Unitarity constraints on ALP interactions

We derive partial-wave unitarity constraints on gauge-invariant interactions of an Axion-Like Particle (ALP) up to dimension-6 from all allowed $2\to2$ scattering processes in the limit of large center-of-mass energy. We find that the strongest bounds stem from scattering amplitudes with one external ALP and only apply to the coupling to a pair of $SU(2)_L$ gauge bosons. Couplings to $U(1)_Y$ and $SU(3)_C$ gauge bosons and to fermions are more loosely constrained.

be present around or below the apparent unitarity violation scale in order to restore the physical behavior of scattering amplitudes. Paradigmatic examples of the use of unitarity relations to derive constraints on the validity of a theory include the seminal work of Lee, Quigg and Thacker [48,49] that imposed an upper bound on the Higgs mass by analyzing the unitarity of the standard model and was used to build a case in favor of the construction of the present generation of colliders. Another classical example are the bounds on new fermions obtained by Chanowitz, Furman and Hinchliffe [50]. On a more formal front, unitarity arguments have also been employed, for example, in connection with the requirement of gauge invariance [51].
In the last decades, partial-wave unitarity has been employed ubiquitously to constrain effective interactions, in particular in the electroweak sector (see for example [52][53][54][55][56][57][58][59]). Recently, Refs. [60][61][62] presented a general systematic study of unitarity bounds for the case of effective interactions in the SMEFT and Higgs EFT (HEFT). Generically, unitarity preservation imposes consistency conditions on the theory such that, for the EFT to be valid up to a given √ S, the effective couplings (scale) need to be smaller (larger) than a certain threshold. Conversely, for given values of the EFT coefficients and scale, unitarity imposes an upper limit on the energy scales at which the EFT can be applied. In that respect unitarity bounds are crucial for the interpretation of actual experiments, which study tails of kinematical distributions, since one can infer unphysical bounds that are too strong if these limits are not respected.
For the case of ALP EFT, the rapid growth of the scattering amplitudes with energy, that leads to partial-wave unitarity violation, is particularly enhanced. The reason for that is the pseudo-Goldstone nature of the ALPs which requires all their interactions to be classically invariant under shifts a(x) → a(x) + α, i.e. to be of the form J µ ∂ µ a. As a consequence, an explicit momentum dependence is present in all ALP couplings.
A partial analysis of unitarity constraints on ALP couplings was presented in Refs. [63,64]. Here we adopt a more systematic approach and derive maximal constraints on all ALP interactions of dimension 5 and 6 from partial-wave unitarity, examining all allowed 2 → 2 scattering processes in the limit of large center-of-mass energy. We adopt a procedure analogous to the one employed in [60][61][62] for the case of effective interactions in the SMEFT and HEFT.
The outline of this article is as follows. We present the relevant Lagrangian employed in Sec. II and briefly discuss the number of relevant operators we consider. The core of the results is contained in Sec. III where we derive first the most general bounds for the ALP couplings to SM gauge bosons which are obtained from the partial-wave analysis of the scattering of boson pairs in Sec. III A. Section III B contains our derivation of the most general independent constraints on ALP couplings to SM fermions which are obtained with the partial-wave analysis of scatterings involving fermion pairs. We briefly discuss the results in Sec. IV. Explicit expressions of the helicity amplitudes for all the relevant processes are presented in the Appendix.

II. ALP EFFECTIVE LAGRANGIAN
We consider the SM extended by the ALP effective Lagrangian [41,42,65] where the effective operators form a complete basis of CP-even ALP interactions up to O(f −3 a ) terms. Here, B µ , W i µ and G a µ are the gauge bosons of the U (1) Y ×SU (2) L ×SU (3) c SM symmetry respectively, and the dual field strengths are defined byX µν = 1 2 ε µνρσ X ρσ . Φ denotes the SU (2) L Higgs doublet whileΦ = iτ 2 Φ * is its dual (being τ i the Pauli matrices). Upon EW symmetry breaking, Φ † Φ = (v + H) 2 /2 with H the physical Higgs boson. The left (right)-handed fermion multiplets are denoted by q, l (u, d, e) and Y u , Y d , Y e are the 3 × 3 Yukawa matrices. All index contractions were left implicit and repeated indices are summed over unless otherwise specified. A mass term m a for the ALP was introduced, which is generically induced in the presence of soft breaking of shift-invariance, such as non-perturbative instanton effects in the case of the QCD axion [6,[66][67][68].
We neglect CP violating effects such that all Wilson coefficients C i are real scalar quantities. Although this is not manifest in Eqs. (2)-(5), all ALP interactions are classically shift-invariant: the interactions to bosons can be written as ∂ µ aJ µ X by integration by parts, where J µ X is the Chern-Simons current associated to the X = {B, W, G} gauge boson. 1 The operators with fermions were taken to follow the minimal-flavor-violation ansatz [69][70][71], i.e. to respect a U (3) 5 global symmetry that is only broken by insertions of the Yukawa couplings. With this flavor structure, they could also be equivalently traded for a set of chirality-conserving ones of the form (∂ µ a)(ψ p γ µ ψ r )δ pr , with p, r flavor indices [44,72].
The operator O aΦ is actually redundant [41,45,65,72]: Nevertheless, it is often retained because the set {OB, OW , OG, O aΦ } forms a complete and non-redundant operator basis at dimension 5 in the bosonic sector that can be of phenomenological interest.
The operator O (2) aΦ has been previously considered in [73][74][75][76] and it is the only shift-invariant operator 2 that can be constructed at dimension 6. SMEFT operators of dimension 6 are neglected: we assume them to be suppressed by a scale Λ SMEFT = f a and work consistently at order (f −2 a Λ 0 SMEFT ). Discussing the interplay of the two expansions is beyond the scope of this work. 3

III. ANALYSIS OF UNITARITY CONSTRAINTS
A. Helicity amplitudes for the scattering of pairs of bosons Consider the two-to-two scattering of bosons V i with helicities λ i where we denote by V either gauge bosons, Higgs or ALP. The corresponding helicity amplitude can be expanded in partial waves in the center-of-mass system as [78] , and θ (ϕ) is the polar (azimuthal) scattering angle. d is the usual Wigner rotation matrix. This expression holds for gauge bosons with λ = 0, ±1, and for scalars (Higgs or ALP) with λ ≡ 0; the fermion case will be addressed below. For further details and conventions see Ref. [60].
In the limit S (M V1 + M V2 ) 2 , partial-wave unitarity for a given elastic channel requires that The most stringent bounds are obtained by diagonalizing T J in the particle and helicity space and then applying the condition in Eq. (9) to each of the eigenvalues. This is the approach which we follow. We start by calculating the scattering amplitudes for all possible combinations of bosons and helicities generated by the SM extended with the Lagrangian in Eq. (1) for a given total electric charge Q = 2, 1, 0 and that give nonvanishing projections on a given partial wave J proportional to some ALP coupling. Conservation of color implies 1 In the G case, only a discrete version of the shift-invariance is preserved due to the presence of non-vanishing instanton configurations. 2 One more operator structure is present at dimension 6, namely (∂µ∂ µ a) 2 . However, applying the ALP equation of motion, this can be fully reabsorbed into a redefinition of the ALP mass. We have checked the completeness of the dimension-6 set with BasisGen [77]. 3 As will become clear from the discussion in Sec. III, the bounds on CW , C f Φ , C aΦ and C (2) aΦ are not expected to change significantly in the presence of dimension 6 SMEFT operators, independently of the interplay between the SMEFT and ALP expansions. This is because all these bounds are dominated by scatterings with one or two external ALPs. that initial or final states with color have to be considered independently of those in a color singlet state. So one is led to consider separately the T J (T J ) amplitude matrices for processes with color singlet (octet) in the initial and final states. One must also take into account that parity conservation at tree level implies the relation and leads to a reduction of the number of independent helicity amplitudes. Time-reversal invariance further reduces the number of helicity amplitudes that need to be evaluated. Altogether, the initial/final states contributing a priori to the T J matrices for each value of Q and J are: and correspondingly the states contributing to the T J matrices are: where upper indices indicate charge and lower indices helicity. We also display in Eqs. (11) and (12) the dimensionality of the particle and helicity matrix for each independent (Q , J) channel. In Eq. (11) the states HH and W ± H are only present when the dimension 6 operator is considered. We list in Tables I-IV, that are shown in the Appendix, the expressions for the most S-divergent part of the amplitudes for the channels which give the dominant contribution to the T J and T J matrices.
Bounds on individual operators. As seen in Tables I-III, for processes with zero or two ALPs as external states, the most energy-divergent amplitudes occur for scattering of transversely polarized gauge bosons, as expected. These amplitudes are all proportional to the product of two axion couplings, therefore, the two powers of their momentum involved in the coupling of ALP to the gauge boson generate the leading S/f 2 a dependence. A good fraction of them contributes to J = 0 matrices, which are, a priori, expected to lead the strongest bounds. Furthermore, for amplitudes with gluon pairs, the strongest bounds are obtained for the gluon pair in the singlet color state 1 Altogether from the diagonalization of the J = 0 matrices and assuming only one non-zero coupling at a time we find that the largest eigenvalues correspond to the Q = 0, T 0 matrix and read respectively. Applying the condition in Eq. (9) to each of these eigenvalues we obtain the bounds |C (2) aΦ | ≤ 101 (2) aΦ is dominated by scattering amplitudes with 2 ALP external states.
Unlike the amplitudes with even number of ALP in the external states, some helicity amplitudes with only one ALP in either the initial or final state have a leading behavior S 3 2 /(f a M 2 W ), as seen in Table IV 4 . These amplitudes involve two longitudinally polarized gauge bosons, whose polarization vectors are proportional to √ S, and one transversely polarized gauge boson whose momentum contributes another power of √ S. This configuration can only be generated by a combination of the SM vertices and those induced by OW and, consequently, the amplitudes involve a single power of the CW coupling and of the SM coupling e, and do not depend on any other Wilson coefficient. As seen from the scattering angle dependence of the amplitudes in Table IV, they contribute only to the T J=1 matrix with either Q = 0 or Q = 1. Diagonalizing these we find that the largest eigenvalue is where c w is the cosine of the weak mixing angle. Therefore, the condition in Eq. (9) implies the constraint on CW Comparing the bounds on CW from J = 0-wave unitarity, Eq. (14), and from J = 1-wave unitarity, Eq. (19), we find that the constraint derived from the J = 1 amplitudes is the strongest for √ S > 260 GeV .
Including multiple operators simultaneously. Fixing C (2) aΦ = 0 (or equivalently barring cancellations between dimension 5 and 6 terms) and allowing multiple dimension 5 operators to vary simultaneously does not alter significantly the bounds reported above. For CW this is obvious, because the leading constraint Eq. (19) is genuinely independent of the other Wilson coefficients. For CB and CG this can be understood considering that CG is dominantly constrained by G ± G ± → G ± G ± scattering in the color singlet channel, which is independent of CB. We have also verified this explicitly by diagonalizing the Q = 0 T J=0 matrix with CB, CG present at the same time. The diagonalization can still be done analytically though the resulting expressions for the eigenvalues are not particularly illuminating. Imposing the unitarity limits on those eigenvalues yields the same bounds as in Eqs. (15) and (16).
Allowing all operators of dimension 5 and 6 to be present simultaneously (i.e. allowing cancellations between both orders) we find that the largest eigenvalues are and correspondingly the unitarity limits on the Wilson coefficients are: These results hold irrespective of whether CW , CB and CG are included simultaneously or individually. It is also worth noting that the bounds on CG and C (2) aΦ are unchanged compared to the individual limits (16) and (17). Considering that CW is always dominantly constrained by Eq. (19), we conclude that only the unitarity constraints on CB depends significantly on whether C (2) aΦ is included or not. Truncating at dimension-5. Finally, it can be interesting to investigate bounds on the dimension-5 interactions only. As we have seen above, the most stringent bounds on CW originates from processes exhibiting just one dimension-5 vertex, therefore, it is not modified when we truncate the EFT expansion to O(f −1 a ). In order to obtain limits on CB and CG independently of assumptions about C (2) aΦ , we can restrict our analysis to a subspace of initial states such that contributions of the dimension-6 operator are negligible for all the scattering amplitudes retained. This is achieved by eliminating "flavor" states in Eqs. (11) and (12) that lead to processes containing two ALP external legs. We can re-derive the constraints on this flavor subspace and we obtain that the largest eigenvalues come from the Q = 0, T 0 matrix and they coincide with those in Eq. (20), leading to the bounds in Eqs. (21)- (23). The result is the same irrespective of whether CB and CG are included simultaneously or individually.

B. Helicity amplitudes involving fermions
ALP couplings to fermions can contribute to processes which can also violate unitarity. In this case the partial-wave expansion is given by In principle, f 1σ 1f 2σ 2 → V 3λ 3 V 4λ 4 amplitudes of a given J partial wave can be incorporated together with the V 1λ 1 V 2λ 2 → V 3λ 3 V 4λ 4 amplitudes in the corresponding T J matrix by extending the basis of states to incorporate the relevant f 1σ 1f 2σ 2 combinations contributing to a given Q; see, for example, Ref. [50]. However, we find that the most energy divergent amplitudes for fermion-antifermion scattering grow at most as √ S and, therefore, the contributions from the ALP-fermion couplings enter with different power of S with respect to the ALP-gauge-boson couplings in the eigenvalues of this generalized T J matrices. Thus, in order to derive independent unitarity constraints on the C f Φ couplings we find it more convenient to follow the alternative procedure presented in Ref. [57] and relate the corresponding f 1σ 1f 2σ 2 → V 3λ 3 V 4λ 4 amplitude to that of the elastic process In this case the unitarity relation is where we take the limit S (M V3 + M V4 ) 2 , (M f1 + M f2 ) 2 . N represents any state which f 1σ 1f 2σ 2 can annihilate into which does not consist of two bosons. Eq. (28) is a quadratic equation for The strongest bounds can be found by considering some optimized linear combinations with the normalization condition f σ |x f σ | 2 = 1, for which the amplitude T J (X → V 3λ 3 V 4λ 4 ) is the largest. In this approach, processes of fermion scattering into one gauge boson and one ALP provide independent constraints on the ALP-fermion coupling. As mentioned above, the most divergent relevant helicity amplitudes grow as √ S  14) and (19). The solid red line indicates the bound on CB in Eq. (22), derived allowing C Because these bounds are inversely proportional to the Yukawa coupling of the fermion and involve larger coefficients, we conclude that the unitarity constraints on the ALP-fermion couplings are orders of magnitude weaker than those on ALP-gauge-boson couplings even for the coupling to the up-quarks. Moreover, the operator O aΦ should only be considered in a scenario where the fermionic operators are absent. In this case, the most stringent unitarity constraint on its coupling originates from Eq. (32).

IV. CONCLUSIONS
We have derived maximal constraints on the effective interactions of Axion-Like-Particles from partial-wave unitarity in 2 → 2 scattering processes. Our results are summarized in Fig. 1. They hold in the kinematic regime where √ S v and the ALP mass was also implicitly taken to be m a √ S. Furthermore the consistency of the ALP EFT expansion requires √ S f a . We find that, for fixed C/f a , the most stringent unitarity bound is imposed on CW in V V → V a scattering processes, while the weakest limits are on ALP-fermion interactions. The constraints exhibit only a limited dependence on whether the effective operators are taken individually or allowed to vary simultaneously, signaling that each of them is dominantly constrained in a class of scattering amplitudes that is nearly orthogonal to the others.
In this respect, let us stress that our results should not be interpreted as strict unitarity constraints on any specific process used in the ALP searches, in the sense that it might be difficult to directly identify the kinematic information available with the subprocess center-of-mass energy of an individual 2 → 2 scattering. Notwithstanding, unitarity bounds must be satisfied in the event generation and, consequently, can affect the shapes of expected distributions used in the searches.
For example, recently, the ATLAS collaboration has searched for axions in events with an energetic jet [92] or a photon [93] and missing transverse momentum. The monojet analysis [92] constrains the axion coupling to gluons to satisfy CG/f a < 0.008 TeV −1 at 95% CL. Using Eq. (23), we find that for the largest allowed coupling in this search unitarity is preserved up to center mass-of-mass energy of 39 TeV, clearly beyond the LHC reach. On the other hand, the mono-photon analysis limits the CW coupling to satisfy CW /f a < 0.12 TeV −1 at 95% CL. From Eq. (19) we read that for CW /f a at the 95% CL boundary, unitarity is violated in subprocesses with center-of-mass energy greater than 1.04 TeV. We conclude that the tail of the expected missing E T distribution should be analyzed cautiously and the unitarity constraints could have an impact in the derivation of the mono-photon search bound.
The unitarity bounds derived in this work would be also relevant in the event that an ALP signal will be detected in the future (independently of the energy regime at which the experimental search is conducted), leading to a defined measurement of one or more ALP couplings. In this case, unitarity bounds would provide an upper limit to the mass scale of the new physics sector the ALP originates from and motivate further searches in this energy region.
Helicity amplitudes at leading order in S We present here the list of unitarity violating amplitudes for all the 2 → 2 scattering processes considered in the evaluation of the unitarity constraints.     λ1 λ2 λ3 λ4 M ×e S 3/2