PHYSICAL REVIEW D 102, 015023 (2020)
Quark flavor phenomenology of the QCD axion
Jorge Martin Camalich ,1,2 Maxim Pospelov,3,4 Pham Ngoc Hoa Vuong ,5 Robert Ziegler ,6,7 and Jure Zupan8 1Instituto de Astrofísica de Canarias, s/n, E-38205, La Laguna, Tenerife, Spain 2Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna, Tenerife, Spain 3School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 4William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 5Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes, CNRS/IN2P3, Grenoble INP, 38000 Grenoble, France 6Theoretical Physics Department, CERN, 1 Esplanade des Particules, 1211 Geneva 23, Switzerland 7Institut für Theoretische Teilchenphysik (TTP), Karlsruher Institut für Technologie (KIT), 76131 Karlsruhe, Germany 8Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA
(Received 20 February 2020; accepted 24 June 2020; published 27 July 2020)
Axion models with generation-dependent Peccei-Quinn charges can lead to flavor-changing neutral currents, thus motivating QCD axion searches at precision flavor experiments. We rigorously derive limits on the most general effective flavor-violating couplings from current measurements and assess their discovery potential. For two-body decays, we use available experimental data to derive limits on q → q0a decay rates for all flavor transitions. Axion contributions to neutral-meson mixing are calculated in a systematic way using chiral perturbation theory and operator product expansion. We also discuss in detail baryonic decays and three-body meson decays, which can lead to the best search strategies for some of the couplings. For instance, a strong limit on the Λ → na transition can be derived from the supernova SN 1987A. In the near future, dedicated searches for q → q0a decays at ongoing experiments could potentially test Peccei-Quinn breaking scales up to 1012 GeV at NA62 or KOTO and up to 109 GeV at Belle II or BES III.
DOI: 10.1103/PhysRevD.102.015023
I. INTRODUCTION gluons [30]. Much activity is being devoted to devise new experimental techniques, and ambitious projects have been The QCD axion is arguably one of the best-motivated proposed for the upcoming years (for a review, see Ref. [31]). particles beyond the Standard Model (SM). Originally In this paper, we reiterate the importance of flavor- predicted by the Peccei-Quinn (PQ) solution to the strong violating transitions for axion searches. Indeed, a QCD CP problem [1–4], i.e., the apparent absence of large CP axion with flavor-violating couplings could well be first violation in strong interactions [5–13], the axion is also an discovered in flavor-physics experiments. While this pos- excellent cold dark matter (DM) candidate in large parts of sibility was already contemplated in the literature for rare the parameter space [14–16]. meson decays and neutral meson mixing [32–35],wego The original electroweak axion models [3,4] were well beyond the state of the art. We show that, in light of strongly disfavored by the nonobservation of flavor- upcoming experiments, there are a number of additional violating Kþ → πþa decays (where a is the axion) [17]. dedicated searches that could and should be performed. Since then, experimental searches for axions have been Besides the two-body meson decays, also the three-body primarily focused on the flavor-conserving axion couplings. and baryonic decays can provide the best sensitivity to Searches using haloscopes [18–25] and helioscopes specific axion couplings. We provide a careful and rigorous [26–29] rely on axion couplings to photons, while searches analysis of the resulting constraints by exploiting the entire using precision magnetometry rely on axion couplings to set of current experimental information. We also improve the predictions for axion-induced neutral-meson oscilla- tions by using effective-field theory methods and derive a Published by the American Physical Society under the terms of new bound from supernova cooling. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Predictions for axion couplings to fermions are model the author(s) and the published article’s title, journal citation, dependent. The only requirement for a successful axion and DOI. Funded by SCOAP3. model is an almost exact global Uð1Þ PQ symmetry that is
2470-0010=2020=102(1)=015023(24) 015023-1 Published by the American Physical Society JORGE MARTIN CAMALICH et al. PHYS. REV. D 102, 015023 (2020) spontaneously broken and anomalous with respect to QCD. from mixing of neutral mesons, Sec. V reviews bounds Therefore, the only coupling shared by all axion models is on flavor-diagonal couplings, and Sec. VI discusses axion the axion coupling to the CP-odd gluon operator, GG˜ , couplings involving the top quark. Finally, in Sec. VII,we arising from the QCD anomaly and responsible for the present the results and experimental projections. Details solution to the strong CP problem. Axion models divide about renormalization of effective axion couplings, exper- into two classes, depending on how the color anomaly imental recasts of two-body meson decays, and hadronic arises. In the KSVZ-type models [36,37] the color anomaly inputs are deferred to the Appendix. is due to a set of heavy fermions that are vectorlike under the SM but chiral under the PQ symmetry. In this case, the II. AXION COUPLINGS TO FERMIONS axion does not couple to elementary SM fermions at tree level. In the DFSZ-type models [38,39], on the other hand, The Lagrangian describing the most general interactions 1 the SM fermions carry PQ charges, and the axion always of the axion with the SM fermions is given by (see also couples to the fermionic currents. Whether these couplings Appendix A) are flavor conserving or flavor violating is a model- ∂ dependent choice. μa ¯ μ V A L ¼ f γ ðc þ c γ5Þf ; ð1Þ aff i fifj fifj j In the original DFSZ model, the PQ charges are taken 2fa to be flavor universal [38,39], so that flavor-violating axion couplings arise only at loop level. In general, though, where f is the axion decay constant; cV;A are Hermitian a fifj flavor-violating axion couplings are present already at matrices in flavor space; and the sum over repeated tree level. This is the case, for instance, in generalized generational indices, i; j ¼ 1; 2; 3, is implied. For future DFSZ-type models with generation-dependent PQ charges convenience, we define the effective decay constants as [40–44], which can also allow one to suppress the axion 2 couplings to nucleons [45–47]. Particularly motivated fa FV;A ≡ : ð2Þ scenarios, which lead to flavor-violating axion couplings fifj V;A cf f at tree level, arise when the PQ symmetry is part of a i j flavor group that shapes the structure of the Yukawa In general, FV;A , i ≠ j, are complex, with ðFV;A Þ ¼ FV;A . sector [32,48]. The PQ symmetry could enforce texture fifj fifj fjfi zeros in the Yukawa matrices [49–51] or be responsible Throughout the paper, we take a to be the QCD axion, so for their hierarchical structure ala` Froggatt-Nielsen that its mass is inversely proportional to fa [83], (FN) [52]. While in the simplest scenario PQ and FN 1012 symmetries are identified [53–55], PQ could also be GeV ma ¼ 5.691ð51Þ μeV : ð3Þ a subgroup of a larger flavor symmetry; see, e.g., fa Refs. [56–65]. Finally, flavored PQ symmetries can arise also in the context of minimal-flavor violation (MFV) For the “invisible” axion, the decay constant is fa ≫ [66,67] or as accidental symmetries in models with gauged 106 GeV [84], in which case the axion is much lighter flavor symmetries [68–71]. than an eV and essentially decoupled from the SM. We will In our analysis, we remain, for the most part, agnostic always be working in this limit, so that in the flavor about the origin of the flavor and chiral structure of axion transitions the axion can be taken as massless for all couplings to SM fermions and simply treat axion couplings practical purposes. to fermions as independent parameters in an effective In this mass range, the axion has a lifetime that is larger Lagrangain. For related studies of axionlike particles with than the age of the Universe and therefore is a suitable DM flavor-violating couplings, see Refs. [72–74] (for loop- candidate. If the PQ symmetry is broken before inflation, induced transitions, see [75–82]). We restrict the analysis to axions are produced near the QCD phase transition and the case of the (practically) massless QCD axion, but our yield the observed DM abundance for axion decay con- 11 13 results can be repurposed for any other light scalar or stants of the order fa ∼ ð10 ÷ 10 Þ GeV [14–16], assum- pseudoscalar with flavor-violating couplings to the SM ing natural values of the misalignment angle. Other fermions, as long as the mass of the (pseudo)scalar is much production mechanisms, e.g., via parametric resonance, smaller than the typical energy release in the flavor allow for axion DM also for smaller decay constants, down 8 transition. to fa ∼ 10 GeV [85]. We will see below that precision The paper is structured as follows. In Sec. II,we flavor experiments are able to test this most interesting introduce our notation for the axion couplings to fermions region of the QCD axion parameter space. and comment on their flavor structure. In Sec. III, we derive the bounds on these couplings from two-body and three- 1Note that diagonal vector couplings are unphysical up to body meson decays, from baryon decays, and from baryon electroweak anomaly terms, which are irrelevant for the purpose transitions in supernovae. Section IV contains bounds of this paper.
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The axion couplings to the SM fermions in the mass V;A simply take cf f to be unknown parameters in an effective basis, cV and cA , are related to the PQ charge matrices i j fifj fifj Lagrangian, which will be constrained solely from data. in the flavor basis, Xf, through III. BOUNDS FROM HADRON DECAYS 1 V;A † † c ¼ ðV Xf Vf V Xq Vf Þ ; ð4Þ Bounds on the vector and axial-vector parts of the flavor- fifj 2 fR R R fL L L ij N violating axion couplings, Eqs. (1) and (2), can be derived from searches for hadron decays with missing energy. where N is the QCD anomaly coefficient of the PQ sym- In this section, we consider the bounds from two-body metry. The unitary rotations V diagonalize the appro- fL;fR decays of pseudoscalar mesons to pseudoscalar and vector † priate SM fermion Yukawa matrices, V y V ¼ ydiag, for mesons, respectively, from three-body meson decays and fL f fR f the “up” and “down” quark flavors, f ¼ u, d. We focus on from decays of baryons. In each case, we first review the axion couplings to quarks and refer the reader to Ref. [86] available and planned experimental measurements and then for present and future prospects for testing lepton flavor– interpret them in terms of the bounds on flavor-violating violating axion couplings. Off-diagonal couplings arise axion couplings. whenever PQ charges, XqL and XfR , are not diagonal in the A. Bounds from two-body meson decays same basis as the Yukawa matrices, yf. Their sizes depend on the misalignment between the two bases, parametrized Because of parity conservation of strong interactions, → by the unitary rotations VfL and VfR (taking XqL and XfR to the P1 P2a decays of a pseudoscalar meson P1 to a be diagonal). pseudoscalar meson P2 are only sensitive to the vector Very different flavor textures of cV;A are possible. couplings of the axion, while the P1 → V2a decays, where fifj V2 is the vector meson, are only sensitive to the axial-vector Provided a suitable set of PQ charges and appropriate flavor couplings (see Appendix C). Searches targeting specifically structures of the SM Yukawa matrices, it is possible for just a þ → πþ V;A the massless axion were performed in K a [89], single off-diagonal coupling to be large c ∼ Oð1Þ, i ≠ j, þ þ þ þ fifj B → K a [90], and B → π a [90] decays. In addition, with all the other off-diagonal couplings zero or very small. searches for SM decays where the invisible final state is V A ∼ 1 For example, one can realize a situation where cbs ¼ cbs a νν¯ pair can be recast to derive limits on the axion V;A 0 1 couplings. This requires that the two-body kinematics of an and all the other cf ≠f ¼ ,bytakingXqL ¼ XuR ¼ , i j (essentially) massless axion is included in the search ðX Þ2 ≠ ðX Þ3, with down Yukawa matrix y such that the dR dR d region, as was the case in the BABAR and CLEO searches only nonvanishing rotation is in the 2-3 right-handed sector, → ð Þνν¯ → πνν¯ → τ → πν¯ ν sRd ∼ 1. Moreover, while one would generically expect axial for B K [91], B [92], and D ð Þ 23 [93]. Note that the corresponding Belle datasets analyzed in couplings cA and vector couplings cV to be of the same fifj fifj Refs. [94,95] cannot be readily used to set bounds on axion X order, the latter can be suppressed in a situation where fR ¼ couplings, because the analyses either cut out two-body − XqL and VfR ¼ VfL , which can arise in models where PQ decays with a massless axion [95] or used multivariate charges are compatible with a grand unified structure [see methods [94] that are difficult to reinterpret for different Ref. [87] for a recent example in SO(10)], and Yukawas are purposes [96] (see also Ref. [97]). Hermitian, positive-definite matrices [see, e.g., Ref. [88] for The available experimental information is summarized in a realization of this scenario in SO(10)]. Table I, where we give the 90% C.L. upper limits on the In the absence of a theory of flavor, we will be agnostic branching ratios for decays involving neutrinos or invisible about the origin of the possible flavor misalignment and massless axions. We include the limits on the decays
TABLE I. Experimental inputs for meson decays; see text for details. We show the 90% C.L. upper bounds on the branching ratios of a pseudoscalar meson P1 to another pseudoscalar (P2) or vector (V2) meson (for sd transitions V2 ¼ ππ instead). The bounds shown are for decays to neutrinos or massless invisible axions. In the latter case, we also show our bounds obtained by recasting related searches for invisible decays (subscript “recast”).
Decay sd cu bd bs −11 −5 −5 BRðP1 → P2 þ aÞ 7.3 × 10 [89] No analysis 4.9 × 10 [90] 4.9 × 10 [90] → −6 −5 −6 BRðP1 P2 þ aÞrecast No need 8.0 × 10 [93] 2.3 × 10 [92] 7.1 × 10 [91] → νν¯ 1 47þ1.30 10−10 0 8 10−5 1 6 10−5 BRðP1 P2 þ Þ . −0.89 × [89] No analysis . × [94] . × [94] −5 BRðP1 → V2 þ aÞ 3.8 × 10 [98] No analysis No analysis No analysis → −5 BRðP1 V2 þ aÞrecast No need No data No data 5.3 × 10 [91] −5 −5 −5 BRðP1 → V2 þ νν¯Þ 4.3 × 10 [98] No analysis 2.8 × 10 [94] 2.7 × 10 [94]
015023-3 JORGE MARTIN CAMALICH et al. PHYS. REV. D 102, 015023 (2020) involving axions that we obtained by recasting the exper- D0 → ρ0a, which results in three charged pions þ MET imental searches for decays involving neutrinos (“recast”). and two displaced vertices. The lack of such analyses Table I shows bounds on the branching ratios for decays to means that there is at present no bound from meson decays þ þ pseudoscalar mesons, P1 → P2a, for K → π a (s → d on axial cu couplings to the axion. Similarly, there is at transition, experimental analysis in Ref. [89]), Dþ → πþa present no publicly available experimental analysis that (c → u transition, our recast of Ref. [93]), Bþ → πþa bounds the B → ρa decays (as discussed above, one cannot (b → d transition, experimental analysis in Ref. [90] and readily use for that purpose the B → ρνν¯ Belle data from our recast of Ref. [92]), and Bþ → Kþa (b → s transition, Ref. [94], while BABAR has not performed such an experimental analysis in Ref. [90] and our recast of analysis). Finally, our recast bounds on B → Kð Þa, B → Ref. [91]). For decays to vector mesons, P1 → V2a, there πa could be easily improved by dedicated experimental is experimental information on the B → K a decay (b → s searches using already collected data. At LHCb, one could transition, our recast of Ref. [91]). In the same section of measure the B → K a and B → ρa branching ratios using 0 Table I, we also include the bounds on Kþ → πþπ a decay ¯0 þ − ¯0 þ − the decay chains such as Bs → K B or B → π B used in Sec. III B below (s → d transition, experimental − − − ¯0 ¯0 followed by B → K ð→ KSπ Þa,orBs → KSB fol- analysis in Ref. [98]). For details on the recast, see lowed by B¯0 → K¯ 0a, ρ0a [105]. One could also attempt Appendix B. more challenging decay chain measurements such as The above analyses could be improved with dedicated Bs → Bsγ, followed by Bs → ϕa or Bs → K a. axion searches applied to existing data or to ongoing We now convert the bounds on the branching ratios in þ → πþ experiments. Sensitivity to the K a decay better Table I to bounds on flavor-violating couplings of axions to than the present world best limit can be achieved at the quarks, Eqs. (1) and (2). The corresponding partial decay NA62 experiment. For a massless axion, an improvement widths are given by by an order of magnitude compared to the BNL result, −11 þ → πþ 7 3 10 8 0 2 BRðK aÞ < . × [89], is expected by 2025 fþð Þ < 2 ;P1 → P2a jFV j [99]. The same flavor transition is probed by KOTO Γ κ ij → π0 ¼ 12 2 ð5Þ searching for the neutral decay mode KL a. The : A0ð0Þ 2 ;P1 → V2a jFA j current limit using data collected in 2015 is BRðKL → ij π0aÞ < 2 × 10−9 and is in the same ballpark as for the decay to the neutrino pair [100]. The KOTO Collaboration with the kinematic prefactor expects the sensitivity to be improved down to the 10−11 M3 M2 3 level [101]. Improved sensitivity in the neutral-kaon κ 1 1 − 2 12 ¼ 16π 2 ; ð6Þ mode can also be expected from the proposed KLEVER M1 experiment [102]. To the best of our knowledge, the future prospects for the where M1 (M2) is the mass of the parent (daughter) meson. → π0 heavy-meson axion decays P1 → P2a or P1 → V2a have Since KL a decay is CP violating, the partial decay not yet been estimated by the experimental collaborations. width in that case is given by Nevertheless, it is clear that improvements over the present 2 V 2 Γ →π0 ¼ κ12f ð0Þ ½Imð1=F Þ ð7Þ situation are justifiably expected. For instance, the exper- KL a þ sd imental error on the Dþ → τþν branching ratio is projected −1 V 0 to be reduced by a factor of 3 with 20 fb integrated and thus vanishes in the CP-conserving limit, ImFsd ¼ ; → π0 þ → πþ luminosity at BESIII [103] compared to the present value cf. Eq. (2). The KL a and K a decay rates obey → π0 ≤ 4 3 þ → [104], indicating a potential significant improvement in the Grossman-Nir bound BRðKL aÞ . BRðK sensitivity to the c → ua transition. The reach on the πþaÞ [106,107]. 2 2 branching ratios for the axionic decay modes of B mesons The form factors fþðq Þ and A0ðq Þ are defined in will be improved with the amount of data expected to be Appendix C, where we also collect the numerical values collected at Belle II, which is roughly a factor of 50 larger used as inputs in the numerical analysis. The resulting V;A compared to the BABAR and Belle samples. bounds on axion couplings Fij are shown in Table III. The Several potentially interesting channels are lacking any implications of these results and future projections will be experimental analyses so far. For example, there is no discussed in Sec. VII. experimental analysis of c → ua transitions that are sensi- tive to the axial-vector coupling; i.e., there are no D → ππ → ρ νν¯ B. Bounds from three-body meson decays Xinv or D Xinv, Xinv ¼ ;a, searches. One could also search for a c → ua signal in D → Ka, D → K a The E787 experiment at Brookhaven performed a search s s 0 decays, all of which could be performed at Belle II and for the three-body Kþ → π πþa decay mediated by the BESIII. Potentially, LHCb could also probe these couplings s → da transition and set the bound BRðKþ → π0πþaÞ ≤ using decay chains, such as B− → D0π− followed by 3.8 × 10−5 at 90% C.L. [98]. The related decay mode
015023-4 QUARK FLAVOR PHENOMENOLOGY OF THE QCD AXION PHYS. REV. D 102, 015023 (2020)
→ π0π0 1 A → 1 A KL a has also been searched for, resulting in the obtained by replacing Reð =FsdÞ Imð =FsdÞ in the rates 0 0 upper limit for light massive axions BRðKL → π π aÞ ≲ of KL. 0.7 × 10−6 [108]. However, this analysis excluded the Better sensitivity to the axial sd axion coupling can be → π0π0 ma ¼ 0 kinematic region and is thus not applicable to achieved by searching for KL a at KOTO [109], the case of the QCD axion [109]. Other decay modes such while a search for Kþ → πþπ0a could be attempted at þ − as KL → π π a or those involving the decays of KS have NA62. Note that Eqs. (8) and (9) can be combined to the not been investigated experimentally. inequality Parity conservation implies that the K → ππa decays are 0 0 þ þ 0 sensitive only to the axial-vector couplings of the axion to BRðKL → π π aÞ ≤ 31BRðK → π π aÞ; ð11Þ quarks (see Appendix C). The form factors entering the τ τ 4 1 predictions are related via isospin symmetry to the form which, besides the ratio of kaon lifetimes L= þ ¼ . [that factors measured in Kþ → πþπ−eþν [110–113], making gives the main effect in the Grossman-Nir bound for precise predictions for K → ππa decay rates possible. The BRðK → πaÞ], also includes the effects of different com- two final-state pions can be only in the total isospin I ¼ 0 binations of Clebsch-Gordan coefficients, form factors, and or the I ¼ 1 state, since the s → da Lagrangian is phase space factors. jΔIj¼1=2, while the initial kaon is part of an isodoublet. The numerical input values for the hadronic matrix Bose symmetry demands the decay amplitude to be elements are described in Appendix C, while the resulting symmetric with respect to the exchange of the two pions. bound on the axion coupling obtained from the current The I ¼ 0 (I ¼ 1) amplitude is even (odd) under this bound on Kþ → πþπ0a [98], is included in Table III. The permutation. The form factors must therefore enter in implications of these results and prospects are discussed combinations which are even (odd) with respect to the further in Sec. VII. exchange of pion momenta, pπ ↔ pπ . The two pions in Other searches using three-body decays could be of 1 2 0 0 0 0 the decay K0 → π0π0a (Kþ → πþπ0a) are in a pure I ¼ 0 interest. Decays such as Bþ → ρ πþa, B → ρ ρ a or → ρ0 → 0ρ0 → ϕρ0 (I ¼ 1) state, and one obtains Bs KS a, Bs K a, Bs a could potentially be even attempted at LHCb, since they result in three or 0 0 2 3 dΓðK → π π aÞ ðm 0 − sÞ four charged particles and a massless invisible particle. L ¼½Reð1=FA Þ 2 K βF2 ð8Þ ds sd 1024π3m5 s Other possible decays of theoretical interest, but probably K only measurable at Belle II, are Bþ → π0πþa, B0 → − 0 − 0 and πþπ a, B → ρþρ a, and Bþ → ρþρ a. Belle II could also access from the ϒð5SÞ run the Bs decays with neutral 0 0 0 þ þ 0 2 − 3 → π → π dΓðK → π π aÞ 1 ðm þ sÞ pions such as Bs KS a, Bs K a, etc. Providing ¼ K β ds jFA j2 1536π3m5 predictions for these decays lies beyond the scope of the sd K present paper, though controlled calculations using QCD 2 β2 2 2β × ðFp þ Gp þ FpGpÞ; ð9Þ factorization and soft-collinear effective theory may be possible [116,117] and could be attempted in the future (see 2 β2 1– 4 2 where s ¼ðpπ1 þ pπ2 Þ and ¼ mπ=s. The functions also related work on two-body form factors [118–121]). Fs, Fp, and Gp correspond to the moduli of the coefficients Until then, one can use as a rough guide the bounds that in the partial-wave expansion of the form factors (which were obtained for the related two-body decays, i.e., Bþ → are complex functions of the kinematic variables of the πþa (on FV Þ and Bþ → ρþa (on FA ), as a ballpark figure − bd bd two-pion system) in the Kþ → πþπ eþν decay channel for what an interesting experimental reach for Bþ → ρ0πþa [110,111,114,115] (see Appendix C). In the first equation, V A (bounding a combination of Fbd and Fbd) might be. we also used the fact that the final state is CP odd, and Finally, one can also use LHCb dimuon data collected therefore the decay occurs through the predominantly for the Bq → μμ analysis to constrain the axial couplings CP-odd component of KL. A A Fbd and Fbs. As long as no vetoes on extra particles in the One can also obtain expressions for other decay modes event are applied in the LHCb analysis, their datasets can be → πþπ− from Eqs. (8) and (9). The amplitude of KL a used to constrain decays with additional particles in the 0 1 contains both I ¼ and I ¼ isospin components, and the final state such as axions. In Ref. [74], the present data decay rate in the isospin limit is A were used to derive constraints on Fbq of the order of 5 −1 − 0 10 GeV, cf. Table III. With 300 fb , the bounds can be ΓðK → πþπ aÞ¼Γ˜ðKþ → πþπ aÞ L strengthened by about an order of magnitude, and could be 0 0 þ 2ΓðKL → π π aÞ; ð10Þ further improved by using also the ATLAS and CMS data on Bq → μμ. The same strategy might also be applied to Γ˜ þ → πþπ0 1 A → where ðK aÞ is obtained by replacing =jFsdj sd transitions using KS → μμ decays, as proposed in 1 A Γ þ → πþπ0 Reð =FsdÞ in ðK aÞ. Rates for the KS decays are Ref. [122].
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C. Bounds from baryon decays D. Supernova bound
Baryon decays, B1 → B2a, are sensitive to both axial and In the core of neutron stars (NS), hyperons coexist vector couplings of the axion. The decay rates are given by in equilibrium with neutrons, protons, and electrons – Λ → [130 133]. The decay na would represent a new 2 2 f1ð0Þ g1ð0Þ cooling mechanism for NS and can thus be constrained ΓðB1 → B2aÞ¼κ12 þ ; ð12Þ jFV j2 jFA j2 by stellar structure calculations and observations. At ij ij exactly zero temperature, the degenerate Λ and neutron distributions must have the same Fermi energy, leaving with the kinematic prefactor κ12 given in Eq. (6). The Λ → 2 2 no phase space for the na decays to occur. The form factors f1ðq Þ and g1ðq Þ are discussed in detail in degeneracy is partially lifted at finite temperature allowing Appendix C. for the Λ → na transitions with a rate that increases At present, there are no published experimental searches with the temperature. The impact of this new cooling → for B1 B2a decays. We therefore set 90% C.L. upper mechanism is maximal during the few seconds after → limits on BRðB1 B2aÞ indirectly. For hyperons, we the supernova (SN) explosion, when a protoneutron star subtract from unity the branching ratios for all channels (PNS) reaches temperatures of several tens of MeV that have been measured so far, adding the experimental [134,135]. Λ errors in quadrature [10].For b decays, we use the SM To estimate the cooling facilitated by the sd-axion prediction for its lifetime at next-to-leading order (NLO) in interaction in this early phase of the supernova evolution, α O 1 s and at ð =mbÞ in heavy quark expansion [123], com- we assume that the PNS is a system of noninteracting (finite pare it to the experimental measurements [10], and ascribe temperature) Fermi gases of neutrons, protons, electrons, Λ → the difference to the allowed value for BRð b B2aÞ.For and Λ baryons that are in thermal and chemical equilib- Λ Λþ → c, we saturate the observed lifetime with the c pa rium. Furthermore, we assume that the neutrinos are decay width. The resulting upper bounds on the branching trapped inside the PNS, while the lepton fraction number, → ratios BRðB1 B2aÞ are collected in Table II and used to relative to baryon number density, is taken to be Y ¼ 0.3 V;A L derive the limits on axion couplings Fij shown in [136]. The occupancy of Λ states is distributed according Table III. Λ 1 1 EΛ−μΛ to the Fermi distribution fp ¼ =ð þ expð T ÞÞ, where The sensitivity could be improved substantially with p is the Λ 3-momentum in the star’s rest frame, EΛ is its dedicated searches for B1 → B2a decays. The BESIII 2 2 2 energy, E p m , and μΛ is its chemical potential. → νν¯ Λ ¼ þ Λ Collaboration plans to measure B1 B2 decays of Neutrons are distributed following an analogous distribu- hyperons by using a sample of hyperon-antihyperon pairs n tion, f 0 , characterized by μ and labeled by the corre- collected at the eþe− → J=ψ peak [127] (see also p n sponding neutron 3-momentum p0, also in the star’s frame. Ref. [128]). At BESIII, one could also study the decays Antiparticles follow identical distributions with the replace- of charmed baryons into axions. Namely, approximately μ → −μ 104 ΛþΛ¯ − 567 −1 ment , so that for the temperatures expected in a c c pairs have been collected in a run with pb ¯ − PNS the densities of Λ and n¯ are negligible. of integrated luminosity at eþe collisions just above the The volume emission rate Q inside the PNS due to the pair-production threshold [129]. Note that bottom baryons − process Λ → na is given by are not produced in the modern eþe machines, since they run at energies below the corresponding pair-production Z 3 Γ Λ → ∞ thresholds, so only LHCb, while challenging, could have mΛ ð naÞ Q ¼ 2 2 2 pdp access to these decays. π ðmΛ − m Þ 0 Z n 0 pmax EΛ − E 0 0 n Λ 1 − n × p dp fp ð fp0 Þ; ð13Þ 0 EΛE TABLE II. The 90% C.L. upper bounds on the branching pmin n fractions for the baryon B1 → B2a decays obtained by adding up the measured branching fractions of the exclusive modes (hyper- 0 0 ons) or by comparing theory predictions for lifetimes with the where pmax (pmin) is the maximal (minimal) neutron measurements (heavy baryons). momentum in the Λ → na decay, if Λ has momentum p ¼jpj (all in the PNS’s rest frame).2 Notice that in the Baryon B1 BRðB1 → B2aÞ90% 0 nonrelativistic limit where p; p ≪ mΛ ∼ mn, and in the Λ 8.5 × 10−3 limit of no Pauli blocking of the final state neutrons this Σþ 4.9 × 10−3 formula reduces to a more familiar form, Ξ0 2.3 × 10−4 Ξ− 6 4 10−4 . × 2In the star’s rest frame Λ is moving in the direction of pˆ. The ’ Λc 1 maximum (minimum) 3-momentum of the neutron in the PNS s −2 rest frame is reached when the neutron recoils in the Λ’s rest Λb 4.1 × 10 frame in the direction (in the direction opposite) of pˆ.
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−mΛ−mn O 2 Q ≃ nnðmΛ − mnÞΓðΛ → naÞe T ; ð14Þ Notice that ða Þ terms that also appear in the trans- formation from Eq. (1) to Eq. (17) do not affect ΔF ¼ 2 where nn is the number density of neutrons. processes and are omitted for that reason. The above L Evaluating the distributions (chemical potentials) for form of aff simplifies somewhat the calculations of the benchmark conditions of T ¼ 30 MeV and nuclear density nonlocal meson mixing matrix elements, Eq. (16). For this, ρ ¼ ρ , using Eq. (13), we obtain for the energy loss per we utilize the appropriate effective-field theories; we use nuc ¯ unit mass ϵ ¼ Q=ρ, chiral perturbation theory (ChPT) for contributions to K−K ¯ mixing, while for the heavy-quark systems, Bd;s − Bd;s and 0 2 0 2 ¯ ϵ 3 6 1038 erg 2 f1ð Þ g1ð Þ D − D, we use the operator product expansion (OPE), ¼ . × GeV V 2 þ A 2 : ð15Þ sg jFsdj jFsdj matching onto local four-quark operators describing the meson mixing. In the following, we use the relativistic Setting as the maximal limit on ϵ the energy lost through 0 0 0 2 δ ⃗− ⃗0 neutrino emission 1 sec after the collapse of the supernova normalization of states, hP ðkÞjP ðk Þi ¼ E ðk k Þ, and 0 − ¯0 SN 1987A, ϵ ≲ 1019 erg=sg[136,137], one obtains bounds the phase convention CPjP i¼ jP i. on jFA j and jFV j in the range 109—1010 GeV. sd sd ¯ Our estimates are afflicted by significant uncertainties. A. K−K mixing Nuclear interactions induce important corrections in the Since ma ≪ 1 GeV, we can use ChPT to describe calculation of the number densities [131,133], and there are contributions from axion exchanges in K−K¯ mixing. For considerable stellar uncertainties stemming from the com- axial couplings, cA , the leading contributions arise at tree plex physics at work in the supernova. Note that the energy fifj level, while for vector couplings, cV , the first nonzero loss per unit mass obtained using the approximate formula fifj in Eq. (14) is independent of the structural details of the contributions are at one loop. PNS, except for the temperature. At T ¼ 30 MeV, this To construct the ChPT Lagrangian in the presence leads to an emission rate that is approximately 40% larger of flavor-violating axions, we use a spurion analysis than in Eq. (15). More than anything, the emission rate [139,140]. In terms of the (pseudo)scalar interactions, suffers from the uncertainty in the temperature of the the Lagrangian for QCD with a flavor-violating axion central region. Variation of this quantity from 20 to can be conveniently written as 40 MeV changes Q by 2 orders of magnitude. Finally, L ¯ ∂ a a − ¯M our bound crucially relies on the validity of the standard QCDþa ¼ qði= þ gs=G T Þq q qq scenario for the SN explosion as applied to SN 1987A, − aq¯ðχ − iχ γ5Þq; ð19Þ which was disputed in a recent publication [138]. S P where we keep only light quarks, q ¼ðu; d; sÞ. The IV. BOUNDS FROM MESON MIXING diagonal mass matrix is Mq ¼ diagðmu;md;msÞ, while χ 3 3 The exchanges of axions with flavor-violating couplings S;P are × Hermitian matrices describing the quark- contribute to ΔF ¼ 2 transitions and can modify meson axion couplings, 0 1 mixing rates from the SM predictions. The contribution from 00 0 axion exchanges to the mixing amplitude of the P0 − P¯0 B − C 00md ms neutral meson system is given by the time-ordered correlator χ B V C S ¼ i@ Fds A; ð20Þ Z − 0 ms md 0 i 4 0 0 FV − L L 0 ¯ sd M12 ¼ 4 d xhP jTf affðxÞ; affð ÞgjP i; ð16Þ mP 0 1 2mu A 00 L Fuu where affðxÞ is the axion-fermion interaction Lagrangian B C B 2 C in Eq. (1). χ − 0 md mdþms P ¼ B FA FA C: ð21Þ @ dd ds A In this section, it will prove useful to use the following 2 0 msþmd ms form of the axion-fermion interaction Lagrangian, A A Fsd Fss a L ¼ −i f¯ ½ðm − m ÞcV aff i fi fj fifj The off-diagonal couplings in (20) and (21) induce kaon 2fa oscillations. The mixing matrix element M12 follows from A þðm þ m Þc γ5 f ; ð17Þ L fi fj fifj j a double insertion of the interaction Lagrangian aff ¼ −aq¯ðχS − iχPγ5Þq, where q ¼ u, d, s; cf. Eq. (16). which is obtained from Eq. (1) with the axion-dependent The Lagrangian for QCD with the flavor-violating field transformation L 3 axion, QCDþa, is formally invariant under SUð ÞR × 3 → χ a V − A a V A SUð ÞL transformations, qR;L gR;LðxÞqR;L,ifa S;P i2f ðcij cijÞ i2f ðcijþcijÞ fi → ½e a PL þ e a PR fj: ð18Þ and Mq are promoted to spurions that transform as
015023-7 JORGE MARTIN CAMALICH et al. PHYS. REV. D 102, 015023 (2020)
→ † s þ ip gRðs þ ipÞgL ð22Þ Expanding (24) in meson fields gives
2 a and that take the values Lð Þ ⊃− f m2 K¯ 0 ChPTþa FA K K ds M χ χ η s ¼ q þ Sa; p ¼ Pa: ð23Þ − iffiffiffi a 2 − 2 ¯ 0 π0 ffiffiffi p V ðmK mπÞK þ p 2 Fds 3 We also define the spurion χ ¼ χ þ iχ that does not 2 S P a 2m contain the axion field and that transforms similarly as þ K K0ðK¯ 0Þ2 þ H:c: ð26Þ † FA 3f χ → g χg . The identification of this symmetry structure ds K R L Here, we kept only the terms relevant for K-K¯ mixing and allows one to build the ChPT Lagrangian, including the pffiffiffi 2 155 6 chiral-symmetry breaking terms. The introduction of the replaced f with the kaon decay constant, fK ¼ . 0 4 spurion χ is needed because the axion cannot be treated . MeV [10], thus capturing part of the SU(3) breaking merely as an external field and enters in the chiral loops that corrects our results at higher orders in the ChPT expansion [139]. We also traded the products B0m for the with two insertions of χS;P. Thus, the ChPT Lagrangian is q also invariant under a Z2 symmetry that transforms meson masses squared [139,140,142]. The two terms in χ → −χ, with all the other fields taken to be Z2 even. (25), expanded in meson fields, are Up to overall normalization factors, the axion-induced ¯ 2 2 contributions to the K−K mixing amplitude have the Lð4Þ ⊃ 2 mK α 2α ν ν ChPTþa A ð 0 þ 1Þ scalings 2m M12 ∼ ðp=ΛχÞ ð1=FÞ F . The integer ν char- F K ds acterizes the usual chiral scaling [139,140] where the 2 − 2 2 mK mπ ¯ 0 2 derivatives of meson fields (and the axion) count as − ðα0 − 2α1Þ ðK Þ þ ; ð27Þ FV OðpÞ and the quark masses count as Oðp2Þ and where ds Λχ ≃ 4πfπ ∼ O 1 fπ the UV cutoff of ChPT is ð GeVÞ ( is where again we only keep the terms relevant for K-K¯ ν the pion decay constant). On the other hand, F counts the mixing. 1 V;A ¯ number of =Fds insertions in the amplitude. Thus, the The axion exchange contributions to the K-K mixing 2 chiral counting of the spurions is Mq ∼ Oðp Þ and νF ¼ 0 amplitude due to axial vector couplings are proportional χ ∼ O 2 ν 1 1 A 2 O 4 and S;P ðp Þ and F ¼ . In the following, we use to ð =FdsÞ and are, up to ðp Þ in the chiral counting, ChPT to calculate the leading axion-exchange contribu- given by ¯ tions to the K−K mixing amplitude including corrections ν ≤ 4 2 2 2 up to NLO corrections in the chiral counting, . This A fK mK mK M ¼ 1 − ðα0 þ 2α1Þ requires two insertions of 1=FV;A and thus ν ¼ 2. 12 FA 2 f2 ds F ds K The leading-order (LO) ChPT Lagrangian, including a 8 2 2 mK 1 − mK as the light degree of freedom, is given by þ 3 16π2 2 log μ2 : ð28Þ fK 2 ð2Þ f μ † L Tr ∂μU∂ U ChPTþa ¼ 4 ð Þ The first term in the parentheses is due to the tree-level 2 axion exchange, Fig. 1(a), and is induced by the first term f † þ B0 Tr½ðs − ipÞU þðs þ ipÞU in the expanded LO ChPT Lagrangian, Eq. (26). The NLO 2 correction is due to the loop diagram in Fig. 1(b), induced 2 1 μ ma 2 by the last term in Eq. (26). We use dimensional regulari- þ ∂μa∂ a − a ; ð24Þ 2 2 zation in the MS scheme and fix the renormalization scale to μ ¼ mK in order to minimize the size of the chiral while the relevant terms in the NLO ChPT Lagrangian are logarithm. The counterterms in Fig. 1(d) which cancel the μ ð4Þ 2 2 † † dependence of the loops are provided by the two terms in L ⊃ B f α0Tr χU Tr χ U 4 ChPTþa 0 ð ½ ½ the expanded Oðp Þ Lagrangian, Eq. (27). While the † 2 † 2 þ α1Tr½ðχU Þ þðχ UÞ Þ: ð25Þ numerical values of the low-energy constants α0;1 are not known, we can estimate their sizes by varying μ in Here, UðxÞ¼expðiλaπa=fÞ is the unitary matrix para- the loop contribution around its nominal value by a factor metrizing the meson fields [139,140]; B0 is a constant of 2, while artificially setting α0;1 to zero. This estimates the related to the quark condensate; B0ðμ ¼ 2 GeVÞ¼ Oðp4Þ contribution in (28) to be ð17% 23%Þ of the 2.666 57 GeV; f is related to the pion decay constant 2 ð pÞffiffiffi Oðp Þ one. f ≃ fπ= 2 ¼ 92.2ð1Þ MeV [141], with normalization The vector couplings of the axions first contribute at − 2 2 4 h0ju¯γμdð0Þjπ ðpÞi ¼ ipμfπ; and α0;1 ∼ f =Λχ are the Oðp Þ through the one loop diagram in Fig. 1(c) with the (unknown) low energy constants. counterterms in Fig. 1(d), resulting in
015023-8 QUARK FLAVOR PHENOMENOLOGY OF THE QCD AXION PHYS. REV. D 102, 015023 (2020)
(a) (b) (c) (d)
FIG. 1. Contributions to K0 − K¯ 0 mixing from exchanges of the flavor-violating axion, up to and including one loop in ChPT.
ΓV 2 2 2 Δ 0 028 6 2 1 A 2 V 12 fK mK mπ mK=mK ¼ . ð Þ GeV Re½ =ðFdsÞ M12 − i ¼ 1 − 2 V 2 2 2 2 Fds mK 0 0018 18 1 V þ . ð Þ GeV Re½ =ðFdsÞ ; ð30Þ 2 1 mK × 2 2 I0ðzπÞþ I0ðzηÞ Δ 2 32π f 3 where we use the relation mK ¼ ReM12 and have set K α α 0 2 2 0 ¼ 1 ¼ . The prefactor of the vector couplings carries mK O 100% þ ðα0 − 2α1Þ : ð29Þ an ð Þ relative uncertainty because of the unknown f2 K contributions of α0;1 coefficients, entering at the same order in ChPT as the loop contribution. The prediction of ΔmK in the SM has large uncertainties stemming from Above, we have rearranged the factors of fK and mK to long-distance contributions. Therefore, to obtain the 1 V 2 make the dependence on ð =FdsÞ as well as the structure bounds on the axion couplings from this observable, we of the chiral corrections more transparent. The two- Δ expt: conservatively saturate the experimental value mK ¼ 2 − 2 μ2 − − −12 point loop function is I0ðzÞ¼ logðmK= Þ z log z 3.484ð6Þ × 10 MeV [10] with the axion-exchange con- þ ð1 − zÞ logðz − 1 − i0 Þ, where the argument is zϕ ¼ α 0 A 2 0 tribution. Assuming 0;1 ¼ , this leads to jFdsj > . × 2 2 V 6 5 mϕ=m . The Γ12 receives a contribution from the disconti- 10 V 5 1 10 K GeV and jFdsj > . × GeV at 90% C.L., for the nuity, ImI0ðzπÞ, i.e., from the on-shell part of the diagram 1 A;V 2 case in which =ðFds Þ are real. These are the bounds Fig. 1(c) with pion and axion running in the loop. The quoted in Table III. Taking into account the estimate for the choice μ ¼ mK again minimizes the log. However, since α A range of values for 0;1 reduces the bound on jFdsj by about the low-energy constants α0 1 are unknown, the predicted ; 10%. Note that without fixing the values of α0 1 there is no V ; M12 is quite uncertain. In the numerical estimates, we use bound on jFV j. Allowing for large cancellations up to 1% α 0 ds 0;1 ¼ and assign 100% uncertainty to the resulting between the loop diagram and the counterterm contribu- V estimate for M12. V tions relaxes the bound on jFdsj by an order of magnitude. As we will see in the next subsection, for heavy quarks, To obtain the bounds on non-SM CP-violating contri- both s-channel and t-channel exchanges of axions lead butions to K-K¯ mixing, we use the normalized quantity to contributions that are parametrically of similar size. However, this is not the case for light quarks. The above ϵSMþa j K j Cε ¼ : ð31Þ ChPT analysis implies that the tree-level s-channel axion K ϵSM 1 A 2 j K j exchange [necessarily proportional to =ðFdsÞ ] is leading in the chiral expansion, while the t-channel contribution is For the theoretical prediction of ϵK, we use the expression subleading. ¯ [143] Note that, in addition to the contributions to K-K mixing from axion exchange, there can be other contributions from ImM12 iϕϵ ϵ ¼ e sin ϕϵ þ ξ ; ð32Þ UV physics which are parametrically of the same order, K Δm ∝ 1 V;A 2 K M12 ð =Fds Þ . In the numerical analysis, we set these UV model-dependent contributions to zero, keeping in where mind that their presence can modify our numerical results. It is also interesting to note that the previously used ImΓ12 ξ ≃ : ð33Þ estimate of the axion-induced meson-mixing amplitude, ΔΓK based on the “vacuum insertion approximation,” is equiv- alent to the LO displayed in Eq. (26), but with a factor 5=6 We take the values for ΔmK ¼ mL − mS, ΔΓK ¼ ΓS − ΓL, instead of 1, and no contribution from the vector cou- and ϕϵ ¼ arctanð2ΔmK=ΔΓKÞ from experiment [10]. With plings [33]. the SM prediction for jϵKj from Ref. [144], and the axion Numerically, Eqs. (28) and (29) give for the contribution contributions to M12 and Γ12 from Eqs. (28) and (29), of the axion to the neutral-kaon mass difference we get
015023-9 JORGE MARTIN CAMALICH et al. PHYS. REV. D 102, 015023 (2020) 2.5ð2Þ × 107 GeV 2 we are calculating the OPE at one single kinematic point δCϵ ¼ Cϵ − 1 ¼ Im K K A over the physical cut of the a þ hadrons intermediate Fds state. We assume the mass of the heavy quark to be large 5 5 105 2 ð Þ × GeV enough so that the energies involved correspond to the þ V ; ð34Þ Fds perturbative regime of QCD and that the violations of quark-hadron duality are small. where in the numerical expressions we set the unknown We first present the results for the B0 − B¯0 system, and low-energy constants to zero, α0 1 → 0, with the quoted 0 ¯0 0 ¯ 0 ; then extend the results to Bs − Bs and D − D systems. errors our estimates of the resulting errors due to this We work at LO in 1=m . The full basis of local operators at 3 Q approximation. this order is given by [146] The global Cabibbo-Kobayashi-Maskawa (CKM) fit by 0 87
3 1∶100 There is a cancellation between the contributions of ¯ V V FIG. 2. Matching of the axion-mediated contributions to B − B ImM12 and ImΓ12 to ϵK in Eq. (34) that makes the prediction of the vectorial axion couplings to this observable even more mixing, diagrams a1 and a2, onto the dimension-6 local operators uncertain. For the experimental inputs, we use the Particle Data in the OPE, diagram b, where we also indicate the operators that
Group (PDG) values [10], using mK0 and mπ0 in (29) for the kaon receive the contributions. Single (double) lines represent the d and pion masses, respectively. quark (b quark).
015023-10 QUARK FLAVOR PHENOMENOLOGY OF THE QCD AXION PHYS. REV. D 102, 015023 (2020)
→ A 2 0 2 3 106 and similarly for the Bs system, with d s. The matching jFdbj > . ð . Þ × GeV; ð45Þ gets corrected at OðαsÞ due to hard-gluon contributions to O Λ the matching, and at ð QCD=mbÞ from corrections to the jFA j > 4.0ð7.7Þ × 105 GeV; ð46Þ heavy-quark limit. sb The matrix elements of the operators in Eq. (37) between when the weak phase of FA;V is aligned with (differs by B0 and B¯0 states are given by hB0jO jB¯0i ∝ f2 B , where f qb i B i B π=2 from) the SM contribution. The first choice corre- is the B meson decay constant and Bi is the appropriate bag parameter. Both of these are well known and have been sponds to MFV-like couplings of the axion, where the CP calculated using lattice QCD [147–150]. Parity conserva- violating phase is the SM one, while the second choice corresponds to generic couplings with new weak phase that tion of strong interactions implies hO˜ i¼hO i so that the i i maximizes the axion exchange contribution to the meson contributions to Δm from the 1= FA FV terms in (38) B ð db dbÞ mixing phase. In deriving the bounds above, we chose in and (39) cancel. Using the lattice results from Ref. [150], each case the sign of the new-physics contribution that we find leads to the weakest bound. For each of the bounds, we also assume that the axion only has axial or vector couplings. Δm 0.053ð5Þ GeV2 0.016ð2Þ GeV2 B ¼ Abs − ; These bounds are also collected in Table III. m ðFA Þ2 ðFV Þ2 0 − ¯ 0 B db db Extending the above OPE results to the D D system Δm 0.077ð8Þ GeV2 0.020ð2Þ GeV2 could be problematic because the violations offfiffiffi quark- Bs − p ¼ Abs ; ð41Þ ≃ 0 A 2 V 2 hadron duality in the a þ hadrons system at s mD mBs ðFsbÞ ðFsbÞ and the αs corrections in the OPE expansion may be large. ðbsÞ Extending naively our results, we obtain for the axion where Δm ¼ 2jM12 j and with the dominant theoretical BðsÞ contribution to the mass difference Λ uncertainty due to power corrections entering at QCD= ∼ 0 1 mb . . The reduced sensitivity to the vectorial couplings Δm 2jMcuj 1 D ¼ 12 ¼ jhD0jLa ð0ÞjD¯ 0ij in this formula was anticipated already in the vacuum- m m m2 eff insertion approximation estimate [33]. The SM predictions D D D 0 12 2 0 034 2 for Δm are consistent, within approximately 10%, with . GeV . GeV Bd;s ¼ Abs − : ð47Þ −10 A 2 V 2 Δ 3 354 22 10 ðFucÞ ðFucÞ the measured values mBd ¼ . ð Þ × MeV and Δm ¼ 1.1688ð14Þ × 10−8 MeV [10]. Comparison with Bs In the last line, we used the lattice QCD results for the bag our predictions immediately shows that we can expect the parameters from Ref. [152] (see also Refs. [163,164]). To bounds on FV;A at the level of 105 to 106 GeV. sb;db obtain a bound on jFA;Vj, we saturate the experimental To derive the precise bounds on the allowed axionic uc value of ΔmD with the axion contribution because the SM contributions, we consider simultaneously both the con- prediction is poorly known. tributions to Δm as well as to the mixing phase. We thus Bd;s A more stringent bound is obtained for CP-violating define axionic contributions from an experimental bound on the D − D¯ mixing phase [126] 0 LSMþa ¯0 2iϕ hBqj eff jBqi C e Bq ; q d; s : 42 Bq ¼ 0 ¯0 ð ¼ Þ ð Þ B LSM B M Γ h qj eff j qi ϕ12 ≡ ϕ2 − ϕ2 ; ð48Þ In the SM, C ¼ 1 and ϕ ¼ 0. From the global fit to Bq Bq where in the most commonly used phase convention ¯ CKM observables, including Bq − Bq mixing observables, 1 05 0 11 ϕM ϕΓ Γ the UTFit Collaboration obtained CBd ¼ . . , 2 ¼ argðM12Þ; 2 ¼ argð 12Þ: ð49Þ ϕ −2 0 1 8 1 110 0 090 ϕ Bd ¼ð . . Þ°, CBs ¼ . . , and Bs ¼ ð0.60 0.88Þ° [124,145] (see also Ref. [151]). The SM The axion contributes at tree level to M12 and only at loop predictions are obtained using the inputs in Table IV and level to Γ12. For present experimental bounds, the SM Ref. [150], and the results for the CKM matrix elements of contributions to ϕ12 are negligible, i.e., ϕ12 at present the “New Physics Fit” of the UTfit Collaboration (summer experimental levels would be induced entirely by the axion 2018) [124,145]. This leads to the 90% C.L. bounds exchanges, and thus
V 1 1 1 3 106 2 a 2 a jFdbj > . ð . Þ × GeV; ð43Þ M ImM12 ImM12 ImM12 ϕ12 ≃ ϕ2 ¼ ≃ ¼ : ð50Þ jM12j ΔmD xΓ V 2 0 3 8 105 jFsbj > . ð . Þ × GeV; ð44Þ cu a Above, we shortened M12 → M12, while M12 is the mixing and matrix element due to the axion exchange.
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Comparison with the experiment, the Heavy Flavor coefficients and thus is expected to be Oð1Þ in the bulk Averaging Group (HFLAV) Moriond 2019 average ϕ12 ¼ of UV axion models. −ð0.25 0.97Þ° [125], and PDG value for mass difference Axion couplings to electrons are constrained from star 8 −1 ΔmD ¼ð95 43Þ · 10 ℏs , gives at 90% C.L. cooling, specifically from their impact on the luminosity function of white dwarfs (WDs) as assessed in Ref. [170], V 7 ≡ 2 ≥ 4 9 4 6 109 jFcuj > 0.2ð2.5Þ × 10 GeV; ð51Þ giving Fe fa=Ce . ð . Þ × GeV at 90 (95)% C.L. Interestingly, there are hints of anomalous energy loss A 7 jFcuj > 0.5ð4.8Þ × 10 GeV; ð52Þ in stars that may be explained by axions with nonzero couplings to electrons (and possibly photons) of the order ≈ 7 109 when saturating ΔmD (ϕ12) with the axion contribution, of Fe × GeV [171,172]. Eq. (47) [Eq. (50)] and where in Eq. (50) we used the Finally, axion couplings to nucleons are constrained by central value of ΔmD. axion emission from the core of supernovae. Converting the At the end of LHCb Upgrade II, the experimental results of Ref. [173] to our notation gives the bound 9 constraints are expected to reach the parametric sizes of FN ≡ 2fa=CN ≳ 1.0 × 10 GeV, where the effective cou- M the SM contributions to the two mixing phases, ϕ2 ∼ pling to nucleons, CN, is given in terms of axion couplings Γ −3 2 2 0 29 2 0 27 ϕ2 ∼ 10 . For the projections of future sensitivities, we to protons and neutrons as CN ¼ Cn þ . Cp þ . CpCn. M thus still use the projected 90% C.L. bound on jϕ2 j < The nucleon couplings are related to the quark couplings 12 2.0 × 10−3 (approximate universality fit projection in (taking a UV scale of 10 GeV and including only QCD Ref. [126]), assuming that the axion saturates the upper running effects) [45,174] bound; i.e., we assume no cancellations with the poorly 0 50 5 A A − 1 − 2δ known SM contribution. This gives for the expected future Cp þ Cn ¼ . ð Þðcuu þ cdd Þ ; ð54Þ V 7 A 8 sensitivities jFcuj > 7.8 × 10 GeV, jFcuj > 1.4 × 10 GeV, Δ 1 − z while the bounds from mD do not change. C − C ¼ 1.273ð2Þ cA − cA − ; ð55Þ Finally, we reiterate that the above mixing bounds on p n uu dd 1 þ z V;A F 0 assume that the axion contribution is not canceled by qq where z m =m 0.48 3 and δ ≡ 0.038 5 cA the UV contributions from heavy particles present in the ¼ u d ¼ ð Þ ð Þ ss þ 0 012 5 A 0 009 2 A 0 0035 4 A UV theory, even though the latter are expected to be . ð Þccc þ . ð Þcbb þ . ð Þctt. Similar to parametrically of the same size. axion-photon couplings, the couplings to protons and neutrons can be suppressed only by tuning; see Refs. [45–47]. However, as discussed in Sec. III D, this V. BOUNDS ON FLAVOR-CONSERVING bound relies on the validity of the standard scenario for AXION COUPLINGS the SN explosion. For completeness, we briefly review the constraints on Direct constraints on flavor-conserving axion couplings flavor-conserving axion couplings that are dominated by to b and c quarks are relatively weak. The bounds on ϒ 1 → γ ψ → γ astrophysical bounds from star cooling. These constrain branching ratios for ð SÞ a and J= a decays A ≳ axion couplings to photons, electrons, and nucleons, [175,176], with a invisible, translate to bounds Fbb 1 A ≳ 0 4 defined by the Lagrangian TeV and Fcc . TeV, respectively [3,177,178]. Besides the radiative effects on nucleon couplings through α ∂μ the parameter δ introduced below Eq. (55), other indirect L em a μν ˜ μν a ¯γ γ ¼ CγF F þ Cee μ 5e constraints arise from loop contributions to the axion- 8π fa 2fa electron coupling. As discussed in the next section, this ∂μa ∂μa þ Cpp¯γμγ5p þ Cnn¯γμγ5n: ð53Þ contribution is largest for the top coupling and of the order 2fa 2fa A 9 of Ftt ≳ 3.4 × 10 GeV. The bound on the diagonal bottom (charm) coupling is weaker by a factor m2=m2 (m2=m2), For a wide range of axion masses, the strongest bound on b t c t 7 and thus of the order of 940 TeV (45 TeV). axion coupling to photons Fγ ≡ fa=Cγ ≥ 1.8 × 10 GeV (95% C.L.) is set both by the CAST experiment [165] and the evolution of horizontal branch (HB) stars in globular VI. AXION COUPLINGS TO TOP QUARKS clusters [166]. The CAST successor, IAXO, is expected Finally, we discuss flavor-violating couplings of the to improve this bound by about an order of magnitude axion that involve the top quark. Direct bounds on flavor- [167,168]. For restricted ranges of DM axion masses close violating axion-top couplings are, in principle, accessible at ∼ 3 μ to ma eV, the ADMX experiment probes the axion- the LHC. Monotop searches [179,180] are sensitive to the 12 photon couplings up to Fγ ≳ 10 GeV [169]. Note that the t → ca; ua flavor-changing neutral currents (FCNC) tran- photon coupling Cγ ¼ E=N − 1.92ð4Þ can be suppressed sitions, and tt¯ þ MET [181] to the diagonal tta¯ couplings. only by tuning the ratio of EM and color anomaly However, these searches bound fa only very weakly, at the
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e.g., DFSZ-type models where all charged lepton couplings are suppressed if the ratio of the Higgs vacuum expectation values is small. Assuming that indeed ΔcijðμÞ ≫ cijðfaÞ, the radiative Δ V Δ A contributions to FCNCs ( csd) and WD cooling ( cee) give stringent bounds on top-axion couplings. V ≳ The strongest constraint on the sd coupling, Fsd 6.8 × 1011 GeV, translates to a bound on the diagonal A ≳ 2 5 107 FIG. 3. Radiative contribution to the flavor mixing of the top-axion coupling Ftt . × GeV, as well as the V;A 8 V;A couplings of the axion to the quark doublets. off-diagonal couplings Ftc ≳ 3.2 ×10 GeV and Ftu ≳ 7.0 ×108 GeV. Here, we have assumed real couplings level of the electroweak scale. Namely, simply restricting for simplicity (purely imaginary couplings result in a the contribution of t → ca to the total width of the top to be slightly different bound), and set in (56) yt ¼ SM 10 O 1 ≳ O yt ðμ ¼ MZÞ, fa ¼ 10 GeV and used the values of smaller than ð GeVÞ gives a bound fa ðvEWÞ, while the CKM elements of the New-Physics fit of the UTfit the monotop searches [180] may lead to a bound on fa that is roughly an order of magnitude stronger. Collaboration (summer 2018) [124,145]. With the same Much more stringent bounds on top couplings to axions numerical inputs, one can derive a bound on the diagonal can be obtained from virtual corrections. Because of the top couplings from WD cooling [33] using the bound ≥ 4 9 109 large top mass, the radiative Yukawa corrections from Fe . × GeV at 90% C.L. This translates to A ≳ 3 4 109 axion-top couplings (Fig. 3) can give sizable contributions Ftt . × GeV, which is about 2 orders of magnitudes to other axion-fermion couplings that are strongly con- stronger than the bound from K → πa. Note that similar strained. The leading log expressions for the radiative radiative contributions are obtained for diagonal light quark contributions are derived in Appendix A, with the couplings which can have an impact in the bounds derived from supernovae in specific models. yt-enhanced contributions collected in Eqs. (A10)–(A14). The most relevant effects are the radiative corrections to V the flavor-violating coupling csd, subject to stringent VII. RESULTS constraints from K → πa (cf. Sec. III A), and the flavor- A In Table III, we summarize the 90% C.L. lower bounds conserving coupling to electrons, cee, which is constrained V;A 2 on the couplings Fij , Eq. (2), obtained using the flavor- by WD cooling (cf. Sec. V). The yt enhanced contributions changing processes discussed in the previous sections. to these couplings are Some of the bounds are afflicted by large theoretical 2 uncertainties which could in principle change the quoted y f Δ V μ t a 2 V A csdð Þ¼ 2 log VtsVtdðctt þ cttÞ numerical values by as much as an order of magnitude. The 64π μ A;V X affected bounds are (i) the bounds on Fsd from supernova − V − A Λ → VksVtdðcukt cuktÞ cooling due to na transition, where the temperature of k the PNS and the interpretation of the SN 1987A neutrino X events are two important sources of potential systematic V A − VtsVkdðctu − ctu Þ ; ð56Þ k k errors; (ii) the meson-mixing bounds from ΔmD and from k V kaon mixing on Fds suffering from poorly known theo- 6y2 f retical predictions; and (iii) bounds on top-axion couplings ΔcA μ t a cA; eeð Þ¼16π2 log μ tt ð57Þ relying on additional, model-dependent assumptions on the absence of cancellations with tree-level contributions. In where μ is the low-energy scale, while on the right-hand Table III all these bounds are flagged by a “†” superscript. side, the couplings are given at the UV scale fa. Future projections for the bounds based on ongoing or These contributions are added to the tree-level sd − a future experiments are given in Table III inside parentheses and ee − a couplings, so from observations, we can bound and will be discussed below. Furthermore, we recall that all only the sum of the loop-induced and tree-level coupling, meson-mixing bounds are sensitive to additional contribu- cijðμÞ¼ΔcijðμÞþcijðfaÞ. Barring cancellations between tions from UV physics. the two contributions, one can thus obtain bounds on the In Fig. 4, we summarize the most relevant bounds for the top quark couplings to axions. The UV values of sd − a different flavor sectors and types of couplings, as well as and/or ee−a couplings can be suppressed in certain the potential reach of ongoing and future experiments. models. A suppressed sd − a coupling arises in scenarios These are compared to the strongest constraints on the where the down-quark sector is aligned or all flavor- diagonal axion couplings to electrons (WD cooling), violating axion couplings are strongly suppressed [76–80]. nucleons (SN 1987A), and photons (HB/CAST), which A scenario with suppressed ee couplings instead arises in, were discussed in Sec. V. For the projected reach on the
015023-13 JORGE MARTIN CAMALICH et al. PHYS. REV. D 102, 015023 (2020) axion photon coupling “Fγðprosp.Þ,” we quote the pros- The K and B meson decays also probe top-quark axion pects for IAXO [167,168]. In Fig. 4, we do not show the couplings indirectly through loop effects; cf. Sec. II and bounds from ADMX, since these depend heavily on the Appendix A. These constraints become important in the assumed axion mass but for particular axion mass ranges scenarios with down-quark alignment or flavor-diagonal can be much stronger than the bounds from helioscopes. (or MFV) couplings in the UV. In particular, the strong bound on the Kþ → πþa branching fraction translates into bounds for the t → c and t → u transitions at the level of A. Present bounds 108–109 GeV. Although we show these constraints in The strongest bound on QCD axion couplings is of the Table III, we did not include them in Fig. 4 as they only order of 1011−12 GeV and due to the stringent constraints constrain the left-handed combination and require the V þ → πþ absence of possible cancellations with tree-level sd − a on Fsd from K a decays; cf. Table III. This bound exceeds even the stellar axion bounds, F ≳ 109 GeV, couplings. e;N – which rely on the diagonal couplings. If the flavor structure Direct probes of flavor-violating up quark axion cou- is not completely generic, the vectorial sd − a couplings plings that are potentially sensitive to relatively high scales could be suppressed, and other probes can become equally are possible with charmed mesons and baryons. The most − or more important (so, clearly, Kþ → πþa should not be sensitive probe of the flavor-violating vectorial cu a þ → πþ interpreted as the strongest constraint on the QCD axion coupling turns out to be the two-body D a decay. þ → τþ → πþν ν¯ independently of the underlying axion model). For in- Recast of the D ð Þ analysis of CLEO and V ≳ O 108 stance, as discussed in Sec. II, the sd − a couplings could BESIII gives the bound Fcu ð Þ GeV. Axial-vector − be strongly suppressed by suitable alignment of PQ-charge cu a couplings are currently probed predominantly 0 A ≳ matrices and SM Yukawas. by D mixing with lower bounds in the range Fcu 6 8 The axial-vector sd − a couplings can be accessed 10 –10 GeV depending on the CP-violating phase of the from three-body kaon decay, Kþ → πþπ0a, from hyperon axion contribution. These couplings can also be directly Λ → decays, and indirectly from K-K¯ mixing, resulting in probed by c pa decays, which in the future could A lower bounds that are in the ∼Oð106−7Þ GeV range. In provide the best bounds on approximately real Fcu (this principle, the best bound on the axial sd − a coupling, case is not included in Fig. 4 for simplicity). A ≳ O 109 Λ → Fsd ð Þ GeV, comes from hyperon na transi- tions in the SN 1987A supernova. In fact, at face value, this B. Future projections is the strongest bound of all the axial-vector axion-quark The sensitivity to the couplings of the flavored axion to couplings in our analysis. However, as pointed out already the quarks can be greatly improved with future experi- above, the supernova bounds should be used with caution ments. Dedicated searches for a massless axion in two- due to difficult to estimate systematics. As discussed below, body kaon decays at NA62 and KOTO are expected to quite impressively, the projected improvements on the reach a sensitivity to the branching fraction better than Λ → na decay branching ratio reach at BESIII will start 10−11 [99] (cf. Sec. III). We thus use to compete with this supernova bound, but using well- controlled transitions measured in the lab. þ → πþ 10−11 BRprojðK aÞ < ; ð58Þ The two-body decays of B mesons probe all the b → s 8 and b → d couplings up to a scale of ∼10 GeV, with the as the (conservative) experimental projection. As shown in A → ρ exception of Fbd due to the absence of dedicated B a Table III, this will allow us to push the lower bounds on searches at the B factories (the related Belle B → ρνν¯ vectorial sd − a couplings beyond a scale of 1012 GeV. 0 0 analysis cannot be readily recast, as discussed in Sec. III). The three-body kaon decays KL → π π a can be The bounds from two-body heavy-baryon Λb decays and potentially searched for at KOTO [109]. However, to this ¯ from Bd;s − Bds mixing are more than 2 orders of magnitude date, there is no analysis of the sensitivity KOTO could less sensitive. Only the bound on the axial-vector bd − a achieve for this decay channel. A direct extension of the ¯ 6 7 → π0π0 ≲ coupling from Λb decays and B − B,inthe10 –10 GeV current experimental sensitivity for BRðKL aÞ ballpark, are phenomenologically relevant constraints. 10−6 [108] to the kinematics of a massless axion would give A ≥ 6 2 108 Note that in this work we have focused on the case of a bound jFsdj . × GeV. A very interesting set of the QCD axion, such that b → qa transitions result in the probes for axial sd − a coupling is also offered by the axion escaping the detector and a missing energy signature. hyperon decays. BESIII has a rich hyperon-physics A more inclusive search strategy, that does not require any program, and as part of it, searches for axions should information about the axion decay modes, is possible using be attempted. We use the projections for the s → dνν¯ → μμ 1 searches for Bs a decays [74], though with a reduced decay modes, estimated for 5 fb integrated luminosity sensitivity to the quark-axion couplings compared to the in Ref. [127], as the projected bounds on the s → da other modes. decays:
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V A TABLE III. 90% C.L. lower bounds on the scales of flavor-violating axion couplings Fij and Fij (in GeV), with future projections in parentheses. Bounds obtained from data with large experimental or theoretical systematic errors are marked with a † superscript. The most relevant constraints in each sector are typeset in bold.
V A Flavors Process Fij (GeV) Fij (GeV) References s → d Kþ → πþa 6.8 × 1011 [89] ð2 × 1012Þ Kþ → πþπ0a 1.7 × 107 [108] (7 × 108) Λ → naðdecayÞ 6.9 × 106 5.0 × 106 [10] ð1 × 109Þ ð8 × 108Þ † † Λ → naðsupernovaÞ 7.4 × 109 5.4 × 109 Σþ → pa 6.7 × 106 2.3 × 106 [10] ð7 × 108Þð3 × 108Þ Ξ− → Σ−a 1.0 × 107 1.3 × 107 [10] Ξ0 → Σ0a 1.6 × 107 2.0 × 107 [10] ð2 × 108Þð3 × 108Þ Ξ0 → Λa 5.4 × 107 1.0 × 107 [10] ð9 × 108Þð2 × 108Þ ¯ 5† 6 K-K (ΔmK) 5.1 × 10 2.0 × 10 [10] 6† 7 (ϵK) 0.9 × 10 4.4 × 10 [124] c → u Dþ → πþa 9.7 × 107 [93] ð5 × 108Þ 5 5 Λc → pa 1.4 × 10 1.2 × 10 [10] ð2 × 107Þ ð2 × 107Þ † † D − D¯ (CP conserving) 2.4 × 106 4.6 × 106 [10] (CP violating) 2.5 × 107 4.8 × 107 [125] ð8 × 107Þ ð1 × 108Þ [126] b → s Bþ;0 → Kþ;0a 3.3 × 108 [91] (3 × 109) Bþ;0 → K þ;0a 1.3 × 108 [91] ð1 × 109Þ 6 6 Λb → Λa 2.1 × 10 1.4 × 10 [10] þ − 5 Bs → μ μ a 2.2 × 10 [74] ð9 × 105) ¯ 5 5 Bs − Bs (MFV) 2.0 × 10 4.0 × 10 (generic) 3.8 × 105 7.7 × 105 [124] b → d Bþ → πþa 1.1 × 108 [92] ð3 × 109Þ Bþ;0 → ρþ;0a (1 × 109) 6 6 Λb → na 3.1 × 10 1.6 × 10 [10] þ − 5 Bd → μ μ a 2.8 × 10 [74] (1.2 × 106) B − B¯ (MFV) 1.1 × 106 2.0 × 106 (generic) 1.3 × 106 2.3 × 106 [124] † † t → u Kþ → πþa (loop) 3 × 108 3 × 108 † † t → c Kþ → πþa (loop) 7 × 108 7 × 108 [89]
Λ → 3 10−7 A ∼ 109 BRprojð naÞ < × ; This will allow to reach scales as high as jFsdj GeV, −7 entering in the range constrained indirectly by the super- BR ðΣþ → paÞ < 4 × 10 ; proj nova emissivity bound. Dedicated searches for such Ξ0 → Σ0 9 10−7 BRprojð aÞ < × ; s → da hyperon transitions could lead to even more Ξ0 → Λ 8 10−7 stringent constraints than the above conservative estimates. BRprojð aÞ < × : ð59Þ
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FIG. 4. Summary of the most important bounds for the different flavor sectors and for vectorial (red) and axial-vectorial (blue) couplings. On the lower axis, we indicate the corresponding values for the effective axion mass defined by mi:eff ≡ 4.69 eV × 6 10 GeV=Fi. Also shown as vertical gray lines are the bounds on axion couplings to electrons Fe (95% C.L.), nucleons FN, and photons Fγ (95% C.L.); see Sec. V for details.
All the limits on heavy-quark transitions from two-body B − B¯ mixing. The expected bound in case of conservative meson decays reported in this paper are obtained recasting scaling is about a factor of 3 smaller. the analyses of “νν¯” decays at BABAR and CLEO. These Finally, the searches for axions in charm-meson and bounds will certainly be improved by dedicated searches in charm-baryon decays could be undertaken at BESIII and the future. Belle II expects to gather 50 ab−1 of data in the Belle II. For mesons, one can rescale the CLEO bound on next five years, roughly a factor 100 larger than the final D → πa by the expected gain in luminosity, which is about integrated luminosity at BABAR. The gain in the bounds a factor 20, assuming that BESIII will collect 20 fb−1, V on the branching ratios depends on the scaling behaviors of which leads to bound on Fcu that is stronger by a factor 5 (2) the backgrounds. In the absence of dedicated experimental for the “optimistic” and conservative scalings, respectively. “ ” 4 þ ¯ − projections, we simply assume an optimistic scaling For baryon decays, approximately 10 Λc Λc pairs have inversely proportional to the increase in luminosity with been produced with 567 pb−1 at BESIII [129],while respect to the integrated luminosities on which the respec- approximately 107 ΣþΣ¯− pairs are expected with 5 fb−1 “ ” tive BABAR analyses were performed. The conservative [127]. BESIII could therefore reach the limit scaling inversely proportional to the square root of the number of total events would result in slightly weaker Λ → 4 10−5 BRprojð c paÞ < × ; ð61Þ bounds. Assuming similar reconstruction efficiencies at Belle II as those achieved at BABAR, one can expect an obtained by naively rescaling with the relative sizes of the V;A V improvement in the sensitivity to Fbs and Fbd by at least samples the projection for the branching fraction of the an order of magnitude; see Table III for the optimistic Σþ → pa decay shown in Eq. (59). This should lead to projections. For the conservative scaling, the expected bounds on the axial cu − a coupling that are comparable bounds are about a factor 5 weaker. with the current bounds from D − D¯ mixing, but with the In case of B → ρa, the future projection can be estimated benefit of no UV model dependence. by using the current Belle bound for the “νν¯” mode in Table I and rescaling with the luminosities. This gives us VIII. SUMMARY AND CONCLUSIONS
−7 In conclusion, in this paper, we explored the phenom- BRðBþ → ρþaÞ < 4 × 10 ; ð60Þ prosp enological implications of the possibility that the QCD axion has flavor-violating couplings to the SM quarks. 9 which would give the sensitivity to fa ≳ 10 GeV from the Although the presence of flavor violation in axion models axial bd − a couplings, about 3 orders of magnitude is very model dependent, this scenario generically arises 1 stronger than the current bound on this coupling from whenever the Uð ÞPQ charges are not family universal. The
015023-16 QUARK FLAVOR PHENOMENOLOGY OF THE QCD AXION PHYS. REV. D 102, 015023 (2020) flavor-violating couplings may even be a necessary ingre- a significant improvement is expected. Precision flavor dient in the UV structure of the theory. This is the case, for facilities can potentially test PQ breaking scales up to the 1 1012 109 instance, if the Uð ÞPQ group that solves the strong CP order of GeV (NA62 and KOTO) and GeV (Belle problem is part of a larger flavor symmetry group, which is II and BES III), if dedicated searches are performed in a frequent feature in models addressing the SM flavor ongoing experiments at strange, charm, and bottom facto- problem with approximate horizontal symmetries. ries. This reach actually falls into the most interesting In this paper, we investigated in detail the flavor region of the parameter space where the axion can account phenomenology of the axion couplings to quarks. We for the observed dark matter abundance or possibly explain advocate to use a model-independent approach, treating various mild hints for anomalous stellar cooling. These every flavor-violating coupling, vectorial or axial-vectorial, expectations strongly motivate a comprehensive experi- as a different parameter. This is very similar, in spirit, to the mental program of searching for axions in rare flavor- global analyses to search for heavy new physics in low- changing transitions, which may well lead to the first energy experiments and the LHC using the SM effective- discovery of the QCD axion. field theory. Throughout this work, we have assumed the limit of a practically massless axion. Much of the frame- ACKNOWLEDGMENTS work developed in this paper can be extended straightfor- wardly to massive axionlike particles with the appropriate We thank Daniel Craik, Marko Gersabeck, Pablo changes due to the different kinematics involved. Goldenzweig, Aida El-Khadra, Martin Heeck, Phil Ilten, The present paper goes beyond previous studies in the Alex Kagan, Hajime Muramatsu, Uli Nierste, Steven same spirit [33–35] in several ways: Robertson, Giuseppe Ruggiero, Luca Silvestrini, Abner (i) We critically examined the bounds that can be Soffer, Emmanuel Stamou, Sheldon Stone, Olcyr derived using two-body decays of heavy mesons Sumensari, Shiro Suzuki, and Javier Virto for useful dis- with final-state axions. By recasting BABAR data of cussions. J. M. C. acknowledges support from the Spanish B → Kð Þνν¯ and B → πνν¯, one can derive limits that MINECO through the “Ramón y Cajal” Program No. RYC- supersede the old CLEO direct searches of B → Ka 2016-20672 and Grant No. PGC2018-102016-A-I00. J. Z. and B → πa. We also find that the Belle analyses on acknowledges support in part by the DOE Grant No. DE- the “νν¯” modes cannot be recast for this purpose. SC0011784. P. N. H. V. acknowledges support from the Thus, there is no bound from two-body decays on Non-Member State CERN Summer Student Programme. A Λ → Fbd, with the best limit currently given by b na decays and B − B¯ mixing. (ii) We derived the strongest direct limit in the up-quark APPENDIX A: RENORMALIZATION sector by recasting data on Dþ → ðτþ → πþν¯Þν as a GROUP EQUATIONS search for the Dþ → πþa decay. Above the electroweak scale, the Lagrangian describing (iii) We provided the theoretical framework and phe- the interactions of the axion with the SM Higgs and the nomenological analysis of new processes not con- fermions take the form → ππ sidered before, such as the three-body K a X decays and baryon decays. We argue that baryon L ⊃ ψγ¯ μ ψ μ † u ¯i j UV i Dμ þðD HÞ DμH þ yijHqLuR decays can in the future give the best sensitivity to ψ several couplings of the axion: the CP-conserving ∂μa cu − a couplings from Λ → pa decay or the axial d ˜ ¯i j e ˜ l¯i j q ¯i γμ j c þ yijHqLdR þ yijH LeR þ H:c: þ cij 2 qL qL vector sd − a coupling from hyperon decays. fa (iv) We derived the strongest limit on axial couplings ∂μa ∂μa ∂μa u ¯i γμ j d ¯i γμ j l l¯i γμlj using a novel interpretation of the supernova SN þ cij uR uR þ cij dR dR þ cij L L 2fa 2fa 2fa 1987A bound, based on the impact of the process ∂ a ∂ a ↔μ Λ → na on the neutrino emissivity of the hot e μ ¯i γμ j μ † þ cij 2 eR eR þ cH 2 iH D H; ðA1Þ protoneutron star. fa fa ↔μ (v) We developed a theoretical framework to extract † † μ † where H D H ¼ H D H − ðDμHÞ H. In the first two lines, reliable limits from neutral-meson mixing. This is — based on the effective-field theories of QCD; chiral- we included the relevant parts of the SM Lagrangian perturbation theory for the case of kaon mixing and the fermion kinetic terms, the Higgs kinetic term, and the operator product expansion for the case of heavy- Yukawa interactions. For axion interactions, we keep only the meson mixing. lowest-dimension operators and allow for general flavor f 3 3 Our main results for present and future constraints on structure. The Yukawa matrices yij are general × matrices, flavor-violating axion couplings are summarized in while the axion interactions are described by Hermitian 3 × 3 Table III and Fig. 4. For several of the considered modes, f matrices cij.
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dc The Yukawa interactions are invariant under the axion- 16π2 H 6 c y y† − c y y† μ ¼ Trð q u u q d dÞ dependent, flavor-diagonal field redefinitions d ln 6 c y†y − c y†y þ Trð d d d u u uÞ aðxÞ aðxÞ iα −iαYψ † † → 2fa ψ → fa ψ H e H; i e i; ðA2Þ þ 2Trðceyeye − clyeyeÞ − 2 3y y† 3y y† y y† cHTrð u u þ d d þ e eÞ: ðA9Þ where α is a free parameter and Yψ denotes hypercharge. Under these transformations, the axion couplings shift as Using these equations, one can express the low-energy couplings in terms of high-energy couplings, diagonal SM → − α ψ → ψ 2α δ cH cH ;cij cij þ Yψ ij: ðA3Þ Yukawas, and the CKM matrix V. Keeping only the effects proportional to the top Yukawa coupling, the radiative Thus, one can always employ the above field redefinition corrections to the axion couplings, ΔcV;A , are given by fifj with α ¼ cH to get rid of the cH operator and shift the axion couplings to fermions as in Eq. (A3), which gives Eq. (1). y2 f Δ V μ t a 2 V A The axion couplings to the gauge fields also do not change, c ð Þ¼ 2 log ½ V3 V3jðctt þ cttÞ didj 64π μ i since the transformations in (A2) are nonanomalous—they correspond to a phase shift of each fermion field that is − V − A − V − A VkiV3jðcukt cuktÞ V3iVkjðctuk ctuk Þ ; proportional to its hypercharge Yψ . When performing the ðA10Þ renormalization group (RG) evolution, this field redefini- μ tion needs to be performed at each scale , in order to y2 f μ 0 Δ A μ − t a 24 Aδ 2 V A keep cHð Þ¼ . c ð Þ¼ 2 log ½ ctt ij þ V3 V3jðctt þ cttÞ didj 64π μ i The RG equations for the couplings in matrix notation, ≠ 0 4 − V − A − V − A keeping cH , are given by (see, e.g., Ref. [182]) VkiV3jðcukt cuktÞ V3iVkjðctuk ctuk Þ ; ðA11Þ c 2 d q 1 † † † 16π ¼ ðyuyu þ ydy Þcq − yucuyu d ln μ 2 d y2 f ΔcV ðμÞ¼ t log a ½2ð3cV − cAÞδ δ 1 uiuj 64π2 μ tt tt it jt c y y† y y† − y c y† þ qð u u þ d dÞ d d d 2 V A V A − ð3ctu þ ctu Þδit − ð3cu t þ cu tÞδjt ; ðA12Þ − y y† − y y† j j i i cHð u u d dÞ; ðA4Þ y2 f Δ A μ t a 24 Aδ − V 3 A δ cu u ð Þ¼ 2 log ½ ctt ij ðctu þ ctu Þ it 2 dcu † † † † i j 64π μ j j 16π ¼ cuyuyu þ yuyucu − 2yucqyu þ 2cHyuyu; d ln μ − V 3 A δ 2 V − 3 A δ δ ðcuit þ cuitÞ jt þ ðctt cttÞ it jt ; ðA5Þ ðA13Þ
2 dc † † † † 6y f 16π2 d ¼ c y y þ y y c − 2y c y − 2c y y ; ΔcA ðμÞ¼− t log a cAδ : ðA14Þ d ln μ d d d d d d d q d H d d eiej 16π2 μ tt ij ðA6Þ The flavor-universal contributions arise from a nonzero cH, which is radiatively generated and then through field dcl 1 † 1 † † † 2 redefinitions absorbed into cfifj at low energies. Note also 16π ¼ yeyecl þ clyeye − yeceye þ cHyeye; d ln μ 2 2 that the high-energy couplings satisfy ðA7Þ cV − cA ¼ V V ½cV − cA : ðA15Þ uiuj uiuj ik jl dkdl dkdl
2 dce † † † † 16π ¼ c y y þ y y c − 2y cly − 2c y y ; d ln μ e e e e e e e e H e e APPENDIX B: DETAILS ON TWO-BODY ðA8Þ RECASTS
For the recasts of P1 → P2νν¯ and P1 → V2νν¯, we use the experimental information in the kinematic regions 4One can verify that these equations transform consistently corresponding to a massless axion, i.e., taking into account under the field redefinitions in Eq. (A3). only the bin that contains events with vanishing invariant
015023-18 QUARK FLAVOR PHENOMENOLOGY OF THE QCD AXION PHYS. REV. D 102, 015023 (2020) mass of the neutrino pair.5 For B → πa, we take from the matrix elements are functions of the momentum exchange 2 0 2 2 BABAR analysis in Ref. [92] for the numbers of observed squared, q ¼ðp − p Þ ¼ ma ≃ 0; i.e., for the predictions, 1 2 and background events in the relevant bin Nobs ¼ and we only need the values of form factors at q ¼ 0. 1 0 Nbg ¼ , respectively, for the total number of B mesons The hadronic matrix elements for P → P a transitions, 8 9 0 1 107 ð0Þ in the data sample Ntot ¼ð . . Þ × and for the with P pseudoscalar mesons, are parametrized by two efficiency ϵ ¼ð6.5 0.6Þ × 10−4. The expected number of sets of form factors μ ϵ → π events is then ¼ Nbg þ NtotBRðB aÞ. To obtain the 0 0 ¯0γμ μ PP0 2 μ PP0 2 90% C.L. upper limits, we follow the statistical treatment hP ðp Þjq qjPðpÞi ¼ P fþ ðq Þþq f− ðq Þ; ðC1Þ in Ref. [91] and use the mixed frequentist-Bayesian μ μ approach described in Refs. [183,184] in order to include where P ¼ðp þ p0Þ . The related matrix element of the 0 μ systematic uncertainties. axial current q¯ γ γ5q vanishes by parity invariance of the In the same way, we proceed for B → Ka and B → K a strong interactions. In the decay, the Lorentz index in (C1) using the BABAR data from Ref. [89] for the two decay is contracted with −iqμ from the derivative of the axion 4 71 0 03 108 channels in each case, and Ntot ¼ð . . Þ × .For field; cf. Eq. (1). The only hadronic inputs needed to þ → þνν¯ 0 → 0νν¯ → 0 PP0 0 the channels fB K ;B K g, we take Nobs ¼ describe the P P a decays are therefore fþ ð Þ. 2 0 0 0 ϵ 9 5 0 5 4 5 0 5 10−4 Kþπþ 0 2 1 1 f ; g, Nbg ¼f ; g and ¼f . . ; . . g× , For fþ ð Þ, we use the Nf ¼ þ þ lattice QCD þ → þνν¯ 0 → 0νν¯ K0π− 0 while for fB K ;B K g, we use Nobs ¼ determination of fþ ð Þ [153], since in the isospin 1 3 1 1 ϵ 1 2 0 1 0 3 0 05 10−4 Kþπþ 0 K0π− 0 f ; g, Nbg ¼f ; g, ¼f . . ; . . g × . symmetric limit fþ ð Þ¼fþ ð Þ. Likewise, we use In each case, the two channels are combined by maximiz- for the axion-induced charm meson decays the lattice QCD ing the likelihood function that is the product of Poisson D0π− 0 calculation of fþ ð Þ reported by the ETM Collaboration 0 probabilities, following Ref. [183]. [156], along with the isospin relation fDþπþ ð0Þ¼fD πþ ð0Þ. Finally, for c → u transitions, we use the search for þ þ For the Bþ → Kþa decay, we use the lattice QCD Dþ → τþν decay in the τþ → πþν¯ channel at CLEO in 2 ≤ 0 05 2 determination of the form factors by the Fermilab/MILC Ref. [93]. In the signal window mmiss . GeV , CLEO Bþ → πþa 11 Collaboration [158], while for the decay, we use observed Nobs ¼ pionlike events, with the expected SM 13 5 1 0 the light-cone sum rule determination of the form factors in background of Nbg ¼ . . , for a total number of ref. [161]. 4 6 105 tagged D decays Ntot ¼ . × and single pion detection The hadronic matrix element for the decays of a ϵ 0 89 efficiency π ¼ . . This results in the 90% C.L. upper pseudoscalar meson P into a vector meson V is given by bound BRðD → πaÞ ≤ 8.0 × 10−6, following the same 0 0 statistical prescription as before. A recast of the recent hVðp ; ηÞjq¯ γμγ5qjPðpÞi Dþ → τþν BESIII analysis [104] using pionlike events qμ 2 from the τþ → πþν¯ channel, right of Fig. 3 in Ref. [104], ¼ iðη · qÞ 2m A0ðq Þ q2 V results in a bound that is about a factor 2 weaker than our η μ recast of the bound from CLEO. μ ð · qÞq 2 þ iðm þ m Þ η − A1ðq Þ B V q2 APPENDIX C: HADRONIC MATRIX ELEMENTS 2p − q μ qμ − η ð Þ − − 2 ið · qÞ ðmB mVÞ 2 A2ðq Þ; ðC2Þ In this Appendix, we give further details on the hadronic mB þ mV q elements describing axion-induced flavor-changing transi- tions. The numerical values of different inputs entering the where η is the polarization vector of the vector meson. predictions are collected in Table IV. Parity conservation implies that the matrix element of the 0 0 vector current, hVðp ; λÞjq¯ γμqjPðpÞi, transforms as an ν 0ρ σ 1. Two-body decays axial vector and must be ∝ ϵμνρσp p η . This gives a We first give the parametrizations of matrix elements vanishing contribution to the decay amplitude upon con- → 0 ð0Þ traction with the derivative interaction of the axion. for two-body hadron decays H H a, where H is a − μ pseudoscalar meson, a vector meson, or a spin-1=2 baryon. Furthermore, contracting Eq. (C2) with iq , and taking The transitions are induced by quark-level transitions of the into account that type q → q0a. The form factors in the resulting hadronic mP þ mV mP − mV A1ð0Þ − A2ð0Þ¼A0ð0Þ; ðC3Þ 2mV 2mV 5Since usually the bin size is wider than the experimental momentum resolution, in this way, we count also background or 0 ≠ 0 one finds that A0ð Þ is the only hadronic input needed to SM events with mνν¯ as axion signals. This renders the → → → ρ resulting bound only more conservative, which will be eventually describe the P Va decays. For the B K and B superseded by a proper recast done by the experimental collab- form factors, we use the light-cone sum rules determina- orations using the full information. tions from Ref. [185].
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TABLE IV. Numerical values for the theoretical inputs used in the analysis as described in Appendix C. The bag parameters for the Bd;s systems are evaluated at the renormalization scale μ ¼ mb [150]. The matrix elements of the corresponding four-quark operators 4 hOii in the D meson system are evaluated at μ ¼ 3 GeV and are shown in units of GeV [152]. Flavors Process Inputs References → → π 0 0 9706 27 s dK fþð Þ¼ . ð Þ [153] 0 00 0 K → ππ fs ¼ 5.705ð35Þ, fs ¼ 0.87ð5Þ, fs ¼ −0.42ð5Þ fp ¼ −0.274ð29Þ, gp ¼ 4.95ð9Þ, gp ¼ 0.51ð12Þ [110,111]
B1 → B2 f1ð0Þ g1ð0Þ Λ → n −1.22ð6Þ −0.89ð2Þ [154,155] Σþ → p −1.00ð5Þ 0.34(1) Ξ− → Σ− 1.00(5) 1.26(5) Ξ0 → Σ0 −0.71ð4Þ −0.89ð3Þ Ξ0 → Λ 1.22(6) 0.24(5) ¯ 155 7 3 K-K fK ¼ . ð Þ MeV [153] → → π 0 0 612 35 c uD fþð Þ¼ . ð Þ [156] Λc → pf1ð0Þ¼0.672ð39Þ, g1ð0Þ¼0.602ð31Þ [157] ¯ D − D hO2i¼−0.1442ð72Þ, hO4i¼0.275ð14Þ [152] → → 0 0 335 36 b sBKfþð Þ¼ . ð Þ [158] B → K A0ð0Þ¼0.356ð46Þ [159] Λb → Λ f1ð0Þ¼0.16ð4Þ, g1ð0Þ¼0.11ð9Þ [160] 230 3 1 3 fBs ¼ . ð . Þ MeV [153] ¯ 0 817 43 1 033 47 Bs − Bs B2 ¼ . ð Þ, B4 ¼ . ð Þ [150] η2 ¼ −2.669ð62Þ, η4 ¼ 3.536ð74Þ [150] → → π 0 0 21 7 b dB fþð Þ¼ . ð Þ [161] B → ρ A0ð0Þ¼0.356ð42Þ [159] Λb → pf1ð0Þ¼0.23ð8Þ, g1ð0Þ¼0.12ð13Þ [162] fB ¼ 190.0ð1.3Þ MeV [153] ¯ B − B B2 ¼ 0.769ð44Þ;B4 ¼ 1.077ð55Þ [150] η2 ¼ −2.678ð62Þ; η4 ¼ 3.547ð74Þ [150]
The hadronic matrix elements for B → B0a decay, with their electric charges and semileptonic decays [154,155]. Bð0Þ the spin-1=2 baryons, are parametrized by For reference, we quote also the uncertainties due to the expected sizes of the SU(3) flavor-breaking effects. 0 0 0 0 hB ðp Þjq¯ γμqjBðpÞi The leading corrections to f1ð Þ vanish because of the 2 2 Ademollo-Gatto theorem [186]. Explicit calculations using 0 0 2 f2ðq Þ ν f3ðq Þ ¼ u¯ ðp Þ f1ðq Þγμ þ σμνq þ qμ uðpÞ; lattice QCD showed that the breaking effects are below 5% M M [187]. For the axial couplings, g1ð0Þ, the leading SU(3) – ðC4Þ flavor breaking corrections can be predicted in chiral perturbation theory by using experimental measurements 0 0 0 hB ðp Þjq¯ γμγ5qjBðpÞi of the isospin-related channels, the semileptonic hyperon 2 2 decays, and lattice QCD results [155]. 0 0 2 g2ðq Þ ν g3ðq Þ ¼ u¯ ðp Þ g1ðq Þγμ þ σμνq þ qμ γ5uðpÞ: M M 2. K → ππa ðC5Þ The hadronic matrix elements needed for the decays μ þ → πþπ0 → π0π0 After contracting with −iq from the derivative on the K a and KL a can be obtained from the − axion field, the decay amplitude involves only two form form factors measured in the decay Kþ → πþπ eþν using factors at q2 ¼ 0, i.e., the vector and axial-vector couplings isospin symmetry. The matrix element of the axial-vector current for the latter process is defined as [114,115] f1ð0Þ and g1ð0Þ. In numerical evaluations, we use for Λb → n form factors the lattice QCD determinations of the Λb → p form factors from Ref. [162] (they are equal in the πþ π− ¯γμγ − h ðpþÞ ðp−Þjs 5ujK ðpÞi isospin-symmetric limit), for Λc → p from Ref. [157] and for Λ → Λ from Ref. [160]. For hyperons, we use the − i μ − μ μ b ¼ ðFðpþ þ p−Þ þ Gðpþ p−Þ þ Rq Þ; ðC6Þ flavor SU(3) symmetry to obtain the form factors from mK
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− − where q ¼ p pþ p− is the 4-momentum of the axion uses the isospin-symmetry relations [112,113] with the and thus q2 ≃ 0. Once the above hadronic matrix element is convention that ðu; dÞ and ð−d;¯ u¯Þ transform as isodoublets, contracted with −iqμ from the derivative on the axion field, pffiffiffi the contribution of R to the decay matrix element vanishes πþπ0 ¯γμγ þ − 2 πþπ− ¯γμγ þ h js 5djK i¼ hð ÞI¼1js 5ujK i; at the physical point q2 ¼ 0. The related matrix element of the vector current completely vanishes due to parity ðC10Þ invariance once it is contracted with the axion-field derivative, similarly to the P → Va decays discussed π0π0 ¯γμγ 0 πþπ− ¯γμγ þ h js 5djK i¼hð ÞI¼0js 5ujK i; ðC11Þ above. The K → ππ form factors depend on three kinematic 2 2 where the subscripts on the right-hand sides indicate that variables which can be chosen to be q , s ¼ðpþ þ p−Þ , we have projected the isospin wave functions of the two θ θ and cos π, where π is the angle between the 3-momenta final pions to either I 0 or I 1. The total amplitude ⃗ − ⃗ ¼ ¼ pπþ and pK in the di-pion rest frame. The form factors are must be even under the exchange of the two pions (Bose complex functions with a strong phase arising due to the symmetry). Therefore, for the isospin symmetric I ¼ 0 rescatterings of the pions. We follow a standard para- (antisymmetric I ¼ 1) wave function, only the s-wave (p- metrization of the form factors using a partial-wave wave) components contribute. A final observation is that expansion of the two-pion system [114,115] and truncate one does not have interference in the total rates between s- it at the p-wave, and p-wave components in these axion decay channels and they are insensitive to the strong phase δ. F ¼ Fs þ Fp cos θπ expð−iδÞ; ðC7Þ 3. Neutral meson mixing G ¼ Gp expð−iδÞ: ðC8Þ Hadronic matrix elements of the four-quark operators δ δ − δ πþπ− Here, ¼ s p is the difference of s- and p-wave (37) involved in the axion contributions to heavy neutral- phase shifts. The prefactors Fs, Fp, and Gp are only meson mixing are conventionally defined in terms of the 2 ¯ functions of q and s. In the experimental analyses of K → so-called bag parameters, Bi. For the case of B − B mixing 2 0 ¯0 ππeν decays, they are often Taylor expanded around q ¼ and shortening hB jOijB i¼hOii, these are defined as 2 0 and s ¼ 4mπ. For the K → ππa decays, we only need 2 0 1 their values at q ¼ , retaining the parameters of the 0 q ¯0 ηq μ 2 2 ðiÞ μ 2 hBqjOi jBqi¼ i ð ÞfB mB BB ð Þ; ðC12Þ expansion in the dimensionless variable s¯ ¼ðs=4mπÞ − 1, 4 q q q around s¯ ¼ 0, ηq ðiÞ with the values for i ðmbÞ and BB ðmbÞ as provided F f f0 s¯ f00s¯2; q s ¼ s þ s þ s in Ref. [150]. These definitions are straightforwardly 0 Fp ¼ fp;Gp ¼ gp þ gps;¯ ðC9Þ extended to the short-distance contributions in the other neutral-meson systems. The values of the different param- which are determined from the experimental data [110,111] eters in these equations are obtained from lattice calcu- and shown in Table IV. lations. In case of the charm-meson oscillations, we use To connect these form factors in the Kþ → πþπ−eþν results in Ref. [152], which directly provides the results in þ þ 0 0 0 channel to those in K → π π a and KL → π π a, one terms of the matrix elements hOii at μ ¼ 3 GeV.
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