The axion-baryon coupling in SU(3) heavy baryon chiral perturbation theory

In the past, the axion-nucleon coupling has been calculated in the framework of SU(2) heavy baryon chiral perturbation theory up to third order in the chiral power counting. Here, we extend these earlier studies to the case of heavy baryon chiral perturbation theory with SU(3) flavor symmetry and derive the axion coupling to the full SU(3) baryon octet, showing that the axion also significantly couples to hyperons. As studies on dense nuclear matter suggest the possible existence of hyperons in stellar objects such as neutron stars, our results should have phenomenological implications related to the so-called axion window.


I. INTRODUCTION
Soon after the discovery of the instanton solution of euclidean Yang-Mills gauge field theories [1], it has been realized that the vacuum structure of theories such as quantum chromodynamics (QCD) is highly non-trivial and that a so-called θ-term appears [2] L QCD = L QCD,0 + θ g 4π which is CP non-invariant as long as no quark is massless and as long as the effective anglē θ = θ + Arg det M is non-zero, where M refers to the quark mass matrix. In Eq. (1), L QCD,0 denotes the usual QCD Lagrangian without the θ-term, g is the QCD coupling constant, G µν the gluon field strength tensor,G µν = µναβ G αβ /2 its dual, and Tr denotes the trace in color space. As a consequence, the neutron acquires an electric dipole moment ∝θ [3].
Theoretical estimations of theθ-induced neutron electric dipole moment (nEDM) roughly vary between |d n | ≈ 10 −16θ e cm and |d n | ≈ 10 −15θ e cm; see Refs. [4][5][6][7]. The extreme small upper limit for the physical value of the nEDM determined in experiments (the most recent result is |d n | < 1.8 × 10 −26 e cm (90 % C.L.) [8]) implies which is a rather unnatural value for a quantity that in principle might take on values between 0 and 2π (where θ = π is a special point [9]). Even thoughθ of O(1) would alter nuclear physics and big bang and stellar nucleosynthesis considerably [10], such anthropic considerations do not constrainθ to such a tiny value. One possible way to explain whyθ ≈ 0 (in other words, to resolve the so-called "strong CP -problem") is the Peccei-Quinn mechanism [11,12], which features a new global chiral symmetry, now usually labeled U (1) PQ , and which automatically leads toθ = 0. The physical relic of this mechanism is not only a CP -conserving QCD Lagrangian, but also a new, very light pseudoscalar pseudo-Nambu-Goldstone boson with zero bare mass called axion [13,14], which soon after its theoretical description also became a serious dark matter candidate [15][16][17][18][19][20][21], possibly forming Bose-Einstein condensates [22]. Both, the fact that the Peccei-Quinn mechanism provides an elegant solution to the strong CP -problem, and that it at the same time might provide a solution to the missing matter problem of the universe, accounts for the unwavering interest in the axion, even though it has not been detected yet and its existence hence remains hypothetical.
The original ("visible") Peccei-Quinn-Weinberg-Wilczek (PQWW) axion with a decay constant at the electroweak scale and a mass in the keV/MeV region seems to be ruled out experimentally [4,[23][24][25], although there are still attempts to make the experimental data compatible with the original model [26,27]. However, these models require a lot of additional assumptions including that these axions restrictively couple to the first generation fermions (up and down quark, electron) and are accidentally pion-phobic. The clearly preferred models are the so-called "invisible" axion models such as the Kim-Shifman-Vainstein-Zakharov (KSVZ) axion model [28,29] (which sometimes is called "hadronic" as it does not couple to leptons) or the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axion model [30,31]. Phenomenologically, the physical properties of such an (almost) invisible axion such as its mass and the couplings to standard model particles (quarks, leptons, gauge bosons) are governed by its expectedly large decay constant, which is traditionally estimated as being [4,15,16,32] 10 9 GeV f a 10 12 GeV , such that the axion mass [33,34] m a ≈ 5. 7 10 12 GeV f a × 10 −6 eV (4) presumably would be somewhere between a few µeV and 0.1 eV. The lifetime of the QCD axion, which may decay into two photons, is incredibly large [33] τ a ≈ 1 eV m a 5 × 10 24 s .
Note that the traditional axion window, Eq. (3), is set in order to match cold dark matter requirements of the canonical invisible axion models that solve the strong CP -problem.
However, axions and axion-like particles have been considered also in the context of several other models with decay constants considerably smaller or larger than the window of Eq. (3).
For example, in the case of hadronic axions a decay constant as small as f a ≈ 10 6 GeV has been proposed [35], while in string theories axions and axion-like particles might appear at the GUT or even the Planck scale, i.e. 10 15 GeV f a 10 18 GeV [36].
The leading order aN coupling has been derived several times since the early days [23,35,[54][55][56]. More recently, this leading order coupling has been reexamined in Ref. [57]. Moreover, such an analysis has been carried out up-to-and-including O(p 3 ) in SU(2) heavy baryon chiral perturbation theory (HBCHPT) in our previous work [58], where p denotes a small parameter (see below in section II.2).
In this paper, we extend these previous studies using SU (3) HBCHPT in two regards: axion also significantly couples to hyperons which has further phenomenological implications, since it has been suggested that hyperons might exist in the cores of neutron stars .
While it is a matter of ongoing research whether the seemingly energetically inevitable existence of hyperons in dense cores are compatible with the observed maximum masses of neutron stars (this question is related to the expected softening of the equation of state and known in the literature as "hyperon puzzle"), it is clear that approaches to constrain axion properties from cooling of neutron stars based on axion-nucleon bremsstrahlung alone might turn out to be insufficient. Depending on the underlying model and parameter sets, particularly the Λ and the Σ − might appear in significant fractions of the total baryon number. As will be shown in this study, the axion coupling to these particular hyperons is of a similar order as that to the nucleons (in fact, it could even be much larger than that to the neutron depending on the axion models), suggesting a revision of axion parameter constraints from stellar cooling.
Before performing the calculations of the axion-baryon couplings, we start with the explicit implementation of our framework and work out its ingredients and building blocks, where q = (u, d, s, c, b, t) T collects the quark fields and a refers to the axion field with decay constant f a . The second term is a remnant of the θ-term of Eq. (1) after the spontaneous breakdown of the Peccei-Quinn symmetry. Depending on the underlying axion model, axions might additionally couple directly to the quark fields, which is here present in the form of the last term. Here, we assume the canonical scenario that the couplings are flavor conserving at tree-level, i.e X q = diag {X q } is a diagonal 6 × 6 matrix acting in flavor space, where the X q 's, q = {u, d, s, c, b, t}, are the coupling constants of the respective axion-quark interactions. These are given, for instance, by for the KSVZ-type axions and DFSZ-type axions, respectively, where x = cot β is the ratio of the vacuum expectation values (VEVs) of the two Higgs doublets within the latter models.
Note that L QCD,0 in Eq. (6) also contains a quark mass termqM q q, with M q = diag {m q } being the real, diagonal and γ 5 -free quark mass matrix.
As usual, it is advisable to perform a transformation on the quark fields, in order to remove the second term of Eq. (6). Choosing which corresponds to vacuum alignment in the θ-vacuum case, we can avoid the leading order mass mixing between the axion and the neutral pseudo-Nambu-Goldstone bosons of the spontaneous breaking of SU(3) chiral symmetry, the π 0 and the η, from the beginning.
Here, . . . denotes the trace in flavor space, z = m u /m d and w = m u /m s . Now the axionquark interaction Lagrangian is given by where q L and q R are the left-and right-handed projections of the quark fields, and Considering only the three-dimensional subspace of flavor space, i. e. q = (u, d, s) T , and introducing we can decompose the matrix X q − Q a into traceless parts and parts with non-vanishing trace, so that where λ 3 and λ 8 refer to the third and eighth Gell-Mann matrices, respectively. From this form of the axion-quark interaction Lagrangian, we can directly read off the required ingredients for the axionic SU(3) heavy baryon chiral Lagrangian, i. e. the external currents Here we have set Note that in contrast to usual chiral perturbation theory, it is also necessary to add isosinglet axial-vector currents a (s) µ,i in order to preserve the full QCD axion interaction. This is possible here, as the subtleties that usually arise due to the U(1) A anomaly are absent, because the model is now anomaly-free.
If one considers also flavor-changing axion-quark couplings at tree-level, X q would be non-diagonal. In this case, it is likewise appropriate to decompose X q − Q a into traceless parts and parts with non-vanishing trace, such that with C (i) depending on the allowed flavor-changing processes and λ i the eight Gell-Mann matrices. This then would cause baryon conversion processes. At this point we note that depending on the underlying model Higgs loops may induce off-diagonal couplings, which would be loop-suppressed but might be relevant due to unsuppressed top Yukawa couplings [87]. As stated above, we here stick to the most prevalent models excluding such flavorchanging axion-quark couplings at tree-level.

II.2. Building blocks of the chiral Lagrangian
The framework we use in this paper is chiral perturbation theory (CHPT) including baryons, meaning the effective field theory of QCD in the low-energy sector. CHPT is used to explore meson-baryon systems based on a systematic expansion in small momenta and quark masses below the scale of the spontaneous breakdown of chiral symmetry, Λ χ ∼ 1 GeV, as first developed by Gasser, Sainio, andŠvarc for the two-flavor case [88] and by Krause for the three-flavor case [89]. However, while baryons can easily be incorporated into CHPT in a consistent manner respecting all symmetries, the power-counting scheme engineered to systematically arrange the infinitely many terms allowed by the fundamental symmetries is spoiled by the fact that the baryon masses m B , which do not vanish in the chiral limit, are roughly of the same order as Λ χ . One way to overcome this problem is to realize that all mass scales need to be assigned to a scaling in the power counting and thus to treat the baryons as extremely heavy, static fermions, which is the idea behind HBCHPT [90][91][92][93]. In this scheme, the four momentum of a baryon field is decomposed as with the four-velocity v µ subject to the constraint v 2 = 1, and p µ a small residual momentum satisfying (v · p) m B . After integrating out the heavy degrees of freedom, the resulting Lagrangian is expressed in terms of the heavy baryon field B, which is now characterized by a fixed velocity v µ (for convenience, we refrain from writing B v to explicitly mark the v dependence). Additionally, it is possible to simplify the Dirac algebra by expressing any Dirac bilinear by means of v µ and the Pauli-Lubanski spin operator which satisfies the relations where the last one is valid in d spacetime dimensions. In Eq. (19) and in what follows, [ , ] refers to the commutator and { , } to the anticommutator.
Any term in the effective Lagrangian consists of a number of elements from a small set of basic building blocks, from which we will present those that are relevant for axionic SU (3) HBCHPT. In this three-flavor case, the ground state baryon octet consisting of the nucleons p and n and the hyperons Σ, Λ, and Ξ are collected in a single 3 × 3 matrix where Equation (20) corresponds to the adjoint representation of SU(3). Due to isospin breaking, the physical Σ 0 and Λ are mixed states made of the Σ 3 and Λ 8 . This mixing is usually parameterized by the mixing angle and one has The pseudoscalar mesons, the pseudo-Nambu-Goldstone bosons of the spontaneous breakdown of chiral symmetry, appear in the Lagrangian in form of a unitary matrix where F p is the pseudoscalar decay constant in the chiral limit and Φ is a Hermitian 3 × 3 matrix given by Furthermore, a scalar external field s is needed to account for the explicit chiral symmetry breaking due to the light quark masses. Setting χ = 2B 0 s, it is then convenient to define which all transform the same way under chiral transformations. Additionally, one defines the chiral covariant derivative where is the chiral connection. Assigning systematically a chiral dimension p to these building blocks in order to establish the power-counting method mentioned above, the effective Lagrangian is constructed by considering all combinations allowed by the underlying symmetries. The terms of the meson-baryon Lagrangian then can be arranged according to the chiral dimension, where the superscript "(i)" denotes the Lagrangian containing all terms of O(p i ). In the case of the purely mesonic Lagrangians L (i) Φ , i is restricted to even numbers. Chiral loops start contributing at O(p 3 ) (see section III.6), which means that L ΦB contains a number of low-energy constants and counter-terms needed for the renormalization (ch. III.4 and III.5).
Note that in non-relativistic HBCHPT the Lagrangian (31) contains an inverse power series in m B , the average octet mass in the chiral limit, which starts contributing at O(p 2 ) (see section III.2). The leading order meson-baryon Lagrangian is given by which in contrast to the two-flavor case contains two isovector axial-vector coupling constants D and F , and in the present case of axionic HBCHPT the coupling constants D i to account for the isoscalar interactions.

II.3. General form of the axion-baryon coupling
For any process B B → B A + a, where B A and B B denote arbitrary baryons and a an axion, the general Feynman rule for the vertex in the baryon rest frame has the form where q µ is the four-momentum of the outgoing axion. The coupling constant G aAB consists of a power series in 1/f a Due to the expected huge value of f a , it is sufficient to determine the leading order term can be expanded according to the power counting rules of HBCHPT. Here in particular where g LO aAB and g N 2 LO aAB refer to the contributions from the leading and next-to-next-to-leading order chiral Lagrangian, i. e. from L ) contributions from the expansion in the average baryon mass in the chiral limit m B , which additionally appear in HBCHPT (see section III.2). In this paper, we will determine each of these contributions to g aAB up to As we work in the physical basis, A and B in G aAB and g aAB can be understood as SU(3) indices running from 1 to 8 that directly relate to the physical baryon fields. G a44 , for instance, thus gives the axion-proton coupling G app , whereas G a66 gives the axion-neutron coupling G ann (see Appendix A). In this basis, the matrix of G aAB is mostly diagonal as we exclude flavor-changing axion-quark couplings, i. e. G aAB = 0 for A = B, with the only which gives where A and B are SU(3) indices in the physical basis (see Appendix A). Here we have The matrix elements of g AB are given in Table IV in Appendix B. As we are working in the physical basis including the Σ 0 -Λ mixing, g AB actually includes effects of O(p 2 ). The pure leading order axion-baryon coupling is found by setting = 0, i.e. by replacingλ A/B →λ A/B , see Eq. (A14) of Appendix A. In this case, λ 3 and λ 8 can be replaced byλ 3 andλ 8 so that the coupling can be expressed by means of the structure constantsf ABC andd ABC : It is hence clear that g (1) aAB in Eq. (35) is simply given by

III.2. Expansion in the baryon mass
The Lagrangian containing the corrections due to the finite baryon masses m B in the heavy-baryon expansion up to O (p 3 ) corresponding to terms proportional to 1/m B and 1/m 2 B is given by [93] Here and where for any four-vector x µ In the present case of axionic HBCHPT, we find with g AB as defined in Eq. (39), so that Let p be the momentum of the incoming baryon, p the momentum of the outgoing baryon, and ω ( ) = v · p ( ) , then the resulting vertex Feynman rule reads which is in complete analogy to the SU(2) case [58]. In the baryon rest frame with p = 0, where q 0 is the relativistic energy of the outgoing axion, we finally find (see Eqs. (35) and (36)) III.3. Next-to-next-to-leading order: Contributions from χ − At next-to-next-to-leading order, there are several contributions that have to be considered. We can differentiate terms with finite low-energy constants (LECs) that are ∝ χ − , terms with LECs having finite and ultraviolet (UV) divergent pieces ∝ χ + , and terms proportional some LECs that have no finite pieces serving as counter-terms to cancel UV divergences from the loop contributions. We start with the former. (11) and (28), can be written as where M 2 π ± is the leading order mass of the charged pions given in Eq. (26). The contributing terms of L ΦB are (here and in the following section, we enumerate the LECs according to the list of terms in Ref. [95]) Inserting Eq. (51) yields The reason for writing the coupling in this way is to match it to the corresponding terms in the SU(2) case, which are given by [58] where N is the nucleon field, and d 18 and d 19 are LECs from the next-to-next-to-leading order SU(2) πN Lagrangian [96]. Apart from a substitution of the LECs, the main difference between the SU(3) case, Eq. (53), and the SU(2) case, Eq. (54), is the explicit effect of the strange quark mass m s in the factor (1 + w/(1 + z)) −1 , which reduces to unity at m s → ∞, i.e. w → 0. In the SU(3) case, the finite value of w accounts for a correction of about two percent.
III.4. Next-to-next-to-leading order: Contributions from χ + From Eqs. (11) and (28), one can readily determine that The relevant terms from L ΦB are [95] The LECs depend on the scale λ and are given by In this equation, d where γ is the Euler-Mascheroni constant. The β-functions are set to cancel the divergences of the one-loop functional, as discussed below. We write the resulting tree-level vertex as where we have set The matrix elements ofd AB are given in Eqs. (B1)-(B9) of Appendix B.
III.5. Next-to-next-to-leading order: Counter-terms The counter-terms needed for the renormalization have been worked out in Ref. [93]. For the present case, we need the terms i = {36, 37, 38, 39} from this paper, which are given by where we have added two terms in order to account for the isoscalar interactions. The LECs have no finite part, i.e.
Their contribution to the tree-level vertex is given by withd c.t.  24) and (A3). Note that the potential diagrams with a aΦBB vertex and a meson line connected to one baryon leg vanish because (v · S) = 0, see Eq. (19), as has been shown in Ref. [58] for the corresponding diagrams in the SU(2) case, which have the same topology. Using the leading order meson-baryon vertex rule, one finds where we have applied dimensional regularization and used the properties in Eq. (19). Moreover, we have defined

III.6.2. Diagram (b)
Expanding u up to order Φ 2 yields so the vertex rule for the aB A B B Φ C Φ C -vertex in the physical basis derived from L ΦB can be written as 1 where we have defined the coupling constant In fact, this vertex and thus diagram (b) is independent of the mixing angle so one might as well substituteλ C →λ C in Eq. (70). Therefore, one can also express the coupling constant by means of the structure constants defined in Eqs. (A9) and (A14), The loops of diagram (b) for all mesons Φ C can then be calculated as

III.6.3. Renormalization
The divergences appearing in the meson loop calculations in dimensional regularization are canceled by setting appropriate β functions for the LECs appearing ind AB (λ), Eq. (60), andd c.t. AB (λ), Eq. (63), of which the β k 's are in accordance with the ones given in Ref. [93], and the β i k 's have been worked out here for the first time. With that, the full renormalized O(p 3 ) contribution reads where we have neglected terms of O (q 0 /M Φ C ). Moreover,d r AB refers to Eq. (60) with renormalized LECs, and The matrix elements of g loop,r AB are given in Eqs. (B10)-(B18) of Appendix B.

IV.1. Leading order axion-baryon coupling
Using the nucleon matrix elements ∆q defined by s µ ∆q = p|qγ µ γ 5 q|p , s µ being the spin of the proton, we set the couplings X c , X b , and X t or only one or two of them were much stronger than X u , X d , and X s , these sea quark terms could not be ignored any longer). Inserting this into the result for the leading order axion-baryon coupling constant g (1) aAB , Eq. (40) (see also aΣ 0 Σ 0 = − ∆u+∆s 2 (1 + z) + w∆d 1 + z + w + ∆u + ∆s 2 (X u + X d ) + ∆dX s , (1 + z) + 2∆u−∆d+2∆s (1 − z) In particular, g (1) app and g (1) ann are exactly the same as in the SU(2) case. Using [97] ∆u = 0.847 (50) , ∆d = −0.407 (34) , and [97] where we also considered corrections from non-vanishing mixing angle related to isospin breaking in the cases of the Σ 0 and the Λ (which is why there also appears a term ∝ X s in aΣ 0 Λ ). In Table I, we list the results for the KSVZ axion, where X q = 0, and the DFSZ axion, where the axion-quark couplings X q depend on the angle β related to the VEVs of the involved Higgs doublets (see above, Eq. (7)). In the KSVZ model, the strongest couplings are hence to be expected for the Σ + and the proton, which is also true for the DFSZ model at small values of sin 2 β (in this region, also the Ξ 0 shows a considerably large coupling with an opposite sign). As noted already in many previous works, the axion-neutron coupling in some scenarios is strongly suppressed and might even vanish in the KSVZ model and the DFSZ model at sin 2 β ≈ 2/3 (corresponding to x = 1/ √ 2, where x is the ratio of the VEVs of the two Higgs doublets). The axion-neutron coupling is also the only baryon conserving coupling that might vanish in the DFSZ model, as it is the only one that changes its sign when varying sin 2 β from zero to unity. At sin 2 β = 1, also the Σ 0 -Λ mixing vertex disappears. At the same value of sin 2 β, the couplings are somehow "harmonized", i.e. the couplings of the axion to particles of the same strangeness S are approximately the same, which is due to flavor symmetry. In case of the neutron and the proton with S = 0 one then has g (1),sin 2 β=1 aAB ≈ −0.14, in case of the Σ particles with S = 1, one has g where AB here denotes particles of the same strangeness, and ∆q 1 83)).

IV.2. Loop corrections and estimation of the NNLO LECs
As stated already, the results for the leading order axion-nucleon coupling in the N f = 3 case are entirely in line with the results of the N f = 2 case, which is also true for the O (p 2 ) and O (p 3 ) corrections stemming from the expansion in 1/m B that appear in the non-relativistic heavy baryon limit, with the only exception that the nucleon mass in the chiral limit m 0 appearing in the SU(2) case is substituted by the average baryon mass in the chiral limit m B . In the limit of soft axions, i.e. q 0 → 0, these terms, see Eq. (50), rapidly vanish.
The more significant corrections stem from the one-meson loop contributions, Eq (74).
For the calculation of the corresponding matrix elements, we use the physical meson masses and decay constants [98,99], M π ± = 139.
where we have set the scale at λ = 1 GeV. The corresponding results for the KSVZ and DFSZ models are displayed in Table II. As expected, the loop contributions are indeed subleading, where the orders of magnitude of the individual terms range between low O (10 −1 ) and O (10 −2 ). Note that just using F π or F p for all the decay constants does not lead to any notable change in these results since the formal difference is of higher order, O(p 5 ), in the chiral expansion.
It is remarkable that the loop corrections to the axion-proton vertex are about one tenth of the full SU(2) result [58], which shows that in this particular case the three-flavor expan- sion works similar to the case of the magnetic moments [100], but different to the baryon masses [101] or weak hyperon decays [102]. Consequently, the largest uncertainty in these calculations is related to the values of the LECs, as discussed next. As for the tree-level contributions from the NNLO Lagrangian (see section III.4), we stated already that the values of the involved LECs are undetermined hitherto. The scaledependent parts ∝ d r k (λ) are expected to compensate the scale-dependence of the loop contributions, such that the actual observable, g aAB , remains scale-independent at O (p 3 ).
For the following estimation of these hitherto unknown LECs, we therefore only consider the scale-independent part such that we can leave aside the scale-dependent loop contributions in our final estimation of the axion-baryon coupling at O (p 3 ). Our understanding of the problem is a Bayesian one: while formally each LEC may take on any arbitrary value, we nevertheless expect that with sufficient probability the LECs are restricted to values that lead to NNLO contributions to g aAB of roughly the same order as the loop contributions discussed above. In other words: we assume that these contributions are indeed sub-leading in comparison to the leading order contributions, which is basically a naturalness argu-ment [103,104]. That this assumption is justified in the present case directly follows from the numerical results of the loop corrections discussed before. This argument is of course not universally valid, but it is nevertheless appropriate in a Bayesian sense, meaning that our results from this ansatz can be used as priors in future determinations of the LECs once suitable experimental or lattice QCD data is available for fitting procedures. Although it is not expected that the axion-nucleon coupling will be measured experimentally with sufficient accuracy in the near future, one may use lattice QCD to compute the relevant couplings by introducing an external isoscalar axial source into the QCD action to mimic the axion, and compute the corresponding form factors.
In practice, we performed a Monte Carlo sampling of the ten involved LECs   N c ); see, e.g., Refs. [105,106]. Therefore, we performed another Monte Carlo sampling for the LECs, where this expectation is taken into account, namely by assigning a larger probability to such sets obeying this expected rule in comparison to sets of LECs deviating from it, which are considered less probable. The result is As expected, the probability distributions of the rather suppressed LECs become considerably thinner, even though it is not excluded that they in reality may achieve higher values.
For the following estimation of the axion-baryon coupling at NNLO, however, we stick to the less rigid estimation of the LECs given in Eq. (85).
The last contribution to the O (q 3 ) axion-baryon coupling stem from terms ∝ χ − discussed in section III.3. From the matching with SU(2), Eq. (54), we deduce that where neither the value of the LEC d 19 from the SU (2) Using this matching with SU(2) and applying the same Bayesian approach as described above, we extrapolate Finally, we can collect all contributions to estimate the full O (p 3 ) axion-baryon couplings given by The corresponding results for the KSVZ and the DFSZ axion are collected in Table III.
Note that while in the leading order case the uncertainties arise from the errors of the quark ratios z and w, and the nucleon matrix elements ∆u, ∆d, and ∆s, the uncertainties in the next-to-next-to-leading order case are dominated by the lack of knowledge of the involved LECs.

V. SUMMARY
In this paper, we have worked out the axion-baryon coupling in SU (3)    In this study, we considered rather "traditional" models. Tree-level axion-quark interactions, if any, are of the same order for all flavors, and there are no flavor-changing processes.
Our calculations, however, can in principle be extended to flavor non-conserving processes by adjusting the matrix X q accordingly and follow the strategy described in Eq. (16). This then would lead to new terms in the axion-baryon interaction Lagrangian including baryonchanging processes.
The other modification of our calculations would be to consider models in which the couplings of the axion to the charm, bottom, and top quark are much stronger than the couplings to the up, down, and strange quarks. Such a model can easily lead to very strong axion-nucleon couplings driven by sea-quark effects (thus avoiding the problem of vanishing axion-neutron coupling), which in the end are balanced by correspondingly larger values of f a , as has bee shown recently in Ref. [108]. Such ideas are entirely compatible with our calculations: While the formulae derived in section III would be unaffected, the only difference would be that the numerical calculations of section IV have to be adjusted as the terms ∝ ∆q i of (77) can not be neglected any more.
Our studies using both SU(2) and SU(3) symmetry have shown that the axion-baryon couplings for rather standard axion models are now known to good precision, but also that in the next-to-next-to-leading order case a higher precision is currently unattainable due to the lack of knowledge of some LECs. In this study, we used a Monte Carlo sampling procedure to extrapolate the most probable values of these unknown LECs, where "probable" here has to be understood in a Bayesian sense. It should thus be clear that our numerical leading order results, Eq. (82), are more solid than the numerical estimations of the NNLO results.
Once the knowledge of the parameters in questions is enhanced, the numerical results of this work can easily be updated. However, the current uncertainties of g aAB are not the major concern considering the fact that the axion window in terms of f a is still very large.
The largest uncertainties regarding the existence of axions is hence strongly linked to the uncertainty of f a and in the end it is not g aAB that counts, but G aAB = −g aAB /f a , see Eqs. (33) and (34). At last, we suggest that G aAB may be computed using lattice QCD by introducing an external isoscalar axial source to mimic the axion.  In the present work, we mainly make use of the physical basis [89], based on a set of traceless, non-Hermitian matricesλ A , A = {1, . . . , 8}, such that the baryon octet matrix B, Eq. (20), and the pseudoscalar meson octet matrix Φ, Eq. (24), can be decomposed as where the baryon fields B A and the meson fields Φ A can directly be equated with the physical particles, i.e. and The explicit form of the generatorsλ A is where is the mixing angle given in Eq. (22). These matrices are related to the common Gell-Mann matrices λ a by a unitary 8 × 8 transformation matrix N viã The non-zero matrix elements of N are (note the sign convention that deviates from the one in [89]) which gives the product rulẽ The structure constants defined in Eq. (A9) can be evaluated using Often, we also need the generators at = 0, and therefore it is convenient to set The properties Eqs. (A8)-(A13) are then of course also valid for theλ A 's withf ABC and d ABC replaced byf ABC andd ABC , respectively. Note that now one haŝ order relations (26) and (27). Moreover, we have set µ Φ C = M Φ C /(4πF p ) for any meson Φ C .
g loop,r g loop,r