IFT-UAM/CSIC-19-3FTUAM-19-1

Flavor constraints on electroweak ALP couplings

M.B. Gavela,1 R. Houtz,1 P. Quilez,1 R. del Rey,1 and O. Sumensari2, ∗

1Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC,Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain

2Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, ItalyIstituto Nazionale Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy

We explore the signals of axion-like particles (ALPs) in flavor-changing neutral current (FCNC) processes.The most general effective linear Lagrangian for ALP couplings to the electroweak bosonic sector is considered,and its contribution to FCNC decays is computed up to one-loop order. The interplay between the differentcouplings opens new territory for experimental exploration, as analyzed here in the ALP mass range 0 < ma

<∼5 GeV. When kinematically allowed, K → πνν decays provide the most stringent constraints for channels withinvisible final states, while B-meson decays are more constraining for visible decay channels, such as displacedvertices in B → K(∗)µ+µ− data. The complementarity with collider constraints is discussed as well.

I. INTRODUCTION

With the Higgs discovery, an era in humankind’s quest forthe fundamental laws of Nature has been completed [1, 2]. Atthe same time, new uncharted territory has been opened: thespin-zero window to the universe. (Pseudo)Nambu-Goldstonescalars (pGBs) are strongly motivated from fundamental prob-lems of the known particle physics laws, that is, of the Stan-dard Model of Particle Physics (SM). They are the generictell-tale of exact, although “hidden” (i.e. spontaneously bro-ken), global symmetries of nature. A paradigmatic exampleis the axion, which results from the dynamical solution tothe strong CP problem of the SM [3–7]. The traditional “in-visible axion” is expected to be extremely light, with massma < 10−2 eV, and its scale fa to be out of direct exper-imental reach, although recently tantalizing axion solutionsto the strong CP problem are being explored with scales aslow as O(TeV) [8–19]. PGBs of deep interest extend wellbeyond axions, though. They appear in a plethora of con-structions which reach beyond the SM (BSM), typically asSM scalar singlets, e.g. in: i) Theories with extra space-timedimensions, ii) Dynamical explanations to the smallness ofneutrino masses: the Majoron [20], iii) String theory mod-els [21], iv) Supersymmetric extensions of the SM [22], andv) Many dynamical flavor theories with hidden global U(1)symmetries, a particular class of which identifies the QCDaxion as a flavon “à la Froggat-Nielsen” (axiflavon or flax-ion) [23–25]. These pGBs are often denoted by the generalname of axion-like particles (ALPs), as anomalous couplingsto gauge currents often appear in addition to purely deriva-tive ones. ALPs may or may not have anomalous couplings togluons, and they are not required to solve the strong CP prob-lem. One practical difference between a generic ALP and trueaxions which solve the strong CP problem is that, for ALPs,

∗ belen.gavela@uam.es; rachel.houtz@uam.es; pablo.quilez@uam.es; ro-cio.rey@uam.es; olcyr.sumensari@pd.infn.it

fa and ma are treated as independent parameters. Outstand-ingly, and as a wonderful byproduct, both axions and ALPsmay be excellent candidates to explain the nature of dark mat-ter (DM) [26–28].

The parameter space for very light ALPs, with masses be-low the MeV scale, is delimited mainly by astrophysical andcosmological constraints. Regarding terrestrial experiments,ADMX has finally entered the critical territory expected foran invisible axion signal if DM is made of axions. In ad-dition, the investment in axion and ALP searches in a largerange of masses is accelerating at present with CAST [29],IAXO [30, 31], and future projects such as Madmax, CASPEr,QUAX, HeXenia, FUNK and electric dipole moment searches(PSI and Co) [32–36]. Also, DM experiments like Xenon [37]and the future Darwin [38] target keV mass ALP dark matter(and solar axions). On the precision arena, flavor experimentsprovide valuable constraints. For instance, NA62 [39] is tak-ing data, and new fixed target facilities (e.g. SHIP [40]) are inpreparation, with sensitivity to MeV-GeVs ALPs and strongcomplementary potential to tackle ALP couplings to gaugebosons and fermions. Belle-II [41] will also have some sensi-tivity to this mass range, as well as the LHC with Mathusla,Faser and CodexB [42–44]. Indeed, ALPs may well show upfirst at colliders [45, 46]. Intense work on ALP signals at theLHC and future colliders is underway [47, 48], and the syn-ergy between collider and low-energy fixed target experimentsis increasingly explored [49]. All couplings must be analyzedcombining fixed-term and accelerator data in a complemen-tary approach.

In this work, we explore ALP contributions to flavor chang-ing neutral current (FCNC) processes, formulating them in amodel-independent approach via the linear realization of theALP effective Lagrangian. The complete basis of bosonicand CP-even ALP couplings to the electroweak sector is con-sidered. That is, the set of gauge invariant and indepen-dent leading-order couplings to the W , Z, photon and Higgsdoublet is discussed. Given that these operators are flavorblind, they may impact flavor-changing data only at looplevel. The couplings of ALPs to heavy SM bosons had been

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largely disregarded until recently, even if a priori they areall expected to be on equal footing with the pure photonicones because of electroweak gauge invariance. In additionto novel collider signatures [47, 48], rare hadron decays pro-vide a superb handle on the ALP couplings to massive vectorbosons [50] for ALP masses below 5 GeV. The one-loop im-pact on FCNC processes of the anomalous ALP-W -W cou-pling was first considered in Ref. [49]: it was shown to induceflavor-changing rare meson decays via W exchange, with theALP radiated from the W boson [49, 54, 55]. The axion canthen either decay in some visible channel or escape the detec-tor unnoticed, and novel bounds were derived in both cases.Given the level of accuracy provided by present flavor experi-ments, it is most pertinent to take into account the competingcontribution of other electroweak ALP couplings leading tothe same final states. In other words, the ensemble of the lin-early independent ALP-electroweak couplings should be con-sidered simultaneously in order to delimitate the parameterspace. Putative anomalous couplings of ALPs to gluons couldalso contribute to flavor-blind decays into visible channels, butnot to FCNC processes other than via pseudoscalar (e.g. ALP-η′ and ALP-pion) mixing in SM flavor-changing decays, andthey are not considered in this paper.

The analysis of two (or more) couplings simultaneously hasthe potential to change the experimental perspective on ALPs.Our theoretical analysis is confronted with the prospects forALP detection in present and upcoming fixed-target experi-ments and B-physics experiments. After the theoretical anal-ysis, the structure of this paper reflects successively the twoalternative scenarios mentioned above, in which the ALP pro-duced in FCNC meson decays can then either decay into vis-ible channels within the detector, or it can be invisible by es-caping the detector (or decaying to a hidden sector). For bothcases, the comparison with data considers first each couplingseparately and then the ensemble in combination, and the re-sulting interference patterns are worked out in detail.

II. BOSONIC ALP LAGRANGIAN

The most general effective Lagrangian describing ALP cou-plings contains – at leading order in the linear expansion –only three independent operators involving electroweak gaugebosons [47, 51–53],

δLeff =∂µa ∂

µa

2− m2

a a2

2

+ caΦOaΦ + cB OB + cW OW ,

(1)

with

OaΦ ≡ i∂µa

faΦ†←→D µΦ ,

OB ≡ −a

faBµνB

µν ,

OW ≡ −a

faW aµνW

µνa ,

(2)

where Φ is the SM Higgs doublet, fa is the ALP decayconstant, ci are real operator coefficients and Φ

←→D µΦ ≡

Φ†(DµΦ

)−(DµΦ

)†Φ. The dual field strengths are defined

as Xµν ≡ 12εµνρσXρσ , with ε0123 = 1.

Upon electroweak symmetry breaking,OaΦ induces a mix-ing between a and the would-be Goldstone boson eatenby the Z. Its physical impact is best illustrated viaan ALP-dependent rotation of the Higgs field, namelyΦ → Φ eicaΦa/fa [51], which trades OaΦ for the followingfermionic couplings:

OaΦ → ia

fa

[QYuΦuR −QYdΦ dR − LY`Φ `R

]+ h.c. ,

(3)

where Yu,d,` denote the SM Yukawa matrices, flavor indicesare omitted, and neutrino masses are disregarded. The ALP-electroweak operators in Eq. (2) are flavor blind, but OaΦ andOW can participate in FCNC processes at one loop via W±

gauge boson exchange. At this order, the parameter spaceof ALP-electroweak couplings in FCNC processes is thus re-duced to two dimensions spanned by the coefficients

{cW , caΦ} . (4)

They may contribute to rare decays as illustrated in the left(caΦ) and right (cW ) panels of Fig. 2. While cW has beendiscussed separately in Ref. [49, 50], and the effective ALP-fermionic interactions have also been considered by them-selves before [48, 54–58], the interplay between cW and caΦ

will be shown below to lead to interesting new features.

III. FCNC ALP INTERACTIONS

The effective interaction between a pGB and left-handedfermions can be expressed in all generality as

Ldi→djeff = −gaij (∂µa) djγµPLdi + h.c. , (5)

where latin indices i, j denote flavor and gaij is an effectivecoupling.

The impact of OaΦ and OW on di → dja (with i 6= j)transitions via one-loop W± exchange induces a left-handedcurrent of the form in Eq. (5), and thus a contribution to raremeson decays. The corresponding Feynman diagrams at thequark level are those contained in the illustration in Fig. 1, aswell as the corresponding self-energy diagrams with the ALPoperator inserted on the quark lines external to the W loop.At the quark level, those one-loop W exchanges result in acontribution to gaij , for i 6= j, given by

gaij = g2∑

q=u,c,t

VqiV∗qj

16π2

[3cWfa

g(xq)−caΦ

4faxq log

(f2a

m2q

)],

(6)where g is the electroweak gauge coupling, and Vqi are theCKM matrix elements. In this equation, mq denotes the mass

3

K+ π+

u

s

W

d

u, c, t u, c, t

acaΦ

K+ π+

u

s

u, c, t

d

W W

acW

FIG. 1. Illustration of diagrams giving one-loop contributions to theprocess K+ → π+a via the interactions defined in Eq. (3).

of a given up-type quark q that runs in the loop, the approxi-mation mdj , mdi � mW has been used, xq = m2

q/m2W , and

the loop function is given by

g(x) =x [1 + x(log x− 1)]

(1− x)2. (7)

It follows that the decay rate for the process K+ → π+acan be expressed as

Γ(K+ → π+a) =m3K |gasd|2

64πf0(m2

a)2λ1/2πa

(1− m2

π

m2K

)2

,

with λπa =[1− (ma+mπ)2

m2K

] [1− (ma−mπ)2

mK2

]. In this ex-

pression, f0 denotes the K → π scalar form factor, whichhas been computed in lattice QCD in Ref. [64]. An analo-gous expression can be obtained mutatis mutandis for the de-cay B → Ka, in which case the relevant form factors can befound in Refs. [65, 66].

In Eq. (6), the contribution proportional to cW is finitedue to the Glashow–Iliopoulos–Maiani (GIM) mechanism, inagreement with the results of Ref. [49]. The caΦ term is in-stead logarithmically sensitive to the ultraviolet scale of thetheory fa, and its contribution is thus approximated by theleading log model-independent component. Furthermore, be-cause g(x) ∼ x + O(x2) for small x, the contributions fromthe up and charm quarks are sub-leading in both terms withrespect to that of the top quark. Also, note that the logarith-mic enhancement of the caΦ term (∝ log (fa/mt)) should beparticularly relevant for large values of fa. This logarithmicdivergence is a consequence of the operator OaΦ being non-renormalizable [55–57], in contrast with renormalizable sce-narios such as two-Higgs doublet models [55, 59–61].

The interplay between caΦ and cW presents interesting fea-tures which depend on their relative sign. Their contributionsto ALP production in rare decays can interfere destructively ifand only if caΦ/cW > 0. Such a cancellation would leave aregion in parameter space which cannot be probed by relyingonly on FCNC decays such as K → πa and B → Ka. Analternative to lift this degeneracy using LHC constraints willbe discussed further below, after deriving the constraints thatfollow from rare meson decays.

In order to determine the detection possibilities for a givenfinal state channel, an important element is whether the ALPcan decay into visible particles within the detector, or whetherit escapes and contributes to an “invisible” channel. We dis-cuss next both cases.

IV. THE INVISIBLE ALP

Let us consider first the scenario of an ALP that does notdecay into visible particles in the detector, which we shall re-fer to as the “invisible ALP”. This situation can arise if a issufficiently light, making a long-lived, or if there are largecouplings of a to a dark sector, making B(a → inv) suffi-ciently large. The analysis performed below is general andapplies to both cases.

The experimental constraints relevant for different ma

ranges are listed next:

• ma ∈ (0,mK −mπ):Searches for the decay K → πνν have been performedat the E787 and E949 experiments. The bounds obtainedcan be directly reinterpreted to limit the parameter space ofnew undetected particles. E787 and E949 experiments takemeasurements in two regions of pion momentum, namelypπ ∈ (140, 199) MeV and pπ ∈ (211, 229) MeV, whichcan be translated into the ALP mass ranges 150 MeV <∼ma

<∼ 260 MeV and ma<∼ 115 MeV, respectively. The

limits reported in these searches are B(K+ → π+νν)exp =(1.73+1.15

−1.05

)× 10−10 [62] and B(K+ → π+νν)exp <

2.2 × 10−9 [63], which lie slightly above the SM predic-tion, B(K+ → π+νν)SM = (9.11± 0.72) × 10−11 [67].Similar searches have been performed at the NA62 experi-ment, which aims at attaining the SM rates in the very nearfuture [68]. In our analysis, we consider the E787 and E949constraints, as summarized in Ref. [62].

• ma ∈ (0,mB −mK):The most constraining experimental limits on B(B →K(∗)+inv) were obtained by the Belle collaboration. Theseare B(B → Kνν) < 1.6 × 10−5 and B(B → K∗νν) <2.7 × 10−5 (90% C.L.) [69], which lie respectively a fac-tor of 3.9 and 2.7 above the SM predictions [70]. In thenear future, Belle-II aims at measuring the SM value witha O(10%) precision [71]. For the new physics scenarioconsidered here, the strongest constraint arises from theB → Kνν result.

We have explicitly checked that ∆F = 2 constraints on the ef-fective couplings gaij are less stringent than the ones presentedabove for most of the ALP parameter space considered here.Nevertheless, those constraints should provide the best boundson caΦ and cW for masses larger than∼ 5 GeV, which are outof reach of rare decays, see Fig. 2. Those observables are notincluded in our analysis, though, since the consistent assess-ment of the corresponding limits would require a completetwo-loop computation, as well as the additional considerationof higher dimension ALP operators, which goes beyond thescope of this paper.

The constraints set on ALP-electroweak coefficients bydata will be analyzed in two steps: first within a one couplingat a time approach, where either only cW or caΦ are switchedon; next, the {caΦ, cW } parameter space spanned by the si-multaneous presence of both couplings will be considered.

Fig. 2 depicts the allowed values of cW (left panel) and caΦ

(right panel) as a function of the ALP mass, when only one of

4

K+→π+a

B→Ka

Belle-II

NA62

fa = 1 TeV

10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

100

ma [GeV]

|cW|

K+→π+a

B→Ka

Belle-II

NA62

fa = 1 TeV

10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

100

ma [GeV]

|caΦ|

FIG. 2. Invisible ALP: constraints on the absolute value of cW (left panel) and caΦ (right panel) as a function of the ALP mass, consideringeach of these couplings separately. The exclusion contours have been derived from the experimental limits on B(K+ → π+ + inv) [62](green) and B(B → K + inv) [63] (blue) by fixing fa = 1 TeV and by setting the other couplings to zero. Projections for NA62 [49] andBelle-II [71] experiments are illustrated by dashed lines.

these two couplings is added to the SM. The constraints ob-tained on the {ma, cW } plane (left panel) coincide with thosederived in Ref. [49]. The constraints on the parameter spacefor {ma, caΦ} (right panel) are a novel contribution of thiswork. The case illustrated corresponds to fa = 1 TeV. Thequantitative similarity of the exclusion limits on the two cou-plings depicted in Fig. 2 is fortuitous; it is easy to check thatthe constraints on caΦ become stronger than those for cW forlarger values of fa, as expected from the logarithmic depen-dence of its contribution, see Eq. (6).

These plots also indicate that kaon constraints are typicallyone order of magnitude stronger than those derived from B-meson decays, although limited to a more restrictedma range.Future prospects from NA62 and Belle-II are also illustratedin Fig. 2 with dashed lines.

When both caΦ and cW are simultaneously considered, aninteresting pattern of destructive interference can take place,as anticipated in Sec. III. Fig. 3 depicts the result of combin-ing the different experimental constraints for fixed values offa and ma

<∼ 0.1 GeV. This shows indeed that when therelative sign of both couplings is positive, a blind directionin parameter space appears. This unconstrained direction isexactly aligned for kaon and B-meson decays. For this rea-son, additional experimental information is then needed tolift the degeneracy. One possibility is to consider the decaysD → π(a→ inv), which are sensitive to a different combina-tion of caΦ and cW , since the up- and down-type quark con-tributions to the term proportional to caΦ have opposite signs,see Eq. 3. These decays, however, suffer from a heavy GIMsuppression, and no such experimental searches have beenperformed to our knowledge. A more promising possibility

is to consider LHC constraints that are sensitive to a specificALP coupling. For example, LHC searches for mono-W fi-nal states are only sensitive to cW 1. In Ref. [47] the authorsderived the current (projected) bounds

|cW |fa

<∼ 0.41 (0.16) TeV−1 , (8)

from 3.2 fb−1 (3 ab−1) of LHC data: these have been su-perimposed in Fig. 3. Similarly, a reinterpretation of pp →tt + MET at the LHC would constrain only caΦ, but suchanalysis goes beyond the scope of this letter. Typically, LHCconstraints are weaker than flavor bounds, except in the regionof parameter space where the flavor signal is suppressed dueto a cancellation between two contributions. In this case, thecomplementarity of low and high-energy constraints becomesan important handle on new physics.

V. THE VISIBLE ALP

We analyze next the case of ALPs produced at loop levelvia rare meson decays, but decaying into visible states via thesame set of bosonic interactions introduced in Eq. (2). For thema range considered in this work, the kinematically accessi-ble decays are a → γγ, a → hadrons and a → ``, with

1 Bounds stemming from mono-Z signals are slightly better, but this finalstate can also be generated by another coupling (cB), which complicatesslightly the reinterpretation in terms of cW and caΦ.

5

FIG. 3. Allowed {cW , caΦ} parameter space for the invisible ALP when those two couplings are simultaneously present. The superpositionof the constraints from K+ → π+ + inv (green) and B+ → K+ + inv (blue) data is shown for an illustrative case with fa = 1 TeV andma

<∼ 100 MeV. The left (right) panel shows the destructive (constructive) interference of the two couplings for cW /caΦ > 0 (cW /caΦ < 0).The red solid (dashed) lines correspond to the current (projected) limits from mono-W searches at the LHC with 3.2 fb−1 ( 3 ab−1) of data [47].

` = e, µ, τ . Both tree-level and loop-level contributions to thedecays are to be taken into account. Indeed, experimental lim-its on ALP couplings to photons, electrons, and nucleons areso stringent that (indirect) loop-induced observables can givestronger constraints than (direct) tree-level ones [48, 50].

At tree level, cW and caΦ contribute respectively to ALPdecays into photons and into fermions. Nevertheless, thecoupling cB may also enter the game for these decays: attree level for the photonic channel and at loop level for thefermionic channel. That is, while the parameter space for theproduction of an ALP via rare meson decays is still the two-dimensional one in Eq. (4), the whole set of ALP electroweakcouplings {caΦ, cW , cB} is relevant for the analysis of visibledecay channels. For consistency, all one-loop contributionsinduced by these three couplings are to be taken into account.

For instance, the partial width for ALP decay into leptons,including one-loop corrections, reads

Γ(a→ `+`−) = |c``|2mam

2`

8πf2a

√1− 4m2

`

m2a

(9)

where αem is the fine structure constant and c`` is given atone-loop order by

c`` = caΦ +3αem

4π

(3 cWs2w

+5 cBc2w

)log

famW

+6αem

π

(cB c

2w + cW s2

w

)log

mW

m`,

(10)

where sw = sin θw, cw = cos θw and θW denotes the weakmixing angle. For the a→ γγ decay, the partial width reads

Γ(a→ γγ) = |caγγ |2m3a

4πf2a

, (11)

where the caγγ coupling is defined at tree level, as

caγγ

∣∣∣tree≡ cB c2w + cW s2

w . (12)

Furthermore, bosonic loops give corrections to caγγ propor-tional to cW . Fermionic loops may also induce nonzero valuesof caγγ at the scale µ = fa, even if the ALP has no tree-levelcouplings to gauge bosons, i.e. cW = cB = 0 [48]. To sumup, both cW and caΦ induce one-loop corrections to the pho-tonic width. Specifically, for ma � ΛQCD,

caγγ

∣∣∣1−loop

=cW

[s2w +

2αem

πB2(τW )

]+ cB c

2w

− caΦαem

4π

(B0 +

m2a

m2π −m2

a

),

(13)

where B0 and B2(τf ) are loop functions, which are detailedin Appendix A. For ma � ΛQCD, the second term in the lastline of the above equation is absent, since it stems from π-amixing which becomes negligible in this mass range.

For hadronic decays, it is pertinent to consider two sepa-rate ma regions: (i) between 3mπ and 1 GeV, and (ii) above3 GeV. In the former region, the dominant hadronic decay is

6

cW

fa= 1 TeV-1

γγ

ee

μμ

had

ττ

10-3 10-2 10-1 100 10110-24

10-20

10-16

10-12

10-8

10-4

ma [GeV]

Γ(a→X)[GeV

]caΦ

fa= 1 TeV-1

γγ

ee

μμ

had

ττ

10-3 10-2 10-1 100 10110-24

10-20

10-16

10-12

10-8

10-4

ma [GeV]

Γ(a→X)[GeV

]FIG. 4. ALP partial decay widths to various two-particle channels as a function of ma, in presence of either cW (left panel) or caΦ (rightpanel), for cW /fa = 1 TeV−1 and caΦ/fa = 1 TeV−1, respectively. The grey shaded areas correspond: i) to the pion mass region, which isexperimentally excluded due to the large π0-a mixing; ii) the interval (1, 3) GeV, in which the hadronic width cannot be fully assessed eitherwith chiral estimates or perturbatively.

a → 3π which can be computed by employing chiral pertu-bation theory [48]. In the region above 3 GeV, the dominantdecays are a → cc and a → bb, which are well described bya perturbative expression analogous to Eq. (9) multiplied bythe color factor Nc = 3.2 In this work we remain agnosticabout the intermediate region ma ∈ (1, 3) GeV, since severalhadronic channels, which are particularly difficult to estimatereliably, open up for these masses.3 In this region, the totalhadronic width Γa will be replaced by its value at the rangefrontier at ma = 3 GeV. Note that this is the most conser-vative choice, since the hadronic width is a continuous andstrictly increasing function of ma.

Fig. 4 illustrates the ALP partial widths as a function ofma, when either only cW (left panel) or caΦ (right panel) arepresent, for the benchmark values cW /fa = 1 TeV−1 andcaΦ/fa = 1 TeV−1. The mass thresholds for each of thefermionic channels are clearly delineated.

In order to analyze the impact of an intermediate on-shellALP on rare meson decays to visible channels, ALP produc-tion via the couplings in Eq. (2) needs to be convoluted withALP decay into SM particles via that same set of couplings.When caΦ and cW are simultaneously present, a very inter-esting pattern of constructive/destructive interference is ex-

2 Note that the decay a → gg is not induced at one-loop level in our setup,since the up- and down-type quark contributions cancel due to the differentsigns in Eq. (3).

3 A first attempt to compute these rates by using a data-driven approach inthis particular ma interval has been proposed in Ref. [72] for the GGacouplings.

pected. We will assume for simplicity cB = cW to illustratethe effect. While a positive sign for cW /caΦ leads to destruc-tive interference in ALP production (see Eq. (6) and Fig. 3),the opposite can occur in the subsequent ALP decay into visi-ble channels. Indeed, the decay into leptons shows destructiveinterference for negative caΦ/cW , see Eq. (9). The expecta-tion for the photonic channel is more involved and depends onthe ALP mass: for ma < mπ the terms in the last parenthesisin Eq. (13) are both real and positive and the interference pat-tern is thus analogous to that for ALP production, while forlarger masses it may differ. Table I summarizes the interfer-ence pattern expected.

cW /caφ Production a→ `+`− a→ γγ

> 0 Destructive Constructive Destructive

< 0 Constructive Destructive Constructive

TABLE I. ALP-mediated rare meson decays: interference pattern be-tween caΦ and cW in ALP production and decay as a function ofcW /caφ sign, by assuming cB = cW . The a→ γγ column assumesma < mπ , see text for details.

Three sets of experimental data that will be considered inorder to constrain the {ma, caΦ, cW } parameter space for avisible ALP: 1) displaced vertices; 2) semileptonic and pho-tonic meson decays; 3) leptonic meson decays.

1. Displaced vertices. Of particular interest are searches forlong-lived scalars, which would result in displaced vertices.

7

B→K(*)a(→μμ)

K+→π+a(→μμ)

K→πγγ

K→πee

SHiP

fa = 1 TeV

10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

100

ma [GeV]

|cW|

fa = 1 TeV

K→πeeK→πγγ

KL→μμ

K+→π+a(→μμ)

B→K(*)a(→μμ)

Bs→μμ

SHiP

10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

100

ma [GeV]

|caΦ|

FIG. 5. Visible ALP: constraints on the absolute value of cW (left panel) and caΦ (right panel) when these couplings are considered separately,as a function of the ALP mass and for fa = 1 TeV. The exclusion contours follow from the experimental limits on K+ → π+ a(→ µµ)

(red) [76], B → K(∗) a(→ µµ) (orange) [73, 74], B(KL → µµ) (green) [77], B(Bs → µµ) (blue) [81, 82], B(K → πee) (purple) [77] andB(K → πγγ) (cyan) [78]. The grey dashed lines are projections for the SHiP experiment [40]. The unconstrained regions in the range of theLHCb bounds correspond to the masses of several hadronic resonances which are vetoed in their analysis.

Two ma ranges are pertinent:

(a) ma ∈ (2mµ,mB −mK)

The LHCb collaboration perfomed searches for long-lived (pseudo)scalar particles in the decays B →K(∗)a, with a → µµ [73, 74]. Limits on B(B →K(∗)a) · B(a → µµ) which vary between 10−10 and10−7 are reported as a function of ma and the properlifetime, τa. For τa < 1 ps, the limit derived is inde-pendent of τa since the ALP would decay promptly.The best constraints are those for values of τa between1 ps and 100 ps, for which the dimuon vertex wouldbe displaced from the interaction vertex. See alsoRef. [75] for a recent reinterpretation of these limits.

(b) ma ∈ (2mµ,mK −mπ)

Similar searches have also been performed by theNA48/2 Collaboration for the decay K+ → π+a,followed by a → µµ [76]. The limits reported onB(K+ → π+a) · B(a → µµ) decrease with ALPlifetime until τa = 10 ps, becoming constant forsmaller values of τa. The best experimental limits areO(10−10) and obtained for τa ≤ 10 ps.

2. Semileptonic and photonic meson decays. Relevant con-straints on ALPs can be inferred from their indirect contri-butions to low-energy meson decays. In particular:

(a) Kaon decays. The measured kaon branching frac-tions B(K+ → π+ee)exp = (3.00 ± 0.09) × 10−7,B(K+ → π+µµ)exp = (9.4 ± 0.6) × 10−8 [77], and

B(K+ → π+γγ)exp = (1.01 ± 0.06) × 10−7 [78]will be taken into account. In order to avoid the uncer-tainty related to the unknown SM long-distance contri-butions, it will be required that the ALP contributionalone does not saturate the 2σ experimental bounds.

(b) B-meson decays. Recently, LHCb observed severaldeviations from the expected values in ratios of B →K(∗)µµ and B → K(∗)ee decays in different binsof dilepton squared mass [79, 80]. If these anomaliesturn out to imply new physics, ALP couplings wouldnot explain them. More precisely, pseudoscalar effec-tive operators induced by a heavy mediator cannot re-produce current deviations due to the constraints de-rived from B(Bs → µ+µ−)exp [81]. On the otherhand, a light ALP with ma

<∼ mB −mK would facestringent limits from LHCb searches for long-lived(pseudo)scalar particles in B → K(∗)a(→ µµ), asmentioned above [73, 74]. For these reasons, we leaveout of our analysis the constraints that would stemfrom the comparison of exclusive B → K(∗)µµ mea-surements with the SM expectation until further clari-fication is provided by the B-physics experiments.

3. Leptonic Bs and KL decays:While the constraints in 1) and 2) above correspond toon-shell ALPs, off-shell contributions are relevant in lep-tonic meson decays. LHCb measured B(Bs → µµ)exp =(3.0 ± 0.6+0.3

−0.2) × 10−9 [81], which agrees with the SMprediction, B(Bs → µµ)SM = (3.65± 0.23)× 10−9 [82].

8

fa = 1 TeV

K→πγγ

K→πee

10-5 10-4 10-3 10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

|cW|

|caΦ|

cW/caΦ > 0, ma = 0.1 GeV

fa = 1 TeV

K→πγγ

K→πee

10-5 10-4 10-3 10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

|cW|

|caΦ|

cW/caΦ < 0, ma = 0.1 GeV

fa = 1 TeV

B→K(*)a(→μμ)

K→πγγ

K+→π+a(→μμ)

10-5 10-4 10-3 10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

|cW|

|caΦ|

cW/caΦ > 0, ma = 0.3 GeV

fa = 1 TeV

B→K(*)a(→μμ)

K→πγγ

K+→π+a(→μμ)

10-5 10-4 10-3 10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

|cW|

|caΦ|

cW/caΦ < 0, ma = 0.3 GeV

FIG. 6. Visible ALP: Allowed parameter space when the couplings {caW , caΦ} are simultaneously present, for fa = 1 TeV and ma =0.1 GeV (upper plots) or ma = 0.3 GeV (lower plots). The different flat directions observed in the figures correspond to the destructiveinterferences of both couplings in ALP production and/or the various ALP channel decays, which depend on the sign of cW /caΦ. See text fordetails.

The ALP contribution to this observable can be computedby a straightforward modification of the expressions pro-vided in Ref. [61]. Similarly, we consider the kaon decayB(KL → µµ)exp = (6.84±0.11)×10−9 [77]. In the lattercase, we impose once again the conservative requirementthat the ALP (short-distance) contribution does not satu-rate the 2σ experimental values. When the complete set

of electroweak couplings in Eq. (2) will be simultaneouslyconsidered for an off-shell ALP, the interference pattern inthe amplitudes can be understood analogously to the sepa-rate discussion on production and decay for on-shell ALPs.

In analogy with the case of the invisible ALP in the previoussection, all of these data will be analyzed first within a one

9

coupling at a time approach, where either only cW or caΦ areswitched on (as cB by itself cannot mediate FCNC processes).In a second step, the simultaneous presence of {caΦ, cW , cB}will be taken into account. We assume cB = cW in the figuresbecause cB has only a modulating role, and this choice doesnot preclude or fine-tune any particular decay channel.

Fig. 5 illustrates the allowed values of |cW | (left panel) and|caΦ| (right panel) in the one-coupling-at-a-time analysis, asa function of the ALP mass and for fa = 1 TeV. Constraintsfrom B(K → πµµ) [77] are not displayed, since they aresuperseded by NA48/2 constraints on long-lived particles inK+ → π+ a(→ µµ) decays. The grey dashed lines are pro-jections for the SHiP experiment [40]. The figure reflects thestringent constraints from LHCb searches for displaced ver-tices in the dimuon channel [73–75] for the large mass rangema ∈ (2µ,mB − mK), see point 1.(a) above. These limitsare more constraining than the analogous searches performedin the kaon sector [76]. Remarkably, this is in contrast to theinvisible scenario discussed in Sec. IV, for which kaon con-straints are considerably stronger than those derived from B-meson decays if K+ → π+a is kinematically allowed. Theseresults, which take into account only one coupling at a time,could be of special interest in specific new physics scenarios.For instance, the case of a non-vanishing caΦ with cB and cWdisregarded (right panel) is motivated by perturbative modelsproducing caΦ at tree level but {cB , cW } only at loop level(e.g. cB ∼ cW ' g2/(16π2) caΦ). Nevertheless, in all gener-ality and for a rigorous approach, the simultaneous presenceof all couplings in the electroweak bosonic basis in Eq. (2)must be considered. This may essentially modify the boundsinferred, as discussed above and illustrated next.

Fig. 6 depicts the bounds resulting when caΦ, cW and cBare simultaneously considered. Once again, in the ALP massregion in which B-physics data on displaced vertices apply,they are seen to be more constraining than the bounds in-ferred from the kaon sector, see Figs. 6c and 6d. Further-more, the four panels in the figure clearly illustrate – for twovalues of ma and cW = cB – the remarkable pattern of con-structive/destructive interference expected from the analysisin Sec. V and Table I. For instance, the two flat directionsin the photonic channel in Fig. 6a result from destructive in-terference in both production and decay for positive cW /caΦ

and ma < mπ . The rest of the figures can be analogouslyunderstood. Once again, the various flat directions in differ-ent channels call for complementarity with collider data andother experimental projects. In particular, the degeneracy inparameter space which induces the flat direction in Fig. 6aand Fig. 6c, common to all rare decay channels discussed inthis work, could be resolved by LHC data. Some of the flatdirections appearing in ALP decays (cf. e.g. Fig. 6d) couldalso be probed by proposed beam-dump experiments such asSHiP, since they can measure ALP decays into both photonsand muons, and because B(a → µµ) and B(a → γγ) donot simultaneously vanish. This is also true for various LHCsearches, and so both experiments could be good handles onremoving flat directions, though a full analysis is beyond thescope of this work.

VI. CONCLUSIONS

The field of axions and ALPs is blooming, with an escala-tion of efforts both in theory and experiment. Theoretically,the fact that no new physics has shown up yet at collidersor elsewhere positions the SM fine-tuning issues as the mostpressing ones and leads to further implications for our per-spective of dark matter. The silence of data is calling for arerouting guided by fundamental issues such as the strong CPproblem and an open-minded approach to hunt for the generictell-tale of global hidden symmetries: derivative couplings asgiven by axions (light or heavy) and ALPs. Experimentally,the worldwide program to hunt specifically for axions andALPs is growing fast. At the same time, other experimen-tal programs are realizing their potential to tackle the axionand ALP parameter space, e.g. the LHC and beam dump ex-periments.

In the absence of data supporting any concrete modelof physics beyond the SM, effective Lagrangians provide amodel-independent tool based on the SM gauge symmetries.Very often the effective analyses rely on considering one ef-fective coupling at a time, though, instead of the complete ba-sis of independent couplings. The time is ripe for further stepsin the direction of a multi-parameter analysis of the ALP ef-fective field theory, and this is the path taken by this work.

We have considered the impact on FCNC processes ofthe complete basis of bosonic electroweak ALP effective op-erators at leading order (dimension 5), taking into accountthe simultaneous action of those couplings. As this basisis flavor-blind, its impact on flavor-changing transitions (e.g.di → dja, with i 6= j) starts at loop level. Indeed, the experi-mental accuracy achieved on rare-decay physics, as well as onlimits of ALP couplings to photons, electrons, and nucleons,is so stringent that loop-induced contributions may providethe best bounds in a large fraction of the parameter space.

We first revisited previous results in the literature, whichhad been derived considering just one operator at a time.We studied next the simultaneous action of the variouselectroweak couplings. An interesting pattern of construc-tive/destructive interference has been uncovered, which de-pends on the relative sign of the couplings and on the chan-nel and mass range considered. In this way, the previousvery stringent bounds stemming from kaon and B-decay dataare alleviated. Furthermore, LHC searches for light pseu-doscalar particles have been highlighted as more importantin regions where deconstructive interference weakens flavorbounds. While they are generally considerably less sensitivethan flavor observables, LHC searches are shown to providecomplementary information to low-energy probes, exploringotherwise inaccessible directions in the ALP parameter space.We have also explicitly illustrated how they can overcomesome of the blind directions on rare meson decays identifiedhere.

We have derived the most up-to-date constraints on theeffective electroweak ALP parameter space for two well-motivated scenarios: (i) an ALP decaying into channels invis-ible at the detector; (ii) an ALP decaying into γγ, ee and/orµµ. The conclusion is that searches for K → πνν decays

10

provide the most stringent constraints in the first case. In con-trast, for the second scenario, the strongest constraints arisefrom searches at LHCb for long-lived (pseudo)scalars (dis-placed vertices) in the decays B → K(∗)a(→ µµ). Thisillustrates beautifully the potential of flavor-physics observ-ables to constrain new physics scenarios. These searches willbe improved in the years to come thanks to the experimentaleffort at NA62, KOTO, LHCb and Belle-II, providing tanta-lizing oportunites to discover new physics, complementary tothe direct searches performed at the LHC.

Much remains to be done to fully encompass the ALP pa-rameter space. For instance, the anomalous ALP gluoniccoupling has not been considered in this work. Even if itcannot mediate FCNC processes, it may impact our resultsfor the visible ALP via the quantitative modification of thebranching ratios. In fact, recent ALP analyses of FCNC de-cays [50] take into account the simultaneous presence of thegluonic coupling and just one electroweak ALP coupling, butno work considers all ALP bosonic couplings together, letalone the complete basis of operators including the most gen-eral fermionic ones. This effort is very involved and will bethe object of future work. In a different realm, note that thetype of effective operators considered above assumes a linearrealization of electroweak symmetry breaking; the alternativeof analyzing ALP FCNC processes via the non-linear effectiveSM Lagrangian is pertinent and also left for future considera-tion.

ACKNOWLEDGMENTS

We acknowledge M. Borsato, F. Ertas, F. Kahlhoefer,J.M. No, V. Sanz, Z. Ligeti and M. Papucci for very usefulexchanges. O. S. thanks the IFT at the Universidad Autónomade Madrid for the kind hospitality. M. B. G, R. dR. and P. Q.thank Berkeley LBNL, where part of this work has been devel-oped. This work has been supported by the European Union’s

Horizon 2020 research and innovation programme under theMarie Sklodowska-Curie grant agreements 690575 (RISE In-visiblePlus) and N◦ 674896 (ITN Elusives). M. B. G andP. Q. also acknowledge support from the the Spanish ResearchAgency (Agencia Estatal de Investigación) through the grantIFT Centro de Excelencia Severo Ochoa SEV-2016-0597, aswell as from the “Spanish Agencia Estatal de Investigación"(AEI) and the EU “Fondo Europeo de Desarrollo Regional"(FEDER) through the project FPA2016-78645-P. The work ofP. Q. was supported through a a La Caixa-Severo Ochoa pre-doctoral grant of Fundación La Caixa.

Appendix A: Loop factors

The loop contributions to the ALP decay into photons andfermions have been computed in Ref. [48]. The loop functionsin Eq. (13) read

B0 =

( ∑f =u,c,t

NcQ2f B1(τf )−

∑f = d,c,b,`−α

NcQ2f B1(τf )

)(A1)

where

B1(τ) = 1− τ f2(τ) ,

B2(τ) = 1− (τ − 1) f2(τ) ,(A2)

with

f(τ) =

{arcsin 1√

τ; τ ≥ 1 ,

π2 + i

2 ln 1+√

1−τ1−√

1−τ ; τ < 1 .(A3)

where τf ≡ 4m2f/m

2a, Qf denotes the electric charge of the

fermion f and Nfc is the color multiplicity (3 for quarks and

1 for leptons.

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