−8 eV. In χPT,thisenhancement resonances a0 and f0. We illustrate this effect in Fig. 4, where we plot the differential decay rate dΓðη → is parametrized as a large disparity in the magnitudes of the þ − þ − Δ ¼ 1 π π Þ j⃗þ − j j⃗þ − j Wilson coefficients of the possible S operators e e =d pe e as a function of pe e for three – ð3Þ different RχT benchmarks—corresponding to different [112,145 147]. Specifically, the coefficient of an SU χ- Δ ¼ 1 2 choices of masses and widths for a0 and f0 within octet ( I = )operatorislargerthanthecoefficientofthe Δ ¼ 3 2 ∼30 uncertainties (see Table I)—as well as for the SM decay leading order 27-plet ( I = ) operator by a factor of . η → πþπ−ðγ → eþe−Þ [107] (labeled as “QED” in Fig. 4). Secondly, phenomenologically, there are at least two choices Δ ¼ 1 While these different benchmark points yield close of S octetoperators that could beresponsiblefor this so- þ − “ ” “Δ ¼ 1 2 ” predictions for Brðη → π π aÞ, their predictions for called octet enhancement (aka I = enhancement ) þ − in kaon decays [148–150],namely, dΓðη → π π aÞ=djp⃗aj differ dramatically, as shown in Fig. 4. In particular, for high enough cuts on the charged ðΔS¼1Þ ¼ 2 ðλ ∂ ∂μ †Þþ ð Þ ⃗ O8 g8fπTr μU U H:c:; 56a momenta pe , the signal acceptance of benchmark ds B1 could be significantly lower than that of B3 (and of the 2 þ − 0ðΔS¼1Þ 0 fπ † † μ † η → π π γ O ¼ −g ðλ 2B0M ðaÞU Þ ð∂μU∂ U Þ SM decay ). Indeed, this could lead to a 8 8 Λ2 Tr ds q Tr variation of as much as an order of magnitude in the expected sensitivity of experimental searches. þ H:c:; ð56bÞ

where λds ≡ ðλ6 þ iλ7Þ=2, Λ ∼ 2mK is a natural χPT cutoff, V. KAON DECAYS and the operators above occur at different orders in the 2 0 4 We conclude our study by exploring signals of the chiral expansion: O8 at Oðp Þ and O8 at Oðp Þ. Fitting piophobic QCD axion in rare kaon decays. Although the existing data on K → ππ and K → πππ, while treating the 0 main focus of ongoing and near-future rare kaon decay coefficients g8 and g8 in (56) on equal footing, yields

055018-14 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021) j þ 0 j ≃ 0 78 10−7 ð 2 − 2Þ g8 g8 . × [151,152]. In order to break this þ þ 0 mK mπ AðK → π π Þ¼3g27 : ð59Þ degeneracyinthefit,onemustinvokethestandardassumption fπ under naïve power counting that ΔS ¼ 1 octet enhancement should appear at lowest order in the chiral expansion, and As alluded to earlier, the hadronic Kþ decay amplitude is 0 −7 therefore, g8 ≪ g8 ⇒ jg8j ≃ 0.78 × 10 .However,thereis not octet enhanced, and its consequent narrow width 0 no first principles derivation of this choice, and it could be relative to that of KS is parametrized by a hierarchy incorrect. For example, in the Resonance Chiral Theory between the 27-plet and octet coefficients, framework discussed in the previous section, it is easy to ≃ 2 5 10−9 ≈ 0 032 j þ 0 j ð Þ speculate that the origin of ΔS ¼ 1 octet enhancement could g27 . × . × g8 g8 : 60 0 be due to the weak interactions inducing a mixing between KS þþ 13 Finally, for the relevant hadronic three-body decay of JPC ¼ 0 σð500Þ 0 andabroad resonance,suchasthe . Upon K , we have [151,153] integration of the low-lying resonances, this effect would be L 0 0 2 2 captured by the operator O8 in (56b), leading instead ðg8 þ g8 þ 2g27Þ m mπ 0 0 −7 Að 0 → πþπ−π0Þ¼ K − 0 to g8 ≪ g ⇒ jg j ≃ 0.78 × 10 . KL 2 g8 2 8 8 3 fπ fπ This ambiguity directly affects predictions for rare kaon 5 2 Δ ¼ 1 0 mπY decays to the piophobic axion, since the S octet þ g8 þ g − g27 ; ð61Þ 0 8 2 2 operators O8 and O8 contribute differently to the ampli- fπ Að þ →πþ Þ Að 0 →π0 Þ Að 0 → ππ Þ tudes K a , KS;L a , and KL a . where Y is one of the standard Dalitz plot variables, In what follows, we will estimate the rates for various 0 0 defined as axionic kaon decays in both scenarios, g8 ≫ g8 and g8 ≪ g8. 0 g8 ≫ g 2 We will show that in the case of 8, all amplitudes Y ≡ ðs3 − s0Þ=mπ; ð62aÞ Að þ → πþ Þ Að 0 → π0 Þ Að 0 → ππ Þ K a , KS;L a , and KL a are octet enhanced, leading to higher axionic kaon decay ≡ ð − Þ2j ð Þ si pK pi i¼1;2;3; 62b rates, and when relevant, more stringent constraints on the mixing angles θ η and θ η . Conversely, in the scenario a ud a s ðs1 þ s2 þ s3Þ ≪ 0 Γð þ → πþ Þ Γð 0 → π0 Þ s0 ≡ ; ð62cÞ with g8 g8, the rates K a and KS;L a 3 are significantly reduced, relaxing the otherwise strong θ with p1 and p2 referring to the four-momenta of the constraints on aηud and offering an exciting prospect for searching for these signals in near-future rare kaon decay charged pions, and p3 the four-momentum of the neutral experiments. daughter ,14 in this case π0. In upcoming subsections, we will normalize the calcu- lated axio-hadronic rates to analogous kaon decay rates in A. K + decays the SM. For later reference, we quote here the dependence 0 of the relevant SM kaon decay amplitudes on g8, g8, as well 1. Axio-leptonic K + decays as g27, the coefficient of the ΔS ¼ 1 27-plet operator at þ þ The amplitude for the axio-leptonic decay K → l νla Oðp2Þ in χPT, which is given by can be easily related to the SM semileptonic amplitudes ðΔS¼1Þ 2 ij † k † μ l via the Ademollo-Gatto theorem [154], which states O ¼ g27fπT ðU ∂μUÞ ðU ∂ UÞ : ð57Þ 27 kl i j that the matrix elements of flavor-changing electroweak ij current operators can only deviate from their SUð3Þχ- Above, Tkl are Clebsch-Gordan coefficients that project the 27-plet, ΔI ¼ 3=2 part of the interaction [146]. The symmetric values to second order in chiral symmetry contributions from (56a), (56b), and (57) to the two-body breaking [155,156]. hadronic kaon decays of interest are [151,153] This implies that, at zero momentum transfer, the following SUð3Þχ relations hold: 2 0 þ − 0 mK pffiffiffi AðK → π π Þ¼2ðg8 þ g8 þ g27Þ μ þ 0 μ þ 2 S hη j¯γ j ij 2 ¼ 3hπ j¯γ j ij 2 þOðϵ Þ ð Þ fπ 8 s u K q ¼0 s u K q ¼0 ; 63a 2 mπ − 2ð þ 2 0 þ Þ ð Þ μ þ 2 g8 g8 g27 ; 58 hη0js¯γ ujK ij 2¼0 ¼ Oðϵ Þ; ð63bÞ fπ q

where ϵ is a measure of SUð3Þχ breaking. Then, since 13 0 ∼ Indeed, a naïve dimensional analysis estimation of g8 j i¼θ jπ0iþθ jη iþθ jη i 2 2 2 −7 a aπ aη8 8 aη0 0 , we have jGF sin θcfπΛ =m0þþ j ∼ Oð10 Þ does not immediately rule out this hypothesis. It is unclear whether a Dalitz plot analysis of 0 → 3π 14 ’ KL data could distinguish it from the alternative descrip- Note that in Sec. VC2, p3 will refer to the axion s tion of octet enhancement. four-momentum.

055018-15 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

FIG. 5. (Disclaimer: not intended as a realistic experimental sensitivity projection.) The dashed lines show the reach of various axionic kaon decays modes in the parameter space of the axion isoscalar couplings, assuming a common branching ratio sensitivity benchmark −8 0 of 10 for all decay channels. The upper (lower) plots assume that octet enhancement in χPT is realized through operator O8 (O8) θ θ defined in (56a) ((56b)). The left (right) plots assume opposite (same) relative sign between aηud and aηs . The orange and yellow bands favored by the 8Be and 4He anomalies are the same as in Fig. 1. The shaded gray regions are excluded by the conservative upper bound ð þ → πþð → þ −ÞÞ ≲ 10−5 0 → þ − Br K a e e and by the observed rate for KL e e , cf. (91).

μ þ þ þ hajs¯γ ujK ij 2¼0 In Fig. 5, we show the hypothetical reach of K → μ νμa pq ffiffiffi 0 μ þ 2 to the axion isoscalar mixing angles, assuming an exper- ¼ðθ π þ 3θ η Þhπ js¯γ ujK ij 2¼0 þ Oðϵ Þ; þ þ a a 8 pffiffiffi q imental sensitivity to branching ratios BrðK → μ νμaÞ ≳ 0 μ þ 2 −8 ¼ðθ þ θ − 2θ Þhπ j¯γ j ij 2 þ Oðϵ Þ 10 . Note that this branching ratio sensitivity figure has aπ aηud aηs s u K q ¼0 : been chosen to facilitate comparison between different ð64Þ axionic kaon decay modes (to be discussed in upcoming Neglecting the difference in phase space, as well as finite subsections) and is not informed by any experimental sensitivity projections. momentum-transfer and SUð3Þχ breaking corrections, which amount to Oð10%Þ [153], it then follows from (64) that 2. Axio-hadronic K + decays

þ þ For axio-hadronic kaon decays, we must first obtain the BrðK → l νlaÞ pffiffiffi contributions from operators (56a), (56b), and (57) to the ≈ jθ þ θ − 2θ j2 ð þ → lþν π0Þ ð Þ amplitudes for Kþ → πþφ (φ ¼ π0; η ; η ). Putting Kþ aπ aηud aηs Br K l ; 65 ud s and πþ on shell, we have In the specific case of a muonic final state, (65) yields pffiffiffi þ þ 0 2 AðK → π π Þ θ π þ θ η − 2θ η ð þ → μþν Þ ≈ 0 84 10−8 a a ud a s Br K μa . × −4 : 2 2 2 5 10 0 × mK mπ pπ ¼ 3g27 − ðg8 þ 4g27Þ þðg8 þ g27Þ ; ð67aÞ ð66Þ fπ fπ fπ

055018-16 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

Að þ → πþη Þ K ud from which it follows that 2 2 2 pη þ þ mK mπ ud ð → π Þj ¼ð2g8 þ 3g27Þ −ðg8 þ 2g27Þ − ðg8 þ g27Þ ; Br K a 27-plet fπ fπ fπ jAð þ → πþ Þj2 ⃗ ≈ K a ð þ → πþπ0Þ pa ð67bÞ þ þ 0 2 Br K jAðK → π π Þj ð71Þ p⃗π þ þ pffiffi AðK → π ηsÞ 2 2 θ π þ θ η þ θ η 2 2 2 ≈ 2 10−8 a a ud 3 a s ð Þ pffiffiffi pffiffiffi pffiffiffi pη × −4 : 72 mK mπ s 5 10 ¼ 2g27 − 2ðg8 þ 2g27Þ þ 2ðg8 þ g27Þ : × fπ fπ fπ In Figs. 1 and 5, we show the excluded parameter space ð67cÞ for the axion’s isoscalar mixing angles—assuming a þ þ ð þ →πþ Þ≲10−5 The axionic decay K → π a is then induced by these conservative experimental bound of Br K a — amplitudes via axion-meson mixing, (see discussion in [46]) for the two assumed scenarios of octet enhancement in χPT which resulted in (70) and (72). þ þ þ þ 0 þ AðK → π aÞ¼θ πAðK → π π Þj 2 2 We also show in Fig. 5 the hypothetical reach of K → a p 0 ¼ma π πþa to the axion isoscalar mixing angles, assuming an þ þ þ þ θ η AðK → π η Þj 2 2 15 ð → a ud ud pη ¼ma experimental sensitivity to branching ratios Br K ud þ −8 þ þ þ þ π aÞ≳10 . The K → π a contours in Fig. 5 make þ θ Að → π η Þj 2 2 ð Þ aη K s pη ¼m : 68 s s a evident that this axionic kaon decay channel is one of the 0 most sensitive in probing the piophobic QCD axion, and that Note that (68) depends on g8 but not on g . This implies 8 þ → πþ þ − that AðKþ → πþaÞ is only octet enhanced in the scenario updating the three-decades-old bounds on K e e 0 with m þ − ≲ 50 MeV [157–159] could cover presently with g8 ≫ g8, i.e., in the standard realization of octet e e unexplored and well-motivated parameter space of light enhancement in χPT via O8. In this case, using (58) and 0 BSM sectors. taking g8 → 0 for simplicity, we can approximate (68) as jAð þ → πþ Þj2j B. K0 decays K a octetenh S pffiffiffi 2 mπ 2 The axio-hadronic decays of the CP-even neutral kaon j2g8θ η þ 2θ η ðg27 −g8 2 Þj 1 a ud a s m ≈ jAð 0 → πþπ−Þj2 K can be estimated via an analogous prescription as the one KS 2 ; Kππ j2ðg8 þg27Þj used in Sec. VA2. First, we obtain the contributions to Að 0 → π0φÞ φ ¼ π0 η η ð69Þ KS , ; ud; s, from operators (56a), 0 π0 (56b), and (57). With KS and on shell, we have ∼ 3 ππ where Kππ corrects for the fact that strong s-wave 2 0 þ − 2 2 → π π 2m mπ pπ0 final state interaction, present in KS , is absent in Að 0 → π0π0Þ¼ð þ 0 − 2 Þ K − − þ þ KS g8 g8 g27 K → π a [62]. With (69) and (60), we finally obtain fπ fπ fπ 2 2 ð þ → πþ Þj 0 mπ pπ0 Br K a octet enh − g8 þ ; ð73aÞ fπ fπ þ þ 2 Γ 0 jAðK → π aÞj K p⃗ ≈ ð 0 → πþπ−Þ S a 0 þ − 2 Br KS 2 2 jAð → π π Þj Γ þ ⃗ m mπ KS ð69Þ K pπ Að 0 → π0η Þ¼−2 K þð þ 2 Þ KS ud g8 g8 g27 2 fπ fπ θ η − 0.032θ η ≈ 0 9 10−5 a ud a s ð Þ 2 . × −4 : 70 5 10 pηud × þðg8 − 2g27Þ ; ð73bÞ fπ 0 þ þ In the scenario with g8 ≪ g , AðK → π aÞ is not octet 8 pffiffiffi 2 pffiffiffi 2 enhanced. Since g8 is then expected to be of the same Að 0 → π0ηÞ¼−4 2 mK þ 2ð þ 2 Þ mπ KS s g27 g8 g27 magnitude as g27, there might be non-negligible interfer- fπ fπ ence between the contributions stemming from O8 and O27. pffiffiffi 2 pηs For simplicity, we will ignore this effect and consider the − 2ðg8 − 2g27Þ : ð73cÞ fπ limiting case of g8 → 0, when O27 provides the dominant contribution to axio-hadronic Kþ decays. In this case, using 0 → π0 The axionic decay KS a is then induced by these (59), (68) can be approximated as amplitudes via axion-meson mixing,

þ þ 2 jAðK → π aÞj j27 15 −8 -plet pffiffiffi This choice of branching ratio sensitivity benchmark of 10 1 2 2 is intended to facilitate comparison between different axionic þ þ 0 2 ≈ jAðK → π π Þj θ π þ θ η þ θ η ; ð71Þ kaon decay modes and is not informed by any experimental a a ud 3 a s Kππ sensitivity projections.

055018-17 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

0 0 0 0 0 Að → π Þ¼θ Að → π π Þj 2 2 K a aπ K p ¼m In prior searches for this decay mode, the published S S π0 a 0 0 analyses by NA31 [161] and NA48 [162] showed the þ θ Að → π η Þj 2 2 aηud KS ud pη ¼m þ − þ − ∼ 0 ud a observed me e distributions down to me e , even though 0 0 þ − 140 þ − 165 þ θ Að → π η Þj 2 2 ð Þ events with me e < MeV (NA31), me e < MeV aη KS s pη ¼m : 74 s s a (NA48) were rejected when extracting upper bounds on 0 0 0 þ − ð → π Þ þ − Note that the only occurrence of g8 in (74) stems Br KS e e . In particular, the me e distribution in from the axion-pion mixing contribution in (73a),and,as Fig. 3 of [162] shows 2 events within the window of θ such, it is suppressed by the small aπ mixing angle. Because 10 MeV

055018-18 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021) and θ ≲ ð3 5–5Þ 10−3 isoscalar mixing angles of aηud . × (for 0 −2 g → 0) and θ η ≲ ð4.5–6.4Þ × 10 (for g8 → 0) are not 2 8 a s θ η − 11θ π þ þ 0 0 −11 a s a → π BrðK → π aÞj27− ≈ 2 × 10 : ð81Þ competitive with limits from K a (see Fig. 5). L plet 2 × 10−3 0 → π0 2. CP-conserving axio-hadronic K0 decays An upper bound on KL a can be inferred from L 0 → a search for light Higgs in the final state KL The CP-conserving axio-hadronic decays of the CP-odd π0ðh → eþe−Þ performed by CERN’s SPS NA31 experi- neutral kaon can be estimated via an analogous prescription ment in 1990 [163]. Assuming that this analysis’ efficiency as the one used in Secs. VA2 and VB. For specificity, we to promptly decaying axions is comparable to that of consider the final state with charged pions, and obtain the ∼17 Að 0 → πþπ−φÞ φ ¼ π0 η η much longer lived Higgses of mass MeV, the bound contributions to KL , ; ud; s, from ð 0 →π0ð → þ −ÞÞ≲ 0 πþ π− on the axionic decay would be Br KL a e e operators (56a), (56b), and (57). Putting KL, , and on ð1–2Þ×10−8. The resulting constraints on the axion shell, we have

0 2 0 2 0 2 2 ð þ þ2 Þ ð þ4 þ11 Þ ð þ þ11 Þp 0 5 Að 0 →πþπ−π0Þ¼ g8 g8 g27 mK − g8 g8 g27 mπ þ g8 g8 g27 π þ þ 0 − mπY ð Þ KL 2 2 2 g8 g8 g27 2 ; 82a 3 fπ 3 fπ 3 fπ 2 fπ 0 2 0 2 0 2 2 ð3g8 þ g − 4g27Þm ð2g8 þ 4g − 3g27Þmπ ðg8 −g − g27Þpη 1 mπY Að 0 → πþπ−η Þ¼− 8 K þ 8 þ 8 ud − 0 þ ð Þ KL ud 2 2 2 g8 g27 2 ; 82b 3 fπ 3 fπ 3 fπ 2 fπ pffiffiffi pffiffiffi pffiffiffi 2 2 2 pffiffiffi 2 2 m 2 mπ 2 pη 3 mπY Að 0 → πþπ−ηÞ¼ ð − 0 Þ K þ ð4 0 − Þ − ð þ 0 − Þ s þ 2 − 0 − ð Þ KL s g8 g8 2 g8 g27 2 g8 g8 g27 2 g8 g8 g27 2 ; 82c 3 fπ 3 fπ 3 fπ 2 fπ

Að 0 → πþπ−π0Þ where the Dalitz plot variable Y has been defined in (62). KL on g8 and g27;see(61). Similarly, the Furthermore, an additional effect contributing to Að 0 → ππ Þ contribution to the axionic amplitude KL a stem- AðK0 → ππaÞ must be taken into account, namely, kinetic 0 − ηð0Þ L ming fromKL mixingbecomes importantinthescenario 0 0 −4 mixing between KL and the neutral pseudoscalar mesons ≫ jθ j jθ j ≲ Oð10 Þ with g8 g8 when aηud , aηs . It can be induced by operators O8 and O27, A straightforwardly obtained by reweighting ηð0Þ→ππa in (53), ðΔS¼1Þ 0 μ 0 Lχ ⊃−2ðg8 þ 2g27Þ∂μK ∂ π CK0 PT L Að 0 → ηð0Þ → ππ Þ¼ L Aðηð0Þ → ππ Þj pffiffiffi KL a a p ð0Þ →p 0 ; 0 μ μ ð0Þ η K þ 2ð − 2 Þ∂ ð∂ η þ 2∂ η Þ ð Þ Cη L g8 g27 μKL ud s : 83 ð Þ 0 − π0 84 In particular, accounting for KL mixing is crucial in order to obtain the correct dependence of the SM amplitude where

 2 2  pffiffiffi m 0 m 0 KL KL 0 C 0 ≡ 2ðg8 − 2g27Þhη þ 2η j Cηjηiþ Cη0 jη i KL ud s 2 2 2 2 mη − m 0 mη0 − m 0 KL KL

≈ 0.02ðg8 − 2g27Þ; ð85Þ

Að 0 → πþπ− Þ and we have used (37) and (54) in obtaining the approxi- The total amplitude KL a is then given by mate equality in (85). the sum of (84) and (82a), (82b), and (82c) reweighted by Unfortunately, the amplitude in (84) is subject to the same axion-meson mixing angles, destructive interference effects, and therefore the same large Aðηð0Þ → ππ Þ uncertainties, as a estimated in Sec. IV B.In AðK0 → πþπ−aÞ¼AðK0 → ηð0Þ → ππaÞ particular, for the region of parameter space where (84) L L 0 þ − 0 þ θ πAðK → π π π Þj 2 2 becomes important, these uncertainties make the estimation a L p 0 ¼ma ð 0 → ππ Þ π of Br KL a unreliable. Nonetheless, for the sake of 0 þ − 0 ð0Þ þ θ η AðK → π π η Þj 2 ¼ 2 − η a ud L ud pη ma illustration, we will include the effects of KL mixing in ud 0 þ − our estimations below by benchmarking the RχT parameters þ θ Að → π π η Þj 2 2 aη KL s pη ¼m : ¼ ¼ 980 Γ ¼50 s s a entering in (84) to ma0 mf0 MeV, a0 MeV, Γ ¼100 b ¼ b ¼ 1 ð86Þ f0 MeV, and cd cm .

055018-19 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

Að 0 → πþπ− Þ Note that KL a is octet enhanced in both Despite the assumptions, simplifications, and uncertain- 0 → πþπ− scenarios, and therefore we can neglect the contributions ties of our estimations, the KL a contours in Fig. 5 0 0 from g8 and g27 when g8 ≫ g8, and likewise neglect the offer a compelling motivation for upcoming kaon experi- 0 0 ≫ þ − ∼ 17 → contributions from g8 and g27 when g8 g8. Despite this ments to search for an me e MeV resonance in KL ð 0 → ππ þ − 18 simplification, obtaining the dependence of Br KL e e final states. If these searches could achieve πþπ−aÞ on the axion-meson mixing angles still involves sensitivities down to branching ratios of Oð10−8Þ, they nontrivial integration of the differential decay width over could almost fully exclude the parameter space favored by the three-body final state phase space. We performed this (or verify the QCD axion explanation of) the 8Be, 4He, and integration numerically for both octet-enhancement scenar- KTeV anomalies. ios under the assumptions stated above, and using (61) for normalization, obtained 0 3. Dielectronic KL decays 0 þ − −8 The last rare kaon decay we shall consider is BrðK → π π aÞj ≫ 0 ≈ 3.54 ×10 L g8 g8 0 → þ − KL e e , whose amplitude can receive a potentially −4 0 ð0Þ þ 10 × ð3.83θaπ −7.42θaη þ 5.715θaη Þ − η − ud s non-negligible contribution from KL a mixing. 2 2 2 þ 1.18θaπ þ 3.92θaη þ 2.60θaη Indeed, using (83) and momentarily neglecting the SM ud s 0 → contribution to the amplitude, it follows that the KL þ θ πð−4.125θ η þ 3.51θ η Þ − 6.14θ η θ η ð87Þ a a ud a s a ud a s þ − 0 − ηð0Þ − e e rate induced by KL a mixing would be and ð 0 → þ −Þj Br KL e e A →0 pffiffiffi SM 0 e pffiffiffi 2 0 þ − 2 1 mK q me BrðK → π π aÞj 0 ≫ ≈ 1.49ðθ π þ θ η þ 2θ η Þ : ≃ L PQ ð2 −4 Þðθ þ 2θ Þ L g8 g8 a a ud a s g8 g27 aη aη Γ 0 8π ud s K fa ð88Þ L pffiffiffi 2 2 g8 −2g27 θ η þ 2θ η ≃0 9 10−11 ðqe Þ2 a ud a s : ð Þ Að 0 → π0π0 Þ . × þ 0 PQ −3 90 We remark that the amplitude KL a is simply g8 g8 10 Að 0 → πþπ− Þ related to KL a by isospin symmetry, which results in The estimate above should be contrasted with the observed 0 → þ − KL e e rate [167], as well as the range of SM 1 – ð 0 → π0π0 Þ¼ ð 0 → πþπ− Þ ð Þ predictions [168 170], Br KL a 2 Br KL a : 89 ð 0 → þ −Þj ¼ð0 87þ0.57Þ 10−11 ð Þ Br KL e e exp . −0.41 × ; 91 In Fig. 5, we show the hypothetical reach of 0 → πþπ− ð 0 → þ −Þj ∼ ð0 3–0 9Þ 10−11 ð Þ KL a to the axion isoscalar mixing angles assum- Br KL e e SM . . × : 92 ing an experimental sensitivity17 to branching ratios ð 0 → πþπ− Þ ≳ 10−8 Br KL a , under both octet-enhancement From (90) and (91), we can extract a nontrivial upper scenarios in χPT. Since the contribution from θaπ is non- bound on the axion isoscalar mixing angles in the octet- 0 negligible for the chosen branching ratio sensitivity bench- enhancement scenario with g8 ≫ g8 (shown in Fig. 5), mark of 10−8, the contours in Fig. 5 implicitly assume the pffiffiffi 1 3 relative sign between θ π and θ η that yields the most −2 = a a ud jθ η þ 2θ η j ≲ 0 5 10 : ð Þ a ud a s . × e 93 0 jq j conservative reach. g8≫g8 PQ It is worth mentioning an exception to our estimations of pffiffiffi 0 þ − e −3 Að → π π Þ For jq ðθ η þ 2θ η Þj ≲ 10 , however, the SM KL a presented in this section. Besides the two PQ a ud a s 0 limiting cases we have been considering of g8 ≫ g8 and contribution to the amplitude cannot be neglected; in this 0 0 g8 ≪ g8, there is also the possibility that g8 and g8 have case, the estimation in (90) is not accurate. Indeed, for a 8 comparable magnitudes. In this case, there would be non- significant part of the parameter space favored by the Be 0 → 4 0 → þ − negligible interference between the amplitudes for KL and He anomalies, the axionic contribution to KL e e þ − 0 π π a originating from operators O8 and O8, which could is subdominant to that of the SM, and in fact it is negligible 0 þ − 0 ð → π π Þ in the octet-enhancement scenario with g8 ≪ g8. A tanta- substantially modify the dependence of Br KL a on the axion-meson mixing angles. lizing possibility remains, however, that once measure- ments and theoretical predictions are improved over (91) 17This choice of branching ratio sensitivity benchmark of 10−8 18 0 → ππ þ − is intended to facilitate comparison between different axionic Dedicated reanalyses of existing data in KL e e final kaon decay modes and is not informed by any experimental states could also be potentially sensitive to the axionic signal. sensitivity projections. See, e.g., [164–166].

055018-20 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021) and (92), a piophobic QCD axion signal could appear in (predominantly) isoscalar magnetic transitions of excited this channel as an excess over the SM expectation. 8Be and 4He nuclei. Unsurprisingly, such signals have long been predicted as smoking gun signatures of the VI. SUMMARY AND DISCUSSION QCD axion. In this paper, we estimated the axionic emission rates of The PQ mechanism was conceived to address a problem 8 4 χ intrinsic to the nonperturbative dynamics of QCD. Yet, the relevant Be and He transitions, taking nuclear and PT presently, the prevalent view is that PQ symmetry breaking uncertainties into account, and showed that the piophobic ≳ 1010Λ QCD axion provides a natural and compelling explanation should take place at scales fPQ QCD. Why should these two scales be so widely separated? PQ cancellation of of the observed data for these two nuclei with quite distinct properties. We also considered in detail potential axionic the strong CP phase would be much more robust against 0 0 ∼ Λ signals in rare decays of the η, η , K , and K mesons. spoiling effects if fPQ QCD. This possibility has long S;L been dismissed due to stringent laboratory constraints on The (often ignored) hadronic and χPT uncertainties the visible QCD axion, in particular on its isovector involved in estimations of these rare meson decays impede couplings. More recently, however, we showed in [46] accurate predictions of axio-hadronic signals; nonetheless, that the Oð10Þ MeV mass range for the QCD axion the ranges we have obtained for several of the processes remains compatible with all existing experimental investigated can be probed in the near future in a variety of 0 constraints if the QCD axion (i) couples dominantly to experimental programs, including η=η and kaon factories. the first generation of SM fermions; (ii) is short lived, ≲4 10−14 þ − decaying with lifetimes × stoe e ; and (iii) is ACKNOWLEDGMENTS piophobic, i.e., has suppressed isovector couplings due to an accidental cancelation of its mixing with the neutral I am grateful to Anna Hayes and Gerry Hale for −4 8 pion, θ π ≲ 10 . These conditions require nontrivial UV explanations of electromagnetic transitions of Be and a 4 completions, but so does any viable QCD axion model, He nuclei. I also thank Maxim Pospelov for discussions whether “heavy” and short lived or ultralight and cosmo- on the experimental sensitivity of the COSY-WASA analy- logically long lived. sis to axionic η decays, and Corrado Gatto for discussions While this possibility forgoes the attractive feature about the REDTOP experiment. This work was supported of explaining the particle nature of dark matter, it by an Early Career Research award from Los Alamos offers a single, consistent explanation for a few persistent National Laboratory’s LDRD program and by the DOE experimental anomalies: the observed rate for π0 → eþe−, Office of Science High Energy Physics under Contract and the “bumplike” excesses in the eþe− spectra of No. DE-AC52-06NA25396.

[1] J. Jaeckel and A. Ringwald, The low-energy frontier of [9] P. deNiverville, M. Pospelov, and A. Ritz, Observing a , Annu. Rev. Nucl. Part. Sci. 60, 405 (2010). beam with neutrino experiments, Phys. [2] J. L. Hewett et al., Fundamental physics at the intensity Rev. D 84, 075020 (2011). frontier,(2012), https://dx.doi.org.10.2172/1042577. [10] B. A. Dobrescu and C. Frugiuele, GeV-scale dark matter: [3] J. Alexander et al., Dark sectors 2016 workshop: Com- Production at the main injector, J. High Energy Phys. 02 munity report, arXiv:1608.08632. (2015) 019. [4] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. [11] C. Frugiuele, Probing sub-GeV dark sectors via high Weiner, A theory of dark matter, Phys. Rev. D 79, 015014 energy proton beams at LBNF/DUNE and MiniBooNE, (2009). Phys. Rev. D 96, 015029 (2017). [5] B. Batell, M. Pospelov, and A. Ritz, Probing a secluded [12] L. Buonocore, C. Frugiuele, and P. deNiverville, Hunt for U(1) at B-factories, Phys. Rev. D 79, 115008 (2009). sub-GeV dark matter at neutrino facilities: A survey of past [6] R. Essig, P. Schuster, and N. Toro, Probing dark forces and and present experiments, Phys. Rev. D 102, 035006 (2020). light hidden sectors at low-energy e þ e− colliders, Phys. [13] Kickoff Meeting of the RF6 SNOWMASS Working Rev. D 80, 015003 (2009). Group, Dark Sectors at High Intensities, https://indico [7] J. D. Bjorken, R. Essig, P. Schuster, and N. Toro, New .fnal.gov/event/44819/ fixed-target experiments to search for dark gauge forces, [14] R. D. Peccei and H. R. Quinn, CP Conservation in the Phys. Rev. D 80, 075018 (2009). Presence of , Phys. Rev. Lett. 38, 1440 (1977). [8] B. Batell, M. Pospelov, and A. Ritz, Exploring portals to a [15] R. D. Peccei and H. R. Quinn, Constraints imposed by CP through fixed targets, Phys. Rev. D 80, conservation in the presence of instantons, Phys. Rev. D 095024 (2009). 16, 1791 (1977).

055018-21 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

[16] S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40, [37] R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. 223 (1978). Watkins, and L. M. Widrow, Solutions to the strong CP [17] F. Wilczek, Problem of Strong p and t Invariance in the problem in a world with gravity, Phys. Lett. B 282, 132 Presence of Instantons, Phys. Rev. Lett. 40, 279 (1978). (1992). [18] J. E. Kim, Singlet and Strong CP [38] M. Kamionkowski and J. March-Russell, Planck scale Invariance, Phys. Rev. Lett. 43, 103 (1979). physics and the Peccei-Quinn mechanism, Phys. Lett. B [19] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can 282, 137 (1992). confinement ensure natural CP invariance of strong [39] S. M. Barr and D. Seckel, Planck scale corrections to axion interactions?, Nucl. Phys. B166, 493 (1980). models, Phys. Rev. D 46, 539 (1992). [20] M. Dine, W. Fischler, and M. Srednicki, A simple solution [40] S. Ghigna, M. Lusignoli, and M. Roncadelli, Instability of to the strong CP problem with a harmless axion, Phys. the invisible axion, Phys. Lett. B 283, 278 (1992). Lett. 104B, 199 (1981). [41] R. Kallosh, A. D. Linde, D. A. Linde, and L. Susskind, [21] J. Preskill, M. B. Wise, and F. Wilczek, of the Gravity and global symmetries, Phys. Rev. D 52, 912 invisible axion, Phys. Lett. 120B, 127 (1983). (1995). [22] L. F. Abbott and P. Sikivie, A cosmological bound on the [42] S. Gori, G. Perez, and K. Tobioka, KOTO vs. NA62 dark invisible axion, Phys. Lett. 120B, 133 (1983). scalar searches, J. High Energy Phys. 08 (2020) 110. [23] M. Dine and W. Fischler, The not so harmless axion, Phys. [43] A. J. Krasznahorkay et al., Observation of Anomalous Lett. 120B, 137 (1983). Internal Pair Creation in Be8: A Possible Indication of a [24] C. E. Aalseth et al., The CERN axion solar telescope Light, Neutral Boson, Phys. Rev. Lett. 116, 042501 (CAST), Nucl. Phys. B, Proc. Suppl. 110, 85 (2002). (2016). [25] S. J. Asztalos et al. (ADMX Collaboration), A SQUID- [44] A. J. Krasznahorkay et al., New evidence supporting the Based Cavity Search for Dark-Matter Axions, existence of the hypothetic , arXiv:1910 Phys. Rev. Lett. 104, 041301 (2010). .10459. [26] D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, [45] E. Abouzaid et al. (KTeV Collaboration), Measurement of and A. Sushkov, Proposal for a Cosmic Axion Spin the rare decay π0 → eþe−, Phys. Rev. D 75, 012004 Precession Experiment (CASPEr), Phys. Rev. X 4, (2007). 021030 (2014). [46] D. S. M. Alves and N. Weiner, A viable QCD axion in the [27] S. K. Lamoreaux, K. A. van Bibber, K. W. Lehnert, and G. MeV mass range, J. High Energy Phys. 07 (2018) 092. Carosi, Analysis of single-photon and linear amplifier [47] D. Aloni, Y. Soreq, and M. Williams, Coupling QCD-Scale detectors for microwave cavity dark matter axion searches, Axionlike Particles to , Phys. Rev. Lett. 123, Phys. Rev. D 88, 035020 (2013). 031803 (2019). [28] E. Armengaud et al., Conceptual design of the [48] A. E. Dorokhov and M. A. Ivanov, Rare decay international axion observatory (IAXO), J. Instrum. 9, pi0− > e þ e−: Theory confronts KTeV data, Phys. T05002 (2014). Rev. D 75, 114007 (2007). [29] A. Arvanitaki and A. A. Geraci, Resonantly Detecting [49] A. E. Dorokhov, E. A. Kuraev, Yu. M. Bystritskiy, and M. Axion-Mediated Forces with Nuclear Magnetic Reso- Secansky, QED radiative corrections to the decay nance, Phys. Rev. Lett. 113, 161801 (2014). pi0− > e þ e−, Eur. Phys. J. C 55, 193 (2008). [30] G. Rybka, A. Wagner, A. Brill, K. Ramos, R. Percival, and [50] P. Vasko and J. Novotny, Two-loop QED radiative cor- K. Patel, Search for dark matter axions with the Orpheus rections to the decay pi0− > e þ e−: The virtual correc- experiment, Phys. Rev. D 91, 011701 (2015). tions and soft-photon , J. High Energy [31] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lind- Phys. 10 (2011) 122. ner, and K. A. van Bibber, Experimental searches for the [51] T. Husek, K. Kampf, and J. Novotný, Rare decay axion and axion-like particles, Annu. Rev. Nucl. Part. Sci. π0 → eþe−: On corrections beyond the leading order, 65, 485 (2015). Eur. Phys. J. C 74, 3010 (2014). [32] Y. Kahn, B. R. Safdi, and J. Thaler, Broadband and [52] D. V. Kirpichnikov, V. E. Lyubovitskij, and A. S. Resonant Approaches to Axion Dark Matter Detection, Zhevlakov, Implication of hidden sub-GeV bosons for 8 4 Phys. Rev. Lett. 117, 141801 (2016). the ðg − 2Þμ, Be- He anomaly, proton charge radius, EDM [33] M. Baryakhtar, J. Huang, and R. Lasenby, Axion and of fermions, and dark axion portal, Phys. Rev. D 102, hidden photon dark matter detection with multilayer 095024 (2020). optical haloscopes, Phys. Rev. D 98, 035006 (2018). [53] J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, [34] I. G. Irastorza and J. Redondo, New experimental ap- T. M. P. Tait, and P. Tanedo, Protophobic Fifth-Force proaches in the search for axion-like particles, Prog. Part. Interpretation of the Observed Anomaly in 8Be Nuclear Nucl. Phys. 102, 89 (2018). Transitions, Phys. Rev. Lett. 117, 071803 (2016). [35] P. Brun et al. (MADMAX Collaboration), A new [54] J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, experimental approach to probe QCD axion dark matter T. M. P. Tait, and P. Tanedo, Particle physics models for the in the mass range above 40 μeV, Eur. Phys. J. C 79, 186 17 MeV anomaly in beryllium nuclear decays, Phys. Rev. (2019). D 95, 035017 (2017). [36] W. A. Bardeen and S. H. H. Tye, Current algebra applied to [55] J. L. Feng, T. M. P. Tait, and C. B. Verhaaren, Dynamical Properties of the light , Phys. Lett. 74B, 229 evidence for a explanation of the ATOMKI (1978). nuclear anomalies, Phys. Rev. D 102, 036016 (2020).

055018-22 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

[56] X. Zhang and G. A. Miller, Can a protophobic vector [77] G. S. Bali et al. (QCDSF Collaboration), Strangeness boson explain the ATOMKI anomaly?, Phys. Lett. B 813, Contribution to the Proton Spin from Lattice QCD, Phys. 136061 (2021). Rev. Lett. 108, 222001 (2012). [57] Y. Kahn, G. Krnjaic, S. Mishra-Sharma, and T. M. P. Tait, [78] M. Engelhardt, contributions to nucleon Light weakly coupled axial forces: Models, constraints, mass and spin from lattice QCD, Phys. Rev. D 86, 114510 and projections, J. High Energy Phys. 05 (2017) 002. (2012). [58] J. Kozaczuk, D. E. Morrissey, and S. R. Stroberg, Light [79] A. Abdel-Rehim, C. Alexandrou, M. Constantinou, V. axial vector bosons, nuclear transitions, and the 8Be Drach, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. anomaly, Phys. Rev. D 95, 115024 (2017). Vaquero, Disconnected quark loop contributions to nu- [59] S. B. Treiman and F. Wilczek, Axion emission in decay of cleon observables in lattice QCD, Phys. Rev. D 89, 034501 excited nuclear states, Phys. Lett. 74B, 381 (1978). (2014). [60] T. W. Donnelly, S. J. Freedman, R. S. Lytel, R. D. Peccei, [80] T. Bhattacharya, R. Gupta, and B. Yoon, Disconnected and M. Schwartz, Do axions exist?, Phys. Rev. D 18, 1607 quark loop contributions to nucleon structure, Proc. Sci., (1978). LATTICE2014 (2014) 141. [61] A. Barroso and N. C. Mukhopadhyay, Nuclear axion [81] A. Abdel-Rehim et al., Nucleon and pion structure with decay, Phys. Rev. C 24, 2382 (1981). lattice QCD simulations at physical value of the pion mass, [62] I. Antoniadis and T. N. Truong, Lower bound for branch- Phys. Rev. D 92, 114513 (2015); , Erratum, Phys. Rev. D ing ratio of Kþ → πþ axion and nonexistence of Peccei- 93, 039904 (2016). Quinn axion, Phys. Lett. 109B, 67 (1982). [82] A. Abdel-Rehim, C. Alexandrou, M. Constantinou, K. [63] W. A. Bardeen, R. D. Peccei, and T. Yanagida, Constraints Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and on variant axion models, Nucl. Phys. B279, 401 (1987). A. V. Avil´es-Casco, Disconnected quark loop contributions [64] L. M. Krauss and M. B. Wise, Constraints on shortlived to nucleon observables using Nf ¼ 2 twisted clover axions from the decay πþ → eþe−eþ neutrino, Phys. Lett. fermions at the physical value of the light quark mass, B 176, 483 (1986). Proc. Sci., LATTICE2015 (2016) 136. [65] M. Davier, Searches for new particles, in Proceedings, 23RD [83] J. Green, N. Hasan, S. Meinel, M. Engelhardt, S. Krieg, J. International Conference on High Energy Physics, Berkeley, Laeuchli, J. Negele, K. Orginos, A. Pochinsky, and S. CA (Berkeley National Laboratory, Berkeley,1986). Syritsyn, Up, down, and strange nucleon axial form factors [66] L. M. Krauss and D. J. Nash, A viable weak interaction from lattice QCD, Phys. Rev. D 95, 114502 (2017). axion?, Phys. Lett. B 202, 560 (1988). [84] F. P. Calaprice, R. W. Dunford, R. T. Kouzes, M. Miller, A. [67] A. Anastasi et al., Limit on the production of a low-mass Hallin, M. Schneider, and D. Schreiber, Search for axion vector boson in eþe− → Uγ,U→ eþe− with the KLOE emission in the decay of excited states of C-12, Phys. Rev. experiment, Phys. Lett. B 750, 633 (2015). D 20, 2708 (1979). [68] D. Banerjee et al. (NA64 Collaboration), Improved limits [85] M. J. Savage, R. D. Mckeown, B. W. Filippone, and L. W. on a hypothetical X(16.7) boson and a Mitchell, Search for a Shortlived Neutral Particle Produced decaying into eþe− pairs, Phys. Rev. D 101, 071101 in Nuclear Decay, Phys. Rev. Lett. 57, 178 (1986). (2020). [86] A. L. Hallin, F. P. Calaprice, R. W. Dunford, and A. B. [69] E. Depero et al., Hunting down the X17 boson at the Mcdonald, Restrictions on a 1.7-MeVAxion From Nuclear CERN SPS, Eur. Phys. J. C 80, 1159 (2020). Pair Transitions, Phys. Rev. Lett. 57, 2105 (1986). [70] Z. Fodor, C. Hoelbling, S. Krieg, L. Lellouch, Th. Lippert, [87] F. W. N. De Boer, K. Abrahams, A. Balanda, H. A. Portelli, A. Sastre, K. K. Szabo, and L. Varnhorst, Up Bokemeyer, R. Van Dantzig, J. F. W. Jansen, B. Kotlinski, and Masses and Corrections to Dashen’s M. J. A. De Voigt, and J. Van Klinken, Search for short- Theorem from Lattice QCD and Quenched QED, Phys. lived axions in a nuclear isoscalar transition, Phys. Lett. B Rev. Lett. 117, 082001 (2016). 180, 4 (1986). [71] R. L. Jaffe and A. Manohar, The G(1) problem: Fact and [88] A. L. Hallin, F. P. Calaprice, R. W. Dunford, A. B. Mcdon- fantasy on the spin of the proton, Nucl. Phys. B337, 509 ald, and G. C. Paulson, Limits on axions from nuclear (1990). decays, Nucl. Instrum. Methods Phys. Res., Sect. B 24/25, [72] M. J. Savage and J. Walden, SU(3) breaking in neutral 276 (1987). current axial matrix elements and the spin content of the [89] F. T. Avignone, C. Baktash, W. C. Barker, F. P. Calaprice, nucleon, Phys. Rev. D 55, 5376 (1997). R. W. Dunford, W. C. Haxton, D. Kahana, R. T. Kouzes, [73] M. Karliner and H. J. Lipkin, Nucleon spin with and H. S. Miley, and D. M. Moltz, Search for Axions from the without hyperon data: A new tool for analysis, Phys. Lett. 1115-kev transition of 65Cu, Phys. Rev. D 37, 618 (1988). B 461, 280 (1999). [90] M. J. Savage, B. W. Filippone, and L. W. Mitchell, New [74] G. K. Mallot, The Spin structure of the nucleon, Int. J. limits on light scalar and pseudoscalar particles produced Mod. Phys. A 15, 521 (2000). in nuclear decay, Phys. Rev. D 37, 1134 (1988). [75] E. Leader, A. V. Sidorov, and D. B. Stamenov, On the [91] V. M. Datar, S. Fortier, S. Gales, E. Hourani, H. Langevin, sensitivity of the polarized parton densities to flavor SU(3) J. M. Maison, and C. P. Massolo, Search for short lived symmetry breaking, Phys. Lett. B 488, 283 (2000). neutral particle in the 15.1-Mev isovector transition of [76] H.-Y. Cheng and C.-W. Chiang, Revisiting scalar and C-12, Phys. Rev. C 37, 250 (1988). pseudoscalar couplings with nucleons, J. High Energy [92] S. Freedman, Search for shortlived axions, Nucl. Instrum. Phys. 07 (2012) 009. Methods Phys. Res., Sect. A 284, 50 (1989).

055018-23 DANIELE S. M. ALVES PHYS. REV. D 103, 055018 (2021)

[93] T. Asanuma, M. Minowa, T. Tsukamoto, S. Orito, and T. [113] H. Osborn and D. J. Wallace, Eta—x mixing, eta → 3pi Tsunoda, A search for correlated e þ e− pairs in the decay and chiral lagrangians, Nucl. Phys. B20, 23 (1970). of AM-241, Phys. Lett. B 237, 588 (1990). [114] J. Gasser and H. Leutwyler, eta → 3pi to one loop, Nucl. [94] S. Pastore, R. B. Wiringa, Steven C. Pieper, and R. Phys. B250, 539 (1985). Schiavilla, Quantum Monte Carlo calculations of electro- [115] A. Kupsc, What is interesting in eta and eta-prime meson magnetic transitions in 8Be with meson-exchange currents decays?, AIP Conf. Proc. 950, 165 (2007). derived from chiral effective field theory, Phys. Rev. C 90, [116] J. Kambor, C. Wiesendanger, and D. Wyler, Final state 024321 (2014). interactions and Khuri-Treiman equations in eta → 3pi [95] A. J. Krasznahorkay, M. Csatlós, L. Csige, D. Firak, J. decays, Nucl. Phys. B465, 215 (1996). Gulyás, Á. Nagy, N. Sas, J. Timár, T. G. Tornyi, and A. [117] B. Borasoy and R. Nissler, Hadronic eta and eta-prime Krasznahorkay, On the Xð17Þ light-particle candidate decays, Eur. Phys. J. A 26, 383 (2005). observed in nuclear transitions, Acta Phys. Pol. B 50, [118] A. V. Anisovich and H. Leutwyler, Dispersive analysis of 675 (2019). the decay eta → 3pi, Phys. Lett. B 375, 335 (1996). [96] D. R. Tilley, J. H. Kelley, J. L. Godwin, D. J. Millener, J. E. [119] J. Bijnens and K. Ghorbani, eta → 3pi at two loops in Purcell, C. G. Sheu, and H. R. Weller, Energy levels of chiral perturbation theory, J. High Energy Phys. 11 (2007) light nuclei A ¼ 8,9,10, Nucl. Phys. A745, 155 (2004). 030. [97] G. G. di Cortona, E. Hardy, J. Pardo Vega, and G. [120] P. Guo, I. V. Danilkin, D. Schott, C. Fernández-Ramírez, Villadoro, The QCD axion, precisely, J. High Energy V. Mathieu, and A. P. Szczepaniak, Three-body final Phys. 01 (2016) 034. state interaction in eta → 3pi, Phys. Rev. D 92, 054016 [98] T. W. Donnelly and R. D. Peccei, Neutral current effects in (2015). nuclei, Phys. Rep. 50, 1 (1979). [121] P. Guo, I. V. Danilkin, C. Fernández-Ramírez, V. Mathieu, [99] X. Zhang and G. A. Miller, Can nuclear physics explain the and A. P. Szczepaniak, Three-body final state interaction in anomaly observed in the internal pair production in the eta → 3pi updated, Phys. Lett. B 771, 497 (2017). Beryllium-8 nucleus?, Phys. Lett. B 773, 159 (2017). [122] G. Colangelo, S. Lanz, H. Leutwyler, and E. Passemar, [100] C.-Y. Chen, D. McKeen, and M. Pospelov, New physics Dispersive analysis of η → 3π, Eur. Phys. J. C 78, 947 via pion capture and simple nuclear reactions, Phys. Rev. D (2018). 100, 095008 (2019). [123] P. Herrera-Siklody, Eta and eta-prime hadronic decays in [101] L. Gan, B. Kubis, E. Passemar, and S. Tulin, Precision tests UðLÞð3Þ ×UðRÞð3Þ chiral perturbation theory, arXiv:hep- of fundamental physics with η and η0 mesons, arXiv: ph/9902446. 2007.00664. [124] B. Borasoy, U.-G. Meissner, and R. Nissler, On the [102] D. J. Mack, Physics and outlook for rare, all-neutral Eta extraction of the quark mass ratio (m(d)- m(u)) / m(s) from decays, EPJ Web Conf. 73, 03015 (2014). Gammaðeta−prime→pi0piþpi−Þ=Gammaðeta−prime→ [103] D. González, D. León, B. Fabela, and M. I. Pedraza, etapiþpi−Þ, Phys. Lett. B 643, 41 (2006). Detecting physics beyond the standard model with the [125] A. H. Fariborz and J. Schechter, Eta − prime → eta pi pi REDTOP experiment, J. Phys. Conf. Ser. 912, 012042 decay as a probe of a possible lowest lying scalar nonet, (2017). Phys. Rev. D 60, 034002 (1999). [104] C. Gatto, B. F. Enriquez, and M. Isabel Pedraza Morales [126] A. M. Abdel-Rehim, D. Black, A. H. Fariborz, and J. (REDTOP Collaboration), The REDTOP project: Rare eta Schechter, Effects of light scalar mesons in eta → three decays with a TPC for optical photons, Proc. Sci., pi decay, Phys. Rev. D 67, 054001 (2003). ICHEP2016 (2016) 812. [127] G. Ecker, J. Gasser, A. Pich, and E. de Rafael, The role of [105] G. Agakishiev et al. (HADES Collaboration), Searching a resonances in chiral perturbation theory, Nucl. Phys. B321, dark photon with HADES, Phys. Lett. B 731, 265 (2014). 311 (1989). [106] M. N. Achasov et al. (SND Collaboration), Search for the [128] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, and E. de process eþe− → η, Phys. Rev. D 98, 052007 (2018). Rafael, Chiral Lagrangians for massive spin 1 fields, Phys. [107] L. G. Landsberg, Electromagnetic decays of light mesons, Lett. B 223, 425 (1989). Phys. Rep. 128, 301 (1985). [129] J. A. Oller and E. Oset, Chiral symmetry amplitudes [108] A. E. Dorokhov, M. A. Ivanov, and S. G. Kovalenko, in the S wave isoscalar and isovector channels and the Complete structure dependent analysis of the decay σ,f0ð980Þ,a0ð980Þ scalar mesons, Nucl. Phys. A620, 438 P− > 1 þ 1−, Phys. Lett. B 677, 145 (2009). (1997); Erratum, Nucl. Phys. 652, 407 (1999). [109] P. Masjuan and P. Sanchez-Puertas, η and η0 decays into [130] V. Cirigliano, G. Ecker, H. Neufeld, and A. Pich, Meson lepton pairs, J. High Energy Phys. 08 (2016) 108. resonances, large N(c) and chiral symmetry, J. High [110] R. Escribano and J.-M. Frere, Study of the eta—eta-prime Energy Phys. 06 (2003) 012. system in the two mixing angle scheme, J. High Energy [131] J. Gasser and H. Leutwyler, Chiral perturbation theory: Phys. 06 (2005) 029. Expansions in the mass of the strange quark, Nucl. Phys. [111] G. Landini and E. Meggiolaro, Study of the interactions of B250, 465 (1985). the axion with mesons and photons using a chiral effective [132] A. Pich, Effective Field theory with Nambu-Goldstone Lagrangian model, Eur. Phys. J. C 80, 302 (2020). modes, Lecture Notes of the Les Houches Summer School [112] J. A. Cronin, Phenomenological model of strong and weak (Oxford University Press, Oxford, 2020), Vol. 108. interactions in chiral Uð3Þ ×Uð3Þ, Phys. Rev. 161, 1483 [133] P. A. Zyla et al. (Particle Data Group), Review of particle (1967). physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020).

055018-24 SIGNALS OF THE QCD AXION WITH MASS OF 17 MeV=c2 … PHYS. REV. D 103, 055018 (2021)

[134] A. Pich, Colorless mesons in a polychromatic world, in [153] J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynam- Phenomenology of large N(c) QCD. Proceedings, Tempe, ics of the standard model, Cambridge Monogr. Part. Phys., USA (World Scientific, Singapore, 2002), pp. 239–258. Nucl. Phys., Cosmol. 2, 1 (1992); 35, 1 (2014). [135] M. F. L. Golterman and S. Peris, The 7=11 Rule: [154] M. Ademollo and R. Gatto, Nonrenormalization Theorem An estimate of m(rho)/f(pi), Phys. Rev. D 61, 034018 for the Strangeness Violating Vector Currents, Phys. Rev. (2000). Lett. 13, 264 (1964). [136] M. Jamin, J. A. Oller, and A. Pich, S wave K pi scattering [155] H. Leutwyler and M. Roos, Determination of the elements in chiral perturbation theory with resonances, Nucl. Phys. V(us) and V(ud) of the Kobayashi-Maskawa matrix, B587, 331 (2000). Z. Phys. C 25, 91 (1984). [137] M. Jamin, J. A. Oller, and A. Pich, Strangeness changing [156] R. F. Lebed and M. Suzuki, Current algebra and the scalar form-factors, Nucl. Phys. B622, 279 (2002). Ademollo-Gatto theorem in spin flavor symmetry of heavy [138] F. Ambrosino et al. (KLOE Collaboration), Measurement quarks, Phys. Rev. D 44, 829 (1991). of the branching ratio and search for a CP violating [157] T. Yamazaki, Search for Exotics in two-body decay of Kþ: asymmetry in the η → πþπ−eþe−ðγÞ decay at KLOE, K(mu2) and K(pi2) revisited, Proceedings of the Second Phys. Lett. B 675, 283 (2009). LAMPF II Workshop (Los Alamos National Laboratory, [139] C. Bargholtz et al. (CELSIUS/WASA Collaboration), Los Alamo, 1982), Vol. II, p. 413. Measurement of the eta → pi þ pi − e þ e− decay branch- [158] T. Yamazaki et al., Search for a Neutral Boson in a Two- þ þ 0 ing ratio, Phys. Lett. B 644, 299 (2007). Body Decay of K >pi X , Phys. Rev. Lett. 52, 1089 [140] M. Berlowski et al. (CELSIUS/WASA Collaboration), (1984). [159] N. J. Baker et al., Search for Shortlived Neutral Particles Measurement of eta meson decays into lepton-antilepton þ pairs, Phys. Rev. D 77, 032004 (2008). Emitted in K Decay, Phys. Rev. Lett. 59, 2832 (1987). [141] P. Naik et al. (CLEO Collaboration), Observation of Eta- [160] J. R. Batley et al. (NA48/1 Collaboration), Observation of ð Þ → 0 þ − Prime Decays to pi+ pi- pi0 and pi+ pi- e+ e-, Phys. Rev. the rare decay K S pi e e , Phys. Lett. B 576,43 Lett. 102, 061801 (2009). (2003). [142] M. Ablikim et al. (BESIII Collaboration), Measurement of [161] G. D. Barr et al. (NA31 Collaboration), Search for the ð Þ → 0 þ − η0 → πþπ−eþe− and η0 → πþπ−μþμ−, Phys. Rev. D 87, decay K S pi e e . CERN-NA031 experiment, 304 092011 (2013). Phys. Lett. B , 381 (1993). et al. [143] E. C. Gil et al. (NA62 Collaboration), The beam and [162] A. Lai (NA48 Collaboration), Search for the decay KðSÞ → pi0 e þ e−, Phys. Lett. B 514, 253 (2001). detector of the NA62 experiment at CERN, J. Instrum. 12, [163] G. D. Barr et al. (NA31 Collaboration), Search for a neutral P05025 (2017). Higgs particle in the decay sequence K0ðLÞ → pi0 H0 and [144] E. Iwai (KOTO Collaboration), Status and prospects of H0 → e þ e−, Phys. Lett. B 235, 356 (1990). J-PARC KOTO experiment, Nucl. Phys. B, Proc. Suppl. [164] J. Adams et al. (KTeV Collaboration), Measurement of the 233, 279 (2012). Branching Fraction of the Decay KðLÞ → pi þ pi − e þ e−, [145] C. W. Bernard, T. Draper, A. Soni, H. D. Politzer, and Phys. Rev. Lett. 80, 4123 (1998). M. B. Wise, Application of chiral perturbation theory to [165] Y. Takeuchi et al., Observation of the decay mode K → 2 pi decays, Phys. Rev. D 32, 2343 (1985). KðLÞ → pi þ pi − e þ e−, Phys. Lett. B 443, 409 (1998). [146] J. Kambor, J. H. Missimer, and D. Wyler, The chiral loop [166] A. Lai et al. (NA48 Collaboration), Investigation of expansion of the nonleptonic weak interactions of mesons, KðL; SÞ → pi þ pi − e þ e− decays, Eur. Phys. J. C 30, Nucl. Phys. B346, 17 (1990). 33 (2003). [147] V. Cirigliano, G. Ecker, H. Neufeld, A. Pich, and J. [167] D. Ambrose et al. (BNL E871 Collaboration), First Portoles, Kaon decays in the standard model, Rev. Mod. Observation of the Rare Decay Mode K0ðLÞ → e þ e−, Phys. 84, 399 (2012). Phys. Rev. Lett. 81, 4309 (1998). [148] J. M. Gerard and J. Weyers, Trace anomalies and the delta [168] L. M. Sehgal, Electromagnetic contribution to the decays ¼ 1 2 I = rule, Phys. Lett. B 503, 99 (2001). KðSÞ → lepton anti-lepton and KðLÞ → lepton anti-lepton, Δ ¼ 1 2 [149] R. J. Crewther and L. C. Tunstall, I = rule for kaon Phys. Rev. 183, 1511 (1969); Erratum, Phys. Rev. D 4, decays derived from QCD infrared fixed point, Phys. Rev. 1582 (1971). D 91, 034016 (2015). [169] G. Valencia, Long distance contribution to KðLÞ→leptonþ [150] A. J. Buras, J.-M. G´erard, and W. A. Bardeen, Large N lepton−, Nucl. Phys. B517, 339 (1998). Δ ¼ approach to kaon decays and mixing 28 years later: I [170] D. G. Dumm and A. Pich, Long Distance Contributions to 1 2 ˆ Δ = rule, BK and MK, Eur. Phys. J. C 74, 2871 (2014). the KðLÞ → u þ mu− Decay Width, Phys. Rev. Lett. 80, [151] J. Kambor, J. H. Missimer, and D. Wyler, K → 2 pi and 4633 (1998). K → 3 pi decays in next-to-leading order chiral perturba- tion theory, Phys. Lett. B 261, 496 (1991). [152] J. Bijnens, P. Dhonte, and F. Persson, K → 3 pi decays in Correction: The terms in Eq. (78) were improperly misaligned chiral perturbation theory, Nucl. Phys. B648, 317 (2003). during the final production stage and have been fixed.

055018-25