The Low-Energy Effective Theory of Axions and ALPs

Axions and axion-like particles (ALPs) are well-motivated low-energy relics of high-energy extensions of the Standard Model, which interact with the known particles through higher-dimensional operators suppressed by the mass scale $\Lambda$ of the new-physics sector. Starting from the most general dimension-5 interactions, we discuss in detail the evolution of the ALP couplings from the new-physics scale to energies at and below the scale of electroweak symmetry breaking. We derive the relevant anomalous dimensions at two-loop order in gauge couplings and one-loop order in Yukawa interactions, carefully considering the treatment of a redundant operator involving an ALP coupling to the Higgs current. We account for one-loop (and partially two-loop) matching contributions at the weak scale, including in particular flavor-changing effects. The relations between different equivalent forms of the effective Lagrangian are discussed in detail. We also construct the effective chiral Lagrangian for an ALP interacting with photons and light pseudoscalar mesons, pointing out important differences with the corresponding Lagrangian for the QCD axion.


Introduction
Axions and axion-like particles (ALPs) are pseudo Nambu-Goldstone bosons, which appear in the spontaneous breaking of a global symmetry and are well motivated new-physics relics in a variety of explicit extensions of the Standard Model (SM) of elementary-particle physics. Their name is derived from the QCD axion, which was introduced by Peccei, Quinn and others to address the strong CP problem [1][2][3][4]. While several explicit models of QCD axions [5][6][7][8] predict a rather strict relation between the axion mass and decay constant, it was realized early on that it is possible to obtain solutions to the strong CP problem with heavier ALPs [9]. Furthermore, supersymmetric and composite-Higgs models can naturally feature light pseudoscalar particles. For example, the R-axion is the pseudo Nambu-Goldstone boson of the R-symmetry breaking in low-energy supersymmetry [10], while non-minimal coset structures in models of compositeness predict pseudo Nambu-Goldstone bosons in addition to the Higgs boson [11]. These models provide ample motivation to search for light ALPs, in particular those with masses in the range between an MeV and tens of GeV, whose couplings are not tightly constrained by existing cosmological [12,13], astrophysical [14,15] and collider bounds [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].
The results of this work apply equally to the cases of the QCD axion and of a more general ALP, and from now on we use the term ALP to represent both options. We use a model-independent approach to connect the ALP couplings to SM particles, which can be probed in low-energy experiments, with the couplings at the fundamental new-physics scale Λ, which we assume to be far above the scale of electroweak symmetry breaking. The leadingorder interactions with SM fields can be parameterized in terms of the Wilson coefficients of dimension-5 operators suppressed by 1/Λ, and hence a heavy new-physics sector corresponds to weak ALP couplings. Starting from the most general effective Lagrangian at dimension-5 order, we calculate the effects of renormalization-group (RG) evolution from the new-physics scale down to the scale of electroweak symmetry breaking and below, systematically including all contributions to the anomalous dimensions arising at two-loop order in gauge couplings and one-loop order in Yukawa interactions. The effects of a redundant operator, in which the ALP couples to the Higgs current, are carefully taken into account. We also calculate the complete one-loop matching contributions at the weak scale, which arise when the top quark, the Higgs boson and the W and Z bosons are integrated out. If the underlying global symmetry is flavor-dependent, the ALP couplings to quarks or leptons can have a non-trivial flavor structure at the scale Λ [33,34]. But even if the underlying global symmetry is flavoruniversal, flavor-violating ALP couplings are inevitably induced radiatively. This opens up the possibility to search for ALPs in rare, flavor-changing decays of mesons and leptons, which could provide information about the structure of a new-physics sector otherwise out of reach of direct searches. We illustrate the numerical effects of RG evolution and weak-scale matching for different values of the new-physics scale Λ. Our study of these effects goes significantly beyond existing studies in the literature, and it is relevant for the case of the QCD axion, too. We also discuss the relations between several equivalent forms of the effective ALP Lagrangian, which differ in the form of the ALP-fermion interactions. Finally, we discuss the matching of the effective Lagrangian at low energies onto a chiral effective Lagrangian describing the couplings of a light ALP to photons and light pseudoscalar mesons, carefully taking into account the presence of a non-zero ALP mass in the effective theory, which gives rise to several important effects.
The results of this work form the basis for precise phenomenological analyses of the physics of a light ALP or axion, connecting low-energy observables in a systematic and accurate way with the couplings of the underlying UV theory.

ALP couplings to the SM
We consider a gauge-singlet, pseudoscalar resonance a, whose couplings to SM fields are, at the classical level, protected by an approximate shift symmetry a → a + c, broken only by the mass term m 2 a,0 . Such a coupling structure arises, for example, if the particle a can be identified with the phase of a complex scalar field.

Choice of the operator basis
The most general effective Lagrangian for this particle including operators of up to dimension 5 reads [35] Here G a µν , W A µν and B µν are the field-strength tensors of SU (3) c , SU (2) L and U (1) Y , and α s = g 2 s /(4π), α 2 = g 2 /(4π) and α 1 = g 2 /(4π) denote the corresponding coupling parameters. B µν = 1 2 µναβ B αβ etc. (with 0123 = 1) are the dual field-strength tensors. The sum in the first line extends over the chiral fermion multiplets F of the SM. The quantities c F are hermitian matrices in generation space. For the couplings of a to the U (1) Y and SU (2) L gauge fields, the additional terms arising from a constant shift a → a + c of the ALP field can be removed by field redefinitions. The coupling to QCD gauge fields is not invariant under a continuous shift transformation because of instanton effects, which however preserve a discrete version of the shift symmetry [3,4]. Above we have indicated the suppression of the dimension-5 operators with the ALP decay constant f , which is related to the relevant new-physics scale by Λ = 4πf . This is the characteristic scale of global symmetry breaking, assumed to be far above the weak scale. It is then a good approximation to neglect contributions from higherdimensional operators, which are suppressed by higher powers of 1/f . 2 Since our effective theory only contains the SM particles and the ALP as degrees of freedom, it would need to be modified in scenarios with a new-physics sector between the weak scale and the scale of global symmetry breaking (v < M NP < 4πf ). Even in this case, the effective Lagrangian (1) offers a model-independent description of the physics below the intermediate scale M NP .
The physical ALP mass is given by the sum of the explicit soft breaking term m 2 a,0 and the contribution to the mass generated by non-perturbative QCD dynamics [6,36,37], such that at lowest order in chiral perturbation theory ( where f π 130 MeV is the pion decay constant. The generalization of this relation to values of m 2 a,0 that are not small compared with m 2 π will be discussed in Section 7. Whereas for the classical QCD axion (with m 2 a,0 = 0) there is a strict relation between the mass and the coupling to gluons, the presence of the additional contribution m 2 a,0 allows for heavier ALPs, which however are still naturally much lighter than the scale f . It is possible to generate this additional contribution dynamically using non-abelian extensions of the SM, in which additional instanton contributions arise [9,[38][39][40][41][42][43][44][45][46][47][48][49][50], or using the recently proposed mechanism of axion kinetic misalignment, in which the axion shift symmetry is explicitly broken in the early universe [51]. It is thus possible to generate an ALP mass significantly larger than the contribution from QCD instantons while preserving the Peccei-Quinn solution of the strong CP problem.
The ALP couplings c F to the SM fermions can, in principle, have a non-trivial structure in generation space, thereby giving rise to flavor-changing neutral current interactions mediated by ALP exchange. The phenomenological constraints on such couplings are very strong, especially for light ALPs, which can be produced in the decays of kaons or B mesons [52][53][54][55][56][57][58], and which can give sizable contributions to flavor-changing transitions in the lepton sector [59][60][61] and to electric dipole moments [62,63]. In extensions of the SM in which the newphysics scale Λ = 4πf is not very far above the TeV scale, the coupling matrices c F must have a hierarchical structure in order to be consistent with these constraints. From the point of view of model building, such a structure can be ensured by imposing the principle of minimal flavor violation [64]. Under this hypothesis, the matrices c Q and c q in the quark sector can be expanded as where counts the order in the spurion expansion. Analogous expressions apply in the lepton sector. The phenomenological implications of these results will be discussed later.

A redundant operator
The form of the effective Lagrangian (1) is not unique. At dimension-5 order one can also write down an ALP coupling to the Higgs doublet φ, given by The operator O φ is redundant, however, because it can be reduced to the fermionic operators in (1) using the field equations for the Higgs doublet and the SM fermions [35]. Indeed, the field redefinitions φ → e ic φ a/f φ and F → e −iβ F c φ a/f F for all chiral fermion multiplets F of the SM, subject to the conditions eliminate the term c φ O φ from the Lagrangian at the expense of shifting the flavor matrices c F by The first three relations in (5) ensure that the SM Yukawa interactions are invariant under the field redefinitions. The fourth relation guarantees that the combination of fermion currents induced by the field redefinitions is anomaly free, and hence no additional contributions to the coefficients of the operators in (1) involving the gauge fields are generated. The conditions (5) define a one-parameter class of field redefinitions, which one can use to eliminate the operator O φ from the effective Lagrangian. One particular solution is given by the choice β u = −1, β d = β e = 1 and β Q = β L = 0, which was adopted in [65,66] and eliminates O φ in favor of a linear combination of operators involving right-handed quark currents. A different solution consists of the choice β F = −2Y F , where Y F denotes the hypercharge of the fermion multiplet F [35,57]. In general, the derivative couplings of the ALP are only defined modulo generators of exact global symmetries of the SM, which include baryon and lepton number. We will see later that physical quantities are independent of the particular choice of β F values as long as the conditions (5) are satisfied.
It follows from this discussion that the redundant operator O φ can be re-expressed in the form where a sum over the generation index i is implied, and the new operator O φ vanishes by the equations of motion. It is a well-known fact that such operators do not need to be included in the renormalization of the basis operators in an effective field theory [67,68]. Hence, it is consistent to leave out the operator O φ from the effective Lagrangian (1). As we will see in Section 3, the original operator O φ is needed as a counterterm to absorb some UV divergences of loop diagrams involving the fermionic operators O F . The correct treatment then consists of projecting O φ back onto our basis using the replacement rule [69][70][71]

Equivalent forms of the effective Lagrangian
Another important freedom in writing down the effective Lagrangian concerns the structure of the ALP couplings to fermions. One can integrate by parts in the third term in (1) and use the SM equations of motion along with the well-known equation for the axial anomaly to put the effective Lagrangian in the alternative form  Figure 1: Contributions to the a → gg decay amplitude involving the ALP-gluon coupling (left) and the ALP couplings to quarks (right). The ALP is drawn as a dotted line. The black circles indicate vertices deriving from the dimension-5 operators in the effective Lagrangian (1).
Here the traces are over generation indices. T F = 1 2 fixes the normalization of the SU (N ) group generators, N c = 3 is the number of colors, and N L = 2 denotes the number of weak isospin components.
and Y e = −1 denote the hypercharge quantum numbers of the SM quarks and leptons. The effective Lagrangians (1) and (9) are equivalent as long as these relations are taken into account. Note, however, that in (9) there is no apparent reason for the complex matricesỸ f to have any particular structure. It is the shift symmetry encoded in the effective ALP Lagrangian (1) that gives rise to the hierarchical structure of these matrices, which results from the appearance of the SM Yukawa matrices in (10). This feature distinguishes an ALP from a generic pseudoscalar boson a. We thus prefer to take the Lagrangian (1) as the starting point of our calculations. Nevertheless, we will see that the combinationsc V V of ALP-boson and ALP-fermion couplings shown in (11) play an important role in phenomenological applications of the effective Lagrangian and in the evolution of the ALP couplings from the new-physics scale Λ down to lower energies.
It is instructive to illustrate the equivalence of the effective Lagrangians (1) and (9) with a concrete example. Consider the decay of an ALP with mass m a Λ QCD into two gluons, which manifest themselves as two jets in the final state. The relevant contributions to the decay amplitude are shown in Figure 1. Calculating the decay rate at one-loop order in perturbation theory, taking into account radiative corrections calculated in [72], one obtains [22] Here n q is the number of light quark flavors with mass below the ALP mass, and where The sum runs over the six quark species of the SM. The parameters c qq (m a ) describe the flavor-diagonal ALP couplings to the quark mass eigenstates and will be defined later in (50). They are connected with the ALP-fermion couplings c q and c Q after these have been transformed into the mass basis of the SM quarks. The above result is obtained based on the effective Lagrangian (1). If instead the calculations are starting from the alternative form of the effective Lagrangian shown in (9), one finds The "−1" inside the bracket accounts for the difference in the fermion loop function, which is a consequence of the difference in the Feynman rules for the ALP-fermion vertices derived from the two Lagrangians. At the same time, the coefficientc GG differs from c GG by the terms shown in the first equation in (11). Because of the trace, the difference between the two parameters is invariant under the unitary transformation to the mass basis, and one finds We thus find that the above two relations for C eff gg are indeed equivalent. It is possible to work with a hybrid form of the effective ALP Lagrangian, in which the ALP-fermion interactions consist of both derivative terms, such as in (1), and non-derivative terms, such as in (4). This is useful, in particular, for low-energy applications in the context of the chiral effective Lagrangian. We will come back to this point in Section 7.
In this work we do not necessarily treat the anomalous couplings of the ALP to gauge fields as being one-loop suppressed, despite the fact that we have factored out α i /(4π) in the definitions of the coefficients c V V . The rationale for this is twofold. Firstly, one can conceive models in which the bosonic couplings are induced by loops involving a parametrically large number N f of new heavy fermions, such that c V V ∝ N f 1 can (at least partially) compensate for the suppression by a loop factor. Secondly, one-loop diagrams involving the bosonic ALP couplings contribute to amplitudes connecting the ALP to fermions (and vice versa), and hence it is important to include these effects in versions of the model in which the genuine fermionic couplings c F in (1) vanish or are very small. We emphasize, however, that in cases where the coefficients c V V and c F are of similar magnitude, one-loop diagrams involving the Wilson coefficients c V V have the same scaling as two-loop diagrams involving the Wilson coefficients c F , see Figure 2. We will include these two-loop contributions in the RG evolution equations of the ALP-fermion couplings and discuss how they can at least partially be included in the matching calculations at the weak scale.

Renormalization-group evolution to the weak scale
The effective Lagrangian (1) is assumed to arise from integrating out some new heavy particles at a scale Λ = 4πf far above the weak scale. Assuming the ALP mass is small -of order 100 GeV or less -we can evolve the Wilson coefficients and operators in the effective Lagrangian down to the scale of electroweak symmetry breaking by solving their RG equations. We now derive the explicit form of these equations, working consistently at two-loop order in gauge couplings and one-loop order in Yukawa interactions. These are the lowest orders at which these interactions contribute to the evolution equations for the ALP couplings. Some technical details of our derivations are relegated to Appendix A. The RG equations for the ALP couplings appearing in the alternative form of the effective Lagrangian in (9) can be derived from the equations below in a straightforward way. They are discussed in Appendix B.

Derivation of the RG evolution equations
Pulling out one factor of α i in the definitions of the ALP couplings to gauge fields in (1) ensures that the Wilson coefficients c V V are scale independent (at least up to two-loop order in gauge couplings), i.e.
For the QCD coefficient c GG this follows from the explicit calculations performed in [73], and an analogous statement holds for c W W and c BB . This is different from the case of a scalar (CP-even) field coupled to two gauge fields, in which the corresponding couplings exhibit a non-trivial RG evolution starting at two-loop order [74,75]. We have checked explicitly that the one-loop diagrams involving the scalar Higgs doublet shown in Figure 3 do not give rise to a scale dependence of the coefficients c W W and c BB either. The Wilson coefficients c F of the ALP interactions with fermions in (1) are scale-dependent quantities and satisfy rather complicated RG equations. At one-loop order there are contributions from Yukawa interactions, which result from the first three graphs shown in Figure 4. While the external-leg corrections (first two graphs) give rise to multiplicative renormalization effects, which in general are not diagonal in generation space, the vertex diagram (third graph) leads to a mixing of the SU (2) L singlet and doublet coefficients c Q and c u,d , as well as c L and c e . Our results for these contributions to the RG equations agree with the corresponding ex-pressions derived in [54,57,76]. The first diagram in the second row of Figure 4 shows a class of UV-divergent one-loop diagrams which require the operator O φ in (4) as a counterterm. As we have discussed in Section 2 this operator is redundant. It is therefore required to map it back onto our operator basis using the replacement rule (8). This gives rise to universal contributions in the RG equations proportional to the parameters β F in (5). In previous studies the operator O φ was included as a basis operator, and its coefficient C φ not only entered the evolution equations for the ALP-fermion couplings, but in fact was assumed to obey an independent RG equation itself [54,57]. Such a treatment gives rise to ambiguous results (see e.g. the discussion in Section 3 of [71]), because as a matter of principle it is impossible to distinguish the matrix elements of O φ from the matrix elements of the fermionic operators O F in (8). 3 In addition, there is a mixing of the Wilson coefficients c V V of the ALP-boson interactions into the coefficients c F , shown by the last diagram in Figure 4. For the case of QCD this mixing has been studied in [73,77], 4 and we agree with the findings of these authors. Note that, owing to our normalization of the coefficients c V V , the corresponding terms in the evolution equations are proportional to α 2 i , and they are diagonal in generation space. Finally, at two-loop order in gauge interactions there are additional generation-independent contributions to the evolution equations, which are proportional to the ALP-fermion couplings. They arise from the second diagram shown in Figure 2 and are diagonal in generation space. We have derived these contributions by generalizing the corresponding results obtained for QCD in [78,79] to the gauge group of the SM. Combining all effects, we obtain (with q = u, d) where C is the eigenvalue of the quadratic Casimir operator in the fundamental representation of SU (N ), and we have abbreviated All quantities on the right-hand side of (18) must be evaluated at the scale µ. Note that the ALP-boson and ALP-fermion couplings entering at O(α 2 i ) appear precisely in the linear combinations already encountered in (11), i.e.
To the best of our knowledge, the contributions proportional to the quantity X, which descend from the redundant operator O φ , as well as the two-loop contributions to the RG evolution equations for the ALP couplings have been derived here for the first time. The appearance of the coefficients β F in the above relations, which are constrained by the conditions (5) but are otherwise arbitrary, appears puzzling at first sight. However, all contributions proportional to the unit matrix in the RG equations give rise to flavor-diagonal contributions after transformation to the mass basis. We will see in Sections 4 and 5 that in predictions for physical quantity any ambiguity in the choice of the β F parameters cancels out. The relations in (17)-(20) form a set of coupled differential equations, from which the scale dependence of the various ALP couplings can be derived. We can simplify the structure of the evolution equations by making use of the freedom to redefine the fermion fields in the SM Lagrangian. The SM Yukawa matrices can be diagonalized by means of bi-unitary transformations, such that If we redefine the fermion fields via then the up-sector and lepton-sector Yukawa matrices are diagonalized, while the down-sector Yukawa matrix is transformed into Here For the purposes of the following discussion we define the matrices c F in this particular basis of fields. Moreover, because of the smallness of the masses of the SM fermions except the top quark, it is a very good approximation to neglect all Yukawa couplings other than y t 1. The RG equations for the ALP-fermion couplings then simplify to where we have defined With our choice of the basis of fermion fields, this quantity will turn out to be the coupling of the ALP to the physical top-quark mass eigenstate (see Section 4 below).

General solution of the evolution equations
Whereas the original ALP-boson couplings c V V are scale independent, this is no longer true for the couplingsc V V , whose definitions contain the scale-dependent ALP-fermion couplings. This fact is discussed in more detail in Appendix A. We find that (in the approximation where only the top-quark Yukawa coupling is kept, see above) the four functionsc GG (µ),c W W (µ), c BB (µ) and c tt (µ) satisfy a closed set of coupled differential equations, which can be solved. We obtainc and is defined in terms of the running top-quark Yukawa coupling. In the solutions (26) we neglect higher-order terms such as those shown in the second line of (27), which is consistent because the effective ALP-boson couplings enter only at two-loop order in (18). Using these solutions, we can now integrate the equations (24) to obtain the evolution of the various ALP-fermion couplings from the new-physics scale Λ down to the weak-interaction scale µ w ∼ 100 GeV. For example, the solution of the first equation takes the form where we have defined, using relation (27), In this way all results can be expressed in terms of U (µ, Λ) and integrals over the running gauge couplings with the ALP-boson couplingsc V V (µ) in (26). The scale evolution of the gauge couplings is governed by the set of coupled differential equations where the β-functions of the three gauge groups are of the form Above the weak scale the relevant one-loop coefficients are β can be any one of the SM coupling parameters. The complete three-loop expressions for the β-functions can be found in [80]. 5 We now present our final expressions for the RG-evolved ALP-fermion couplings, beginning with flavor non-diagonal effects, which are insensitive to the β F parameters. We find (with as well as The last two relations show how a possible flavor non-universality of the diagonal couplings [c Q,u (Λ)] ii at the new-physics scale, which is allowed even under the assumption of minimal flavor violation, evolves to low energies. Before presenting our solutions for the generation-diagonal couplings we return to the question of the β F dependence of the evolution equations (24), which hints at a redundancy of our results. In all physical quantities the dependence on these parameters cancels out. It follows that only certain linear combinations of the flavor-diagonal ALP-fermion couplings are physical. In particular, we find that the differences 5 The coupling parameter α 1 in this work differs from our α 1 by a factor 5/3. (34) are independent of the β F parameters once the relations (5) are taken into account. For i = 3 the first relation reduces to (27). We will see in Section 4 that the solutions (32)-(34) are sufficient to calculate arbitrary physical processes involving ALPs, where however the second relation in (34) gets modified when one transforms the left-handed down-quark fields to the mass basis. We can push further and obtain an explicit approximate expression for the quantity U (µ w , Λ) in (28). At leading order in perturbation theory the top-quark Yukawa coupling and the strong coupling obey the coupled system of equations where we neglect the small effects of the weak interactions. The exact solution of this system exhibits a "quasi fixed point", where the running of α t (µ) tracks the evolution of α s (µ). One finds [81,82] where µ 0 is some reference scale. Using this result in (28), we find after a straightforward calculation where the dots refer to terms of order y 4 t ln 2 (Λ 2 /µ 2 ) and higher. In this expression the large logarithms of the scale ratio Λ 2 /µ 2 are resummed to all orders of perturbation theory. This explicit result implies that The numerical impact of the evolution effects on the ALP-fermion couplings will be discussed in Section 5.

Transformation to the mass basis
Once the effective Lagrangian has been evolved to the weak scale µ w , it is appropriate to express it in terms of fields defined in the broken phase of the electroweak symmetry, which correspond to the mass eigenstates of physical particles. This leads to where s w ≡ sin θ W and c w ≡ cos θ W denote the sine and cosine of the weak mixing angle, and we have defined [22] All coupling parameters and operators in (39) are now defined at the weak scale µ w . Recall that the Wilson coefficients c V V are scale independent.
To obtain the ALP interactions with fermions contained in L ferm we must transform the fermion fields to the mass basis, in which the Yukawa matrices are diagonalized, see (21). Under the corresponding field redefinitions the flavor matrices c F transform into new hermitian matrices Note that the two matrices k U and k D are connected via the CKM matrix V , such that and are therefore not independent. Likewise, the ALP couplings to the neutrinos are identical to those to the left-handed charged leptons, i.e. k ν = k E . In terms of these matrices we obtain The matrices k F and k f are evaluated at the scale µ w . The corresponding expressions can be obtained from the results compiled in Section 3.2 by recalling that these relations have been derived in a basis for which all transformation matrices are equal to the unit matrix except It is instructive to study what the hypothesis of minimal flavor violation [64] implies for the structure of the ALP-fermion couplings after electroweak symmetry breaking. Transforming the expressions (3) to the mass basis, we obtain The only non-diagonal contributions are those involving the CKM matrix. To very good approximation we can set the diagonal entries of the Yukawa matrices to zero for all quarks other than the top quark. In this approximation Higher-order terms in have the effect of generating more complicated functions of the top-quark mass, while the dependence on CKM parameters remains unchanged. We thus find that, under the hypothesis of minimal flavor violation and to very good approximation, flavor-violating couplings only arise in the couplings k D to left-handed down-type quark currents. The leptonic couplings k E and k e are proportional to the unit matrix in this approximation.
Several important weak-scale processes involving ALPs have been discussed in the literature [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. Their rates can be calculated in terms of the couplings entering the effective weak-scale Lagrangian (39). To mention three prominent examples, we briefly consider the decay a → γγ of a heavy ALP (with mass of order the weak scale) as well as the exotic decay modes Z → γa and h → Za of the Z boson and the Higgs boson. Calculating the corresponding decay amplitudes at one-loop order, and setting the matching scale µ w equal to the mass of the decaying particle, one finds [22,65,66] The coefficients C eff γγ and C eff γZ in the first two cases are given by with f (τ ) as defined in (14). The function B 1 ≈ 1 for all light fermions with mass m f m a , while B 1 ≈ − m 2 a 12m 2 f for heavy fermions (m f m a ). Thus, each electrically charged fermion lighter than the ALP adds a potentially large contribution to the effective Wilson coefficient C eff γγ , while fermions heavier than the ALP decouple. Similarly, one finds that B 3 ≈ 1 for all fermions much lighter than the Z boson (irrespective of the ALP mass), while for the top quark |B 3 | 1 as long as the ALP is lighter than the top-quark mass. In the third decay rate in (46) we have defined the phase-space function λ(x, y) = (1 − x − y) 2 − 4xy and the parameter integral where d[xyz] ≡ dx dy dz δ(1 − x − y − z). Throughout this paper m t ≡ m t (m t ) denotes the running top-quark mass in the MS scheme evaluated at µ = m t . The quantity F is numerically close to 1 for ALP masses below the weak scale. Finally, we have introduced the parameters which contain the relevant ALP couplings to fermions and will play an important role in our discussion below. This definition generalizes relation (25) for the top quark to other ALPfermion couplings. The scale evolution of these quantities from the new-physics scale Λ to the electroweak scale can be derived from (34). For up-type quarks and charged leptons, the parameters c f f are equal to the differences of ALP-fermion couplings considered in this result, and we have For down-type quarks one finds that where V is the CKM matrix. Hence, the result given in (34) is not directly applicable. Instead, we obtain If the matrix k U (Λ) is diagonal, as required under the hypothesis of minimal flavor violation, see (45), then the terms shown in the third line vanish.

Matching contributions at the weak scale
Let us now assume that the ALP is significantly lighter than the weak scale, and that we are interested in low-energy processes at energies E 100 GeV. We can then integrate out the heavy SM particles -the top quark, the Higgs boson and the weak gauge bosons W ± and Z 0at the scale µ w and match the effective Lagrangian (39) onto a low-energy effective Lagrangian in which these degrees of freedom are no longer present as propagating fields. Just below the scale µ w , this Lagrangian takes the form where L ferm is given by (43) but with the top-quark fields t L and t R removed. In general, the Wilson coefficients c GG , c γγ , k F and k f in this effective Lagrangian differ from the corresponding coefficients in the effective Lagrangian above the weak scale by calculable matching contributions, which arise when the weak-scale particles are integrated out. We now discuss the calculation of the relevant matching conditions at one-loop and partial two-loop order.

Matching contributions to the ALP-boson couplings
One-loop matching corrections to the ALP-gluon and ALP-photon couplings c GG and c γγ could in principle arise from loop graphs containing top quarks and heavy electroweak gauge bosons. Two representative diagrams are shown in Figure 5. The corresponding effects were calculated in [22], and it was shown that for a light ALP these effects decouple like m 2 a /m 2 t and m 2 a /m 2 W , respectively. For a light ALP far below the weak scale there are thus no matching contributions to the effective low-energy Lagrangian (54) from these loops, i.e.
Matching corrections of order m 2 a /m 2 t or m 2 a /m 2 W , which arise from the Taylor expansions of the functions B 1 (τ ) and B 2 (τ ) in (48)  Wilson coefficients of dimension-7 operators in the low-energy effective theory below the weak scale, which we neglect for simplicity. As a side remark, let us mention briefly that the situation would be different if we were to perform the calculations based on the alternative form of the effective Lagrangian shown in (9). In this case there are non-vanishing matching contributions from top-quark loop diagrams, which lead to Recall that, according to (11) and (40), the coefficientsc GG andc γγ above the weak scale are related to the corresponding unprimed coefficients bỹ where the sum in the first (second) equation runs over all quark (fermion) species in the SM. When crossing the weak scale, one needs to add the matching contributions given above, and this has the effect of removing the contributions from the top quark in these relations. We thus obtaiñ The same procedure repeats itself as µ is evolved to lower energies and one crosses the threshold of other heavy fermions.

Matching contributions to the ALP-fermion couplings
One-loop matching corrections to the ALP-fermion couplings arise from graphs containing heavy electroweak gauge bosons. Some representative diagrams are shown in Figure 6. Loop Figure 6: One-loop matching contributions to the ALP-fermion couplings. In the second diagram (V 1 V 2 ) = (W W ), (ZZ), (Zγ) or (γZ). In the last two diagrams V = W, Z, but in the sum of all contributions only the W -boson graphs with internal top quarks (plus the corresponding graphs with Goldstone bosons) give rise to non-zero contributions.
diagrams involving Higgs bosons give contributions proportional to the Yukawa couplings of the external fermions. Since the top quark is integrated out in the effective theory below the weak scale, these graphs are proportional to y 2 f for some light SM fermion f and hence can be neglected. The first diagram in Figure 6 arises from ALP mixing with the Z boson via a top-quark loop. The second graph gives rise to matching contributions proportional to the ALP-boson couplings. The corresponding effects were calculated in [22] for the case where the external fermions are leptons. Here we generalize these results to the case of quarks, where however contributions involving virtual top quarks require a special treatment. The remaining diagrams contain vertex and external-leg corrections from loops involving heavy W and Z bosons. We have calculated these diagrams in a general R ξ gauge, finding that the sum of all contributions yields a gauge-invariant answer. Moreover, the sum of all contributions involving Z bosons and their Goldstone bosons vanishes. For the diagrams involving W bosons a non-zero contribution remains, which arises from graphs containing internal top quarks. These diagrams contribute to the couplings k D (µ w ) in the left-handed down-quark sector only, and they are the only source of flavor off-diagonal effects. Combining all terms, we find the matching contributions (with F = U, D, E, ν and f = u, d, e) These contributions must be added to the RG-evolved coefficients at µ = µ w , so that one obtains k F,f (µ w ) + ∆k F,f (µ w ) for the ALP-fermion couplings just below the weak scale. All scale-dependent parameters on the right-hand side of the above relations are evaluated at the scale µ w . RG invariance requires that the ALP-boson couplings entering in these relations must appear in the form of the couplingsc V 1 V 2 , at least in the coefficients of the ln(µ 2 w /m 2 W,Z ) terms. Hence, via the substitution c V 1 V 2 →c V 1 V 2 we can account for an important subclass of two-loop matching contributions. The scheme-dependent constant δ 1 arises from the treatment of the Levi-Civita symbol in d dimensions. We obtain δ 1 = − 11 3 in a scheme where µναβ is treated as a d-dimensional object, and δ 1 = 0 if it is instead treated as a 4-dimensional quantity.
The non-trivial flavor structure is captured by the quantity where x t = m 2 t /m 2 W . These matching contributions are sources of flavor-changing ALP interactions even if the underlying UV theory does not contain new sources of flavor or CP violation beyond those present in the SM. We have neglected the Yukawa couplings of the light quarks and leptons. In this approximation there are no flavor off-diagonal matching contributions in the up-quark and lepton sectors.

Flavor-diagonal couplings
The flavor-diagonal ALP-fermion interactions in (43) can be expressed in terms of vector and axial-vector currents. The vector currents are conserved below the weak scale and thus do not contribute to physical matrix elements. It follows that we can rewrite this Lagrangian in the equivalent form (for µ µ w ) where the sum runs over all charged fermion species in the low-energy theory (the quarks u, d, s, c, b and the leptons e, µ, τ ). The couplings c f f have been defined in (50) in terms of the diagonal elements of the matrices k f and k F . Note that the ALP-neutrino interactions can be dropped in the low-energy Lagrangian (but not in the theory above the weak scale, where they contribute at one-loop order to the ALP couplings to W and Z bosons). Using integration by parts, the derivative on the neutrino axial-vector current vanishes because the neutrinos are massless in the SM. At the matching scale µ w , the coefficients c f f (µ w ) are given by the sum of the contributions from RG evolution, shown in (51) and (53), and weak-scale matching, see (59) and (60). In this sum the dependence on the matching scale µ w partially cancels out; however, some scale dependence remains and cancels when the evolution below the weak scale is taken into account (see Section 6 below). In order to get a feeling for the magnitude of the radiative corrections we choose the new-physics scale Λ = 4πf with f = 1 TeV and evaluate the coefficients c f f (µ) in the vicinity of µ w = m t . We find numerically We use the two-loop expression for the running coupling α s (µ) and the one-loop approximations for the couplings α 1 (µ) and α 2 (µ), and we evaluate the function U (µ w , Λ) using the explicit form (37). For the couplings c d i d i in the down-quark sector we work under the assumption of minimal flavor violation and have approximated |V tb | 2 ≈ 1 and |V td | 2 ≈ |V ts | 2 ≈ 0. From (20), the matching conditionsc V V (Λ) can be written in the form where the sums run over all quark and fermion flavors. We observe that electroweak radiative corrections are generally very small, while the contributions proportional to c tt from the Yukawa interactions as well as QCD effects can be sizable. For example, in scenarios where the ALP-boson couplings at the UV scale are an order of magnitude larger than the ALPfermion couplings, the corrections induced byc GG can give contributions to c qq (µ w ) of about 7%, whereas the contributions ofc W W andc BB are negligible. The logarithms of the ratio (m 2 t /µ 2 w ) in the above expressions show the remaining dependence on the weak matching scale. This dependence cancels out when evolution effects below the weak scale are included.
The numerical results shown in (62) are relevant for an ALP which is part of a new-physics sector at a scale Λ ∼ 10 TeV. For the QCD axion, one typically considers much higher scales in the vicinity of f ∼ 10 12±3 GeV. This gives rise to significantly enhanced evolution effects. For example, choosing Λ = 4πf with f = 10 12 GeV and setting µ w = m t we find It is very useful to derive a simple, approximate expression for the ALP-fermion couplings at the scale µ w , in which one neglects the small two-loop electroweak evolution effects as well as the two-loop contributions proportional to the ALP-fermion couplings themselves. This yields (for q = t and µ µ w ) which as we will see in the next section continues to hold below the weak scale. Note that only the last term in the first line is scale-dependent in this approximation, and one needs to adjust the value of β 0 whenever one crosses a quark threshold. In the first relation T u 3 = 1 2 and T 3 d = − 1 2 denotes the weak isospin. In the above expressions large logarithms of the scale ratio Λ/µ w are resummed to all orders in perturbation theory. The most striking effect is the universal admixture (weighted only by weak isospin) of a contribution proportional to c tt (Λ) to all ALP-fermion couplings, even those involving the charged leptons. When one re-expands the above expressions to first order in couplings, one obtains This effect was noted previously in [57], where the opposite sign was obtained and in the argument of the logarithm the scale µ was used rather than m t . Note, however, that this effect is due to the first diagram in Figure 6, which no longer contributes below the scale of the top quark. Also, the resummation effects included here can be numerically very important. With f = 10 12 GeV, for instance, formula (66) would predict ±0.84 c tt (Λ), overshooting the effect by more than a factor of 2. The evolution effects in (65) are of potentially large importance not only for ALPs, but also for the classical QCD axion. In order to illustrate this fact we consider the DFSZ model [7,8], in which the ALP couplings at the UV scale Λ = 4πf satisfy [83] where tan β = v u /v d is the ratio of the vacuum expectation values of the two Higgs doublets, with a phenomenologically motivated range spanning 0.28 < tan β < 140 [84]. The axion mass is given by relation (2) with m 2 a,0 = 0, i.e. it is uniquely determined by the decay constant f . Assuming that the masses of the additional Higgs bosons are larger than Λ, we can evolve these coupling parameters down to the weak scale. Figure 7 shows the axion-electron coupling at the high scale (red line) and the RG-evolved couplings c ee (m t ) at the electroweak scale for different axion masses. The smaller the axion mass, the larger are the evolution effects because the corresponding values of Λ increase proportional to 1/m a , ranging from Λ 7.3 TeV for m a = 1 keV to Λ 7.3 · 10 9 TeV for m a = 1 µeV. The figure shows that in the DFSZ model the axion-electron coupling can be enhanced through evolution effects by up to an order of magnitude for small values of tan β.

Flavor-changing couplings
The flavor-changing ALP-fermion couplings in (43) can be integrated by parts without introducing additional contributions to the Wilson coefficients c V V . This gives (for µ µ w ) where throughout this discussion i = j. The fermion masses and coupling parameters must be evaluated at the scale µ. This form of the Lagrangian makes explicit that flavor-changing amplitudes are suppressed by the masses of the fermions involved. (The same is true for the flavor-conserving interactions in (61), but in this case integrating by parts generates additional contributions to the ALP-gluon and ALP-photon couplings.) At the weak scale µ w , the generation off-diagonal coefficients [k f (µ w )] ij and [k F (µ w )] ij are again given by the sum of the contributions from RG evolution and weak-scale matching. Recall that generation off-diagonal matching contributions are captured by the quantity∆k D (µ w ) in (60). For all coefficients other than k D , one finds from (32) and (33) that flavor-changing interactions at the weak scale are inherited from the UV scale Λ. We find Note that for k u and k U we only need the entries where i, j = 3, since the top quark has been integrated out in the effective theory below the weak scale. If the UV theory respects minimal flavor violation, then all these couplings vanish. For the off-diagonal elements of the coefficient k D we find the more interesting result where the integral I t (µ w , Λ) has been defined in (30). If the original ALP Lagrangian (1) at the new-physics scale respects the principle of minimal flavor violation, the matrix k U is diagonal, as shown in (45). In this case the above expression simplifies significantly, and we find where (again setting the new-physics scale to Λ = 4πf with f = 1 TeV) Note that under the hypothesis of minimal flavor violation the matrix k U is diagonal but not necessarily proportional to the unit matrix in generation space, see (45). The first term on the right-hand side of (71) thus accounts for the possibility that [(k U )(Λ)] 33 = [(k U )(Λ)] 11 . If this is the case, then the off-diagonal matrix elements (73) at the new-physics scale can be non-zero, providing a UV source of flavor violation. Evolving the coefficients to the weak scale µ w = m t , we obtain numerically The matching contributions proportional toc GG andc W W are very small. Relation (71) shows explicitly how flavor-changing effects are generated through RG evolution from the new-physics scale Λ to the weak scale (first line) and matching contributions at the weak scale (second and third lines). These loop-induced effects should be considered as the minimal effects of flavor violation present in any ALP model, even if the matrix k D is diagonal at the new-physics scale Λ (which would be a stronger assumption than minimal flavor violation). The terms proportional to c W W in (71) agree with a corresponding expression derived in [53]. Our results for the evolution effects and the contribution proportional to c tt (µ w ) are new. The logarithm of (µ 2 w /m 2 t ) in the coefficient of c tt (but not the x t -dependent remainder) was derived in [56]. The more general expressions shown above, and in particular the results (60) and (70), which do not assume minimal flavor violation, are derived here for the first time.
In the sum of the contributions from scale evolution and weak-scale matching, the dependence on the matching scale µ w drops out. This is obviously true for the coefficients in (69), but it also holds for the sum of all terms on the right-hand side of (70). In fact, we will see in Section 6 that the flavor off-diagonal Wilson coefficients do not run below the weak scale (in the approximation where the Yukawa couplings of the light quarks are put to zero). Hence, the expressions shown in (69) and (70) hold for all values µ < µ w .
6 Renormalization-group evolution below the weak scale Now that we have obtained the values of the Wilson coefficients at the weak scale, we should evolve these coefficients down to lower scales, so that they can be used in calculations of lowenergy observables. Compared with (18) the evolution equations simplify significantly, because the Yukawa interactions mediated by Higgs exchange are absent in the low-energy theory, as are diagrams including the heavy weak gauge bosons. The only remaining contributions to the evolution equations result from the second diagram in Figure 2 and the last diagram in Figure 4, where the gauge bosons can be gluons or photons. We obtain where Q = U, D and q = u, d. Below the weak scale the scale dependence of the effective coefficientsc GG andc γγ is very weak, since it only arises at two-loop order. At next-toleading logarithmic order, it is consistent to neglect this effect. Note also that the evolution effects below the weak scale are diagonal in generation space, and hence the flavor-changing couplings are scale-independent in the low-energy theory, as stated above. For the flavordiagonal couplings only the parameters c f f defined in (50) are physical. At next-to-leading logarithmic order, their scale evolution is given by In the low-energy theory below the weak scale the relevant β-function coefficients are β QCD 0 = 11 − 2 3 n q for QCD and β QED Here n q denotes the number of light quark flavors with mass below the scale µ, and the sum over f includes all light fermions with mass below µ. Note that the dependence on the matching scale µ w cancels when the expressions given in (62) are used in the above relations.
According to (58), the effective Wilson coefficientsc GG (µ w ) andc γγ (µ w ) contain the c f f parameters of all light fermions in the effective theory below the scale µ w . Generalizing these results to lower scales, we definẽ Like the β-function coefficients β QCD 0 and β QED 0 , the effective couplings change by discrete amounts whenever one crosses a flavor threshold, and an appropriate matching must be performed in the usual way. In other words, one first evolves the coefficients from the weak scale to the scale µ b m b , then eliminates the bottom quark from the list of light fermions, then evolves from the b-quark scale to the scale µ τ m τ , then eliminates the τ lepton from the list of light fermions, and so on. In each step the coefficients of the β-functions as well as the values ofc GG andc γγ need to be adjusted. Concretely, at values of µ just below the scale and similarly for c (µ). In the two last terms of the first line the ALP-boson couplings and the β-function coefficients are evaluated with n q = 5 active quark flavors, whereas in the second line they are evaluated with n q = 4 flavors. The numerical impact of these low-scale evolution effects is very small. For example, with µ w = m t and µ 0 = 2 GeV we find It is instructive to compare the above results with analogous expressions derived for the quark coefficients c qq in [85]. In this paper only QCD evolution effects were included. The results obtained there are in agreement with our findings when we ignore the terms proportional to the electromagnetic coupling α in the first line of (76). However, in [85] the same equation was used to account for evolution effects above the electroweak scale. This ignores the by far dominant contributions from the top-quark Yukawa interactions in (65), which as we have discussed have an important impact on all ALP-fermion couplings.
The scale-dependent ALP-boson couplingsc V V defined in (77) are not only relevant in the context of the evolution equations for the ALP-fermion couplings, but they are also closely related to some observables of phenomenological interest. In (12) and (46) we have given explicit expressions for the a → gg and a → γγ decay rates, the latter of which plays a pivotal role in the phenomenology of a light ALP. The fermion loop function entering these expressions satisfies B 1 (τ ) ≈ 1 for τ 1 (corresponding to "light" fermions with m f m a ) and B 1 (τ ) ≈ 0 for τ 1 (corresponding to "heavy" fermions with m f m a ). Moreover, the loop function B 2 (4m 2 W /m 2 a ) ≈ 0 for a light ALP with mass m a m W . Let us now apply an MS-like approximation scheme, in which we treat the "light" fermions as (approximately) massless and the "heavy" fermions as infinitely heavy. We then obtain where the effective couplings on the right-hand side are precisely those defined in (77).

Matching onto the chiral Lagrangian
Using the results derived in the previous sections, the effective ALP Lagrangian (54) can be evolved down to scales far below the scale of electroweak symmetry breaking. When one reaches energies of order 1-2 GeV, only the three light quark flavors u, d, s remain as active degrees of freedom. In order to study the low-energy interactions of a light ALP with hadrons, one should match this Lagrangian onto a chiral effective Lagrangian incorporating the ALP couplings to the light pseudoscalar mesons (π, K, η). In order to find the bosonized form of the ALP-gluon interaction, one eliminates the aGG term in favor of ALP couplings to quark bilinears, whose chiral representation is well known. To this end, one performs the chiral rotation [35,83,86,87] where q is a 3-component vector in flavor space, κ q is a real matrix, which we choose to be diagonal in the quark mass basis. Under the chiral rotation the measure of the path integral is not invariant [88,89], and this generates extra terms adding to the anomalous ALP couplings to gluons and photons. Imposing the condition Tr κ q = κ u + κ d + κ s = 1 (82) ensures that the ALP coupling to GG is eliminated from the Lagrangian, at the expense of modifying the ALP-photon and ALP-fermion couplings as well as the quark mass matrix. At a scale µ χ ∼ 1-2 GeV, this leads to the effective Lagrangian where m q = diag(m u , m d , m s ), and the dots represent the ALP-lepton couplings and possible flavor-changing ALP interactions, both of which are irrelevant to this discussion. The quantitiesĉ with q = u, d, s, are the modified ALP-fermion and ALP-photon couplings, whose explicit expressions in terms of the ALP couplings at the UV scale can directly be obtained from the results compiled in the previous sections. The effective Lagrangian (83) is equivalent to the original Lagrangian (54) evolved to the low scale µ χ and it describes the same physics, even though the ALP coupling to gluons has been removed at the Lagrangian level. The ALP interactions with quarks now enter in two places: in the derivative couplings proportional to the parametersĉ qq , and through the phase factors multiplying the quark mass matrix. Note that the choice of the κ q parameters is completely arbitrary as long as the constraint (82) is satisfied. Below we will demonstrate with two examples that the results obtained for physical observables are indeed independent of the κ q parameters. As a side remark, let us mention that the effective ALP-pion Lagrangian can also be derived starting from the alternative form of the effective Lagrangian shown in (9). This Lagrangian differs from the original one in (1) by a chiral rotation of the same form as that shown in (81), but with a different choice of the κ q parameters not subject to the condition (82). A second chiral rotation is then required to eliminate the ALP-gluon coupling. The resulting ALP-pion Lagrangian is equivalent to the one derived here.
Let us now discuss the matching of the effective Lagrangian (83) onto a chiral effective Lagrangian, working consistently at lowest order in the chiral expansion. The Dirac Lagrangian for the quark fields matches onto the standard form of the Gasser-Leutwyler Lagrangian [90], but with the mass matrix replaced by the ALP-field dependent matrix Next, the axial-vector currents in the derivative couplings of the ALP to quark fields can be matched onto chiral currents using the replacement rules In this way, one obtains [35,86,87] L ALP χP T = where fπ λ a π a (x) contains the pseudoscalar meson fields (λ a are the Gell-Mann matrices), and the parameter B 0 ≈ m 2 π /(m u + m d ) is proportional to the chiral condensate. The covariant derivative is defined as For the case of the QCD axion (with m 2 a,0 = 0), the chiral effective ALP Lagrangian was first introduced in [35] and has recently been studied in great detail in [85]. In general, the last term in the first line of (87) gives rise to a mass mixing of the ALP with the pseudoscalar mesons π 0 and η 8 . In order to eliminate this mixing, one chooses the matrix κ q in such a way that κ q m q ∝ 1. When combined with the condition (82) this implies With this choice, the modified ALP-photon coupling takes the form where we have used the ratios m u /m d = 0.49±0.02 and 2m s /(m u +m d ) = 27.4±0.1 [91]. Nextto-leading order corrections in the chiral expansion toĉ γγ have been worked out in [85]. They reduce the coefficient in front of c GG to −(1.92 ± 0.04). At lowest order in the chiral expansion one finds that, with the above choice of κ q values, there are no additional contributions to the a → γγ decay amplitude beyond those governed byĉ γγ in (89). QCD dynamics generates a mass for the ALP, thereby breaking the continuous shift symmetry of the classical Lagrangian.
a Z a t Figure 8: Leading-order contributions to the a → γγ decay amplitude in the chiral expansion. The π 0 γγ coupling is obtained from the Wess-Zumino-Witten term not shown explicitly in (87).
Expanding the terms in the first line of (87) to quadratic order in the pion and ALP fields, one finds [6,36,37] up to higher-order corrections in the chiral expansion and independent of the choice of the individual κ q values. Here f a = −f /(2c GG ) is the axion decay constant. More generally, one finds that the axion potential is a periodic function of the axion field, which is invariant under the discrete shift transformation a → a + 2nπf a . Let us now discuss the structure of the effective chiral Lagrangian for a light ALP in the presence of a non-vanishing mass term m 2 a,0 . As we will show, in this case some non-trivial complications arise from the mixing of the ALP with the pseudoscalar mesons, which give rise to an additional contribution to the a → γγ decay amplitude [22]. For simplicity, we will consider the case of two light flavors u and d. The generalization to three flavors is straightforward, but the additional contributions one finds are suppressed by m u,d /m s . In our discussion we do not impose the relations κ u = m d /(m u + m d ) and κ d = m u /(m u + m d ), which would be the analogue of (88) for the case of two flavors. Physical quantities are independent of the choice of the κ q parameters, and it is instructive to trace in detail how the dependence on these parameters, which enters via the matrixm q (a) and via the coupling parametersĉ qq and c γγ in (84), cancels out. As an important example, we consider the decay a → γγ. As shown in Figure 8, at leading order in chiral perturbation theory there are two contributions to the decay amplitude: one involving the couplingĉ γγ and one involving the mixing of the ALP with the neutral pion, followed by the decay π 0 → γγ mediated by the axial anomaly. The latter coupling can be implemented in the chiral Lagrangian through the Wess-Zumino-Witten term [92]. Combining the two contributions, we find where p 2 a = m 2 a , and the last factor in the second line arises from the π 0 → γγ vertex. 6 Adding up the various terms inside the bracket in the first line we find that any dependence on the parameters κ q cancels out, and one is left with the combination which was identified in [22] as the effective ALP coupling to photons in the chiral effective theory. Note that there are additional contributions from the charged leptons, which have been given in (46) but are not included here. In the limit of a very light ALP (m 2 a m 2 π ) the above relation reduces to (89) in the approximation where the strange-quark mass is decoupled. For a heavier ALP, however, the additional contributions can be important. For m 2 a m 2 π we obtain C eff γγ c γγ − 1.67c GG + 0.5(c uu − c dd ), which is now explicitly dependent on the quark couplings. The contribution proportional to the mass difference of the up and down quarks in (92) results from the coupling of the neutral pion to G a µνG µν,a . The corresponding matrix element can be derived using the anomaly equation and assuming isospin invariance of the pion matrix elements of axial-vector and pseusoscalar currents [94,95]. One finds The pion then decays into two photons via the axial anomaly. The contribution 5/3 in (92) arises from an analogous coupling to the flavor-singlet meson ϕ 0 (the analogue of η 1 in flavor SU (3)) [96]. The result (92) implicitly assumes that the ALP is lighter than the flavor-singlet mesons, because these have been integrated out from the chiral Lagrangian. In the opposite limit one should use the perturbative expression shown in (47) for the effective ALP-photon coupling. Finally, the contribution proportional to the c qq parameters is due to the kinetic mixing of the ALP with the neutral pion. This effect introduces a dependence of the effective ALP-photon coupling on the parameters c qq , which is absent for the QCD axion. Note that the difference of the ALP couplings to up and down quarks receives an important contribution from RG evolution. From (65) we find the approximate expression In order to avoid the presence of ALP-pion mixing contributions in perturbative calculations, one needs to diagonalize the quadratic terms in the effective chiral Lagrangian. Upon expanding the Lagrangian to quadratic order in fields, we obtain where the symmetric matrices accounting for kinetic and mass mixing are given by For the special choice κ u = m d /(m u +m d ) and κ d = m u /(m u +m d ), the quantity δ κ vanishes and relation (99) reduces to a result derived in [22]. But this choice does not eliminate the ALP-pion mixing. Instead, in the presence of a non-vanishing ALP mass the optimal choice of the κ q parameters is In the limit where m 2 a /m 2 π → 0 this reduces to the default choice usually adopted in the literature, but for generic ALP masses the additional contributions introduce important corrections. With the choice (101) the physical neutral-pion state does not contain an admixture of the ALP at first order in f π /f , the parameterĉ γγ in the effective Lagrangian (87) agrees with the effective ALP-photon coupling C eff γγ shown in (92), and the parametersĉ qq satisfy the relation Finally, with this choice the physical ALP mass can be expressed as This result generalizes relation (2) to arbitrary values of the Lagrangian parameter m 2 a,0 . When the effective chiral Lagrangian (87) is expressed in terms of the physical states given in (97), one finds (now for general κ q parameters) where D µ π ± = (∂ µ ∓ ieA µ )π ± , and for simplicity we have suppressed the subscript "phys" on the fields. The coefficient in front of the ALP-photon coupling, which is the sum of the coefficientĉ γγ and a contribution from the Wess-Zumino-Witten term, is nothing but the  Figure 9: Dependence of the phase-space functions g 00 (r) (blue) and g +− (r) (red) on the ALP mass (with r = m 2 π /m 2 a ).
quantity C eff γγ given in (92). It is independent of the choice of the κ q parameters. The remaining dependence, which enters via the quantity δ κ , drops out when one calculates physical matrix elements. Indeed, using integration by parts it can be seen that the coefficient of δ κ vanishes when the equations of motion for the pion fields are used. They can thus be dropped from the effective Lagrangian. It follows that a single parameter ∆c ud governs the leading-order interactions of the ALP with pions, and we obtain the final expression 4∂ µ a π 0 π + D µ π − + π 0 π − D µ π + − 2π + π − ∂ µ π 0 + m 2 a a 2π + π − π 0 + π 0 3 + O This generalizes the effective axion-pion Lagrangian derived in [97] to the case of an ALP with non-zero mass parameter m 2 a . As an important application of the Lagrangian (104) we consider the decays of an ALP into three pions, which is allowed if the ALP mass is larger than 3m π . We obtain the decay amplitudes where m 2 +− = (p π + + p π − ) 2 is the invariant mass squared of the charged pion pair. These expressions agree with corresponding results derived in [22]. In this reference also the differential distributions in the Dalitz plot were derived. Note that the chiral expansion makes sense only in the region of phase space where the pion momenta are small compared with the scale of chiral symmetry breaking, 4πf π 1.63 GeV. This requires the ALP to be lighter than about 3 GeV. For the total decay rates one finds where (with 0 ≤ r ≤ 1/9) Both functions are normalized such that g ab (0) = 1, and they vanish at the threshold r = 1/9. The dependence of these two functions on the ALP mass is shown in Figure 9. Interestingly, the two decay rates are almost of equal size, despite of the fact that the rate of the a → 3π 0 mode contains a symmetry factor 1/6. From a phenomenological point of view the a → 3π decay rates can be important. For m a = 1 GeV, we find that Γ(a → 3π)/Γ(a → γγ) 4.6 (∆c ud /C eff γγ ) 2 , where the ratio of couplings is naturally of O(1), see (92) and (98).

Conclusions
Axions and axion-like particles are well-motivated new-physics candidates in extensions of the SM with a spontaneously broken global symmetry. In these models the mass scale of the new physics sector is set by the scale at which the global symmetry is broken, whereas the mass of the associated pseudo Nambu-Goldstone boson can be significantly smaller. The coupling structure of the ALP is therefore determined at the UV scale, while experimental searches are performed at energies comparable to its mass. The couplings at this low scale dictate the most relevant interactions of the ALP, its branching ratios and the most promising search strategies.
In this work we have derived the low-energy ALP couplings by starting from the most general Lagrangian including all leading-order dimension-5 operators at the UV scale, systematically running the Wilson coefficients and matching to the theory below the electroweak scale and finally to the chiral Lagrangian. The corresponding equations represent a complete framework to calculate couplings of an ALP or axion to SM particles at any given scale.
At the UV scale the ALP Lagrangian can be defined in equivalent bases, which make manifest either the derivative character of the ALP couplings or the suppression of the ALPfermion couplings by the fermion masses, respectively. We have demonstrated the equivalence of these two bases explicitly, using the example of the one-loop ALP decay widths into gauge bosons.
We have presented the renormalization-group equations from the UV scale to the electroweak scale and given their analytic solutions. Operators coupling the ALP to gauge bosons can be defined to be scale invariant. The anomalous dimensions of the ALP-fermion couplings have contributions from Yukawa interactions and gauge interactions, the latter of which we have calculated up to two-loop order.
At the electroweak scale, we have expressed the Lagrangian in terms of fields in the broken phase and integrated out the top quark, Higgs boson and W and Z bosons. We have included all matching contributions at one-loop order and partially accounted for two-loop gauge contributions. There is an important flavor-universal contribution to the ALP couplings to fermions from the ALP mixing into the neutral SM Goldstone boson. This coupling is induced through top-quark loops and generates, for example, sizable ALP-electron interactions even if only ALP couplings to quark doublets or right-handed up-type quarks are present in the UV theory. Depending on the ALP mass, experimental searches for a → e + e − decays, astrophysical constraints or precision spectroscopy searches that are sensitive to the simultaneous presence of ALP-electron and ALP-nucleon couplings can discover an ALP with these properties even if the ALP does not interact with leptons at the UV scale. This argument extends to the QCD axion, and we have shown that the DFSZ axion has sizable couplings to electrons even at low tan β, where the tree-level coupling is suppressed.
Running below the electroweak scale is solely determined by the ALP-gluon and ALPphoton couplings. At the QCD scale we have matched the effective Lagrangian onto the chiral Lagrangian extended with an ALP field. We have discussed ALP-pion mixing and determined the optimal choice of the chiral rotation to eliminate the ALP admixture to the physical π 0 state. We stress that this choice is different for an axion without explicit mass term and an ALP which has a mass even in the absence of a dynamical mass term. We have shown explicitly that the ALP-photon coupling and the ALP couplings to pions are independent of the parameters of the chiral rotation.
The results of this paper form the basis for precise phenomenological analyses of the physics of a light ALP or axion, connecting low-energy observables in a systematic and accurate way with the couplings of the underlying UV theory.

A Scale dependence of thec V V parameters
Contrary to the original ALP-boson couplings c V V , the quantitiesc V V defined in (11) are no longer scale independent at two-loop order, but they satisfy the evolution equations Here q = u, d, s, c, b, t and = e, µ, τ run over the various fermion flavors of the SM. We have derived these equations using the evolution equations (17) and (18). It is a very good approximation to drop all Yukawa couplings other than y t , in which case only the ALP-fermion coupling c tt enters on the right-hand side of the equations. The evolution equation for this quantity can be derived from (18 Relations (A.1) and (A.2) form a coupled system of equations, which can be solved to obtain the scale-dependent coefficients c tt (µ) andc V V (µ). The solutions, in the approximation needed in this work, have been given in (26) and (27).

B Evolution equations for the effective Lagrangian (9)
The effective Lagrangian (9) provides an alternative description of the ALP interactions with SM fields. Here we present the RG evolution equations for the coupling parametersc V V and Y f in this Lagrangian. The evolution equations for the quantitiesc V V have already been given in (A.1). These parameters are not scale independent, in contrast to the couplings c V V appearing in the original Lagrangian (1). This fact may seem puzzling at first sight, because the operators describing the ALP-boson interactions are the same in the two forms of the effective Lagrangian. However, under renormalization the ALP-boson operators mix with the derivative couplings of the ALP to fermions in (1). In the Lagrangian (9) these derivative couplings must be decomposed into linear combinations of the non-derivative ALP-fermion interactions and the ALP-boson interactions. This decomposition introduces a non-trivial scale evolution of the parametersc V V . We have derived these equations for the coupling matricesỸ f starting from (10) and the evolution equations for the parameters c V V and c F given in (17) and (18), as well as the well-known RG equations of the SM Yukawa matrices [99,100]. We obtain and X as given in (19).