JHEP08(2018)117 Springer June 19, 2018 ), as well as August 8, 2018 August 9, 2018 : γ August 21, 2018 : : ( : a 2 ` → Revised , Received 1 Accepted ` Published c , which affects the meson oscillation 0 s Published for SISSA by B https://doi.org/10.1007/JHEP08(2018)117 [email protected] and , 0 B , 0 D and Stephen F. King , b 0 K ) decays involving a flavourful axion. We also calculate the mixing . 3 Eung Jin Chun B , a conversion. Finally we describe the phenomenology of a particular “A problem, and non-standard axion-like particles (ALPs). We give the D e 1806.00660 , The Authors. − Beyond Standard Model, Cosmology of Theories beyond the SM, CP CP K c µ

We present a comprehensive discussion of the phenomenology of flavourful , [email protected] and eee → Seoul 02455, Korea School of Physics andSO17 Astronomy, 1BJ University Southampton, of United Southampton, Kingdom E-mail: [email protected] INFN Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044Korea Frascati, Institute Italy for Advanced Study, b c a Open Access Article funded by SCOAP violation, Global Symmetries ArXiv ePrint: to Z” Pati-Salam model, insymmetry. which PQ Here symmetry all arisespredictions accidentally axion and due couplings to correlations discrete between are flavour flavour-dependent fixed observables. by a fitKeywords: to flavour data, leading to sharp between axions and heavy mesons probability and mass difference.axion Mixing decays also into contributes two to photons,flavour-violating meson and decays decays may into involving be axions final relevantµ and for state ALPs. axions We of discuss charged the lepton form Abstract: axions, including both standardto Peccei-Quinn the (PQ) strong axions,flavourful associated axion-fermion and with axion-photon the couplingsheavy solution and meson calculate ( the branching ratios of Fredrik Bj¨orkeroth, Flavourful axion phenomenology JHEP08(2018)117 problem” is to ] or to extend the CP 5 , 4 22 5 21 25 – 1 – 16 9 ]. The resulting QCD axion provides a candidate for 8 7 , 3 6 15 9 4 18 5 7 20 17 3 ]. The two most common approaches to realising such a PQ symmetry 12 3 1 – 1 19 symmetry. The most popular resolution of this so-called “strong 6.1 Mass matrices6.2 and parameters Predictions 6.3 Decay correlations 4.1 Parametrisation of4.2 mixing Meson mass4.3 splitting Axion-pion mixing and ALPs 2.1 Lagrangian 2.2 Physical axion2.3 basis Decay constant and axion-photon coupling postulate a Peccei-Quinn (PQ) symmetry, namely awhich is QCD-anomalous broken global spontaneously, U(1) leading symmetry to athe pseudo-Nambu-Goldstone QCD boson axion (pNGB) [ called is either to introduceHiggs heavy sector vector-like (the quarks DFSZ (the model) KSVZ [ model) [ 1 Introduction One of the puzzlesCP of the Standard Model (SM) is why QCD does not appear to break C Couplings in the A toD Z Pati-Salam Couplings model in A to Z: numerical fit 7 Conclusion A Axion-meson mixing B Heavy meson decay branching ratio 6 A to Z Pati-Salam model 4 Axion-meson mixing 5 Lepton decays 3 Heavy meson decays Contents 1 Introduction 2 Axion couplings to matter JHEP08(2018)117 0 s ], B 24 ], and and 38 0 symmetry continuous B , 0 PQ and radiative D a , 2 0 ` problem, and non- K → ] in the “A to Z” Pati- describes the flavourful 1 CP ` 2 38 ]. Alternatively, attempts 21 – 15 ] to unify the U(1) ]. The resultant axion is variously 48 , 49 conversion. Finally we describe the 30 involving a flavourful axion. Moreover, e , ]). For a complementary analysis of ALP B 29 − , and all axion couplings are fixed by a fit ], which otherwise presents a stronger case 50 µ 40 CP ]. We present the flavourful axion-fermion and , and – 2 – and D 51 ], as recently discussed [ , 37 ], or GUT models where specific Yukawa structures have been K eee – 44 – 34 [ → 41 µ ]. In this regard, it is appealing to consider an approximate 14 ]. It is also possible that PQ symmetry could arise accidentally – ]. problem [ 33 12 11 – CP 25 ]. It is possible that PQ symmetry arises from flavour symmetries [ , as well as ] within the allowed window of the axion (or PQ symmetry-breaking) GeV [ 23 ], where quarks and lepton are unified. This is difficult to achieve in a Recent efforts have been made [ , 10 aγ 12 2 – 39 1 invariance, it is enough to protect the PQ symmetry up to some higher- 22 ` − 8 ]. 9 47 – → CP 45 1 = 10 ` a f discrete flavour symmetries The layout of the remainder of the paper is as follows. Section In this paper we focus on the phenomenology of flavourful axions, including both It has also been realised that the PQ axion need not emerge from an exact global These ideas should not be confused with alternatives to PQ symmetry, such as Nelson-Barr type 1 axion-fermion and axion-photon couplings both for the standardresolutions axion to and the for strong proposed ALPs. [ In phenomenology of the A toshow Z how Pati-Salam unification model, leads which to predicts correlationsas a the between flavourful down-type different axion quark flavour [ and dependent charged observables, leptonarises couplings from are very the similar. same Notably,field as flavon the content axion is fields necessary that toto dictate solve quark fermion the and strong Yukawa lepton structures, masses no and additional mixing. and its consequences, whichlead has to new not contributions beenaxion to discussed decays neutral in into meson the two massdecays photons splitting, literature involving which meson before. final may decays into be statedecays These axions relevant can axions, and for including ALPs. two-body We decays also discuss lepton signatures and bounds ataxion-photon the LHC, couplings see both [ forquite naturally the are standard non-diagonal. axionfor We use two-body and these decays for of couplings ALPs, heavy towe mesons calculate and calculate the the show branching mixing that ratios between they axions and neutral hadronic mesons with a Froggatt-Nielsen-like U(1) flavourdubbed symmetry a [ “flaxion” or “axiflavon”; we shall refer simplystandard to PQ a “flavourful axions, axion”. associatedstandard axion-like with particles the (ALPs) (see solution e.g. to [ the strong flavour symmetries [ from Salam model [ grand unified theory (GUT) based onfor SO(10) unification. [ dimensional operators [ PQ symmetry guaranteed by discreteto (gauge) link symmetries [ PQ symmetrywere made protected in by [ continuouslinking gauge the symmetries axion to scale the tobeen the flavour made flavour problem symmetry-breaking to scale, incorporate and such various attempts a have flavourful PQ symmetry as a part of such scale U(1) symmetry, but couldmetry result leading from to some an discreteof accidental symmetry global strong- or U(1) continuous symmetry. gauge Considering sym- the observed accuracy dark matter [ JHEP08(2018)117 (2.2) (2.3) (2.1) ) are shows L,R C , in terms X tabulates ii ) D Rf U which transform f , . M Rf the axion derivative   U † ∂ Lf , , , j L are proportional to the Rf Rf U f µν Lf U U  written here as (diagonal) ˜ F , R R U 5 = ( i f f 2 γ f µν x x f i ) Lf,Rf f ij i , F m x † † Rf Rf A a m a f U U − − focusses on the phenomenology of π α + − f 8 ij / 6 ∂ anomaly ( V i aγ Lf Lf ¯ L  f c U U µ + L L + γ . The axion-photon coupling is discussed f f i ∂ ¯ defined in terms of the PQ-breaking scale x x u,d,e f L X = DW aµν † † are defined by Lf Lf f ˜ + (as well as chiral coupling matrices N G gives more details about axion-meson mixing. DW U U f i u,d,e + – 3 – the fermion mass terms,   X m f = a µν m 2 A , and unitary matrices /N f L 1 2 1 2 ) A m G f discusses the mixing between axions and neutral a a a L + PQ a µ µ f M PQ is the PQ charge in the “SUSY basis” where the superpotential 4 v ∂ ∂ v ) = ) = s c ( and π kin f R R α 8 1 2 = − x f the QCD and electromagnetic anomalies. In the physical L X X a V = = = f = + − ∂ m L L L L L , where X X anomaly R ( ( + f L 1 2 1 2 anomaly x are real, L kin discusses lepton decays. Section = = L R ≡ − 5 f f f concludes. Appendix c x A f V 7 x , L f x below. The physical masses details the calculation the heavy meson branching ratios. Appendix contains the kinetic terms, . Then c B 2.3 f are the fermion PQ charges in the left-right (LR) basis, we apply these couplings to calculate the branching ratios of heavy meson decays kin 3 R L f x and anomaly (or domain wall) number In this formulation, the implications of flavour structure are clear. If all generations of Note that right-handed particles in supersymmetric theories typically manifest as left-handed an- , 2 L f PQ a fermion couple equally to the axion, the chargetifermions matrices is defined. x matrices. As Hermitian. in section of the mass matrix inleft- the and weak basis, right-handed fields, respectively. The vector and axial couplings are given by with the axion decayv constant couplings to matter, and (mass) basis below the electroweak symmetry-breaking scale, we have Relevant to a discussion on axion-fermion interactions is the Lagrangian where the derivation of the couplingsthe in numerical the fit A to to flavour Z data. Pati-Salam model and2 appendix Axion couplings to2.1 matter Lagrangian involving a flavourful axion.mesons while Section section the A to Z model,ables, which and predicts section correlations betweenAppendix different flavour dependent observ- section JHEP08(2018)117 a (2.5) (2.6) (2.7) (2.8) (2.4) is then and the mesons. no longer a , a R η f   /f x s 1 ) − .. 5 ∝ c and γ . = a ]. Let us briefly 0 ∗ sd m L c π f 54 + h – x −  a a which provide further 52 f 0∗ sd c d 0 for the physical axion, as ( boson. We still need to β η ) is transformed into the µ f + 0 π Z = 0), the Goldstone field . A ¯ . dγ f π ), ]. The axion mass generated 2.2 − + eV 11  ), the axion-pion mixing term , i DW µ and d d q ∗ e ) . Lf,Rf ] gives us N f u, d, s i 5 m 2 m m π U  γ R V 55 f 2 a , and there is no flavour violation. In = , we still get flavour-violating (vector d problem is not solved, and + 2 π sd = 3 f 3 L q I c I u ¯ q m ) d GeV / a h ∝ R − 2 m (a good approximation to leading order), f CP f ) d ( R 12 s d / x 0 sd ,f d m 2 c π L m m − 10 ( f f m − µ L 2 π x ) as the physical axion and extract its mass, u  + q, β f   – 4 – 5 m x sγ u m a ) is conveniently calculated by rotating away the γ a ( 2.5 f a 1 2 a m u,d + ¯ u f = ( β 2.2 q q 2 m = β + i 5 i = 0 π γ R f ) affects also the axion-quark couplings. For example e 70(6)(4) f π µ 2 a . d A  in eq. ( i L , m 2.4 → ¯ ¯ qγ e . For d a 3 q 1 h = 5 i I q c ) − s R a . This leads to a low-energy effective Lagrangian below the R = u 1 m f m L transformation is not chiral ( − d x u,d,s i ; this is generally not true in flavoured axion models. Meanwhile ¯ + u X = R f h m + 1 However, as long as q u PQ u − d L A 3   + f L m m ¯ x u 1 a ( h µ − u PQ + 1 2 ∂ v 1 m ⊃ − − u = quarks, the axion-quark Lagrangian in eq. ( ≈ f eff m s ⊃ − , the U(1) L 1 V 0 ∂ = R − ∗ f 1 x and m − ∗ → L d = m ∂ , L The transformation in eq. ( u L f The mass of the ALP no longer arises from the QCD vacuum, and the relation 3 x holds. We don’t specify any particular mass generation scheme here. for physical basis, There remains additional mixingsmall corrections. with A heavier precise mesons calculation such performed in as [ chiral symmetry-breaking scale, Using the relation vanishes. We identify the state anomaly via chiral transformations of light quarks ( where we have axion does not mixidentify with the the physical longitudinal axionOne component can at of then low determine the energysummarize the how as it canonical the works, axion following stateby mass the the orthogonal prescription and QCD e.g. to anomaly couplings in coupling [ [ in eq. ( and axial) interactions due to weak mixing2.2 encoded in Physical axionThe basis above Lagrangian describes an interactingThe axion, off-diagonal not couplings necessarily to in fermions its arewe mass nevertheless will eigenstate. see. Unlike standard DFSZ models with PQ-charged Higgs doublets, our flavoured standard axion models, e.g. DFSZ,axion charges couples can only be via assignedif such that doesn’t couple to theinterpreted QCD as anomaly, an ALP. the strong identity, i.e. JHEP08(2018)117 ) 2 q DW ( . If (2.9) (3.1) 2 N , and φ + ), the 0 v f d 21 q 2 φ 0 75. V x . P 0 φ q = ≈ , but can be P 0 sd , where π = (¯ c f aγ 0 , = c 2 DW 2, P | / ≫ s . For completeness, 2 PQ /N (0) β v d a 21 1. + f PQ V f v . | ≡ ) into DW 0 In more general models, 3 ≡ φ q 3, giving N 56 x and . /  4 P d . sd 0 0 q + π φ 2 P V 2 P v ≈ = 8 d 22 m m φ m = (¯ d u x A − ) is given in terms of the electro- m P DW m ≪ = = 1 . The kinetic mixing contribution is s a 2.2 =  . It depends on a form factor c is defined by symmetry is broken by the VEV of a A PQ m 3 P E/N relate to the constituent quarks, e.g. a v a . B 2, f m 0 PQ / 6 2 d P

q β 0 , z P P ) ) q q 2 PQ may lead to observable decays of heavy mesons z z DW P f v q – 5 – ∂ N ], which is applicable to our flavourful axion. V L

+ 56 ) with coupling strength defined in eq. ( and appendix 2(4 + 3(1 + of a heavy meson P d 11 ˜ 4 ¯ F a da A − 1 Γ( 0 . π , we simply have P = aF → sd φ 16 c DW E d x s ). Its indices → c to within 1%. We will encounter exactly this scenario when N , through the coefficient i 2.3 E P 2, ) = φ = / a v 0 ) is provided in appendix u i aγ φ β P contribute to symmetry breaking, we define c x 3.1 φ → DW ≈ can be removed by normalising all charges such that N P φ PQ + x v Br( dominates, we recover to good approximation the one-field relation; if, say, , u 11 with PQ charge i i A φ decay proceeds by ¯ φ φ v v a ), needs to be further diagonalized away to obtain the physical axion basis. This 1 = . We see that the diagonal couplings are modified by an amount proportional to as defined in eq. ( . , the ratio of anomaly numbers is fixed to + 0 u 21 d f π 2.8 6 c A . V , whereas the off-diagonal couplings are unchanged. Physically, this is a consequence For a two-body decay The axion-photon coupling The above discussion identifies the physical axion basis in the limit of no kinetic mixing → i Its PQ charge = 6= 4 j + DW φ sd with K a rederivation of eq. ( Nambu-Goldstone boson was made in [ branching ratio is given by 3 Heavy meson decays The flavour-changing vector couplings in into axions. A general study of such flavour-changing processes involving a (massless) In unified models,section such as the A to Z model with Pati-Salam unification presented in one VEV v discussing the A to Z model presented in section magnetic anomaly number 2.3 Decay constant andIn axion-photon standard coupling axion scenarios,is the the decay QCD anomaly constant single number. field Provided the U(1) where several fields between the axion and heavierin eq. mesons. ( Such mixing,will induced be by discussed the in effective detailnegligibly Lagrangian in small section for theimportant standard for QCD an axion ALP. with c N of the QCDinteractions anomaly that being contribute to flavour-conserving, and unable to mediate flavour-violating where JHEP08(2018)117 ] , , < ¯ a ν at → 58 ν ) + 8 0 0 a L π − ) π 0 with a K 10 → → a K ) ( × + 0 0 L 6 K π . K 0. For kaon K ( ]. 2 decays of the π → 62 ]. The current → < B 0 , consistent with → ) 2 65 decays. ¯ B ν q – 10 B ν − pairs throughout its B 0 63 [ π . This was done in the 0, they set Br( 10 B ¯ ν 11 and ]. Rare B × ν → ' − and Br( decays at the LHC, which + 7 0 70 L D 15 05 0 at 90% CL [ . . 5 π 10 , 1 B X ]. K − +1 − 12 K × → m 69 10 − 73 + is the momentum transfer to the . 2) × ], is expected to observe over 100 10 . 0 9 ] for K 0 . P 61 × 4 p 57  0 ) = 1 . < (0) − ¯ 1 9 ν 1 . ]. An analysis has also been performed + ) ν P f a < p + event [ ) 67 [ ) π 0.74(6)(4) 0.78(5)(4) 0.68(4)(3) 0.27(7)(5) 0.32(6)(6) 0.23(5)(4) ]. The current NA62 experiment at CERN,  ¯ ν = a 8 ν ]. A bound on axion decays is also provided: K 59 a + − → ) = (2 ( + p π ¯ – 6 – ν 60  π + 10 [ ν π K K π K K π = π 0 → K at the CERN SPS [ × → q π 10 → → → → → → → + → ¯ 1 ν − . + ν s s Decay 5 K →  (0) extracted from [ 0 B B 10 D D of arbitrary mass. For K K B D + π B 0 L . < f 0 × at 90% CL [ K ) 1 → X ¯ ν 10 0 10) L ν . − 0 0 K ]. More recent and powerful experiments, namely BaBar and π physics has a rich phenomenology, and is recently of particular 10  71 was made by CLEO, which collected 10 × → B a 84 . 0 L 73 ]. Detector upgrades and additional data taken since 2015 are expected . . Form factors K 0 , while generally not as tightly constrained as those for kaons, may also for a boson 68 [ ¯ ν 1 to good approximation. For heavier mesons, we use results from lattice < The canonical example of this type of flavour-violating decay is 0 ν 8 Searches have also been performed for the neutral kaon decay ) ) − ≈ X a . 0 K decays. ]) yield a measurement Br( 10 + . ( ]. Its successor KOTO has been constructed at J-PARC. A pilot run in 2013 at 90% CL [ a π Table 1 π π a (0) s × 59 + 5 66 0 + 5 ], summarised in table B − π → f π → → . 57 10 0 L + ) B → 0 → K × K and 0 + X L 3 0 . type provide insights into newlight physics. invisible A particle dedicated searchlifetime. for It decays provides the like limits5 Br( to significantly improve thesebeen bounds. proposed to An measure additional experiment dubbedB KLEVER has interest due to persistent anomaliesmay in be observed evidence semileptonic for charged lepton flavour violation (cLFV) [ best limit is set by90% the CL E391a [ experiment atyielded KEK, a giving Br( limit Br( for π which recently recorded their first events, reaching a sensitivity of Br( K for which the SM predicts Br( K which can be constrained by searchesE949 for and the E787 rare experiments, decay which(see observed also in total [ seven events.the Combined SM analyses prediction [ (0 Br( encapsulating hadronic physics, where axion. The lightnessdecays, of the axion meansQCD we [ can safely take the limit JHEP08(2018)117 = K PQ N (3.2) (3.3) /v | 0 , which P q c P pair. The f q ¯ V decays have ν | ν s 1 allows us to B < ) . a and ) 2 | , which are typically s K ¯ , ν ( D ν 2 (0) π , )  + ]; assuming the sensitivity D f K | → ( 3 72 π D GeV  0 PQ → 2 P v 2 P 12 m m B 10 . −  2 2 1

0  P q 2 ), we define a branching ratio coefficient , predicted by flavoured axion models, has P f a q , where available. 0 decays, which have corresponding branching 3.1 , which are a background to the axion signal. V π

¯ ν , along with experimental limits on the branch- 3 P 0 PQ GeV) B – 7 – ν v 2 P m 0 → 12 π → 0 ) improvement in branching ratio limits. P 2 and B c → (10 ) 0 K (10 , or the corresponding decays involving a are allowed, but no meaningful experimental information P a ) = ˜ B O may be easily calculated using the same formulas for a  a 0 (100) TeV, but without an experimental probe, little more Γ( 1 ) a 0 ) K P O π η ( K 16 η of ( → → decays are of course welcome, they are not expected to be more π  s = P → PQ D 0 D → 0 v s Little is said in the literature about decays of charmed mesons of the P ). These are also given in table Br( ), an improvement of approximately one order of magnitude. The B are tabulated in table → or 13 D 0 2 P − a ˜ c P decays can probe the up-type quark Yukawa matrix. 0 , and → –10  a D P π 0 c 14 , we may expect an pairs, improving the limits on many rare decays [ − decays. K Ultimately the experimental data can be used to constrain the ratio N → B s 0 √ (10 , → B D ) (see table  0 O s decays, given below. The trivial requirement that Br( 5 , which depends only on hadronic physics, by 10 0 D B − B It is worth noting that the decay P 10 and → (10 × P ˜ are all 4 Axion-meson mixing In this section wethe discuss impact the on mixing the between meson axions oscillation and probabilities. neutral Such hadronic a mesons, mixing and effect can also lead to The values of ˜ ing ratio and the correspondingno bound experimental on constraints, however we can compute the numerical coefficients ˜ c i.e. sensitive to flavoured axionsscenarios than only other decays. OnBounds. the other hand,for in a flavoured given axion decay. Collecting terms in eq. ( and place weak bounds on can be said. As webe will rather show small below, the when predictedratios compared approximately branching to three ratios and are one anyway order expectedexperimental of to probes magnitude greater. of In conclusion, while further is available. D form branching ratio for not been analysed explicitlyfrom by searches experiments. for the However, some SMGenerically, process information any bound may on be the gleaned SMon decay the will two-body translate decay into to aform an bound axion. as strong Finally, (or we stronger) remark on the fact that also decays of the experimental reach by theO stated limits on theupgraded decays experiment Belle-II at5 SuperKEKB is expected toscales collected as approximately Belle, have not yet provided limits on this exact process. However we may estimate their JHEP08(2018)117 | | | | | | | | | | | d d d d d d d u u u d 21 21 21 31 32 31 32 21 21 21 31 (4.1) V V V V V V V V V V V | | | | | | | | | | | GeV , which 8 8 8 8 5 5 5 5 / ¯ 11 12 10 ν ν 10 10 10 10 10 10 10 10 0 PQ 10 10 10 . Relevant v × × × × × × × × P × × × 0 2 3 1 3 1 1 0 ...... 9 9 7 → . . . 1 1 2 1 3 2 2 6 2.2 6 5 2 P > > > > > > > & > > > ,   0 s 5 11 11 13 13 13 13 14 13 13 14 P γ − − − − − − − − − − → µ P 10 10 10 10 10 10 10 10 10 10 c ¯ dγ × × × × × × × × × × ∗ sd c 51 67 30 26 92 74 11 33 38 64 ...... font marks the current best limit 4 + d 5 ] 3 ] ] ] ] ] 3 ] ] 5 ] 7 ] ] 6 ] ] ] ] ] γ µ 68 62 66 74 76 59 58 71 71 73 71 77 75 75 75 75 Bold ¯ sγ sd c E787 [ + CLEO [ CLEO [ CLEO [ KOTO [ q 5 – 8 – γ µ ¯ E949 + E787 [ E949 + E787 [ qγ q c ) BaBar [ ) Belle [ ) BaBar [ ) Belle [ ) Belle [ ) Belle [ ) E391a [ u,d,s 8 5 5 5 * NA62 (future) [ 1 1 1 4 1 4 1 3 ) mark the expected reach of current or planned experiments. X 10 10 8 4 4 5 5 5 5 )* Belle-II (future) [ 10 = ∗ − − − − 10 6 q − − − − − − − − − − − −   10 10 10 10 10 10 10 10 10 10 10 10 10 10 a 10 10 × × × × × × × × × × × × × µ PQ × 5 9 9 3 × × 2 6 0 4 3 9 9 3 ∂ . . . v ...... , while parentheses mark the bound on the rare decay 59 5 4 4 5 1 . 2 1 1 1 1 . 0 1 a 73 01 0 . . 0 1 < < < < < < < < 0 0 P ( ( ⊃ − < < ( < → 0 ∂ from flavour-violating meson decays. L P PQ v )( )( )( ¯ )( )( ν a < ¯ ν ¯ ν ) ¯ ¯ ν ν a < ν a < problem, it may be relevant for non-standard axions such as ALPs. Readers who ν a < a < a < ν a < S a < ν a ν a <   0 0  0 ( + 0 0   0 0  0 . Branching ratios (upper limits) and corresponding bounds (lower limits) on the PQ- π π π π π K K π π K K π π K K CP Decay Branching ratio Experiment ˜ → → → → → → → → → → → → → → → 0 0 0 0 0 0 s 0 0 L L +       s B B B B B D K K B B B B D D K ( ( ( ( ( Axion-quark couplings in theto mass-diagonal meson basis mixing were are discussed the in terms section strong are not interested in ALPsbounds may on skip PQ this axions. section, since it will not4.1 lead to any Parametrisation competitive of mixing from searches for should be comparable. Asterisks ( new contributions to bothAlthough meson the decays mixing into effect axions will and turn axion out decays into to two be photons. negligible for PQ axions which solve the Table 2 breaking scale JHEP08(2018)117 , , c 8 d c − ]. (4.4) (4.2) (4.3) − 10 81 2, and u × / c s 8 β = . DW | 0 0 π of data [ N . We quote the c K 1 . As an example, η + | − A in addition to that d 22 axion , 0 ) A , and 2 P P 0 P GeV. Similar results for η = 6 m K . P containing a small admix- ), there arises in particular , ]. The error is dominated s . This kinetic mixing can 10 0 P a c 0 0 P 78 P 4.2 , there is almost no impact on m ∗ K × π +(∆ η c ,K 2, 0 − 2 P 0 mass difference, experimentally / 2 f P 2 PQ 1 d SM & P, 0 = f S v ) β µ 2 q  MeV [ P K | , η, η ) and ( sd PQ 0 a a∂ P c m + DW 12 v − f c µ π | − 4.1 N ∂ 0 L P ]; near-future lattice calculations aim to = = 10 = P K + 79 and PQ 0 = (∆ → f P v × K P P d 11 P c P m c A 2 . Belle-II is expected to improve the sensitivity m ) and its antiparticle m | s – 9 – 3 P 006) P = B .  η . This induces axion contributions to any process X ], with further improvements from next-generation 0 ,P a d , and − c s 2 80 P ' |  m c η , or = 2, P + B / 484 u axion eff aP . d , P a ) c β L vice versa (20%) [ D P − + O , m 1 DW = (3 u K N c q to (∆ exp 1) corresponds to the bound + K = = ) → . This is naturally generalised to include also mesons containing . 0 K m ≈ u 11 a 0 η P PQ 0 c A ( m π K , /v 0 c s = P c P f 2 u P c mixing are tabulated in table c − mixing by about one order of magnitude with the full 50 ab s d is the meson decay constant for 0 ≡ c B . These derivative couplings translate into effective axion-meson couplings D P P + 21 d f η is not larger than the experimental central value. We then have u − quarks. For a QCD axion with A and c 0 b P = B = D , m η sd plings of axions.lead In to this interesting consequences. subsection, we Asaxion-pion shown will in kinetic see eqs. mixing ( that as eventure a of flavour-diagonal consequence the couplings of nominal can axion thenormally and involving physical D of 4.3 Axion-pion mixingWe and have ALPs seen thatthereby axion-meson the kinetic mass mixing difference) can of affect neutral heavy the mesons, oscillation arising probability from (and off-diagonal quark cou- by the theory uncertainty, whichreduce may the be error large on [ ∆ machines. As a conservative estimate,∆ we shall only demandwhich the (assuming axion contribution to any The total mass difference is thenconsider given the by ∆ effect ofmeasured axion-kaon mixing to on be the (∆ Axions and ALPs withneutral off-diagonal quark meson couplings willfrom mediate mixing weak between interactions. a An heavy additional explicit contribution calculation, to showing mesonresult, how mass namely axion that splittings, interactions yield is an given in appendix and the standard meson dynamics.the However, the effect results may are be valid detectable. for generalised ALPs,4.2 where Meson mass splitting c be diagonalised by the transformations where c where where again JHEP08(2018)117 Φ, 2 Φ (4.6) (4.7) (4.8) (4.5) M T Φ 1 2 − a. | | | | 0 0 0 0 s   B B D 0 K 0 a. c c c 2 c π π GeV | | | 2 a | η 5 5 η / 6 6 , 2 a m π . − 10 10 θ 10 10 PQ ) m       − 1 0 v 0 0 × × × 2 × π 0 π 2 π cos η 2 π q η 1 8 4 2 . This is subsequently diago- η 0 are extracted from global averages m − 0 & & − & & π 1 − 0 2 π DW η π − P η 1 (1 θ f 2 a 0 /N ), the physical basis accounting also − 8 q π 0 10 12 12 m − sin 1 π − − − 0 4.1 c , where − 2 → π   10 MeV q π 0 10 10 10 ) η 0 0 / 0 θ 2 π ≡ 0 1, its contribution to the physical pion is − 2 × π π 2 η π × × × − η η 0 0 η exp 0 0 ) 1 π ) π  2 π 2 π − c 70 90 P η . . 0 1 + m − – 10 – 2   013) 006) π 0014) , π m . . 2 )) induces mass couplings of the form . 0 0 +2 − η 0 0 , q 2 a 0 0 0 2 (1 π 2 π π 0 π 0 η 4.3 η m 25 2 a   π 2 π m  . π , with ˜ 0 − π θ and m a m π 2       (6 θ 0 1 0 333 484 0 π 2 π /f − m . . = 2 π sin η 0 1688 q π 0 . m π f π ]. Meson decay constants (3 (3 2 π m η 0 − θ + (1 + sin π 78 m 1  = c π a θ 2 a Φ + π 0 q 0 0 s 0 = ˜ tan 2 θ a m M K D from contributions to neutral meson mass differences. Measured values B B cos , we have 0 − cos → − PQ − −   π PQ 0 0 η 0 0 s a v /v → → B B D K System (∆ π f ) and 0 2 rotation in terms of an angle a 0 0 π π × c a, π = ]. 0 82 are given in the PDG [ π . Limits on η P The axion-meson mixing effect discussed above can modify decays of heavy mesons to Kinetic diagonalisation (as in eq. ( m vanishingly small. However, this mixingthe may mass be and interesting decay for constant more are general not ALPs, necessarily where correlated. lighter mesons plus an axion,idea as is well very as simple: to the in decay the of standard an hadronic axion decay of to a two photons. heavy meson The into basic two pions, one of To leading order in For a QCD axion with Starting from the canonicalfor physical kinetic basis mixing in is eq. thus ( obtained by field transformations where nalised by a 2 where Φ = ( Table 3 of ∆ given in [ JHEP08(2018)117 . ), 3 a in m (4.9) 67%, γγ . 4.13 2 aγ (4.12) (4.13) (4.14) (4.15) (4.16) (4.11) g → π 1 a 64 )=20 0 ) and ( π ) = + 4.9 π ] γγ → 83 → . + . . . ) a followed by ) K 0 for the whole range of a γγ π + 1 GeV + π − → π .  0 1 → 110 MeV π → − and Br( , gives Γ( GeV + . , Γ( ˜ . + 2 F 1 63 eV K a 10 3 a . 1 − K f aγ GeV − 7  m aF g 2 a 0 TeV (4.10) Γ( a 10 10 ' π aγ 2 a 2 m −  g GeV m , perhaps the most interesting processes m 2 for C  1 4 < 2 a 2 m 5 a π − 10 = 3 colours, we have [ 0 2 a  f − N 2 a 0 m π 1 3 0 2 2 a C 2 π m ˜ c aγ − π 3 10 π m g . Considering only the mixing-induced effect, −    N 0 m m − listed in table m 2 a 0 0 π 2 a × 2 2 π 0 γγ ˜ c π π α 576 2 a 5 m 2 a 2 π 10 m – 11 – α GeV η m − 7 m ' → m m . − − 0  − 0 0 10 π a 0  π ) = 4 2 π η 2 π ˜ c 10 . ). mix mix ) ) ' & m ) m γγ × a ) and aγ a  aγ  ]. Combining the two expressions in eqs. ( + a 8 ) corresponds to 4.14 g f a . g → 7 ( π + ( 5 ' 85 + 0 π = 10–96 MeV) the bound 10 ) π 110 MeV. Similarly, one finds the axion decay to photons π → . 4.10 a + × → . γγ Γ( → m K 2 a mix + + ) → & m K K a aγ a ) to an ALP, still denoted by g Γ( f ( Γ( 4.8 = 5–100 MeV [ a m ]) place very strong bounds for 84 100 MeV, which translates to Applying eq. ( . a the range of the E787 result gives (for which is less stringent than eq. ( m Let us finally note that the E787 experiment searched for Therefore the bound in eq. ( Extensive studies of ALPsure over a 1 wide of range [ of parameter space (summarised in e.g. fig- The standard form of theWe axion-photon may then coupling, write the mixing-induced axion-photon coupling as In the SM with massless valence quarks and which is applicable for Taking the ballpark of Br( we find a mass-dependent bound above, leading to aneutral final pion state into consisting two photons of can an also axion. mediate theinduced Similarly, decay by the mixing on are standard an decay axionwe into of have two a photons. the neutral pions in the final state can convert into an axion via the mixing effect discussed JHEP08(2018)117 ; 1 × 1) ηe  5  ϑ 975, . . (5.4) (5.1) (5.2) (5.3) 5 = 0 − as the = 1, this e decays. > , µ λ ηz A τ i ) P ϑ | decays are θ > ∼ − )) e 21 a e = µ k + A θ P | · e is given to good η by µ → (or 2 2( ` | + 1 . For a more precise e ` − | e µ 21 η C µ | V . , | / 3 2 m / ˆz  ! ]( · , 2 1 ∗ PQ 2 ` 2 ` a ) η v follow analogously to meson , for the three possible lepton e m m 21 m = a GeV . V 2 PQ PQ ( 2 − ` v decays, which are not generally v

ηz 12 ). Assuming 2 e 21 1 and ϑ ` 3 10 1 → A

ea e π e ` in a muon ensemble, but as we shall 2  1 cos A m `

2 η →

' 2 (192 , for which the SM background consists + ` = 0) and maximally anisotropic ( 1 / µ a µ 1 3 ` 2 e ` GeV) 5 µ . The muon decay width Γ

P + A ¯ 2 ν m C e )) + 2Re[ m `

12 ν e 1 . e 2 F ` 2 k + ` V – 12 – 4 → · ], corresponding to ) decaying into a positron with helicity e G , i.e. (10

→ η ] has performed a broader search, accommodating η ˆz 1 ) 86 + , ' ( ` [ → 1 = e c µ 87 ` ) ), the branching ratio may once again be written in 2 6 λ + ¯ ν

2 − Γ( 1 2 µ ν = (0 3.3 ` ) = ˜ 1 π + − 10 e ` a η e 2 µ )–( 16 C × `

m → 6 ( 3.2 = . decay, → . We can describe the degree of muon polarisation , by 2 2 + 2 | β 2 . 1 ` µ ` ηe ` e 21 < 4 → ϑ → Γ( 1 C ) 1 ` as well as massive bosons, but are less sensitive for isotropic decays | ` a Br( ˜ c h c ' cos + 1 A | µ e e  , as well as an axion. Neglecting k = e X e → k λ onto the beam direction −| + 3 µ Two-body lepton decays of the form η with a polarisation µ = 2 PQ m v e + . k µ a is the positron emission angle; in this region SM three-body decays are strongly = · 2 2 ` η θ Let us sketch the angular dependence of The most interesting of these is GeV. They searched specifically for decays with an angular acceptance cos → |M| 9 1 where projection of treatment one should considerassume the all distribution muons of areis highly sufficient polarised for opposite our purposes. to the TWIST measures beam the direction, positron i.e. emission angle Consider and momentum non-zero anisotropy in the massless limit. Thedecays limits are for given isotropic in ( table isotropic, as these would relate to TWIST; the formulas generalise immediately to isotropic, i.e. the decay isthe purely limit vectorial Br( (or axial),10 the experiment at TRIUMF provides where suppressed. The TWIST experiment [ These are evaluated, with correspondingdecays. limits placed The on results are tabulated in table almost entirely of ordinary approximation by Γ where As done for mesonsterms in of eqs. a ( coefficient ˜ ` decays, with the notablethe difference decaying that particle both has axial non-zero and spin. vector We couplings define contribute, a since total coupling 5 Lepton decays JHEP08(2018)117 ] , A | | | | | | | 54 88 e =0 e e e e e e 32 31 21 21 21 21 21 − − from a (5.6) (5.7) (5.5) V C C C C C C ). | GeV | | | | | | 10 9 m V / 6 6 9 9 9 11 PQ 5.6 = ∼ v GeV [ 10 10 10 10 10 10 PQ 2 10 ) v ` × 10 × × × × × × 5 m eγ 4 8 2 8 9 , . 10 . . . . . 1 (discussed be- 5 ) 1 1 1 2 1 & and the photon → & , the differential > xy > > > > > 2 e µ , which, if experi- ` eee 1) PQ λ − γ v 2 2 − y ` 14 14 11 ` → − − − y → , − 1 ) µ = 1, particularly in the ` → 10 10 10 θ + c interaction). The signal 1 A )(2 x × × × in the limit of between ` ( x A cos 2 , i.e. (twice) the fraction of γ 87 92 82 1 2 µ y aγ . . . − ` + θ 2 ` (1 AP V /m ] ] 7 ] ] ] ] 4 ] 4 → γ . − ] 1 . Summing over 89 89 87 86 88 87 87 E ` ∗ θ ) = ) (1 GeV implies the signal is very small e 21 3 µ 1 (corresponding to an SM-like 12 cos = 2 2 x, y 2 PQ V | = 1 (a ( m ( − v µ can be related to the formula in eq. ( y 10 e 21 2 P e 21 | A , = C π A A | 1 ARGUS [ ARGUS [ ∼ e 21 ' ` 32 A C 1) TWIST [ | – 13 – , f ηe /m ) PQ − 2Re[ ϑ = 1) TWIST [ = 0) TWIST [ 2 v ' ` , and tentatively probe scales of = − A A E x, y µ , giving ( ( ea 2 ( A 1. Moreover, the angle = e ( 21 f = 0) Jodidio et al. [ , giving V πm 1 ≥ → A = 2 3 ` A e |M| 21 − ( y µ 32 x m V + 2 2 PQ =

= 2 2 5 5 5 6 v * Mu3e (future) [ = x 2 9 2 − − − − − − ` θ 1 e 21 − π e ` e 21 10 10 10 10 10 10 A 10 C 32 dΓ A

× × × × × × × 1 and d cos 6 5 8 0 1 6 α ...... 5 1. The A to Z model, discussed below, predicts exactly this scenario, 2 1 5 1 2 2 ≤ = . < < < ∼ y Additionally, we may examine decays with an associated photon, i.e. Γ θ d x, y 2 x d . ) is a function of d a < a < a < aγ . These can be studied in experiments searching for + 2 + + x, y ( ` µ e e . Branching ratios (upper limits) and corresponding bounds (lower limits) on aγ f 2 ` → → → Finally, the Mu3e experiment, primarily designed to look for Decay Branching ratio Experiment ˜ → + + + → τ τ µ 1 1 where invariant mass carried away by thevation lighter requires lepton and photon, respectively. Energy conser- ` mentally measured, are unequivocal signscertainly of unobservable. new The physics; differential decay in ratemay the for be SM, expressed Br( by despite the enhancement. low), can also be used toby test for the end of its run. ` The limiting cases are current interaction), or strength with respect toregion the with SM cos background isalthough maximised the for high predicted PQ scale where we define the anisotropy for highly polarised muons,decay we rate have is cos given by Table 4 two-body cLFV decays. The assumed anisotropy JHEP08(2018)117 , ] ]. f  we 86 R 96 efγ 6 (5.8) (5.9) (5.10) → µ 14(stat) annihilation- . 0 . − e 2  + we summarise x ) e θ . 5 03 , i.e. c γ . γ 2 ) + ] ] ] ] 2 . − ` θ θ γ c 93 92 95 95 2 → ` )(1 − . cos ], in particular as they 1 ) = (6 ) x ` → (1 90 ≡ ¯ . In table νγ − x | 1 x, y θ GeV. We see that the limit ` MEG [ . ( e c eν 21 8 BaBar [ BaBar [ (1 − ◦ C 1 yf | 10 → . 8 140 d and ) µ × x > decays [ 10 y ) = x d 2 θ . eγ γ × − 1 θ 2 Z 4 ` x . x, c & xy π 9 ( α ]. Unlike the TRIUMF experiment [ − | 2 40 MeV, in agreement with the SM [ → > e 8 8 µe * MEG-II (future) [ 94 13 = 1 − − > C 2(1 14 38 MeV, ` − | ) − / ) 10 10 γ > , can give information on decays to axions. The , f 10 a GeV ) – 14 – aγ E e 10 2 × × 0). In practice, appropriate cuts are made on the PQ / θ 2 ` ¯ × νγ v ` 3 4 ,E × ], would be welcome. An explicit limit on . . = 1 + ' 2 γ PQ . eν 3 4 x, c → 6 ] experiments. The region of interest for MEG is for v γ γ E 4 → 93 ( 2 2 1 f . 1 92 θ θ → ` ` 1 at 90% CL [ 3 ` µ , where the SM background disappears. However, decays 9 cos Br( m Br( π − 2 2 PQ

v 45 MeV and γ < = 2 10 γ < γ < ' 2 ` − 1 2 + − are under control. A comprehensive experimental study of such π θ e ` ` > × e e µ c 1 5 , yields a limit ` C 32 1 e . Experimental upper limits on cLFV decays

8 . R → → → ] and MEG [ 1 E − α Decay Branching ratio Experiment decay itself, + − − 91 10 < = τ τ µ β for × ) θ 8 c 1 Table 5 Γ − d 2 efγ . is stronger by approximately a factor 40. x d 10 011, yielding the bound . ) d 0 → × ea is a light scalar or pseudoscalar, is given by the Crystal Box experiment, which ' eaγ µ → 1, or equivalently f f → µ 1) and a collinear divergence ( R ' Also radiative The radiative decay possesses two divergences: an IR divergence due to soft photons The primary sources of background are radiative muon decay (RMD)The and accidental Crystal Box analysis uses the cuts µ ' 5 6 53(sys)) . x Requiring the axion decayBr( to not significantly exceedfrom the error on this measurement, i.e. in-flight (AIF). For large photon energies and increased stopped muon rate, AIF dominates over RMD. find current and future experimental limits on branching ratios of most precise measurement comes from0 MEG, giving Br( the induced backgrounds signals, e.g. for the MEG-IIwhere upgrade [ sets Br( discussed above, this limit does not assume isotropic decays. Using the same cuts Such cuts were discussedrelated to in LAMPF the [ contextx, y of with an associated flavouredvanishes axion for very are soft axions. also suppressed One might in consider this a broader limit, region of i.e. phase the space, integral provided is fixed by kinematics to ( minimum photon energy and angular acceptance well away from the IR-divergent region. Alternatively one can write the decay rate in terms of We may relate the branching ratios of decays with and without a radiated photon by JHEP08(2018)117 ]. 17 → − The + 100 10 (5.12) (5.11) µ 7 ]. × 99 , 6 2 | < conversion in aN axion [ µe C . e ) , 4 R ] currently under − conversion is only  ! A,Z 98 µ e ( N . To lowest order in 4 15 S | − 16 and the axion-nucleon 2 GeV | − − µ PQ e 21 v e 21 2 e 12 nucleophobic 2 µ 10 , although the presence of C C m | m 10 × eee 2 N is model-dependent, given in ] for standard formulae), but  1 ln 2 m ], based on the rather successful |

4 53 PQ . → means these processes are again suggests 2 µ v aN e 2 21 ) 38 | µ m C C e | 4 PQ 21 PQ is the total nucleon spin of a nucleus ) 2 eee v | C 3 cap ) | /v ) 4 PQ − 2 e 11 ) is qualitatively given by A,Z | → v ( µ A,Z A bound. The upcoming experiments Mu2e Γ | ( e 11 N αZ µ ( 2 S A e 41 Al, and both aim to probe A, Z | π ]. The Mu3e experiment [ 3 − − ) (1) couplings and form factors, SINDRUM-II 2 e ) µ 97 10 3 µ → O 2 a – 15 – Γ( → m × µ m 5 µ 3 2 e π m − We may also consider processes without an axion in Al m 43 . 2 − 16 1 q improvement over the SINDRUM result. µ ( ≈ ≈ 4 ) ∼ + . The numerical factor e )) (1). The suppression by − PQ e O + /v A, Z e family symmetry. The family symmetry is completely broken by ( ) N conversion. , set by SINDRUM [ ]. Assuming again − 5 cap conversion is a spin-dependent process which was discussed in [ m → e 12 − Z ], a factor 10 e A,Z e ( µ − + 101 aN [ Γ × → − − µ C 10 ] and are GeV. 4 ], which seeks to resolve the flavour puzzle by way of Pati-Salam unification 103 µ − , is the momentum-transfer and µ 13 6 GeV, comparable to the × A µ Br( = − 39 2 µ 6 0 100 10 . (1) couplings, we see that such decays are only reachable by experiment pro- 102 Γ( 10 m 1 10 . O aN and ≡ × g ≈ < & ) 7 2 ) PQ q + v eee PQ ). Not accounted for here are nuclear spin and structure form factors, which were e < A,Z v ( µe As the axion (or ALP) also couples to quarks, one may consider − It is even possible to suppress the nucleon couplings entirely, yielding a , the branching ratio for the axion-mediated decay is given by 7 e R → 2 e Au µe + A, Z We present here a recently proposedA to QCD Z axion model model [ [ coupled to an sets and COMET are bothat looking 90% for CL [ 6 A to Z Pati-Salam model where ( discussed in [ realistically detectable in ALP scenarios.R The current best limit comes from SINDRUM-II: axion-mediated The conversion-to-capture ratio in a nucleus ( nuclei, mediated by the axion.coupling The relevant couplings areterms now of flavour-diagonal couplingsare of essentially the given up byin and the more down quark general quarks. PQ scenarios In such charges as standard (see a cases e.g. flavoured these [ axion, these can deviate significantly. Assuming vided sensitivity by four ordersm of magnitude, i.e. Br( the final state. Axiontwo mediation axion will vertices induce and theonly additional decay interesting suppression for by ALPs. 1 e The current upperdevelopment bound is on expected the to branching start ratio taking is data Br( in 2019, and will significantly improve the µ JHEP08(2018)117 . is D x (6.1) (6.2) prob- ], and 39 CP (1) ratio of . O are correlated    (1) factor 0 b ], and assuming 0 s 0 y O y , 39 0 s is an 0    xy and 1 meV. A fit of these B 0 b 3) 0 0 we provide the best fit . y < 0 0 d d / 0 0 0 0 0 0 0 0 1 0 s , 0 d (1), we may infer generic D y 0 d    y By By ( y O iξ  − e c we derive explicitly the axion- 0 b    y m C = + e correspond closely to the three up-  and couple to left-handed SM fields. are phases.    c 0 d 4 ξ y , A ,Y η 1. Meanwhile and    1 4 2 4 16 8 2 8 4 b 0 b ∼ 0 y ,    c ], with central results collected in appendix a 0 s 0 0 0 iη – 16 – y e 38  b 0 0 d d b 0 d m y By By  + ].    a    = 104 d 0 0 0 0 1 1 0 1 1    ,Y a    , which are triplets under m 1 are small perturbations of a flavon VEV. The c c φ = 13 23  ν b  b c 3 b  m i 4 2 , predicting the lightest neutrino with a mass of  c 0 a a m     are real, with dimensions of mass and ], we updated and improved the numerical fit to flavour data, as well as demon- = b i u 38 m m Y Note that the scales of the various free parameters are constrained by the model itself. In [  a a Clebsch-Gordan factor, introducedGeorgi-Jarlskog by mechanism. additional In Higgs the multipletson neutrino in sector, which a the the variation principle modelm of of relies sequential the dominance demands aparameters to normal data ordering has and been strong performed [ mass hierarchy, with By rather simple assumptions aboutall the flavon dimensionless VEVs, discussed couplings fully inproperties in [ the of renormalisable the theory parameters.type are quark Parameters Yukawa couplings, i.e. with the down-type quarkcouplings, Yukawa couplings, and i.e. where All parameters are dimensionlessately and made in real general by complex, anthe overall although light rephasing three Majorana of can the neutrinos be three (after immedi- Yukawa seesaw) matrices. is The mass matrix of The charged fermion Yukawa matrices are given at the GUT scale by mass matrices of the SMmatter fermions. couplings from However in the appendix parameters Yukawa superpotential. for the In A appendix to Z model and6.1 corresponding axion couplings. Mass matrices and parameters strating that, with smalllem. adjustments, the The A axion to thencouplings; Z emerges model in from can other the resolve words,Moreover, same the no as strong flavons additional that all field are Yukawaare content couplings responsible is known are for necessary exactly, fixed SM to with Yukawa byaxion realise no the couplings a additional fit to PQ matter, free to axion. we data, parameters. limit also our As the discussion the axion primarily focus to couplings the of resultant this Yukawa and work is on However, information about thestructure underlying of symmetry the remains flavons.structures in The based initial the on viability the particular of so-called vacuum CSD(4)leptogenesis the vacuum was model, alignment, considered was which demonstrated in predicts in [ certain [ Yukawa gauge singlet flavons JHEP08(2018)117 , ae g and (6.3) f V and the , PQ v    i , we calculate P 015 . 4 0 7 − − − ) is proportional to e u 2 5 10 10 φ − is fixed by × × ). Recalling that GeV. 10 0 . 3 aγ . 6 12 × g 2.3 7 − 7 10 . 1 − i ∼ i u 2 we give the model predictions for 05 φ . , or the axion-electron coupling 4 6 0 − , , aγ − using eqs. ( ' h e g 10 5       3 f . 5 5 − 0 × PQ A − − by an MCMC analysis. Bayesian credible v − 0 10 . 10 10 8 6 0065 0057 0085 0013 × ⇒ . . . . and × × − 0 0 0 0 – 17 – 3 in this model. Here, . may be further probed by increased sensitivity in f 5 5 . . − − − − ` V GUT i δ PQ i v 015 /M 05 . . 0 i and 51 25 72 0 0013 7 0057 7 073 is determined primarily by the largest VEV among the e . . . . . u 2 e GeV, all processes involving two axion vertices, includ- . 5 , are heavily suppressed. For all mesons 0 0 3 φ ` 23 − 12 − 0 4 θ − − . PQ eee 1 10 v ∼ h 10 10 = 6. In other words, although the charge assignments are very b × × → ∼ 7825 0 0 99 0 0085 7 0065 073 0 . . . 3 . . . , which in turn is dominantly responsible for the charm quark . . µ 0 0 0 DW 0 1 0 u 0 4 , which is only loosely constrained by naturalness arguments to be PQ N − − − Y v PQ          v in ' ' ' . The details of how the flavons and parameters are related are given in b e d u 3 A − A A − , showing that − GeV when computing the branching ratio. GeV. In principle, any two measurements of either flavour violation (as dis- − 10 carrying PQ charge. The VEV of this flavon (named ] they perform the running of low-scale experimental results (from global fits) up to the GUT = C 12 12 = = × φ e d u 4 10 105 . , such as haloscopes and helioscopes. In table V V V = 10 ∼ Predictably, as We may immediately compute the branching ratios for all aforementioned meson The PQ-breaking scale aγ are Hermitian, we have In [ g = 3 8 f | PQ PQ b ing meson mixing and scale, assuming the MSSM; theyand provide CKM GUT-scale mixing values for parameters. quark and charged lepton Yukawa couplings, would be sufficient todomain overconstrain wall number different, the A to Zto model will resemble the originalsome DFSZ of model the in experiments mostv sensitive phenomenologically interesting experimental probes. We explicitly set and lepton decaysis and the neutral axion meson scale v mass splittings.cussed The in only this paper), remaining the parameter axion-photon coupling 6.2 Predictions Once the fermion mixingthe matrices vector are and known axial from couplingA the matrices fit, we can immediately determine the parameter Yukawa coupling; as thethis third is generation the heaviest largely relevant| does fermion in not the couple flavouredappendix to axion theory. the The PQ numerical symmetry, fit gives intervals are also provided, showingtensions that in despite the a predictions for largecurrent number and of future free neutrino parameters, experiments. small flavons The model is fitted to experimental results JHEP08(2018)117 1; , is (6.4) PQ x > v ), essen- 45 ; detecting − + µ (10 O 5. Now consider , typically . e and 4 Y . 0 L x ∼ ∼ K √ s (KOTO) 8 d . 1 8 (1) factor. Y /m − − O µ e (Mu3e future) 10 (NA62 future) m 31 9 × 12 − is given by ≈ 5 − 10 2 2 | | < 10 × e d µ/K 21 21 is well approximated empirically by × 38. Should both of these decays be 5 V V . R the branching ratio is | | Experimental sensitivity 2 1 0 | . d . 21 45 ≈ . , which is determined by the fit and acts eee V | 4 x / . With all other parameters held constant, 2 → , which are the most experimentally promis- , up to diagonal Clebsch-Gordan factors. No- 2 ' | 13 12 12 | e µ/K a e µ ) − − − e 21 ) – 18 – 21 + Y ; the NA62 experiment is expected to be able to R a V a e V a GeV) | 10 10 10 | + to be slightly lower than the natural prediction. + ∼ + 1; at the GUT scale, π e 12 × × × π d 88, → = . > 3 PQ = 2 Y . → → r 29 19 + v → s 8 . . 2 µ 1. Similar ratios can be considered for other decays of + = 10 2 2 | + = 5 + µ e /m 21 K < K x µ Branching ratio PQ C GeV, which should be true up to an | and v Br( m ( Br( MeV, while for 12 . a µ/K ∼ + ≡ 24 PQ and R a π a a − x v = 10 0 + of the ratio 2 + | π π e µ/K → d x –10 21 PQ R v V → → → + 23 | − 0 L + + K Process . µ K 10 K x √ ∼ 8 . 1 − mesons and charged leptons. However, as this requires direct observation of both . Predictions for axion-induced processes in the A to Z model. Branching ratios are e axion ) 9 B . P We then find that the ratio of branching ratios In summary, we find that evidence for or against the A to Z model must come primarily 6 or m ≈ information about the high-scale parameters.generically For one models expects where K decays, which are suppressed ingeneral both ALPs. sectors, these are realistically feasible only for more For the model bestmeasured fit experimentally, such point aa ratio, valuable which statistic is for independent constraining of the the flavour axion sector scale of the model, giving immediate ing among flavoured axionby decays. the couplings Theirthe branching dependence ratios are on determined,r respectively, and leptons. In thistably, A the to (2,2) Z model, entriesas differ a by necessary a Clebsch-GordanNaturally, parameter factor one to expects distinguish thethe strange two quark decays and muon masses. However, two-body decays maysometimes be placing powerful stronger channels constraints for thanthe excluding those strongest other limits from flavour on astrophysics, models, which typically give 6.3 Decay correlations The prominent feature of unified models is correlations between Yukawa couplings of quarks from the (non-)observation of exclude most of the model’sthe parameter model space. definitively. A next-generation Secondarythe experiment sources could A of exclude to interest are Z decays model of would require Table 6 computed assuming (∆ tially undetectable. JHEP08(2018)117 and its problem, 0 s B ), as well as CP γ ( a and 2 ` 0 B → , 1 0 ` D , 0 K is an ideal channel for looking at these πa involving a flavourful axion. We have also → B – 19 – K , and ], and shown how unification leads to correlations be- D 38 , K problem, and all axion couplings are fixed by a fit to quark and lepton conversion. e CP − µ and GeV or less. eee 6 problem but which may generically arise from spontaneously broken symmetries and In conclusion, flavourful axions can appear naturally in realistic models and have a Correlations between observables may arise in specific flavourful axion models. To il- → Acknowledgments The authors thank Joergand Jaeckel, Andreas Robert J¨uttner,Enrico Ziegler Nardi, for GiovanniConsolidated Marco helpful Grant ST/L000296/1 Pruna discussions and and the comments. European Union’s SFK Horizon acknowledges 2020 Research the and STFC quark Yukawa structures whichQCD determine axions their some masses offlavourful ALPs the and many flavour-violating of mass processes them ratios. mayis we be 10 consider important, especially Although are if for not the symmetry-breaking competitive, scale for vant processes involving flavourful axions,charged including lepton meson flavour-violating decays processes.such and processes mixing, For a as are well very QCDtypes as weak. axion, of typically However, decays, the especially boundsexactly in this specific from type models of flavour-violating such couplingviolating as is the processes the largest. for A both By to comparing quarks Z multiple and flavour- Pati-Salam leptons, model, one where may experimentally probe lepton and solve the strong masses and mixing. rich phenomenology beyond thatwe of have the standard attempted KSVZ/DFSZ to paradigms. provide In the this first paper comprehensive discussion of a number of rele- lustrate this, we have describedpredicts the phenomenology a of flavourful the QCD A axiontween to different [ Z flavour-dependent Pati-Salam observables, model, as which couplings the down-type are quark very and similar. chargedfields Within lepton this that model, dictate since fermion the Yukawa axion structures, arises no from additional the field same flavon content is necessary to contributions to neutral meson massinto splitting, two photons meson which decays may intoflavour-violating be axions processes and relevant involving for axion final ALPs. decays stateµ We axions, have of also the discussed form charged lepton couplings both for the standardare axion non-diagonal. and Using for these ALPs,body couplings, and decays we shown have of that calculated they heavycalculated the quite mesons the branching naturally ratios mixing for betweenconsequences, two- axions which has and not hadronic been discussed mesons in the literature before. These can lead to new In this paperincluding we both standard have PQ reviewed axions, associatedand and with also the extended for solution to the non-standard theCP axion-like strong phenomenology particles of (ALPs) which flavourful domultiple axions, not scalar care fields. about the We strong have presented the flavourful axion-fermion and axion-photon 7 Conclusion JHEP08(2018)117 , 0 D (A.7) (A.4) (A.5) (A.6) (A.1) (A.2) (A.3) − 0 , 2 D , we have P 2 , , µ 0 1 P K . a∂ ) , violation, a. µ − 1 0 2 ∂ 0 η P K ∗ P µ K CP η 2 K − 2 ε η − 1 a ∂ √ + µ P . . p . η ) ∂ ) 2 ) (odd) by in terms of . 0 ∗ ∗ P P 2 2 − + K 2 η η | P P ( 1 2 P 0 kin P 2 P − − | P η − − L µ | 0 K 1 0 P ε η → P P | P a∂ ( ( µ ( = 2 1 2 µ 2 2 2 i ∂ 1 ∗ P 1 √ a ∂ 1 + √ η ∗ √ P µ η , where P − 2 p ∂ = (even) and η = a, P 0 m + P in the kinetic mixing term. This can be √ = 0 2 1 2 = η L P i η characterising indirect 2 P P η = 2 L − + 3 1 − 2 2 − 0 η − η 2 1 0 kin P P 10 + – 20 – µ ,P , . L p µ , η ) ) are close (but not exactly equal) to the physical 0 ∂ 2 1 ∂ ) 2 0 ∼ η ,K 0 + 2 P 2 ∗ P , P ) P 0 1 1 K η P P 2 ≡ ε µ µ + P P eigenstates + K K 2 + ∂ ∂ 2 P 1 0 η K . 1 2 1 2 P P → m 2 P ε ( 2 η ( CP 1 − ( + 2 P , defined by − 1 2 + 1 2 2 diagonal kinetic and mass terms; these can be made canonical 1 2 P a , √ 2 P 1 1 √ 1 µ 2 √ a µ m η K ,P 2 a P = ∂ ( = 2 + a ∂ = 1 0 2 m 2 η 1 µ | 1 P 1 , so defined by having definite lifetimes in weak decays. They are 1 2 P ∂ P P µ η K L − a 1 2 ε ∂ − 2 P | 1 K , which introduces a factor 1 2 1 2 m = = p + − iP 1 + 0 m a 2 and 0 kin µ L → a are strong eigenstates. The superscript 0 signifies we are not in a diagonal p L → 2 a S a 0 a∂ 2 = m K µ P P 1 2 ∂ S , 1 2 ) is described by the Lagrangian 0 − K 0 P = = B We diagonalise the kinetic Lagrangian by transformations 0 m − 0 kin L 0 L We also define a “total” coupling Note the wrong sign of the by letting absorbed in new couplings We will neglect such a contribution in this work. Rewriting eigenstates given in terms of a small parameter Inversely, In the case of the kaon, the states where (physical) basis. We define the B 674896 and InvisiblesPlus RISE No.No. 690575. 690575. EJC is FB supported is also supported by in InvisiblesPlus part RISE by theA INFN “Iniziativa Specifica” TAsP-LNF. Axion-meson mixing Kinetic mixing between the axion and neutral mesons (any of the pairs Innovation programme under Marielodowska-Curie grant Sk agreements Elusives ITN No. JHEP08(2018)117 (B.4) (B.1) (B.2) (B.3) (A.8) (A.9) (A.11) (A.10) ) is .. 2.2 c . +h # . ) 2 P P m 2 ) and 2 η | 2 . P + ,P η . 1         1 i 2 P 1 η ' | P . , η | a, P ( 2 µ 2 P 0  − ) η 1 2 P 0 . 1 q η P a − µ p p − (0) 2 P γ 2 1 | + + P 2 m ), the amplitude may be written f 0 P ¯ q , m η ) p P q η ) | 0 0 | 0 p 2 1 − 2 P + , 10 0 1 P η η P 2 q µ 1 h m a γ (0)( µ µ p 1 + 2 ) −  = (¯ + 0 0, wherein the form factor is defined by the q Φ, where Φ = ( eV. defined by the Lagrangian in eq. ( (1 + 2 f 0 2 2 P 2 2 P p Φ µ η q 2 P 2 p η P  1 → f 2 q = m PQ 2 P 2 2 2 M q ( µ η 2 v P η − η η – 21 – − i V T 0 m q γ i 1 P − P m P Φ 1 2 1 − − | q + q 1 1 2 η η p 0 − PQ 2 a P )¯ 1 1 ( f + ) into q v 0 , m P 0 − a | ' ) m V q q P p p 2 a µ 2 P 2 2 i q µ ) encapsulating hadronic physics. The lightness of the P ∂ = P  η PQ 2 P m γ q m f q v q = m m P ( − V )+         ¯ q L i 2 + = (¯ 2 | − 0 2 P M (1 f − P 1 2 a P P P m h by + = , we conclude that m is the momentum transfer to the axion. For a two-body decay 2 m 1 2 = η | P 2 2 P , corresponding to the physical squared masses, are given to good M Φ ( p η 2 Φ 2 P | | ≡ | M − P M m differ by approximately 3 1 " = 2 m p L 1 2 2 ∆ K | η = − a = p and m of a heavy meson = S a 0 q K → L P We have not taken into account a mass difference from SM physics, such as for kaons, 0 m → L relation such that It depends on aaxion form means factor we can safely take the limit where P where B Heavy meson decay branchingThe Feynman ratio rule for the vertex ( Recalling that The eigenvalues of approximation for small In matrix form, we may write The mixing is transferred to the mass matrix, giving JHEP08(2018)117 (B.5) (B.6) (B.7) (C.1) (C.2) (C.3) triplet ). 4 , symmetry A 0 GUT 2 P , consists of a M P m c 1 ( CP i m − F O ϕ d 2 2 P h h i . ) d m 2 u 1 c 2 , | ) Σ 0 φ h ϕ F P · u x (0) m i h ϕ F + ) ≈ u v (  f u 2 | a Σ φ , . h ud 3 m · ( λ i ≡  2) 0) 2 0 , , ϕ F / + 2 P h 4 1 2 P ( 1 , , , c  u 2 m m ) 2 F 2 λ dΩ is ) , we let − | d 15 a , 0 = (1 = (0 φ i + given by , arriving at 1 h 2 P P ϕ d P ) m u 1 u 1 | π c  1 d 2 m p ϕ ϕ /v Σ a | − h F φ ϕ (1) and assumed real by a x x 3 P 0 p 2 u | · ia P O i m h e singlet. ) u F = 2 i m

( u 1 ) ( | L 0 |M| Σ 0 d φ ϕ h P 2 − – 22 – 2 P · q ρ , , 2 PQ λ π 2 P p P 1 f v P | q F These are the familiar SM fermions as well as a set + 1) 0) ( + m 32 V , , i m

9 u c 1 0 )( 2 1 ϕ 1 , , 2 π h F λ ) ( 1 d a 16 i 2 h + dΓ = space. The VEVs are aligned according to the CSD(4) 1 ) m = (0 = (1 d 15 c 3 √ d 1 4 + Σ u 1 u 1 F φ 0 ) = h The fields Σ acquire high-scale VEVs which give dynamical ) ϕ ϕ · A = a P 3 0 x x i h F m P ϕ in 10 ( · , which are naturally ( , respectively. d λ c 1 − F x i → → ( λ f 2 P i 3 The central actors in the flavoured axion model are the are Weyl fermions, by definition transforming as left-handed fields. In other words, , P The effective Yukawa superpotential below the GUT scale, once mes- φ λ + m c f i ( Γ(  = messengers in the renormalisable theory, expected to be f, f = eff X Y | a W p | have been integrated out, is given by . Taking only the scalar part of superfields = and direction X | φ 0 are denoted v P c i The differential decay rate in the rest frame of p To be precise: This is rather imprecise but tolerable, as the Higgs sector is not relevant to the PQ mechanism, and | F 9 are the left-handed components of a weak SU(2) 10 , c i f fields are anyway replaced by their VEVs eventually. where we havescale expanded around theprescription, flavon i.e. VEVs, noting that each masses to the Goldstone field. flavons with explicit couplings at high scale. InF the corresponding Lagrangian, theof right-handed fermion neutrinos. part The of lightcorresponding the Higgs chiral superfield. scalar superfields doublets keep the same notation as their C Couplings in theSuperpotential. A to Z Pati-Salamsengers model Integrating over the solid angle Ω yields a factor 4 with the momentum of decay products JHEP08(2018)117 ), , i.e. (C.7) (C.8) (C.4) (C.5) (C.6) Σ ) and Q, L v ( L     a , d a d 1 a L a → u 1 ϕ d 1 u 1 ϕ u PQ ϕ PQ PQ ϕ PQ ix v f v v ix v ix ix − −     exp exp exp exp c 1 c 1 1 1 f .. f R d by R explicit: c d , . d h d h ) d a ) ) c ) i ∗ Σ i ∗ . i i i d 1 v + h u 1 2 d 1 ϕ in the charged lepton sector, 1 u ϕ v o f, f ϕ ϕ  ϕ λ 2 ϕ ·h ·h ·h a ud x ·h i → f u 1 ˜ f L λ d ). With all above considerations ( L ¯ ϕ ( , with some small mixing assumed d ¯ i ϕ d PQ i Σ ( d ( → X h v ix d v d d 15 C.6 d d also provides Clebsch-Gordan factors  d d Σ Σ − . This amounts to nothing more than v Σ further decompose into ( ud Σ ≡ 1 ud v h a v h d →  , v λ d 15 λ u 1 2 1 2 d 2 ,λ 2 ud a. L L,R ϕ PQ λ d 2 Σ PQ d 15 √ √ λ ν v √ 2 ϕ √ v exp ix v ˜ λ + + , h 1 PQ + +  ϕ d   v R and   x → h a a u i → a a , d ) d 2 u 2 , and , v exp d 2 u 2 – 23 – 2 Q = ∗ u d 15 ϕ ϕ PQ c ϕ ϕ PQ i 1 ϕ . v PQ v PQ ϕ Σ v u 1 ,λ f are not independent, but related by the single U(1) a ix L,R v ix h ix ix v PQ a d ud u e ϕ ϕ  1 ϕ → v −  ˜ λ and adjoint Σ h v − , ·h ˜ a λ )  u ,  ϕ d i It is also convenient to work in the left-right (LR) basis, L h u → x u 1 exp u L,R → exp Σ c (¯ 2 ϕ d c exp d The Yukawa Lagrangian may thus be written as 2 v exp ϕ f ud u 1 , f ·h 2 u 2 X λ Σ u v d 15 R f R ,λ v u ( h h u ≡ d L,R 2 1 ) R i → i ) ) ) i u e i λ ∗ u u a ∗ √ u 2 i d 2 i u Σ Σ u 2 , and d 2 ϕ 1 ϕ → h + h d .. ϕ λ 3 ϕ ·h 2 2 ·h R c . R ·h ˜ λ ·h f f are very heavy and phenomenologically uninteresting, so will be ne- √ f ( ,d ( L L ) i i ¯ L ϕ u d + ∗ → u d e (¯ ( ρ i c 3 )+ h d d u 3 u d Σ Σ f u ϕ d 2 2 2 h h h 3 Σ Σ → v v ρ λ λ λ 2 2 v v ( d u ·h L , 2 2 2 2 ¯ fh d ). Below the EWSB scale, √ √ f d O λ 3 λ c i ( 1 √ √ n 3 λ + + + ˜ λ , d λ + + + c i ⊃ ), respectively. In addition, u → = Y ( L d Y 1 , e → λ −L L c −L i ν taking the Hermitian conjugatetaken of into the account, the terms Lagrangian in becomes eq. ( which are different forquarks quarks and charged and leptons, leptons. weso reparametrise the To couplings account for the splitLagrangian between (left-right basis). down-type in terms of Weyl fermions between Higgs bi-doublets to giveare the negligible. MSSM The 2HDM; fields we Σ assume acquire the real VEVs, effects with of magnitudes this generically mixing written The interplay between the singlet Σ Let us makef the SM( field components of the PS fields Component fields are given by Lagrangian (SUSY basis). glected henceforth. The phase fields rephasing symmetry. We identify the Goldstone (or axion) field The radial fields JHEP08(2018)117 , 3 h (C.9) (C.13) (C.11) (C.12) (C.10) , triplet    0 b 4 0 symmetry at y A . For consistency, 0 s Rj . R 0 d f u 1 d 1 CP xy ) ) x ϕ ϕ , , Li u u ¯ v v is uncharged under d (   ( d 3) 0 0 i ) ) 3 1 ud d c 0 0 d d / ij ≡ − 3 d d Θ λ 0 d Θ λ ( ( c F i i y M , f By By ) ( x = = Θ Θ Ri − ∗ 3 ∗ 3 Rj 3 f d i λ λ    x 3 3 Ri j j f d − δ δ x v ,  ,B Li ) ) d , + + a ] for a fuller discussion on Higgs PQ = u d x v (  ( ( ∗ i ∗ i e i ) i i 3 3 .. 39 e c u 1 u 1 a u PQ . Θ Θ ( v ϕ ϕ 3 3 i i h h → λ λ e Θ ,M + h are real by an assumed ∗ 3 ∗ ud ∗ ud Ri + = = λ    ˜ λ λ λ 3 0 Rj b 0 b 1 1 0 j j j e y Rj δ δ δ Collecting all free parameters, we have u Li 0 s 0 0 0 ¯ + , c e y , y + + Li – 24 – , f u 2 ∗ i u d 2 ¯ d e u ∗ i ∗ i ij 0 0 d d i ϕ ϕ i i Σ Σ Li 0 d u u j ij v d j d j v v v f y M We perform an axion-dependent rotation of the u d ) ϕ 2 By By 2 ϕ ϕ 2 1 M h Li h h ) f √ √ λ λ Rj    x e ∗ ju ∗ jd ∗ jd d Rj x ) has only marginal relevance for axion physics. We refer Moreover, all λ ˜ v u λ λ = = a − PQ x    v 0 s , f , such that we may write the couplings as matrices, and 2) i 11 Li = − 3 i d d u e e d h x Li Σ Σ Σ ϕ ( u u d d v v v , b , y x → v v v · h 2 2 2 ( a u 1 u d 1 d PQ v ϕ ϕ Σ Σ √ √ √ Q Li i ,M a v PQ v v v f e v u d 2 ), we let 2 i = = = 1 1    e + c c √ √ λ λ Ri e u d ij ij ij 13 23 = / ∂f = = M M M Y 0 d b  b c a Ri b  ¯ y 4 2 f −L 0 a a + ) is a function taking into account the VEV alignment of the    f Li ( u i v / ∂f , the exact form of Θ( , and fixes the third column of the Yukawa matrices. As 3 = Li ¯ R PQ f The above expressions, while precise, are not very illustrative. In explicit matrix form, u f ( triplet products like Note that under hermitian conjugation, the PQ charges change sign, i.e 3 f 11 4 M ¯ P we will always specify which Weyl fermion we are referring to. Lagrangian (derivativefermion basis). fields tois replace also the linear induced. couplings with Extending derivative ones; the the Lagrangian anomaly to term include the fermion kinetic terms, with dimensionless parameters defined by mixing and the origin of the third family couplings. we have where Θ as well as mixingfh effects between various HiggsU(1) doublets. It tracesthe its interested origin reader to to the the term original “A to Z” paper [ The coupling matrices are given exactly by A noting that each term mustwith be those PQ-invariant, of allowing the us SM tohigh fermions. replace scales. the flavon PQ charges Lagrangian (condensed linear basis). This rather hefty expression can be put in a more conventional format by 1) expanding the JHEP08(2018)117 , . 7 L 9 are d Q f U x (C.17) (C.18) (C.14) (C.15) (C.16) →  L Ri d f , masses. µ ., L γ ., τ c u . c . Ri Q ¯ f U + h and Ri , . + h f i b   → x R . f f Rj . f e d V V L + µ 3 matrices. We define the R R u U Li γ f f † Li f, u ¯ ) e × x x ) f f U ij † † 5 µ f f . V δ γ γ V V c e i ≡ R f . f ) were set to zero, and choosing Li m + − b A ¯ x f + anomaly η † f f f , and note that, since charges + h . + − Li c f V CKM U U f . ( f V V L L x Rj Rj V f f R  R e ¯ d ( f f x x + h µ † † Li f f x Li ¯ ¯ † e + d f U U ¯ u,d,e Rj are diagonal 3 fγ , and e V L ij   X e kj = u f δ f R 1 2 1 2 M µ Li ≡ U d k f – 25 – ¯ ]. They parametrise threshold corrections by a e a u,d,e γ x R ) m X + µ ≡ PQ = e ij , f ∂ v f ) = ) = ik X L 105 Q . All but one (¯ ) U Rj M R R f i a L U − d η x µ PQ f X X − ,
∂ v Li x and f ¯ CKM † d + − f Ri V − Rj f V d U f L L ij d U ( / ∂f = . In terms of Dirac spinors, X X L L Li M M + ( ¯ ( ( f ¯ † ∂ Ri f R d † f ¯ x 1 2 1 2 h f + L The axion-fermion derivative couplings become d † X U Rj ij f + u U = = Rj M , such that the mass terms become = ≡ u,d,e Li u f f Li R ≡ X f ¯ R = − u V f A Li f / L f ij m ∂f X ¯ u are vectors and δ Rj V a X Li u i u ij u µ ¯ PQ R f ∂ v m f → Li M and (quarks). The corresponding best fit input parameters are given in table ¯ u , − R † 8 L = → L u f ij f u,d,e = X , X m = M ∂ L f L = e 24 to account for the small GUT-scale difference between L i . − L L 0 U = X − We rotate to the mass basis by unitary transformations L → = b L ¯ (leptons) and We fit the modelperformed, to assuming data the at MSSM,series in the of [ GUT dimensionless scale.η parameters The running from low to high scale was D Couplings in A toThe Z: best numerical fit fit parameters, as well as a Bayesian 95% credible interval, are given in tables where real, where now coupling matrices where by definition Derivative couplings. e resulting in JHEP08(2018)117 9 0 =1 TeV . . = 1 TeV 23 11 14 19 1 . . . . . 882 015 144 643 762 616 234 815 998 ...... 24 11 34 8 40 249 1 2 3 7 2 0 8 51 60 70 334 1 . . 643 379 966 558 560 957 753 1577 SUSY ...... SUSY 3 5 2 0 1 4 75 1 0 13 → → → → → → → → → → → → → → → → M M 4 0 2 . . . → → → → → → → → → → 72 35 59 82 . . . . 326 911 093 569 049 459 022 400 892 =5, ...... 38 . = 5, 00 94 β . . − 569 273 181 530 354 098 645 β ...... 1368 . 10.4 272.1 218 0 242.6 231 . ] 59.20 58 32 32.88 32 10 39.27 37 . . 610 8.611 8 010 1.006 0 132 2.116 2 625 3.607 3 690 7.413 7 564 2.540 2 ...... 31 68.85 63 07 13.04 12 34 8 43 312 1 2 3 7 2 106 . . 731 3.607 3 118 4.733 4 722 2.296 2 551 0.545 0 510 1.432 1 906 3.038 1 727 1.689 1 1524 0.1463 0 ...... 3 5 2 0 1 3 72 1 0 13 range Best fit Interval → → → → → → → → → 0 230 [ σ range Best fit Interval . → → → → → → → → → → 81 40 . . – 26 – 310 998 106 588 330 484 < σ ...... 12 99 . . 500 594 183 537 408 057 673 ...... 1418 . Data Model Data Model 33.57 32 8.460 8 41.75 40 261.0 202 1.004 0 2.119 2 3.606 3 7.510 7 2.524 2 3.616 3 4.856 4 2.453 2 0.544 0 1.459 1 2.982 2 69.22 66 1.700 1 13.03 12 0.1471 0 Central value 1 Central value 1 2 2 eV eV 5 3 6 5 4 3 − − − − − − 24. The model interval is a Bayesian 95% credible interval. 24. The model interval is a Bayesian 95% credible interval. 3 2 . . 5 10 10 ◦ ◦ ◦ 0 0 − − 10 /meV − 10 / / ◦ 10 10 / / / / / − − /meV 1.940 1 / / / i ◦ ◦ ◦ 10 10 10 21 2 2 31 . Model predictions in the lepton sector, at the GUT scale. We set tan . Model predictions in the quark sector at the GUT scale. We set tan /meV 0.187 0 /meV 8.612 8 /meV 50.40 49 q q q b s d t c u q 23 13 12 ◦ / / / / / = = / m y y y y y y δ Observable θ θ θ / 1 2 3 ββ m m 21 31 b b ` ` ` e µ τ ` 12 13 23 η η Observable θ θ θ δ y y y ∆ ∆ m m m P α α m Table 8 and ¯ and ¯ Table 7 JHEP08(2018)117 B B 120 (1979) (2010) 646 935 151 592 039 . . . . . Phys. Rev. 43 2 2 , 82 (1992) 132 Phys. Rev. , (in Russian), ]. ]. 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