Flavourful Axion Phenomenology
arXiv:1806.00660v2 [hep-ph] 15 Aug 2018 Abstract: Björkeroth, Fredrik phenomenology axion Flavourful JHEP to submission for Prepared aorvoaigdcy novn nlsaeain fthe of W axions ALPs. state for relevant final be involving may decays and flavour-violating photons, to two contributes into also decays Mixing axion difference. mass and probability ymty eealainculnsaefie yafi oflvu d flavour to fit Keywords: a by ob fixed flavour-dependent are between couplings correlations accident and axion arises predictions all symmetry PQ Here which in symmetry. model, Pati-Salam Z” to aoru xo-emo n xo-htnculnsadca ( and meson heavy couplings axion-photon and axion-fermion flavourful µ ewe xosadhaymesons heavy and axions between xos nldn ohsadr eciQin(Q axions, strong (PQ) the Peccei-Quinn to standard both including axions, a c b NNLbrtr ainl iFact,VaE em 0 04 Fr 00044 40, Fermi E. Via Frascati, di Nazionali Laboratori INFN E-mail: Southampton, of University Astronomy, Kingdom and Physics Korea 02455, of Seoul School Study, Advanced for Institute Korea [email protected] → eee and [email protected] epeetacmrhniedsuso ftephenomenology the of discussion comprehensive a present We xos xo-ieprils aorvoain aiysy family violation, flavour particles, axion-like axions, µ K CP − , e D rbe,adnnsadr xo-ieprils(Ls.We (ALPs). particles axion-like non-standard and problem, ovrin ial edsrb h hnmnlg fapart a of phenomenology the describe we Finally conversion. a , B ugJnChun, Jin Eung easivligaflvuflain eas aclt h m the calculate also We axion. flavourful a involving decays ) K 0 , b D tpe .King F. Stephen 0 , B 0 , and [email protected] B s 0 hc ffcstemsnoscillation meson the affects which , c servables. clt h rnhn aisof ratios branching the lculate form eo easit xosand axions into decays meson O71JSuhmtn United Southampton, 1BJ SO17 soitdwt h solution the with associated sai Italy ascati, lydet iceeflavour discrete to due ally ℓ ics hre lepton charged discuss e 1 , mty unification mmetry, t,laigt sharp to leading ata, → ℓ 2 a ( γ ) fflavourful of swl as well as , clr“A icular iethe give ixing Contents
1 Introduction 1
2 Axion couplings to matter 3 2.1 Lagrangian 3 2.2 Physical axion basis 4 2.3 Decay constant and axion-photon coupling 5
3 Heavy meson decays 5
4 Axion-meson mixing 8 4.1 Parametrisation of mixing 9 4.2 Meson mass splitting 9 4.3 Axion-pion mixing and ALPs 10
5 Lepton decays 12
6 A to Z Pati-Salam Model 16 6.1 Mass matrices and parameters 16 6.2 Predictions 17 6.3 Decay correlations 18
7 Conclusion 19
A Axion-meson mixing 20
B Heavy meson decay branching ratio 21
C Couplings in the A to Z Pati-Salam Model 22
D Couplings in A to Z: numerical fit 26
1 Introduction
One of the puzzles of the Standard Model (SM) is why QCD does not appear to break CP symmetry. The most popular resolution of this so-called “strong CP problem” is to postulate a Peccei-Quinn (PQ) symmetry, namely a QCD-anomalous global U(1) symmetry which is broken spontaneously, leading to a pseudo-Nambu-Goldstone boson (pNGB) called the QCD axion [1–3]. The two most common approaches to realising such a PQ symmetry is either to introduce heavy vector-like quarks (the KSVZ model) [4, 5] or to extend the Higgs sector (the DFSZ model) [6, 7]. The resulting QCD axion provides a candidate for
– 1 – dark matter [8–10] within the allowed window of the axion (or PQ symmetry-breaking) 9 12 scale fa = 10 − GeV [11]. It has also been realised that the PQ axion need not emerge from an exact global U(1) symmetry, but could result from some discrete symmetry or continuous gauge sym- metry leading to an accidental global U(1) symmetry. Considering the observed accuracy of strong-CP invariance, it is enough to protect the PQ symmetry up to some higher- dimensional operators [12–14]. In this regard, it is appealing to consider an approximate PQ symmetry guaranteed by discrete (gauge) symmetries [15–21]. Alternatively, attempts to link PQ symmetry protected by continuous gauge symmetries to the flavour problem were made in [22, 23]. It is possible that PQ symmetry arises from flavour symmetries [24], linking the axion scale to the flavour symmetry-breaking scale, and various attempts have been made to incorporate such a flavourful PQ symmetry as a part of such continuous flavour symmetries [25–33]. It is also possible that PQ symmetry could arise accidentally from discrete flavour symmetries [34–37], as recently discussed [38] in the “A to Z” Pati- Salam model [39], where quarks and lepton are unified. This is difficult to achieve in a grand unified theory (GUT) based on SO(10) [40], which otherwise presents a stronger case for 1 unification. Recent efforts have been made [29, 30, 48] to unify the U(1)PQ symmetry with a Froggatt-Nielsen-like U(1) flavour symmetry [49]. The resultant axion is variously dubbed a “flaxion” or “axiflavon”; we shall refer simply to a “flavourful axion”. In this paper we focus on the phenomenology of flavourful axions, including both stan- dard PQ axions, associated with the solution to the strong CP problem, and non-standard axion-like particles (ALPs) (see e.g. [50]). For a complementary analysis of ALP signatures and bounds at the LHC, see [51]. We present the flavourful axion-fermion and axion-photon couplings both for the standard axion and for ALPs, and show that they quite naturally are non-diagonal. We use these couplings to calculate the branching ratios for two-body decays of heavy mesons K, D, and B involving a flavourful axion. Moreover, we calculate 0 0 0 0 the mixing between axions and neutral hadronic mesons K , D , B and Bs and its con- sequences, which has not been discussed in the literature before. These can lead to new contributions to neutral meson mass splitting, meson decays into axions and axion decays into two photons which may be relevant for ALPs. We also discuss lepton decays involving final state axions, including two-body decays ℓ ℓ a and radiative decays ℓ ℓ aγ, as 1 → 2 1 → 2 well as µ eee and µ e conversion. Finally we describe the phenomenology of the A to Z → − Pati-Salam model, which predicts a flavourful axion [38], and show how unification leads to correlations between different flavour dependent observables, as the down-type quark and charged lepton couplings are very similar. Notably, as the axion arises from the same flavon fields that dictate fermion Yukawa structures, no additional field content is necessary to solve the strong CP , and all axion couplings are fixed by a fit to quark and lepton masses and mixing. The layout of the remainder of the paper is as follows. Section 2 describes the flavourful axion-fermion and axion-photon couplings both for the standard axion and for ALPs. In
1 These ideas should not be confused with alternatives to PQ symmetry, such as Nelson-Barr type resolutions to the strong CP problem [41–44], or GUT models where specific Yukawa structures have been proposed [45–47].
– 2 – Section 3 we apply these couplings to calculate the branching ratios of heavy meson decays involving a flavourful axion. Section 4 discusses the mixing between axions and neutral mesons while Section 5 discusses lepton decays. Section 6 focusses on the phenomenology of the A to Z model, which predicts correlations between different flavour dependent observ- ables, and Section 7 concludes. Appendix A gives more details about axion-meson mixing. Appendix B details the calculation the heavy meson branching ratios. Appendix C shows the derivation of the couplings in the A to Z Pati-Salam model and Appendix D tabulates the numerical fit to flavour data.
2 Axion couplings to matter
2.1 Lagrangian Relevant to a discussion on axion-fermion interactions is the Lagrangian
= + + + , (2.1) L Lkin Lm L∂ Lanomaly where contains the kinetic terms, the fermion mass terms, the axion derivative Lkin Lm L∂ couplings to matter, and the QCD and electromagnetic anomalies. In the physical Lanomaly (mass) basis below the electroweak symmetry-breaking scale, we have 1 + = (∂ a)2 + f¯(∂/ m )f , Lkin Lm 2 µ i − i i f=Xu,d,e ∂µa ¯ µ f f ∂ = fiγ (Vij Aijγ5)fj, (2.2) L −vPQ − f=Xu,d,e αs a a ˜aµν α a ˜µν anomaly = Gµν G + caγ Fµν F , L 8π fa 8π fa with the axion decay constant fa = vPQ/NDW defined in terms of the PQ-breaking scale vPQ and anomaly (or domain wall) number NDW . The axion-photon coupling is discussed f f f in Section 2.3 below. The physical masses mi are defined by mi = (ULf† M URf )ii, in terms f of the mass matrix in the weak basis, M , and unitary matrices ULf , URf which transform left- and right-handed fields, respectively. The vector and axial couplings are given by
f 1 1 V = (XL + XR)= ULf† xfL ULf + URf† xfR URf , 2 2 (2.3) f 1 1 A = (X X )= U † x U U † x U . 2 L − R 2 Lf fL Lf − Rf fR Rf 2 xfL , xfR are the fermion PQ charges in the left-right (LR) basis, written here as (diagonal) f f matrices. As xfL , xfR are real, V and A (as well as chiral coupling matrices XL,R) are Hermitian. In this formulation, the implications of flavour structure are clear. If all generations of a fermion couple equally to the axion, the charge matrices xLf,Rf are proportional to the
2 Note that right-handed particles in supersymmetric theories typically manifest as left-handed an- c c c tifermions f . Then xf ≡−xfR , where xf is the PQ charge in the “SUSY basis” where the superpotential is defined.
– 3 – identity, i.e. V f = 1 (x +x )I , Af = 1 (x x )I , and there is no flavour violation. In 2 fL fR 3 2 fL − fR 3 standard axion models, e.g. DFSZ, charges can be assigned such that x = x and the fL − fR axion couples only via Af ; this is generally not true in flavoured axion models. Meanwhile if xfL = xfR , the U(1)PQ transformation is not chiral (NDW = 0), the Goldstone field a doesn’t couple to the QCD anomaly, the strong CP problem is not solved, and a is then interpreted as an ALP.3 However, as long as x / I , we still get flavour-violating (vector fL,fR ∝ 3 and axial) interactions due to weak mixing encoded in ULf,Rf .
2.2 Physical axion basis The above Lagrangian describes an interacting axion, not necessarily in its mass eigenstate. The off-diagonal couplings to fermions are nevertheless V f and Af for the physical axion, as we will see. Unlike standard DFSZ models with PQ-charged Higgs doublets, our flavoured axion does not mix with the longitudinal component of the Z boson. We still need to identify the physical axion at low energy as the state orthogonal to π0 and η mesons. One can then determine the canonical axion mass and couplings [52–54]. Let us briefly summarize how it works, following the prescription e.g. in [11]. The axion mass generated by the QCD anomaly coupling in Eq. 2.2 is conveniently calculated by rotating away the anomaly via chiral transformations of light quarks (q = u, d, s),
βq a m i 2 γ5 q e fa q, βq = ∗ , (2.4) → mq 1 1 1 1 where m− = mu− + md− + ms− . For mu,d ms (a good approximation to leading order), ∗ 1 1 1 ≪ we have m− mu− + md− . This leads to a low-energy effective Lagrangian below the ∗ ≈ chiral symmetry-breaking scale,
π0 a π0 a i +βu i +β m u¯ u e fπ fa m d¯ d e − fπ d fa +h.c.. (2.5) Leff ⊃− u h L Ri − d h L Ri Using the relation u¯ u = d¯ d = m2 f 2/(m + m ), the axion-pion mixing term h L Ri h L Ri π π u d vanishes. We identify the state a in Eq. 2.5 as the physical axion and extract its mass, 2 2 2 mumd mπfπ ma = 2 2 . (2.6) (mu + md) fa
There remains additional mixing with heavier mesons such as η′ which provide further small corrections. A precise calculation performed in [55] gives us 1012 GeV m = 5.70(6)(4) µeV. (2.7) a f a The transformation in Eq. 2.4 affects also the axion-quark couplings. For example for u, d and s quarks, the axion-quark Lagrangian in Eq. 2.2 is transformed into the physical basis,
∂µa µ µ ¯ µ ∂ ∂′ cqqγ¯ γ5q +sγ ¯ (csd′ csdγ5)d + dγ (csd′∗ csd∗ γ5)s , (2.8) L →L ⊃−vPQ − − q=Xu,d,s 3 The mass of the ALP no longer arises from the QCD vacuum, and the relation ma ∝ 1/fa no longer holds. We don’t specify any particular mass generation scheme here.
– 4 – u d d d where cu = A11 + NDW βu/2, cd = A11 + NDW βd/2, cs = A22 + NDW βs/2, csd′ = V21, and d csd = A21. We see that the diagonal couplings are modified by an amount proportional to NDW , whereas the off-diagonal couplings are unchanged. Physically, this is a consequence of the QCD anomaly being flavour-conserving, and unable to mediate flavour-violating interactions that contribute to csd. The above discussion identifies the physical axion basis in the limit of no kinetic mixing between the axion and heavier mesons. Such mixing, induced by the effective Lagrangian in Eq. 2.8, needs to be further diagonalized away to obtain the physical axion basis. This will be discussed in detail in Section 4 and Appendix A. The kinetic mixing contribution is negligibly small for the standard QCD axion with ma ≪ mπ and fa ≫ fπ, but can be important for an ALP.
2.3 Decay constant and axion-photon coupling
In standard axion scenarios, the decay constant fa is defined by vPQ/NDW , where NDW is the QCD anomaly number. Provided the U(1)PQ symmetry is broken by the VEV of a 4 single field φ with PQ charge xφ, we simply have vPQ = xφvφ. In more general models, 2 2 2 where several fields φ contribute to symmetry breaking, we define vPQ = φ xφvφ. If one VEV v dominates, we recover to good approximation the one-field relation; if, say, φi P v . 0.1v , v x v to within 1%. We will encounter exactly this scenario when φj6=i φi PQ ≈ φi φi discussing the A to Z model presented in Section 6. The axion-photon coupling aF F˜ defined in Eq. 2.2 is given in terms of the electromag- netic anomaly number E, through the coefficient
E 2(4 + z) mu caγ = , z = 0.56. (2.9) NDW − 3(1 + z) md ≈ In unified models, such as the A to Z model with Pati-Salam unification presented in Section 6, the ratio of anomaly numbers is fixed to E/N = 8/3, giving c 0.75. DW aγ ≈
3 Heavy meson decays
The flavour-changing vector couplings in may lead to observable decays of heavy mesons L∂ into axions. A general study of such flavour-changing processes involving a (massless) Nambu-Goldstone boson was made in [56], which is applicable to our flavourful axion. For a two-body decay P P a of a heavy meson P = (¯q q ) into P = (¯q ′ q ), the → ′ P ′ ′ P ′ branching ratio is given by
f 2 2 3 1 VqP qP ′ 3 mP ′ 2 Br(P P ′a)= 2 mP 1 2 f+(0) , (3.1) → 16πΓ(P ) vPQ − mP | | f with V as defined in Eq. 2.3. Its indices qP qP ′ relate to the constituent quarks, e.g. a + + ¯ d d K π a decay proceeds by s¯ da with coupling strength Vsd V21. For completeness, → → ≡ 2 a rederivation of Eq. 3.1 is provided in Appendix B. It depends on a form factor f+(q )
4 Its PQ charge xφ can be removed by normalising all charges such that xφ ≡ 1.
– 5 – encapsulating hadronic physics, where q = p = p p ′ is the momentum transfer to the a P − P axion. The lightness of the axion means we can safely take the limit q2 0. For kaon → decays, f (0) 1 to good approximation. For heavier mesons, we use results from lattice + ≈ QCD [57], summarised in Table 1.
Decay f+(0) K π 1 → D π 0.74(6)(4) → D K 0.78(5)(4) → D K 0.68(4)(3) s → B π 0.27(7)(5) → B K 0.32(6)(6) → B K 0.23(5)(4) s →
Table 1. Form factors f+(0) extracted from [57] for K, D and B decays.
K+ → π+a The canonical example of this type of flavour-violating decay is K+ π+a, which can → be constrained by searches for the rare decay K+ π+νν¯. This was done in the E949 → and E787 experiments, which observed in total seven events. Combined analyses [58] (see also [59]) yield a measurement Br(K+ π+νν¯) = 1.73+1.15 10 10, consistent with → 1.05 × − the SM prediction (0.84 0.10) 10 10 [60]. A bound on− axion decays is also provided: ± × − Br(K+ π+a) < 0.73 10 10 at 90% CL [59]. The current NA62 experiment at CERN, → × − which recently recorded their first K+ π+νν¯ event [61], is expected to observe over 100 → events, reaching a sensitivity of Br(K+ π+a) < 1.0 10 12 at 90% CL [62]. → × − 0 0 KL → π a Searches have also been performed for the neutral kaon decay K0 π0νν¯, for which the L → SM predicts Br(K0 π0νν¯) = (2.9 0.2) 10 11 [63–65]. The current best limit is set L → ± × − by the E391a experiment at KEK, giving Br(K0 π0νν¯) < 2.6 10 8 at 90% CL [66]. L → × − Its successor KOTO has been constructed at J-PARC. A pilot run in 2013 yielded a limit 0 0 8 0 0 0 Br(KL π νν¯) < 5.1 10− [67]. An analysis has also been performed for KL π X → 0 × 0 0 0 → 8 for a boson X of arbitrary mass. For m 0 0, they set Br(K π X ) . 5 10 X ≃ L → × − [68]. Detector upgrades and additional data taken since 2015 are expected to significantly improve these bounds. An additional experiment dubbed KLEVER has been proposed to measure K0 π0νν¯ at the CERN SPS [69]. L →
B and Bs decays B physics has a rich phenomenology, and is recently of particular interest due to persistent anomalies in observed semileptonic B decays at the LHC, which may be evidence for charged lepton flavour violation (cLFV) [70]. Rare B decays of the type B π(K)νν¯, while → generally not as tightly constrained as those for kaons, may also provide insights into new
– 6 – physics. A dedicated search for decays like B π(K)a with a light invisible particle a was → made by CLEO, which collected 107 BB pairs throughout its lifetime. It provides the limits Br(B π (K )a) < 4.9 10 5 and Br(B0 π0(K0)a) < 5.3 10 5 at 90% CL [71]. ± → ± ± × − → × − More recent and powerful experiments, namely BaBar and Belle, have not yet provided limits on this exact process. However we may estimate their experimental reach by the stated limits on the decays B π(K)νν¯, which are typically (10 5) (see Table 2), an → O − improvement of approximately one order of magnitude. The upgraded experiment Belle-II at SuperKEKB is expected to collected approximately N = 5 1010 BB pairs, improving × the limits on many rare decays [72]; assuming the sensitivity scales as √N, we may expect an (102) improvement in branching ratio limits. O It is worth noting that the decay B0 π0a, predicted by flavoured axion models, has → not been analysed explicitly by experiments. However, some information may be gleaned from searches for the SM process B0 π0νν¯, which are a background to the axion signal. → Generically, any bound on the SM decay will translate into a bound as strong (or stronger) on the two-body decay to an axion. Finally, we remark on the fact that also decays of the form B0 K0a and B0 η(η )a are allowed, but no meaningful experimental information s → s → ′ is available.
D and Ds decays Little is said in the literature about decays of charmed mesons of the form D ,0 π ,0a ± → ± or D K a, or the corresponding decays involving a νν¯ pair. The branching ratio for s± → ± D π(K)a may be easily calculated using the same formulas for K and B decays, given → below. The trivial requirement that Br(D π(K)a) < 1 allows us to place weak bounds on → v of (100) TeV, but without an experimental probe, little more can be said. As we will PQ O show below, the predicted branching ratios are anyway expected to be rather small when compared to K and B decays, which have corresponding branching ratios approximately three and one order of magnitude greater. In conclusion, while further experimental probes of D decays are of course welcome, they are not expected to be more sensitive to flavoured axions than other decays. On the other hand, in flavoured axion scenarios only D decays can probe the up-type quark Yukawa matrix.
Bounds f Ultimately the experimental data can be used to constrain the ratio Vq q ′ /v for a | P P | PQ given decay. Collecting terms in Eq. 3.1, we define a branching ratio coefficient c˜P P ′ , → which depends only on hadronic physics, by
2 12 2 f 10 GeV ′ Br(P P ′a) =c ˜P P VqP q ′ , (3.2) → → P vPQ i.e. 3 2 3 1 mP mP ′ 2 c˜P P ′ = 1 f+(0) . (3.3) → 16π Γ(P ) (1012 GeV)2 − m2 | | P The values of c˜P P ′ are tabulated in Table 2, along with experimental limits on the branch- → ing ratio and the corresponding bound on vPQ, where available. D, Ds and Bs decays have
– 7 – no experimental constraints, however we can compute the numerical coefficients c˜, which are all (10 14 10 13). These are also given in Table 2. O − − −
Decay Branching ratio Experiment c˜P P ′ vPQ/GeV → K+ π+a < 0.73 10 10 E949 + E787 [59] 3.51 10 11 > 6.9 1011 V d → × − × − × | 21| < 0.01 10 10* NA62 (future) [62] > 5.9 1012 V d × − × | 21| < 1.2 10 10 E949 + E787 [58] × − < 0.59 10 10 E787 [73] × − K0 π0a < 5 10 8 KOTO [68] 3.67 10 11 > 2.7 1010 V d L → × − × − × | 21| (K0 π0νν¯) (< 2.6 10 8) E391a [66] L → × − B π a < 4.9 10 5 CLEO [71] 5.30 10 13 > 1.0 108 V d ± → ± × − × − × | 31| (B π νν¯) (< 1.0 10 4) BaBar [74] ± → ± × − (< 1.4 10 4) Belle [75] × − B K a < 4.9 10 5 CLEO [71] 7.26 10 13 > 1.2 108 V d ± → ± × − × − × | 32| (B K νν¯)(< 1.3 10 5) BaBar [76] ± → ± × − (< 1.9 10 5) Belle [75] × − (< 1.5 10 6)* Belle-II (future) [77] × − B0 π0a 4.92 10 13 → × − (B0 π0νν¯) (< 0.9 10 5) Belle [75] & 2.3 108 V d → × − × | 31| B0 K0 a < 5.3 10 5 CLEO [71] 6.74 10 13 > 1.1 108 V d → (S) × − × − × | 32| (B0 K0νν¯) (< 1.3 10 5) Belle [75] → × − D π a < 1 1.11 10 13 > 3.3 105 V u ± → ± × − × | 21| D0 π0a < 1 4.33 10 14 > 2.1 105 V u → × − × | 21| D K a < 1 4.38 10 14 > 2.1 105 V u s± → ± × − × | 21| B0 K0a < 1 3.64 10 13 > 6.0 105 V d s → × − × | 31| Table 2. Branching ratios (upper limits) and corresponding bounds (lower limits) on the PQ- breaking scale vP Q from flavour-violating meson decays. Bold font marks the current best limit from searches for P P ′a, while parentheses mark the bound on the rare decay P P ′νν¯, which → ∗ → should be comparable. Asterisks ( ) mark the expected reach of current or planned experiments.
4 Axion-meson mixing
In this section we discuss the mixing between axions and neutral hadronic mesons, and the impact on the meson oscillation probabilities. Such a mixing effect can also lead to new contributions to both meson decays into axions and axion decays into two photons. Although the mixing effect will turn out to be negligible for PQ axions which solve the strong CP problem, it may be relevant for non-standard axions such as ALPs. Readers who are not interested in ALPs may skip this section, since it will not lead to any competitive bounds on PQ axions.
– 8 – 4.1 Parametrisation of mixing Axion-quark couplings in the mass-diagonal basis were discussed in Section 2.2. Relevant to meson mixing are the terms
∂µa µ µ ¯ µ ∂′ cqqγ¯ γ5q + csdsγ¯ γ5d + csd∗ dγ γ5s , (4.1) L ⊃−vPQ q=Xu,d,s u d d where again cu = A11 + NDW βu/2, cd = A11 + NDW βd/2, cs = A22 + NDW βs/2, and d csd = A21. These derivative couplings translate into effective axion-meson couplings
eff fP µ aP = cP ∂µa∂ P, (4.2) L − vPQ XP 0 0 0 where f is the meson decay constant for P = π ,η,η , K , K , and c 0 = c c , c = P ′ π u − d η c +c 2c , c ′ = c +c +c , and c 0 = c = c . This kinetic mixing can be diagonalised u d− s η u d s K sd ∗K0 by the transformations a η a a , P P + P , (4.3) → 1 η2 → 1 η2 − P P − P P q P q P where η c f /v . This is naturally generalised to include also mesons containing c P ≡ P P PQ and b quarks. For a QCD axion with m m and f f , there is almost no impact on a ≪ P a ≫ P the standard meson dynamics. However, the results are valid for generalised ALPs, where the effect may be detectable.
4.2 Meson mass splitting Axions and ALPs with off-diagonal quark couplings will mediate mixing between a heavy 0 0 neutral meson P (P = K, D, B, or Bs) and its antiparticle P in addition to that from weak interactions. An explicit calculation, showing how axion interactions yield an additional contribution to meson mass splittings, is given in Appendix A. We quote the result, namely that 2 2 2 fP 0 (∆mP )axion ηP mP = cP 2 mP . (4.4) ≃ | | | | vPQ
The total mass difference is then given by ∆mP = (∆mP )SM +(∆mP )axion. As an example, consider the effect of axion-kaon mixing on the K0 K0 mass difference, experimentally L − S measured to be (∆m ) = (3.484 0.006) 10 12 MeV [78]. The error is dominated K exp ± × − by the theory uncertainty, which may be large [79]; near-future lattice calculations aim to reduce the error on ∆m to (20%) [80], with further improvements from next-generation K O machines. As a conservative estimate, we shall only demand the axion contribution to any 8 ∆mP is not larger than the experimental central value. We then have ηK0 . 8 10− , 6 | | × which (assuming c 0 1) corresponds to the bound v & 2 10 GeV. Similar results for K ≈ PQ × D, B and Bs mixing are tabulated in Table 3. Belle-II is expected to improve the sensitivity of D0 D0 mixing by about one order of magnitude with the full 50 ab 1 of data [81]. − −
– 9 – System (∆mP )exp/MeV vPQ/GeV 0 0 12 6 K K (3.484 0.006) 10− & 2 10 cK0 0 − 0 ± +2.70 × 12 × 6| | D D (6.25 2.90) 10− & 4 10 cD0 0 − 0 − × 10 × 5| | B B (3.333 0.013) 10− & 8 10 cB0 0 − 0 ± × 8 × 5| | B B (1.1688 0.0014) 10 & 1 10 c 0 s − s ± × − × | Bs |
Table 3. Limits on vP Q from contributions to neutral meson mass differences. Measured values of ∆mP are given in the PDG [78]. Meson decay constants fP 0 are extracted from global averages given in [82].
4.3 Axion-pion mixing and ALPs We have seen that axion-meson kinetic mixing can affect the oscillation probability (and thereby the mass difference) of neutral heavy mesons, arising from off-diagonal quark cou- plings of axions. In this subsection, we will see that even flavour-diagonal couplings can lead to interesting consequences. As shown in Eqs. 4.1 and 4.2, there arises in particular axion-pion kinetic mixing as a consequence of the physical π0 containing a small admix- ture of the nominal axion and vice versa. This induces axion contributions to any process normally involving π0. Kinetic diagonalisation (as in Eq. 4.3) induces mass couplings of the form 1 ΦT M 2 Φ, − 2 Φ where Φ = (a, π0) and
2 2 ma ηπ0 ηπ0 2 + 2 mπ0 1 ηπ0 2 2 2 − 1 ηπ0 MΦ = m 0 − , (4.5) π ηπ0 q 1 2 1 η 0 − π q where η 0 = c 0 f /v =c ˜ 0 f /f , with c˜ 0 c 0 /N . This is subsequently diago- π π π PQ π π a π ≡ π DW nalised by a 2 2 rotation in terms of an angle θ , where × π 2 2 2m 0 η 0 1 η 0 π π − π tan 2θπ = 2 2 2 2 . (4.6) m 0 (1 2η 0 )q m (1 η 0 ) π − π − a − π Starting from the canonical physical basis in Eq. 4.1, the physical basis accounting also for kinetic mixing is thus obtained by field transformations cos θ a + sin θ π0 a π π , → 2 1 η 0 − π q (4.7) 0 sin θπηπ0 0 cos θπηπ0 π cos θπ + π sin θπ a. → 2 − − 2 1 η 0 1 η 0 − π − π q q To leading order in ηπ0 , we have
0 2 2 ηπ mπ0 0 0 0 ηπ0 ma a a + 2 2 π , π π 2 2 a. (4.8) → m 0 m → − m 0 m π − a π − a
– 10 – 0 For a QCD axion with m m and η 0 1, its contribution to the physical pion is a ≪ π π ≪ vanishingly small. However, this mixing may be interesting for more general ALPs, where the mass and decay constant are not necessarily correlated. The axion-meson mixing effect discussed above can modify decays of heavy mesons to lighter mesons plus an axion, as well as to the decay of an axion to two photons. The basic idea is very simple: in the standard hadronic decay of a heavy meson into two pions, one of the neutral pions in the final state can convert into an axion via the mixing effect discussed above, leading to a final state consisting of an axion. Similarly, the standard decay of a neutral pion into two photons can also mediate the decay on an axion into two photons. Applying Eq. 4.8 to an ALP, still denoted by a, perhaps the most interesting processes induced by mixing are K+ π+a and a γγ. Considering only the mixing-induced → → effect, we have 2 2 + + ηπ0 ma + + 0 Γ(K π a) 2 2 Γ(K π π ). (4.9) → ≃ m 0 m → π − a Taking the ballpark of Br(K+ π+a) . 10 10 listed in Table 2 and Br(K+ π+π0) = → − → 20.67%, we find a mass-dependent bound
2 c˜π0 m f & 4 a TeV (4.10) a m2 m2 π0 − a which is applicable for ma . 110 MeV. Similarly, one finds the axion decay to photons
2 2 3 ηπ0 ma ma 0 Γ(a γγ) 2 2 Γ(π γγ). (4.11) → ≃ m 0 m m 0 → π − a π
In the SM with massless valence quarks and NC = 3 colours, we have [83]
2 3 2 0 α mπ0 NC Γ(π γγ)= 3 2 7.63 eV. (4.12) → 576π fπ ≃
The standard form of the axion-photon coupling, 1 g aF F˜, gives Γ(a γγ)= 1 g2 m3. 4 aγ → 64π aγ a We may then write the mixing-induced axion-photon coupling as
2 α c˜π0 ma 1 (gaγ )mix 2 2 . (4.13) ≃ π m 0 m fa π − a Therefore the bound in Eq. 4.10 corresponds to
7 1 (g ) . 5.8 10− GeV− for m . 110 MeV. (4.14) aγ mix × a Extensive studies of ALPs over a wide range of parameter space (summarised in e.g. Fig. 1 10 1 of [84]) place very strong bounds for gaγ < 10− GeV− for the whole range of ma . 100 MeV, which translates to
2 10 1 c˜π0 m 10− GeV− & 7 a fa 2 10 2 2 GeV. (4.15) × m 0 m gaγ π − a
– 11 – Let us finally note that the E787 experiment searched for K+ π+a followed by a γγ → → in the range of m = 5 100 MeV [85]. Combining the two expressions in Eqs. 4.9 and a − 4.13, the E787 result gives (for m = 10 96 MeV) the bound a − 5 1 (g ) . 5 10− GeV− , (4.16) aγ mix × which is less stringent than Eq. 4.14.
5 Lepton decays
ℓ1 → ℓ2a Two-body lepton decays of the form ℓ ℓ a follow analogously to meson decays, with 1 → 2 the notable difference that both axial and vector couplings contribute, since the decaying e particle has non-zero spin. We define a total coupling Cℓ1ℓ2 by
e 2 e 2 e 2 Cℓ1ℓ2 = Vℓ1ℓ2 + Aℓ1ℓ2 . (5.1)
As done for mesons in Eqs. 3.2– 3.3, the branching ratio may once again be written in terms of a coefficient c˜ℓ1 ℓ2 , by → 12 2 e 2 10 GeV Br(ℓ1 ℓ2a) =c ˜ℓ1 ℓ2 Cℓ1ℓ2 , (5.2) → → v PQ where 3 2 3 1 mℓ1 mℓ2 c˜ℓ1 ℓ2 = 1 . (5.3) → 16π Γ(ℓ ) (1012 GeV)2 2 1 − mℓ1 !
These are evaluated, with corresponding limits placed on vPQ, for the three possible lepton decays. The results are tabulated in Table 4. The most interesting of these is µ+ e+a, for which the SM background consists → almost entirely of ordinary β decay, µ+ e+νν¯. The muon decay width Γ is given to good → µ approximation by Γ Γ(µ+ e+νν¯) G2 m5 /(192π3). Assuming µ+ e+a decays are µ ≃ → ≃ F µ → isotropic, i.e. the decay is purely vectorial (or axial), the experiment at TRIUMF provides the limit Br(µ+ e+a) < 2.6 10 6 [86], corresponding to v / V e (or Ae ) > 5.5 → × − PQ | 21| | 21| × 109 GeV. They searched specifically for decays with an angular acceptance cos θ > 0.975, where θ is the positron emission angle; in this region SM three-body decays are strongly suppressed. The TWIST experiment [87] has performed a broader search, accommodating non-zero anisotropy A as well as massive bosons, but are less sensitive for isotropic decays in the massless limit. The limits for isotropic (A = 0) and maximally anisotropic (A = 1) ± decays are given in Table 4. Let us sketch the angular dependence of µ ea decays, which are not generally → isotropic, as these would relate to TWIST; the formulas generalise immediately to τ decays. Consider µ+ with a polarisation η = (0, η) decaying into a positron with helicity λ = 1 e ± and momentum ke, as well as an axion. Neglecting me and ma, 3 2 mµ e 2 e e = C (m 2λ (η k )) + 2Re[A (V )∗] (m 2(η k )) , (5.4) |M| v2 | 21| µ − e · e 21 21 µ − · e PQ λe= 1 X± h i
– 12 – where η k = k cos ϑ . We can describe the degree of muon polarisation P as the · e −| e| ηe µ projection of η onto the beam direction ˆz, i.e. P cos ϑ = η ˆz/ η . For a more precise µ ≃ ηz · | | treatment one should consider the distribution of η in a muon ensemble, but as we shall assume all muons are highly polarised opposite to the beam direction, i.e. P 1, this µ ∼ − is sufficient for our purposes. TWIST measures the positron emission angle θ = ϑ ϑ ; ηz − ηe for highly polarised muons, we have cos ϑ P cos θ. Summing over λ , the differential ηe ≃ µ e decay rate is given by
2 e 2 3 dΓ C21 mµ = |M| | | 2 (1 APµ cos θ), (5.5) d cos θ 32πmµ ≃ 32π vPQ − where we define the anisotropy 2Re[Ae (V e ) ] A = 21 21 ∗ . (5.6) − Ce 2 | 21| The limiting cases are Ae = V e , giving A = 1 (corresponding to an SM-like V A current 21 21 − − interaction), or Ae = V e , giving A = 1 (a V + A interaction). The signal strength with 21 − 21 respect to the SM background is maximised for A = 1, particularly in the region with cos θ 1. The A to Z model, discussed below, predicts exactly this scenario, although ∼ the high predicted PQ scale v 1012 GeV implies the signal is very small despite the PQ ∼ enhancement. Finally, the Mu3e experiment, primarily designed to look for µ eee (discussed below), → can also be used to test for µ ea, and tentatively probe scales of v & 1010 GeV [88] by → PQ the end of its run.
Decay Branching ratio Experiment c˜ℓ1 ℓ2 vPQ/GeV → µ+ e+a < 2.6 10 6 (A = 0) Jodidio et al [86] 7.82 10 11 > 5.5 109 V e → × − × − × | 21| < 2.1 10 5 (A = 0) TWIST [87] > 1.9 109 Ce × − × | 21| < 1.0 10 5 (A = 1) TWIST [87] > 2.8 109 Ce × − × | 21| < 5.8 10 5 (A = 1) TWIST [87] > 1.2 109 Ce × − − × | 21| . 5 10 9* Mu3e (future) [88] & 1 1011 Ce × − × | 21| τ + e+a < 1.5 10 2 ARGUS [89] 4.92 10 14 > 1.8 106 Ce → × − × − × | 31| τ + µ+a < 2.6 10 2 ARGUS [89] 4.87 10 14 > 1.4 106 Ce → × − × − × | 32|
Table 4. Branching ratios (upper limits) and corresponding bounds (lower limits) on vP Q from two-body cLFV decays. The assumed anisotropy A can be related to the formula in Eq. 5.6.
ℓ1 → ℓ2aγ Additionally, we may examine decays with an associated photon, i.e. ℓ ℓ aγ. These can 1 → 2 be studied in experiments searching for ℓ ℓ γ, which, if experimentally measured, are 1 → 2 unequivocal signs of new physics; in the SM, Br(µ eγ) 10 54, certainly unobservable. → ∼ − The differential decay rate for ℓ ℓ aγ in the limit of m = m = 0 may be expressed 1 → 2 ℓ2 a by 2 e 2 3 d Γ α Cℓ1ℓ2 mℓ1 (1 x)(2 y xy) = 2 2 f(x,y), f(x,y)= −2 − − , (5.7) dx dy 32π v PQ y (x + y 1) −
– 13 – where f(x,y) is a function of x = 2Eℓ2 /mℓ1 , y = 2Eγ /mℓ1 , i.e. (twice) the fraction of invariant mass carried away by the lighter lepton and photon, respectively. Energy conservation requires x,y 1 and x + y 1. Moreover, the angle θ between ℓ and the ≤ ≥ 2γ 2 photon is fixed by kinematics to 2(1 x y) cos θ =1+ − − . (5.8) 2γ xy Alternatively one can write the decay rate in terms of x and c cos θ , i.e. θ ≡ 2γ 2 e 2 3 2 d Γ α Cℓ1ℓ2 mℓ1 1 x(1 cθ)+ x = 2 2 f(x, cθ), f(x, cθ)= − − . (5.9) dx dcθ 32π v PQ (1 x)(1 cθ) − − We may relate the branching ratios of decays with and without a radiated photon by Br(ℓ ℓ aγ) α = 1 → 2 = dx dyf(x,y). (5.10) Rℓ1ℓ2 Br(ℓ ℓ a) 2π 1 → 2 Z The radiative decay possesses two divergences: an IR divergence due to soft photons (x 1) and a collinear divergence (θ 0). In practice, appropriate cuts are made on the ≃ 2γ ≃ minimum photon energy and angular acceptance well away from the IR-divergent region. Such cuts were discussed in the context of ℓ ℓ γ decays [90], in particular as they 1 → 2 related to LAMPF [91] and MEG [92] experiments. The region of interest for MEG is for x,y 1, or equivalently c π, where the SM background disappears. However, decays ≃ θ ≃ with an associated flavoured axion are also suppressed in this limit, i.e. the integral f vanishes for very soft axions. One might consider a broader region of phase space, provided R the induced backgrounds5 are under control. A comprehensive experimental study of such signals, e.g. for the MEG-II upgrade [93], would be welcome. An explicit limit on µ efγ, → where f is a light scalar or pseudoscalar, is given by the Crystal Box experiment, which sets Br(µ efγ) < 1.1 10 9 at 90% CL [94]. Unlike the TRIUMF experiment [86] → × − discussed above, this limit does not assume isotropic decays. Using the same cuts6 we find f 0.011, yielding the bound v /GeV > 9.4 108 Ce . In Table 5 we summarise ≃ PQ × | 21| current and future experimental limits on branching ratios of ℓ ℓ γ. R 1 → 2 Decay Branching ratio Experiment µ+ e+γ < 4.2 10 13 MEG [92] → × − . 6 10 14* MEG-II (future) [93] × − τ e γ < 3.3 10 8 BaBar [95] − → − × − τ µ γ < 4.4 10 8 BaBar [95] − → − × −
Table 5. Experimental upper limits on cLFV decays ℓ1 ℓ2γ. → Also radiative β decay itself, µ eννγ¯ , can give information on decays to axions. The → most precise measurement comes from MEG, giving Br(µ eννγ¯ ) = (6.03 0.14(stat) → ± ± 5 The primary sources of background are radiative muon decay (RMD) and accidental e+e− annihilation- in-flight (AIF). For large photon energies and increased stopped muon rate, AIF dominates over RMD. 6 ◦ The Crystal Box analysis uses the cuts Eγ ,Ee > 38 MeV, θeγ > 140 .
– 14 – 0.53(sys)) 10 8 for E > 45 MeV and E > 40 MeV, in agreement with the SM [96]. × − e γ Requiring the axion decay to not significantly exceed the error on this measurement, i.e. Br(µ eaγ) . 1 10 8, yields a limit v / Ce & 1.2 108 GeV. We see that the limit → × − PQ | µe| × from µ ea is stronger by approximately a factor 40. → µ → eee and µ − e conversion We may also consider processes without an axion in the final state. Axion mediation will induce the decay µ eee, although the presence of two axion vertices and additional → suppression by 1/vPQ means these processes are again only interesting for ALPs. The current upper bound on the branching ratio is Br(µ+ e+e e+) < 1.0 10 12, set by → − × − SINDRUM [97]. The Mu3e experiment [98] currently under development is expected to start taking data in 2019, and will significantly improve the sensitivity by four orders of magnitude, i.e. Br(µ eee) . 1 10 16. To lowest order in m2, the branching ratio for → × − e the axion-mediated decay is given by
2 3 e 2 e 2 2 + + + memµ A11 C21 mµ 15 Br(µ e e−e ) 3 | |4| | ln 2 , → ≈ 16π Γ(µ) vPQ me − 4 ! (5.11) 12 4 41 e 2 e 2 10 GeV 1.43 10− A C . ≈ × | 11| | 21| v PQ Assuming (1) couplings, we see that such decays are only reachable by experiment pro- O 6 vided vPQ . 10 GeV. As the axion (or ALP) also couples to quarks, one may consider µ e conversion in e − nuclei, mediated by the axion. The relevant couplings are now C21 and the axion-nucleon coupling gaN = CaN mN /vPQ. The numerical factor CaN is model-dependent, given in terms of flavour-diagonal couplings of the up and down quarks. In standard cases these are essentially given by the quark PQ charges (see e.g. [53] for standard formulae), but in more general scenarios such as a flavoured axion, these can deviate significantly.7 The axion-mediated µ e conversion is a spin-dependent process which was discussed in [100]. − The conversion-to-capture ratio in a nucleus (A, Z) is qualitatively given by
Γ(µ e (A, Z)) m5 (αZ)3 m2 m2 R(A,Z) − − µ µ N Ce 2 S(A,Z)C 2, (5.12) µe →(A,Z) 2 2 2 (A,Z) 4 21 N aN ≡ Γ ∼ (q ma) π2 Γ vPQ | | | | µ−cap − µ−cap where q2 m2 is the momentum-transfer and S(A,Z) is the total nucleon spin of a nucleus ≈ µ N (A, Z). Not accounted for here are nuclear spin and structure form factors, which were discussed in [100] and are (1). The suppression by v4 suggests µ e conversion is only O PQ − realistically detectable in ALP scenarios. The current best limit comes from SINDRUM-II: RAu < 7 10 13 [101]. Assuming again (1) couplings and form factors, SINDRUM-II µe × − O sets v & 106 GeV, comparable to the µ 3e bound. The upcoming experiments Mu2e PQ → and COMET are both looking for µ Al e Al, and both aim to probe R < 6 10 17 − → − µe × − at 90% CL [102, 103], a factor 104 improvement over the SINDRUM result.
7 It is even possible to suppress the nucleon couplings entirely, yielding a nucleophobic axion [99].
– 15 – 6 A to Z Pati-Salam Model
We present here a recently proposed QCD axion model [38], based on the rather successful A to Z model [39], which seeks to resolve the flavour puzzle by way of Pati-Salam unification coupled to an A Z family symmetry. The family symmetry is completely broken by 4 × 5 gauge singlet flavons φ, which are triplets under A4 and couple to left-handed SM fields. However, information about the underlying symmetry remains in the particular vacuum structure of the flavons. The initial viability of the model, which predicts certain Yukawa structures based on the so-called CSD(4) vacuum alignment, was demonstrated in [39], and leptogenesis was considered in [104]. In [38], we updated and improved the numerical fit to flavour data, as well as demon- strating that, with small adjustments, the A to Z model can resolve the strong CP problem. The axion then emerges from the same flavons that are responsible for SM Yukawa couplings; in other words, no additional field content is necessary to realise a PQ axion. Moreover, as all Yukawa couplings are fixed by the fit to data, also the axion couplings are known exactly, with no additional free parameters. As the focus of this work is on axion couplings to matter, we limit our discussion primarily to the resultant Yukawa and mass matrices of the SM fermions. However in Appendix C we derive explicitly the axion-matter couplings from the Yukawa superpotential. In Appendix D we provide the best fit parameters for the A to Z model and corresponding axion couplings.
6.1 Mass matrices and parameters The charged fermion Yukawa matrices are given at the GUT scale by
0 0 0 b ǫ13c yd 0 0 (yd/3) 0 0 u d 0 0 e − 0 0 Y = a 4b ǫ23c , Y = Byd ys 0 , Y = Byd xys 0 . (6.1) a 2b c By0 0 y0 By0 0 y0 d b d b All parameters are dimensionless and in general complex, although three can be immediately made real by an overall rephasing of the three Yukawa matrices. The mass matrix of the light Majorana neutrinos (after seesaw) is
0 0 0 1 4 2 0 0 0 ν iη iξ m = ma 0 1 1 + mbe 4 16 8 + mce 0 0 0 , (6.2) 0 1 1 2 8 4 0 0 1 where mi are real, with dimensions of mass and η, ξ are phases. Note that the scales of the various free parameters are constrained by the model itself. By rather simple assumptions about the flavon VEVs, discussed fully in [39], and assuming all dimensionless couplings in the renormalisable theory are (1), we may infer generic O properties of the parameters. Parameters a, b and c correspond closely to the three up-type quark Yukawa couplings, i.e. a b c 1. Meanwhile y0, y0 and y0 are correlated ≪ ≪ ∼ d b s with the down-type quark Yukawa couplings, i.e. y0 y0 y0. B is an (1) ratio d ≪ b ≪ s O of couplings, and ǫ 1 are small perturbations of a flavon VEV. The (1) factor x is i3 ≪ O a Clebsch-Gordan factor, introduced by additional Higgs multiplets in a variation of the
– 16 – Georgi-Jarlskog mechanism. In the neutrino sector, the principle of sequential dominance on which the model relies demands a normal ordering and strong mass hierarchy, with m m m , predicting the lightest neutrino with a mass of < 1 meV. A fit of these a ≫ b ≫ c parameters to data has been performed [38], with central results collected in Appendix D. The model is fitted to experimental results8 by an MCMC analysis. Bayesian credible intervals are also provided, showing that despite a large number of free parameters, small ℓ ℓ tensions in the predictions for θ23 and δ may be further probed by increased sensitivity in current and future neutrino experiments.
The PQ-breaking scale vPQ is determined primarily by the largest VEV among the u flavons φ carrying PQ charge. The VEV of this flavon (named φ2 ) is proportional to the parameter b in Y u, which in turn is dominantly responsible for the charm quark Yukawa coupling; as the third generation largely does not couple to the PQ symmetry, this is the heaviest relevant fermion in the flavoured axion theory. The numerical fit gives b = 3.4 3 | | × 10− . The details of how the flavons and parameters are related are given in Appendix C, showing that b φu /M v φu 1012 GeV. ∼ h 2 i GUT ⇒ PQ ≃ h 2 i∼ 6.2 Predictions Once the fermion mixing matrices are known from the fit, we can immediately determine the vector and axial coupling matrices V f and Af using Eqs. 2.3. Recalling that V f and Af are Hermitian, we have
3 0.05i 5 0.015i 1.0 4.3 10− e− 1.7 10− e− u u 3 0.05i × − × 4 V = A 4.3 10− e 0.5 6.0 10− , − ≃ × 5 0.015i − 4 − × 7 1.7 10− e 6.0 10− 7.3 10− − × − × × 0.78 0.25 0.0065 d d − V = A 0.25 0.72 0.0057 , (6.3) − ≃ − 5 0.0065 0.0057 7.5 10− − − × 0.99 0.073 0.0085 e e − V = A 0.073 0.51 0.0013 , − ≃ − 5 0.0085 0.0013 7.5 10− − − × We may immediately compute the branching ratios for all aforementioned meson and lepton decays and neutral meson mass splittings. The only remaining parameter is the axion scale v , which is only loosely constrained by naturalness arguments to be v 1012 PQ PQ ∼ GeV. In principle, any two measurements of either flavour violation (as discussed in this paper), the axion-photon coupling gaγ , or the axion-electron coupling gae, would be sufficient to overconstrain vPQ in this model. Here, gaγ is fixed by vPQ and the domain wall number NDW = 6. In other words, although the charge assignments are very different, the A to Z model will resemble the original DFSZ model in experiments sensitive to gaγ , such as haloscopes and helioscopes. In Table 6 we give the model predictions for some of the most
8 In [105] they perform the running of low-scale experimental results (from global fits) up to the GUT scale, assuming the MSSM; they provide GUT-scale values for quark and charged lepton Yukawa couplings, and CKM mixing parameters.
– 17 – 12 phenomenologically interesting experimental probes. We explicitly set vPQ = 10 GeV when computing the branching ratio.
Branching ratio Process 12 Experimental sensitivity (vPQ = 10 GeV)
K+ π+a 2.19 10 12 . 1 10 12 (NA62 future) → × − × − K0 π0a 2.29 10 12 < 5 10 8 (KOTO) L → × − × − µ+ e+a 8.3 10 13 . 5 10 9 (Mu3e future) → × − × − Table 6. Predictions for axion-induced processes in the A to Z model. Branching ratios are 12 computed assuming vP Q = 10 GeV, which should be true up to an (1) factor. O
Predictably, as v 1012 GeV, all processes involving two axion vertices, includ- PQ ∼ ing meson mixing and µ eee, are heavily suppressed. For all mesons P , we calculate → (∆m ) 10 23 10 24 MeV, while for µ eee the branching ratio is (10 45), P axion ∼ − − − → O − essentially undetectable. In summary, we find that evidence for or against the A to Z model must come primarily from the (non-)observation of K+ π+a; the NA62 experiment is expected to be able to → exclude most of the model’s parameter space. A next-generation experiment could exclude 0 + the model definitively. Secondary sources of interest are decays of KL and µ ; detecting the A to Z model would require vPQ to be slightly lower than the natural prediction. However, two-body decays may be powerful channels for excluding other flavour models, sometimes placing stronger constraints than those from astrophysics, which typically give the strongest limits on vPQ.
6.3 Decay correlations
The prominent feature of unified models is correlations between Yukawa couplings of quarks and leptons. In this A to Z model, Y d Y e, up to diagonal Clebsch-Gordan factors. ∼ Notably, the (2,2) entries differ by a parameter x, which is determined by the fit and acts as a necessary Clebsch-Gordan factor to distinguish the strange quark and muon masses. Naturally, one expects x m /m > 1; at the GUT scale, m /m 4.5. Now consider ∼ µ s µ s ∼ the two decays K+ π+a and µ+ e+a, which are the most experimentally promising → → among flavoured axion decays. Their branching ratios are determined, respectively, by d 2 e 2 e 2 the couplings V21 and C21 = 2 V21 . With all other parameters held constant, the | | | | | e |2 d 2 dependence on x of the ratio r = V21 / V21 is well approximated empirically by r 1.8√x | | | | ≈ 6.9 e− .
We then find that the ratio of branching ratios Rµ/K is given by
+ + e 2 Br(µ e a) V21 1.8√x R → 4.45 | | 31 e− . (6.4) µ/K ≡ Br(K+ π+a) ≃ V d 2 ≈ → | 21| For the model best fit point x = 5.88, R 0.38. Should both of these decays be µ/K ≈ measured experimentally, such a ratio, which is independent of the axion scale vPQ, is a
– 18 – valuable statistic for constraining the flavour sector of the model, giving immediate infor- mation about the high-scale parameters. For models where Y d Y e, typically x > 1; ∼ generically one expects Rµ/K < 1. Similar ratios can be considered for other decays of K or B mesons and charged leptons. However, as this requires direct observation of both decays, which are suppressed in both sectors, these are realistically feasible only for more general ALPs.
7 Conclusion
In this paper we have reviewed and extended the phenomenology of flavourful axions, including both standard PQ axions, associated with the solution to the strong CP problem, and also for non-standard axion-like particles (ALPs) which do not care about the strong CP problem but which may generically arise from spontaneously broken symmetries and multiple scalar fields. We have presented the flavourful axion-fermion and axion-photon couplings both for the standard axion and for ALPs, and shown that they quite naturally are non-diagonal. Using these couplings, we have calculated the branching ratios for two- body decays of heavy mesons K, D, and B involving a flavourful axion. We have also 0 0 0 0 calculated the mixing between axions and hadronic mesons K , D , B and Bs and its consequences, which has not been discussed in the literature before. These can lead to new contributions to neutral meson mass splitting, meson decays into axions and axion decays into two photons which may be relevant for ALPs. We have also discussed charged lepton flavour-violating processes involving final state axions, of the form ℓ ℓ a(γ), as well as 1 → 2 µ eee and µ e conversion. → − Correlations between observables may arise in specific flavourful axion models. To illustrate this, we have described the phenomenology of the A to Z Pati-Salam model, which predicts a flavourful QCD axion [38], and shown how unification leads to correlations between different flavour-dependent observables, as the down-type quark and charged lepton couplings are very similar. Within this model, since the axion arises from the same flavon fields that dictate fermion Yukawa structures, no additional field content is necessary to solve the strong CP problem, and all axion couplings are fixed by a fit to quark and lepton masses and mixing. In conclusion, flavourful axions can appear naturally in realistic models and have a rich phenomenology beyond that of the standard KSVZ/DFSZ paradigms. In this paper we have attempted to provide the first comprehensive discussion of a number of relevant processes involving flavourful axions, including meson decays and mixing, as well as charged lepton flavour-violating processes. For a QCD axion, typically the bounds from such processes are very weak. However, K πa is an ideal channel for looking at these types of decays, → especially in specific models such as the A to Z Pati-Salam model, where exactly this type of flavour-violating coupling is the largest. By comparing multiple flavour-violating processes for both quarks and leptons, one may experimentally probe lepton and quark Yukawa structures which determine their masses and mass ratios. Although for QCD axions some of the flavour-violating processes we consider are not competitive, for flavourful ALPs many of them may be important, especially if the symmetry-breaking scale is 106 GeV or less.
– 19 – Acknowledgments
The authors thank Joerg Jaeckel, Andreas Jüttner, Enrico Nardi, Giovanni Marco Pruna and Robert Ziegler for helpful discussions and comments. SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation programme under Marie Skłodowska-Curie grant agreements Elusives ITN No. 674896 and InvisiblesPlus RISE No. 690575. EJC is supported also by InvisiblesPlus RISE No. 690575. FB is supported in part by the INFN “Iniziativa Specifica” TAsP-LNF.
A Axion-meson mixing
Kinetic mixing between the axion and neutral mesons (any of the pairs K0 K0, D0 D0, − − B0 B0) is described by the Lagrangian 0 + 0 , where − Lkin Lm
0 1 µ 1 0 µ 0 µ 0 µ 0 kin = ∂µa ∂ a + ∂µP ∂ P ηP ∂µa ∂ P ηP∗ ∂µa ∂ P , L 2 2 − − (A.1) 1 0 = m2a2 m2 P 0P 0. Lm −2 a − P where P 0, P 0 are strong eigenstates. The superscript 0 signifies we are not in a diagonal (physical) basis. We define the CP eigenstates P1 (even) and P2 (odd) by
1 0 0 1 0 0 P1 = (P + P ), P2 = (P P ). (A.2) √2 √2 − Inversely, 0 1 0 1 P = (P1 + P2), P = (P1 P2). (A.3) √2 √2 −
In the case of the kaon, the states K1,2 are close (but not exactly equal) to the physical eigenstates KS and KL, so defined by having definite lifetimes in weak decays. They are given in terms of a small parameter ε 10 3 characterising indirect CP violation, K ∼ − 1 1 KS = (K1 + εKK2), KL = (K2 + εK K1). (A.4) 1+ ε 2 1+ ε 2 | K | | K | We will neglect suchp a contribution in this work. Rewritingp 0 in terms of P , we have Lkin 1,2
0 1 µ 1 µ 1 µ ηP + ηP∗ µ ηP ηP∗ µ = ∂µa ∂ a + ∂µP1 ∂ P1 ∂µP2 ∂ P2 ∂µa ∂ P1 − ∂µa ∂ P2, Lkin 2 2 − 2 − √2 − √2 1 0 = m2a2 m2 P 2 + m2 P 2. Lm −2 a − P 1 P 2 (A.5)
Note the wrong sign of the P2 diagonal kinetic and mass terms; these can be made canonical by letting P iP , which introduces a factor i in the kinetic mixing term. This can be 2 → 2 absorbed in new couplings η1,2, defined by 1 i η1 = (ηP + η∗ ), η2 = (ηP η∗ ). (A.6) √2 P −√2 − P
– 20 – We also define a “total” coupling η2 η2 + η2 = 2η η = 2 η 2. ≡ 1 2 P P∗ | P | We diagonalise the kinetic Lagrangian by transformations
a η1 η2 a , P1 P1 + a, P2 P2 + a. (A.7) → 1 η2 → 1 η2 → 1 η2 − − − The mixing is transferredp to the mass matrix,p giving p
2 2 2 2 0 1 2 2 2 ma + mP η 2 mP a m m = mP (P1 + P2 )+ 2 a + (η1P1 + η2P2) +h.c.. L →L −2 " 1 η 1 η2 # − − (A.8) p In matrix form, we may write = 1 ΦT M 2 Φ, where Φ = (a, P , P ) and Lm − 2 Φ 1 2 m2 η2 η η a + 1 2 2 1 η2 2 2 mP 1 η 1 η η1 − − − M 2 = m2 1 0 . (A.9) Φ P 2 p p 1 η η−2 0 1 p1 η2 − 2 p The eigenvalues of MΦ, corresponding to the physical squared masses, are given to good approximation for small η by
m2(1 η2), m2 (1 + η2), m2 . (A.10) a − P P Recalling that η2 = 2 η 2, we conclude that | P | ∆m m m m 1 + 2 η 2 1 η 2m . (A.11) | P | ≡ | P1 − P2 |≃ P | P | − ≃ | P | P We have not taken into account a mass differencep from SM physics, such as for kaons, where KS and KL differ by approximately 3 µeV.
B Heavy meson decay branching ratio
µ The Feynman rule for the vertex (∂µa)¯q1γ q2 defined by the Lagrangian in Eq. 2.2 is
f Vq1q2 µ i qµγ , (B.1) − vPQ where q = p = p p is the momentum transfer to the axion. For a two-body decay a 1 − 2 P P a of a heavy meson P = (¯q q ) into P = (¯q ′ q ), the amplitude may be written → ′ P ′ ′ P ′ f VqP qP ′ µ = i (pP pP ′ )µ P ′ q¯P γ qP ′ P . (B.2) M − vPQ − h | | i
2 It depends on a form factor f+(q ) encapsulating hadronic physics. The lightness of the axion means we can safely take the limit q2 0, wherein the form factor is defined by the → relation µ µ P ′ q¯ γ q ′ P = f (0)(p + p ′ ) , (B.3) h | P P | i + P P
– 21 – such that f VqP qP ′ 2 2 = i (mP mP ′ )f+(0). (B.4) M vPQ − The differential decay rate in the rest frame of P is
1 2 pP ′ dΓ = 2 | 2 |dΩ, (B.5) 32π |M| mP with the momentum of decay products p ′ = p given by | P | | a| 2 2 2 2 1/2 2 2 (m (m ′ + m ) )(m (m ′ m ) ) (ma m ′ ) P P a P P a ≪ P mP mP ′ pP ′ = pa = − − − − , | | | | 2mP ≈ 2mP (B.6) Integrating over the solid angle Ω yields a factor 4π, arriving at
f 2 2 3 1 VqP qP ′ 3 mP ′ 2 Γ(P P ′a)= 2 mP 1 2 f+(0) . (B.7) → 16π vPQ − mP | | C Couplings in the A to Z Pati-Salam Model
Superpotential The effective Yukawa superpotential below the GUT scale, once messengers X have been integrated out, is given by
u c u c eff c (F φ1 )huF1 (F φ2 )huF2 WY = λ3(F h3)F3 + λ1u · + λ2u · · Σu Σu d c h i d d c h i u c (C.1) (F φ1)hdF1 (F φ2)h15F2 (F φ1 )hdF1 + λ1d · d + λ2d · + λud · , Σ Σd Σd h 15i h i h i with explicit couplings λ, which are naturally (1) and assumed real by a CP symmetry O at high scale. In the corresponding Lagrangian, the fermion part of the chiral superfields c c 9 F , Fi are denoted f, fi , respectively. These are the familiar SM fermions as well as a set of right-handed neutrinos. The light Higgs scalar doublets keep the same notation as their corresponding superfield.10 The fields Σ acquire high-scale VEVs which give dynamical masses to the X messengers in the renormalisable theory, expected to be (M ). O GUT Goldstone field
The central actors in the flavoured axion model are the A4 triplet flavons φ. Taking only the scalar part of superfields φ, we let
φ ϕ = 1 ( ϕ + ρ ) eiaϕ/vϕ , ϕ v x , (C.2) i → i √2 h i ϕ i h i≡ ϕ ϕ 9 c To be precise: f, fi are Weyl fermions, by definition transforming as left-handed fields. In other words, c fi are the left-handed components of a weak SU(2)L singlet. 10 This is rather imprecise but tolerable, as the Higgs sector is not relevant to the PQ mechanism, and fields are anyway replaced by their VEVs eventually.
– 22 – where we have expanded around the flavon VEVs, noting that each ϕ consists of a scale v h i and direction x in A4 space. The VEVs are aligned according to the CSD(4) prescription, i.e. xϕu = (0, 1, 1), xϕu = (1, 4, 2), 1 1 (C.3) u u xϕ1 = (1, 0, 0), xϕ1 = (0, 1, 0).
The radial fields ρϕ are very heavy and phenomenologically uninteresting, so will be ne- glected henceforth. The phase fields aϕ are not independent, but related by the single U(1) rephasing symmetry. We identify the Goldstone (or axion) field a by v a a x ϕ ϕ , v2 x2 v2 . (C.4) ≡ ϕ v PQ ≡ ϕ ϕ ϕ PQ ϕ X X Component fields are given by xϕvϕ aϕ = a. (C.5) vPQ
Lagrangian (SUSY basis) The Yukawa Lagrangian may thus be written as
u c λ1u u c ixϕ1 a Y λ3fh3f3 + (f ϕ1 )huf1 exp −L ⊃ √2 Σ · h i vPQ h ui λ ixϕu a λ ixϕd a + 2u (f ϕu )h f c exp 2 + 1d (f ϕd )h f c exp 1 2 u 2 d 1 d 1 √2 Σ · h i vPQ √2 Σ · h i vPQ h ui h 15i ix d a u λ2d d d c ϕ2 λud u c ixϕ1 a + (f ϕ2 )h15f2 exp + (f ϕ1 )hdf1 exp √2 Σ · h i vPQ √2 Σ · h i vPQ h di h di + (ρ )+ h.c.. O ϕ (C.6)
c c Let us make the SM field components of the PS fields f,fi explicit: f (Q,L), fi c c → → (ui , di ). Below the EWSB scale, Q and L further decompose into (uL, dL) and (νL, eL), respectively. In addition, h v , h , hd v , with some small mixing assumed be- u → u d 15 → d tween Higgs bi-doublets to give the MSSM 2HDM; we assume the effects of this mixing are negligible. The fields Σ acquire real VEVs, with magnitudes generically written vΣ, i.e.
Σ v , Σd v , Σ v , (C.7) h ui→ Σu h 15i→ Σd h di→ Σd d The interplay between the singlet Σd and adjoint Σ15 also provides Clebsch-Gordan factors which are different for quarks and leptons. To account for the split between down-type quarks and charged leptons, we reparametrise the couplings λ in the charged lepton sector, so λ λ˜ , λ λ˜ , and λ λ˜ . 1d → 1d 2d → 2d ud → ud Lagrangian (left-right basis)
It is also convenient to work in the left-right (LR) basis, in terms of Weyl fermions uL,R, dL,R, eL,R, and νL,R. This amounts to nothing more than taking the Hermitian conjugate
– 23 – of the terms in Eq. C.6. With all above considerations taken into account, the Lagrangian becomes ix u a ¯ λ1uvu u ϕ1 Y = λ3(f h3 ∗)fR3 + (¯uL ϕ1 ∗)uR1 exp − −L · h i √2v · h i vPQ Σu λ v ixϕu a λ v ixϕd a 2u u u 2 1d d ¯ d − 1 + (¯uL ϕ2 ∗)uR2 exp − + (dL ϕ1 ∗)dR1 exp √2v · h i vPQ √2v · h i vPQ Σu Σd λ v ixϕd a λ v ixϕu a 2d d ¯ d − 2 ud d ¯ u 1 + (dL ϕ2 ∗)dR2 exp + (dL ϕ1 ∗)dR1 exp − √2v · h i vPQ √2v · h i vPQ Σd Σd + d e , d e , λ λ˜ , λ λ˜ , λ λ˜ + h.c.. L → L R → R 1d → 1d 2d → 2d ud → ud n o (C.8) This rather hefty expression can be put in a more conventional format by 1) expanding the A triplet products like Q ϕ , such that we may write the couplings as matrices, and 2) 4 · h i noting that each term must be PQ-invariant, allowing us to replace the flavon PQ charges with those of the SM fermions.11 Moreover, all λ are real by an assumed CP symmetry at high scales.
Lagrangian (condensed linear basis) Collecting all free parameters, we have a a i (xu xu ) i (xd xd ) vP Q Li − Rj u vP Q Li − Rj d ¯ Y = e Mij u¯LiuRj + e Mij dLidRj −L a (C.9) i (xe xe ) vP Q Li − Rj e + e Mij e¯LieRj +h.c.. The coupling matrices are given exactly by
u vu u M = λ ϕ ∗ + δ λ Θ (u) , ij ju∗ j i j3 3∗ i √2vΣ h i u d vd d u ∗ ∗ Mij = λjd∗ ϕj i + δj1λud∗ ϕ1 i + δj3λ3∗Θi(d) , (C.10) √2vΣ h i h i d e vd d u ˜ ∗ ˜ ∗ Mij = λjd∗ ϕj i + δj1λud∗ ϕ1 i + δj3λ3∗Θi(d) , √2vΣ h i h i d where Θi(f) is a function taking into account the VEV alignment of the A4 triplet h3, as well as mixing effects between various Higgs doublets. It traces its origin to the term ¯ c fh3fR3, and fixes the third column of the Yukawa matrices. As F3 is uncharged under U(1)PQ, the exact form of Θ(h3,f) has only marginal relevance for axion physics. We refer the interested reader to the original “A to Z” paper [39] for a fuller discussion on Higgs mixing and the origin of the third family couplings. The above expressions, while precise, are not very illustrative. In explicit matrix form, we have 0 0 0 b ǫ13c yd 0 0 (yd/3) 0 0 u d 0 0 e − 0 0 M = vu a 4b ǫ23c , M = vd Byd ys 0 , M = vd Byd xys 0 , (C.11) a 2b c By0 0 y0 By0 0 y0 d b d b 11 c Note that under hermitian conjugation, the PQ charges change sign, i.e xf ≡−xfR . For consistency, we will always specify which Weyl fermion we are referring to.
– 24 – with dimensionless parameters defined by
u u λ1uvϕ1 λ2uvϕ2 Θi(u) a = , b = , c = λ3Θ3(u), ǫi3 = √2vΣ √2vΣ Θ3(u) u u (C.12) λ1dv d λ1dv d u 0 ϕ1 0 ϕ2 0 λudvϕ1 yd = , ys = , yb = λ3Θ3(d), B = . √ √ λ v d 2vΣd 2vΣd 1d ϕ1 Lagrangian (derivative basis) We perform an axion-dependent rotation of the fermion fields to replace the linear couplings with derivative ones; the anomaly term is also induced. Extending the Lagrangian to include ¯ ¯ the fermion kinetic terms, f (fLi∂f/ Li + fRi∂f/ Ri), we let
a a P i xf i xf f e vP Q Li f , f e vP Q Ri f , (C.13) Li → Li Ri → Ri resulting in
¯ ¯ ∂µa ¯ µ ¯ µ = i fLi∂f/ Li + fRi∂f/ Ri xfLi fLiγ fLi + xfRi fRiγ fRi L − vPQ f=u,d,e f=u,d,e (C.14) X X M u u¯ u M d d¯ d M e e¯ e +h.c. + anomaly. − ij Li Rj − ij Li Rj − ij Li Rj We rotate to the mass basis by unitary transformations u U u , d U d , L → Q L L → Q L e U e , f V f , such that the mass terms become L → L L R → f R u d e m = M u¯LiuRj + M d¯LidRj + M e¯LieRj +h.c. L ij ij ij (C.15) muδ u¯ u + (V ) mdδ d¯ d + meδ e¯ e +h.c., → i ij Li Rj CKM ik k kj Li Rj i ij Li Rj f f where by definition m U †M V , U U , and V Uu†U . ≡ f f Q ≡ u CKM ≡ d Derivative couplings The axion-fermion derivative couplings become
∂µa ¯ µ ¯ µ ∂ = fL(Uf†xfL Uf )γ fL + fR(Vf†xfR Vf )γ fR +h.c., (C.16) L −vPQ f=Xu,d,e h i where now f , f are vectors and x , x are diagonal 3 3 matrices. We define the L R fL fR × coupling matrices X U †x U and X V †x V , and note that, since charges x are L ≡ f fL f R ≡ f fR f f real, XL = XL† and XR = XR† . In terms of Dirac spinors,
∂µa µ ∂ = fγ¯ (Vf Af γ5)f, (C.17) L −vPQ − f=Xu,d,e where 1 1 Vf = (XL + XR)= Uf†xfL Uf + Vf†xfR Vf , 2 2 (C.18) 1 1 A = (X X )= U †x U V †x V . f 2 L − R 2 f fL f − f fR f
– 25 – D Couplings in A to Z: numerical fit
The best fit parameters, as well as a Bayesian 95% credible interval, are given in Tables 7 (leptons) and 8 (quarks). The corresponding best fit input parameters are given in Table 9. We fit the model to data at the GUT scale. The running from low to high scale was performed, assuming the MSSM, in [105]. They parametrise threshold corrections by a series of dimensionless parameters ηi. All but one (η¯b) were set to zero, and choosing η¯ = 0.24 to account for the small GUT-scale difference between b and τ masses. b − Data Model Observable Central value 1σ range Best fit Interval θℓ / 33.57 32.81 34.32 32.88 32.72 34.23 12 ◦ → → θℓ / 8.460 8.310 8.610 8.611 8.326 8.882 13 ◦ → → θℓ / 41.75 40.40 43.10 39.27 37.35 40.11 23 ◦ → → δℓ / 261.0 202.0 312.0 242.6 231.4 249.9 ◦ → → y /10 5 1.004 0.998 1.010 1.006 0.911 1.015 e − → → y /10 3 2.119 2.106 2.132 2.116 2.093 2.144 µ − → → y /10 2 3.606 3.588 3.625 3.607 3.569 3.643 τ − → → ∆m2 /10 5 eV2 7.510 7.330 7.690 7.413 7.049 7.762 21 − → → ∆m2 /10 3 eV2 2.524 2.484 2.564 2.540 2.459 2.616 31 − → → m /meV 0.187 0.022 0.234 1 → m /meV 8.612 8.400 8.815 2 → m /meV 50.40 49.59 51.14 3 → m /meV < 230 [106] 59.20 58.82 60.19 i → α 10.4 38.0 70.1 P21 − → α 272.1 218.2 334.0 31 → m /meV 1.940 1.892 1.998 ββ →
Table 7. Model predictions in the lepton sector, at the GUT scale. We set tan β =5, MSUSY =1 TeV and η¯b = 0.24. The model interval is a Bayesian 95% credible interval. −
– 26 – Data Model Observable Central value 1σ range Best fit Interval θq / 13.03 12.99 13.07 13.04 12.94 13.11 12 ◦ → → θq / 0.1471 0.1418 0.1524 0.1463 0.1368 0.1577 13 ◦ → → θq / 1.700 1.673 1.727 1.689 1.645 1.753 23 ◦ → → δq / 69.22 66.12 72.31 68.85 63.00 75.24 ◦ → → y /10 6 2.982 2.057 3.906 3.038 1.098 4.957 u − → → y /10 3 1.459 1.408 1.510 1.432 1.354 1.560 c − → → y 0.544 0.537 0.551 0.545 0.530 0.558 t → → y /10 5 2.453 2.183 2.722 2.296 2.181 2.966 d − → → y /10 4 4.856 4.594 5.118 4.733 4.273 5.379 s − → → y 3.616 3.500 3.731 3.607 3.569 3.643 b → →
Table 8. Model predictions in the quark sector at the GUT scale. We set tan β =5, MSUSY = 1 TeV and η¯b = 0.24. The model interval is a Bayesian 95% credible interval. −
Parameter Value Parameter Value Parameter Value
5 4.047i 3 2.434i a /10− 1.246 e ǫ13 /10− 6.215 e ma /meV 3.646 3 2.080i 2 3.867i b /10− 3.438 e ǫ23 /10− 2.888 e mb /meV 1.935 2.777i c 0.545 B 10.20 e mc /meV 1.151 0 5 − 4.816i yd /10− 3.053 e x 5.880 η 2.592 0 4 2.097i ys /10− 3.560 e ξ 2.039 0 2 yb /10− 3.607
Table 9. Best fit input parameter values.
– 27 – References
[1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (1977) 1440. [2] F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [3] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223. [4] J. E. Kim, Phys. Rev. Lett. 43 (1979) 103. [5] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 166 (1980) 493. [6] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B (1981) 199. [7] A. R. Zhitnitsky, Sov. J. Nucl. Phys. 31 (1980) 260 [Yad. Fiz. 31 (1980) 497]. [8] J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. B 120 (1983) 127 [Phys. Lett. 120B (1983) 127]. [9] L. F. Abbott and P. Sikivie, Phys. Lett. B 120 (1983) 133 [Phys. Lett. 120B (1983) 133]. [10] M. Dine and W. Fischler, Phys. Lett. B 120 (1983) 137 [Phys. Lett. 120B (1983) 137]. [11] J. E. Kim and G. Carosi, Rev. Mod. Phys. 82 (2010) 557 [arXiv:0807.3125 [hep-ph]]. [12] R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. Watkins and L. M. Widrow, Phys. Lett. B 282 (1992) 132 [hep-ph/9203206]. [13] M. Kamionkowski and J. March-Russell, Phys. Lett. B 282 (1992) 137 [hep-th/9202003]. [14] S. M. Barr and D. Seckel, Phys. Rev. D 46 (1992) 539. [15] E. J. Chun and A. Lukas, Phys. Lett. B 297 (1992) 298 [hep-ph/9209208]. [16] M. Bastero-Gil and S. F. King, Phys. Lett. B 423 (1998) 27 [hep-ph/9709502]. [17] K. S. Babu, I. Gogoladze and K. Wang, Phys. Lett. B 560 (2003) 214 [hep-ph/0212339]. [18] A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D 69 (2004) 015007 [hep-ph/0210172]. [19] A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D 67 (2003) 095008 [hep-ph/0211107]. [20] A. G. Dias, E. T. Franco and V. Pleitez, Phys. Rev. D 76 (2007) 115010 [arXiv:0708.1009 [hep-ph]]. [21] K. Harigaya, M. Ibe, K. Schmitz and T. T. Yanagida, Phys. Rev. D 88 (2013) no.7, 075022 [arXiv:1308.1227 [hep-ph]]. [22] C. Cheung, JHEP 1006 (2010) 074 [arXiv:1003.0941 [hep-ph]]. [23] L. Di Luzio, E. Nardi and L. Ubaldi, Phys. Rev. Lett. 119 (2017) no.1, 011801 [arXiv:1704.01122 [hep-ph]]. [24] F. Wilczek, Phys. Rev. Lett. 49 (1982) 1549. [25] K. S. Babu and S. M. Barr, Phys. Lett. B 300 (1993) 367 [hep-ph/9212219]. [26] M. E. Albrecht, T. Feldmann and T. Mannel, JHEP 1010 (2010) 089 [arXiv:1002.4798 [hep-ph]]. [27] A. Celis, J. Fuentes-Martin and H. Serodio, Phys. Lett. B 741 (2015) 117 [arXiv:1410.6217 [hep-ph]]. [28] Y. H. Ahn, Phys. Rev. D 91 (2015) 056005 [arXiv:1410.1634 [hep-ph]]. [29] Y. Ema, K. Hamaguchi, T. Moroi and K. Nakayama, JHEP 1701 (2017) 096 [arXiv:1612.05492 [hep-ph]]. [30] L. Calibbi, F. Goertz, D. Redigolo, R. Ziegler and J. Zupan, Phys. Rev. D 95 (2017) no.9, 095009 [arXiv:1612.08040 [hep-ph]]. [31] K. Choi, S. H. Im, C. B. Park and S. Yun, JHEP 1711 (2017) 070 [arXiv:1708.00021 [hep-ph]]. [32] F. Arias-Aragon and L. Merlo, JHEP 1710 (2017) 168 [arXiv:1709.07039 [hep-ph]]. [33] M. Linster and R. Ziegler, arXiv:1805.07341 [hep-ph]. [34] S. F. King and C. Luhn, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340 [hep-ph]]. [35] S. F. King, J. Phys. G 42 (2015) 123001 [arXiv:1510.02091 [hep-ph]]. [36] S. F. King, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413 [hep-ph]].
– 28 – [37] S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, New J. Phys. 16 (2014) 045018 [arXiv:1402.4271 [hep-ph]]. [38] F. Björkeroth, E. J. Chun and S. F. King, Phys. Lett. B 777 (2018) 428 [arXiv:1711.05741 [hep-ph]]. [39] S. F. King, JHEP 1408 (2014) 130 [arXiv:1406.7005 [hep-ph]]. [40] F. Björkeroth, F. J. de Anda, I. de Medeiros Varzielas and S. F. King, Phys. Rev. D 94 (2016) no.1, 016006 [arXiv:1512.00850 [hep-ph]]. [41] A. E. Nelson, Phys. Lett. 136B (1984) 387. [42] A. E. Nelson, Phys. Lett. 143B (1984) 165. [43] S. M. Barr, Phys. Rev. Lett. 53 (1984) 329. [44] S. M. Barr, Phys. Rev. D 30 (1984) 1805. [45] F. Björkeroth, F. J. de Anda, I. de Medeiros Varzielas and S. F. King, JHEP 1506 (2015) 141 [arXiv:1503.03306 [hep-ph]]. [46] S. Antusch, M. Holthausen, M. A. Schmidt and M. Spinrath, Nucl. Phys. B 877 (2013) 752 [arXiv:1307.0710 [hep-ph]]. [47] S. Antusch, C. Gross, V. Maurer and C. Sluka, Nucl. Phys. B 877 (2013) 772 [arXiv:1305.6612 [hep-ph]]. [48] Y. H. Ahn, arXiv:1804.06988 [hep-ph]. [49] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147 (1979) 277. [50] J. Jaeckel, Phys. Lett. B 732 (2014) 1 [arXiv:1311.0880 [hep-ph]]. [51] I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey and V. Sanz, Eur. Phys. J. C 77 (2017) no.8, 572 [arXiv:1701.05379 [hep-ph]]. [52] W. A. Bardeen and S.-H. H. Tye, Phys. Lett. 74B (1978) 229. [53] M. Srednicki, Nucl. Phys. B 260 (1985) 689. [54] W. A. Bardeen, R. D. Peccei and T. Yanagida, Nucl. Phys. B 279 (1987) 401. [55] G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, JHEP 1601 (2016) 034 [arXiv:1511.02867 [hep-ph]]. [56] J. L. Feng, T. Moroi, H. Murayama and E. Schnapka, Phys. Rev. D 57 (1998) 5875 [hep-ph/9709411]. [57] A. Al-Haydari et al. [QCDSF Collaboration], Eur. Phys. J. A 43 (2010) 107 [arXiv:0903.1664 [hep-lat]]. [58] A. V. Artamonov et al. [BNL-E949 Collaboration], Phys. Rev. D 79 (2009) 092004 [arXiv:0903.0030 [hep-ex]]. [59] S. Adler et al. [E949 and E787 Collaborations], Phys. Rev. D 77 (2008) 052003 [arXiv:0709.1000 [hep-ex]]. [60] A. J. Buras, D. Buttazzo, J. Girrbach-Noe and R. Knegjens, JHEP 1511 (2015) 033 [arXiv:1503.02693 [hep-ph]]. [61] R. Marchevski, talk given at Moriond EW, 11 March 2018. [URL: https://indico.in2p3.fr/event/16579/contributions/60808/] [62] R. Fantechi [NA62 Collaboration], arXiv:1407.8213 [physics.ins-det]. [63] M. Gorbahn, M. Patel and S. Robertson, arXiv:1104.0826 [hep-ph]. [64] J. Brod, M. Gorbahn and E. Stamou, Phys. Rev. D 83 (2011) 034030 [arXiv:1009.0947 [hep-ph]]. [65] M. Antonelli et al., Phys. Rept. 494 (2010) 197 [arXiv:0907.5386 [hep-ph]]. [66] J. K. Ahn et al. [E391a Collaboration], Phys. Rev. D 81 (2010) 072004 [arXiv:0911.4789 [hep-ex]]. [67] B. Beckford [KOTO Collaboration], arXiv:1710.01412 [hep-ex].
– 29 – [68] J. K. Ahn et al. [KOTO Collaboration], PTEP 2017 (2017) no.2, 021C01 [arXiv:1609.03637 [hep-ex]]. [69] M. Moulson [NA62-KLEVER Project Collaboration], J. Phys. Conf. Ser. 800 (2017) no.1, 012037 [arXiv:1611.04864 [hep-ex]]. [70] D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, JHEP 1711 (2017) 044 [arXiv:1706.07808 [hep-ph]]. [71] R. Ammar et al. [CLEO Collaboration], Phys. Rev. Lett. 87 (2001) 271801 [hep-ex/0106038]. [72] S. Cunliffe, arXiv:1708.09423 [hep-ex]. [73] S. Adler et al. [E787 Collaboration], Phys. Rev. Lett. 88 (2002) 041803 [hep-ex/0111091]. [74] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 94 (2005) 101801 [hep-ex/0411061]. [75] J. Grygier et al. [Belle Collaboration], Phys. Rev. D 96 (2017) no.9, 091101 Addendum: [Phys. Rev. D 97 (2018) no.9, 099902] [arXiv:1702.03224 [hep-ex]]. [76] P. del Amo Sanchez et al. [BaBar Collaboration], Phys. Rev. D 82 (2010) 112002 [arXiv:1009.1529 [hep-ex]]. [77] T. Abe et al. [Belle-II Collaboration], arXiv:1011.0352 [physics.ins-det]. [78] C. Patrignani et al. [Particle Data Group], Chin. Phys. C 40 (2016) no.10, 100001. [79] J. Brod and M. Gorbahn, Phys. Rev. Lett. 108 (2012) 121801 [arXiv:1108.2036 [hep-ph]]. [80] Z. Bai, N. H. Christ and C. T. Sachrajda, EPJ Web Conf. 175 (2018) 13017. [81] L. K. Li, Int. J. Mod. Phys. Conf. Ser. 46 (2018) 1860062. [82] J. L. Rosner, S. Stone and R. S. Van de Water, [arXiv:1509.02220 [hep-ph]]. [83] R. Miskimen, Ann. Rev. Nucl. Part. Sci. 61 (2011) 1. [84] J. Jaeckel, P. C. Malta and J. Redondo, arXiv:1702.02964 [hep-ph]. [85] P. Kitching et al. [E787 Collaboration], Phys. Rev. Lett. 79 (1997) 4079 [hep-ex/9708011]. [86] A. Jodidio et al., Phys. Rev. D 34 (1986) 1967 Erratum: [Phys. Rev. D 37 (1988) 237]. [87] R. Bayes et al. [TWIST Collaboration], Phys. Rev. D 91 (2015) no.5, 052020 [arXiv:1409.0638 [hep-ex]]. [88] A.-K. Perrevoort, Sensitivity Studies on New Physics in the Mu3e Experiment and Development of Firmware for the Front-End of the Mu3e Pixel Detector, PhD thesis, Heidelberg University, 2018 [unpublished]. [89] H. Albrecht et al. [ARGUS Collaboration], Z. Phys. C 68 (1995) 25. [90] M. Hirsch, A. Vicente, J. Meyer and W. Porod, Phys. Rev. D 79 (2009) 055023 Erratum: [Phys. Rev. D 79 (2009) 079901] [arXiv:0902.0525 [hep-ph]]. [91] J. T. Goldman et al., Phys. Rev. D 36 (1987) 1543. [92] A. M. Baldini et al. [MEG Collaboration], Eur. Phys. J. C 76 (2016) no.8, 434 [arXiv:1605.05081 [hep-ex]]. [93] A. M. Baldini et al. [MEG II Collaboration], Eur. Phys. J. C 78 (2018) no.5, 380 [arXiv:1801.04688 [physics.ins-det]]. [94] R. D. Bolton et al., Phys. Rev. D 38 (1988) 2077. [95] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 104 (2010) 021802 [arXiv:0908.2381 [hep-ex]]. [96] A. M. Baldini et al. [MEG Collaboration], Eur. Phys. J. C 76 (2016) no.3, 108 [arXiv:1312.3217 [hep-ex]]. [97] U. Bellgardt et al. [SINDRUM Collaboration], Nucl. Phys. B 299 (1988) 1. [98] A. Blondel et al., arXiv:1301.6113 [physics.ins-det]. [99] L. Di Luzio, F. Mescia, E. Nardi, P. Panci and R. Ziegler, Phys. Rev. Lett. 120 (2018) no.26, 261803 [arXiv:1712.04940 [hep-ph]]. [100] V. Cirigliano, S. Davidson and Y. Kuno, Phys. Lett. B 771 (2017) 242 [arXiv:1703.02057 [hep-ph]].
– 30 – [101] W. H. Bertl et al. [SINDRUM II Collaboration], Eur. Phys. J. C 47 (2006) 337. [102] R. Donghia [Mu2e Collaboration], Nuovo Cim. C 40 (2017) no.5, 176. [103] C. Wu [COMET Collaboration], Nucl. Part. Phys. Proc. 287-288 (2017) 173. [104] P. Di Bari and S. F. King, JCAP 1510 (2015) no.10, 008 [arXiv:1507.06431 [hep-ph]]. [105] S. Antusch and V. Maurer, JHEP 1311 (2013) 115 [arXiv:1306.6879 [hep-ph]]. [106] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589 [astro-ph.CO]].
– 31 –