Quantization of Axion-Gauge Couplings and Non-Invertible Higher Symmetries

We derive model-independent quantization conditions on the axion couplings (sometimes known as the anomaly coefficients) to the Standard Model gauge group $[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_q$ with $q=1,2,3,6$. Using these quantization conditions, we prove that any QCD axion model to the right of the $E/N=8/3$ line on the $|g_{a\gamma\gamma}|$-$m_a$ plot must necessarily face the axion domain wall problem in a post-inflationary scenario. We further demonstrate the higher-group and non-invertible global symmetries in the Standard Model coupled to a single axion. These generalized global symmetries lead to universal bounds on the axion string tension and the monopole mass. If the axion were discovered in the future, our quantization conditions could be used to constrain the global form of the Standard Model gauge group.


Introduction
Axions have long been a major target in particle phenomenology.Originally emerging as an elegant solution to the strong CP problem [1,2,3,4], axions have since been identified as one of the most well-motivated dark matter candidates [5,6,7,8], appearing in a number of extensions of the Standard Model (SM).They have motivated dozens of unique experimental searches bridging a wide variety of disciplines such as collider physics, astrophysics, condensed matter physics, and quantum optics [8].Axion-like particles can provide a natural candidate for the inflaton [9,10], and may also play a central role in a variety of important cosmological effects such as the cosmological constant problem [10].Axion-like particles also exist ubiquitously in string theory [11,12,13,14,15,16].See, for example, [17,10,18,19,20,21] for various reviews.Symmetry and topology have always been at the center stage in axion physics.In recent years, there has been a transformative development in our understanding of symmetries in theoretical physics, motivated by advancements in high energy physics, condensed matter physics, and mathematical physics.Global symmetries have been generalized in several different directions, including the higher-form symmetries, higher-group symmetries, non-invertible symmetries, and more.(See [22,23,24,25,26,27] for reviews.)In particular, the axion shift global symmetry, a classical U (1) global symmetry that suffers from the quantum Adler-Bell-Jackiw (ABJ) anomaly [28,29], is recently recognized to be an exact non-invertible global symmetry in the axion-Maxwell theory [30,31,32,33] (which build on earlier works in [34,35]).This is a new kind of symmetry that does not have an inverse.Higher-group symmetries for axions are also found in [36,37,38,39,40,41], as well as non-invertible higher-form symmetries [32,33].These new symmetries lead to universal inequalities on the energy scales where various emergent global symmetries are broken in the UV completion of axion models [40,32].In particular, for axion-electrodynamics, it was shown in [32] that the axion string tension T and the monopole mass m monopole are bounded from below by the mass of the lightest electrically charged particle m electric , i.e., m electric ≲ min m monopole , √ T . 1   In this paper, we extend the previous analysis in [40,32] to the full SM coupled to a single axion field θ(x).The three coupling constants K 3 , K 2 , K 1 (defined in (2.1)) between the axion field and the instanton number densities of the su(3) × su(2) × u(1) Y gauge fields are quantized by imposing the periodic identification θ(x) ∼ θ(x) + 2π.The precise quantization conditions depend on the global form of the SM gauge group, which we explain below.It is known that a Z 6 center subgroup of SU (3) × SU (2) × U (1) Y acts trivially on all the SM particles.Therefore, the SM gauge group need not be a product group, but can be one of the following: (3) × SU (2) × U (1) Y ]/Z q , q = 1, 2, 3, 6 . (1.1) More physically, the global form of G SM depends on what gauge charges are allowed or introduced in physics beyond the Standard Model (BSM).For example, in the SU (5) and many other Grand Unified Theories (GUT), all new particles carry vanishing Z 6 charges, and q = 6. 2 In other UV models for the axion (such as the simplest version of the KSVZ model where the heavy fermion is in the fundamental of SU (3) but neutral under SU (2) × U (1) Y [42,43]), the heavy fermions may carry nontrivial Z 6 charges, and the corresponding global form of G SM is different. 3See [47] for 1 Note that the electrically and magnetically charged particles are not on the same footing, i.e., there is no electromagnetic duality.This is because we assume at low energies the effective theory contains an axion field that couples to the gauge field as θF ∧ F . 2 Said differently, the SM gauge group is embedded as a [SU (3) × SU (2) × U (1) Y ]/Z 6 ⊂ SU (5) subgroup, but not as a product group SU (3) × SU (2) × U (1) Y . 3The q = 6 case is in many ways phenomenologically preferred.In particular, if q ̸ = 6, the Z 6 -charged BSM particles cannot decay into the SM particles and form stable relics in cosmology.See Section 3 for more discussions.For the same reason, we assume that the hypercharge is never smaller than 1/6 and the gauge group U (1) Y is compact.The q = 6 case can also be realized from F-theory [44,45,46].more discussions.
We provide several bottom-up derivations of the quantization conditions of the axion-gauge couplings K i 's for each version of the global form of the SM gauge group.Below the electroweak symmetry breaking (EWSB) scale, this gives the quantization conditions of the axion-gluon coupling N and (bare) axion-photon coupling E (defined in (2.2)). 4These quantization conditions are satisfied by every UV consistent axion models, but our derivation is universal and modelindependent.We summarize our quantization conditions in Table 1.
Our quantization conditions give interesting constraints on the effective axion-photon coupling g aγγ , which is expressed in terms of the ratio E/N .We show that, in the q = 6 case, any QCD axion model lying to the right of the E/N = 8/3 line (realized by one of the DFSZ models [48,49] and many GUT models) on the |g aγγ |-m a plot must necessarily have more than one axion domain wall in a post-inflationary scenario.Our result provides an invariant meaning for the ratio E/N = 8/3, which is taken as a primary target for experimental detection.
We further analyze the generalized global symmetries, including the higher-group and noninvertible symmetries.We summarize the generalized global symmetries, which depend on the number theoretic properties of the quantized axion-gauge couplings K i , in Table 2.In many cases, we find that one symmetry is subordinate to the other, meaning that the former cannot exist without the latter.This hierarchy constrains the energy scales where these symmetries are broken, from which we derive universal inequalities on the axion string tension and the hypercharge monopole mass, generalizing [40,32].
The rest of the paper is organized as follows.In Section 2, we provide several different derivations for the model-independent quantization on the couplings between the axion and the SM gauge fields.In Section 3, we apply the quantization conditions to constrain the effective axion-photon coupling g aγγ .Section 4 discusses the generalized global symmetries of the SM coupled a single axion field, and how they can be used to bound the axion string tension and the monopole mass.In Appendix A, we provide more details on the higher-group and non-invertible symmetries in the SM coupled to an axion.Finally, appendix B discusses fractional axion-gauge couplings in the presence of additional topological degrees of freedom.

Quantization of the axion couplings to gauge fields
The axion field, denoted as a, is a periodic scalar field whose periodicity is given by a ∼ a + 2πf , where f is the axion decay constant.We consider a generic coupling of the axion field to the SM gauge fields, Here, g i 's are the gauge coupling constants, and G a µν , F a µν , and B µν are the SU (3), SU (2) and U (1) Y field strengths, respectively.The coupling constants K i 's can take only a quantized set of values, and the precise quantization conditions will be determined below.The dual field strengths are defined as G a µν = 1 2 ϵ µνρσ G a,ρσ and similarly for the SU (2) and U (1) Y field strengths.The U (1) Y hypercharge gauge field is normalized such that Q Y ∈ Z/6. 5 Without loss of generality, we assume K 3 ≥ 0, which can always be achieved by a field redefinition a → −a.
Below the EWSB scale, (2.1) reduces to where • • • denotes the coupling between the axion and the massive gauge fields.The electric coupling e = g 1 cos θ W = g 2 sin θ W is fixed by the coupling g 1 , g 2 and the weak mixing angle Here we follow the standard convention in the literature [50] to denote the axion-gluon coupling by N . 6It is related to K 3 (which equals the number N DW of axion domain walls) as 7 3) The (bare) axion-photon coupling E is related to K 1 , K 2 as This can be obtained from the standard relations, Here A a µ and B µ are the SU (2) and U (1) Y gauge fields, respectively.
For our purposes, it is convenient to absorb the gauge couplings as well as the axion decay constant by rescaling the fields as 5 For instance, the covariant derivative acting on the left-handed quark doublet is 6 In the literature, the term "axion-like particle" (ALP) is sometimes used instead of "axion" if N = 0. We will not make any such distinction. 7It is somewhat awkward that N can sometimes be a half integer, while K 3 = N DW is always a positive integer as we will discuss later.In contrast, the bare axion-photon coupling E is generally not an integer.where the fields on the right-hand side are normalized to have the canonical kinetic term such as − 1 4 F µν EM F EM,µν .By an abuse of terminology, we refer to both θ and a as an axion.θ has the periodicity of 2π, θ ∼ θ + 2π, which should be viewed as a gauge symmetry.The SU (3), SU (2), and U (1) Y gauge fields in this normalization convention are denoted as A a 3,µ , A a 2,µ , and A 1,µ , respectively.

q G
In terms of the differently normalized fields (2.5), the axion coupling (2.1) can be equivalently written as where we have also adopted the differential form notation.The traces are taken in the fundamental representation, and the Lie algebra generators are normalized such that Tr(T a T b ) = 1 2 δ ab .Similarly, the coupling after EWSB (2.2) can be written as (2.7) We summarize the quantization conditions in Table 1. 8 We present them both in terms of K i 's as well as in terms of E and N .The latter are more natural below the EWSB scale and are commonly used in the literature.They are also directly relevant for the axion-photon coupling g aγγ discussed in Section 3. Interestingly, we find that the quantizations of different K i 's, E, and N are sometimes correlated depending on the global form of the SM gauge group.In the rest of this section, we explain how the conditions are derived.We provide several independent (but related) derivations of the quantization conditions: 1. Assuming the PQ mechanism, we derive the quantization conditions from analyzing the quantum numbers of the PQ fermions.
2. Fractional instanton numbers.This extends the analysis in [52] for T 4 to general spacetime manifolds.
The first three derivations will be presented in the rest of this section, while the last one is discussed in Appendix A.1.
In some axion models, such as the DFSZ model [48,49], the axion coupling (2.2) below the EWSB scale is a valid description, but at higher energy scales the UV theory is different from the one we consider in (2.1).In such cases, only the couplings E and N are meaningful whereas K i 's are not.Our results on the quantization conditions of E and N are still valid even in those cases.
We work under the minimal setup where the only degrees of freedom are the SM fields and a single axion field.However, in the presence of additional topological degrees of freedom, the axion-gauge couplings K i can be further fractionalized [53], which we review in Appendix B.Here we assume that there aren't such topological degrees of freedom.

Quantum numbers of Peccei-Quinn fermions
We begin by demonstrating the quantization conditions in Table 1 by assuming that the couplings between the axion and the SM gauge fields (2.1) are generated by the spontaneous breaking of a classical U (1) P Q global symmetry in a UV model with additional heavy fermions.The axion couplings to the gauge fields are then given by the ABJ triangle anomaly coefficients, from which we derive the quantization conditions.Even though this derivation is intuitive, it is not universal and is subject to the above assumption on the UV completion.In later subsections, we provide more rigorous, model-independent derivations that reproduce the same results.
We denote an SU (2) irreducible representation by its spin J = 0, 1  2 , 1, • • • , and an SU (3) representation by {ℓ 1 , ℓ 2 } with ℓ 1 ≥ ℓ 2 .Here ℓ i is the number of boxes in the i-th row of the Young diagram.For instance, 3 = {1, 0}, 3 = {1, 1}, and 8 = {2, 1} are the fundamental, anti-fundamental, and adjoint representations, respectively.The hypercharge is normalized so that Q Y ∈ Z/6.Let X (R) ∈ Z be the U (1) P Q charge of a heavy left-handed Weyl fermion R whose quantum numbers are (ℓ i (R), J(R), Q Y (R)).Then, the triangle anomaly coefficients are given by (2.8) Here, T (X) and d(X) denote the (Dynkin) index and dimension of a representation X, respectively.If the SM gauge group G SM is given by [SU (3) × SU (2) × U (1) Y ]/Z q , only those representations R with trivial charge under Z q ⊂ SU (3) × SU (2) × U (1) Y are allowed.See [54] for related discussions.Depending on the allowed set of representations, K i 's in (2.8) can take different sets of quantized values.
We first begin with the q = 1 case, where there is no restriction on the set of allowed representations.The index and dimension of arbitrary SU (2) and SU (3) representations are given by (2.9) It is straightforward to show that the index is always an integer or a half integer, for instance, by induction.From (2.8), we obtain In terms of the axion couplings below the EWSB scale, this leads to Consider now the q = 2 case, where the SM gauge group becomes Only the representations on which the Z 2 subgroup generated by (1 3×3 , −1 2×2 , −1) acts trivially are allowed.The element −1 2×2 ∈ SU (2) is represented by (−1) 2J times the identity matrix in the spin J representation, whereas an element e iα ∈ U (1) Y acts as e iα(6Q Y ) on a hypercharge Q Y representation.Hence, the condition on the allowed representations for q = 2 is 2J = 6Q Y mod 2 . (2.12) Combined with (2.8) and (2.9), this leads to The conditions on individual K i 's are easy to see.On the other hand, the last condition follows from the fact that 2 × 2T (J) + (6Q Y ) 2 d(J) is divisible by 4 for every representation satisfying (2.12).After EWSB, we have , and the allowed representations are those on which the ) acts trivially.The element e 4πi/3 1 3×3 ∈ SU (3) is represented by e 4πi(ℓ 1 +ℓ 2 )/3 times the identity matrix on the {ℓ 1 , ℓ 2 } representation.Therefore, the allowed representations satisfy It follows that q = 3 : The last condition is due to the fact that 6 × 2T (ℓ i ) + (6Q Y ) 2 d(ℓ i ) is divisible by 9 for every representation satisfying (2.15). 9After EWSB, we have (2.19) 9 One way to check this is as follows.Using (2.15), we first write 6Q Y = (ℓ 1 + ℓ 2 ) + 3x where x is an integer.Then, from (2.9), we have where mod 9, we need to prove f (ℓ i ) = g(ℓ i ) = 0 mod 3 for all ℓ i .This can be proved by, for instance, first explicitly checking that f (ℓ i ) = 0 mod 3 for ℓ 2 = 0, 1, 2, then noting that f (ℓ 1 , ℓ 2 + 3) = f (ℓ 1 , ℓ 2 ) mod 3. The same argument applies to g(ℓ i ).
Finally, for the q = 6 case, the gauge group is ).This requires (2.20) All the particles in the SM, and also in the SU (5) and various other GUT models, obey (2.20).This leads to Below the EWSB scale, we have The correlated quantization condition between N and E, which comes from 24K 3 + 18K 2 + K 1 ∈ 36Z, will play an important role in putting constraints on the effective axion-photon coupling g aγγ discussed in Section 3. Proving To check these conditions from (2.8), it is sufficient to consider the contribution from a single heavy PQ fermion whose quantum numbers (ℓ i , J, Q Y ) are subject to the condition (2.20).Indeed, it is straightforward to show that (2.23) The first condition follows from 2J = 6Q Y mod 2, and the second from ℓ 1 + ℓ 2 = 6Q Y mod 3. Note that (2.21) is also equivalent to imposing the q = 2 and q = 3 conditions (2.13), (2.16) simultaneously.

Fractional instantons
We now provide a rigorous, model-independent derivation of the quantization conditions from the fractional instanton numbers.This derivation is universal and does not require any assumption about the UV origin of the axion.The fractional instantons on T 4 of the SM gauge group have been previously studied in [52] by imposing the 't Hooft's twisted boundary conditions [55,56].
The periodic identification of the dynamical axion field θ ∼ θ + 2π is a gauge symmetry, and the exponentiated action e iS should be gauge-invariant.In other words, e iS should be single-valued under θ ∼ θ +2π.In the presence of the axion coupling (2.6), e iS transforms under θ ∼ θ +2π by a phase exp(2πi i K i n i ), where n i 's are the instanton numbers (see (2.24) below).The quantization of the instanton numbers (which depends on the global form of the SM gauge group) then gives the quantization of the K i 's, and therefore E, N .
We begin with q = 1.The SU (3), SU (2), and U (1) Y instanton numbers in this case are not correlated, and they separately satisfy10 Therefore, e iS is single-valued under the identification θ ∼ θ + 2π for all possible instanton configurations if and only if the quantization condition (2.10) is satisfied.
For q = 2, since the SM gauge group is 2), the two gauge fields A 2 and A 1 combine into a single U (2) gauge field given by A U (2) ≡ A 2 + A 1 1 2×2 .It follows that the instanton numbers n 2 and n 1 are generally fractional, with the fractional parts correlated as follows: To derive (2.25), recall that the first and second Chern classes for the where F U (2) is the U (2) field strength.The Chern classes satisfy 11 from which (2.25) follows straightforwardly.By rewriting (4n 1 ), we find that e iS is single-valued if and only if the quantization condition (2.13) is satisfied.
There is an alternative way to find the allowed values of fractional instanton numbers (2.25).First, we may view A 2 and A 1 separately as SO(3) = SU (2)/Z 2 and U (1)/Z 2 gauge fields, respectively.By saying that A 1 is a U (1)/Z 2 gauge field, we mean that 2A 1 is a properly normalized U (1) gauge field satisfying the usual quantization of magnetic fluxes 2F 1 2π ∈ Z.The fact that the two gauge fields combine into the U (2) gauge field A U (2) = A 2 + A 1 1 2×2 further means that [37,59] c 1 (F 1 ) = w 2 (A 2 ) mod 2 . (2.28) Here the first Chern class c 1 (F 1 ) = 2F 1 2π is viewed as an element of the cohomology group H 2 (X, Z) where X is the spacetime manifold.w 2 (A 2 ) ∈ H 2 (X, Z 2 ) is the second Stiefel-Whitney class for A 2 , which measures the obstruction to lift an SO(3) bundle to an SU (2) bundle. 12The relation (2.28) guarantees that the transition functions of the SO(3) and U (1)/Z 2 bundles are correlated in such a way that they consistently combine into a U (2) bundle.Moreover, the fractional part of the instanton number for the SO(3) gauge field A 2 can be determined from the second Stiefel-Whitney class as [60,62,63,64,37,59] ) is the Pontryagin square, which is reviewed, for instance, in [59, Appendix B].On spin manifolds, we have P(w 2 (A 2 )) = 0, 2 mod 4. From these, we can recover (2.25) as follows.First, the fact that c 1 (F 1 ) = 2F 1 2π has integral periods implies that n 1 = Next, (2.29) together with the fact that P(w 2 (A 2 )) = 0, 2 mod 4 gives us n 2 ∈ 1 2 Z. Lastly, taking the Pontryagin square of both sides of (2.28) and integrating them over the spacetime manifold leads to the correlated condition n 2 − 2n 1 ∈ Z.
We now move on to the q = 3 case.The SM gauge group is , and the two gauge fields A 3 and A 1 combine into a U (3) gauge field A U (3) ≡ A 3 + A 1 1 3×3 .The instanton numbers n 3 and n 1 can take fractional values and they satisfy Similar to before, (2.30) is most easily understood in terms of the Chern classes of the U (3) gauge field.We have (2.31) The quantization (2.27) then leads to (2.30).By rewriting (9n 1 ), we see that e iS is single-valued if and only if (2.16) is satisfied.
Alternatively, to derive (2.30), we may first view A 3 and A 1 as P SU (3) = SU (3)/Z 3 and U (1)/Z 3 gauge fields, respectively.The condition that they combine into a U (3) gauge field implies that where the first Chern class c 1 (F 1 ) = 3F 1 2π for A 1 is again viewed as an element of H 2 (X, Z).Similarly, the fractional part of the P SU (3) instanton number is determined by w 2 (A 3 ) as [60,62,63,64,37,59] From these, the quantization of the individual fractional instanton numbers, n 3 ∈ 1 3 Z and n 1 ∈ 1 9 Z, follows immediately.The correlated quantization condition n 3 − 6n 1 ∈ Z is also easily obtained by taking the square on both sides of (2.32) and then integrating them over the spacetime manifold.
Finally, for q = 6, the allowed fractional values of the instanton numbers and their correlations are: (The last condition 2n 3 + n 2 + 6n 1 ∈ Z is redundant and can be obtained from the other conditions, but we keep it for later convenience.Z are obtained by taking (Pontryagin) squares of the two sides of equations in (2.35) and then integrating them over the spacetime manifold, similar to the q = 2 and q = 3 cases.To obtain the quantization condition of K i 's in (2.21), we can rewrite

36
(36n 1 ).We see that the exponentiated action e iS is single-valued if and only if (2.21) is satisfied.
As a side remark, the conditions on the instanton numbers can be used to derive the periodicities and identifications of the θ-angles in the SM (not coupled to an axion): where θ 3 , θ 2 , and θ 1 are respectively the SU (3), SU (2), and U (1) Y θ-angles (normalized the same way as in (2.6)).Many of these periodicities were derived in [47].Note that depending on the global form of the SM gauge group, θ-angles can have extended as well as correlated periodicities.Upon EWSB, θ-angle for the electromagnetic gauge field is given by θ EM = (θ 1 + 18θ 2 )/36, whose (correlated) periodicity can be inferred from (2.36).The normalization for θ EM is such that the Lagrangian contains θ EM e 2 16π 2 F EM,µν F µν EM where F EM,µν is the canonically normalized EM gauge field strength (2.5).

Chern-Simons gauge theory
Next, we provide an alternative (but related) derivation of the quantization conditions, which is based on the relationship between the 4d axion couplings and 3d Chern-Simons gauge theories [65].More specifically, we rephrase the quantization condition on K i 's as the vanishing condition of an 't Hooft anomaly of a certain 1-form global symmetry [63] in a 3d Chern-Simons theory.Related discussions can be found, for instance, in [37].We emphasize that this derivation is conceptually equivalent to the one in Section 2.2, but phrased in terms of the anyon spins of a 3d theory.
The main point is that the invariance of the axion couplings to the gauge fields under θ ∼ θ+2π is equivalent to requiring that the 3d Chern-Simons theory based on the gauge group [SU (3) × SU (2) × U (1) Y ]/Z q is well-defined and gauge-invariant.The Chern-Simons Lagrangian (2.37) is not a globally well-defined differential form, as the gauge fields themselves are not.To define the Chern-Simons action in a mathematically precise way, one extends the Chern-Simons gauge fields to a 4-manifold whose boundary is the spacetime 3-manifold.
One then considers a 4d bulk theory with Lagrangian dL CS , which is a well-defined differential form on the 4-manifold [66].The quantization condition on K i 's follows from requiring that the theory does not depend on the choice of the auxiliary 4-manifold, which in turn is equivalent to requiring exp(2πi i K i n i ) = 1 on every closed 4-manifolds.This is analogous to the quantization of the coefficient for the Wess-Zumino-Witten term in the pion Lagrangian [67].Alternatively, the Chern-Simons coupling (2.37) arises from the circle compactification of the axion coupling (2.6) in the presence of a nontrivial winding number of the axion field around the internal circle.
To find the conditions on K i 's from (2.37), we begin from q = 1.In this case, the Chern-Simons couplings are well-defined for arbitrary integer K i 's. 13 The theory has a Z (1) 1-form center global symmetry. 14The Z We obtain a Chern-Simons theory based on the gauge group [SU (3)×SU ( 2)×U (1) Y ]/Z q with q = 2, 3, 6 by gauging a Z (1) q subgroup of the 1-form symmetry.For instance, consider the q = 2 case, where the A 2 and A 1 gauge fields combine into a U (2) = [SU (2) × U (1) Y ]/Z 2 gauge field.Starting from the q = 1 Chern-Simons theory, we need to gauge the to obtain the q = 2 theory [68].For this to make sense, first of all, we require K 1 ∈ 2Z, in accordance with (2.13).Furthermore, to be able to gauge a 1-form symmetry, the corresponding 't Hooft anomaly must vanish.The 't Hooft anomaly of a Z (1) N 1-form symmetry in 3d QFTs is measured by the topological spin of the generating line operator modulo 1/2 (on spin manifolds) [64].In our case, the topological spin of the Wilson line Requiring this to vanish, we recover the quantization condition (2.13).
For q = 3, the two gauge fields A 3 and A 1 now combine into a U (3) = [SU (3) × U (1) Y ]/Z 3 gauge field.Therefore, starting from the q = 1 Chern-Simons theory, we obtain the q = 3 theory by gauging the Z . This requires K 1 ∈ 3Z.The 't Hooft anomaly, i.e., the topological spin of the Wilson line (2.39) The anomaly vanishes if and only if the quantization condition (2.16) is satisfied. 13On non-spin manifolds, we instead need to require K 1 ∈ 2Z, whereas K 2 , K 3 ∈ Z is not affected. 14In the presence of the Chern-Simons coupling K1 4π A 1 dA 1 in 3d, magnetic monopoles carry electric charges which are multiples of K 1 , and they break the U (1) (1) center symmetry down to Z Finally, to obtain the q = 6 theory, we need to gauge the Z , which first requires K 1 ∈ 6Z.The topological spin of this Wilson line is and it vanishes if and only the quantization condition (2.21) is satisfied.

Constraints on the effective axion-photon coupling g aγγ
We now discuss model-independent constraints on the axion physics from the quantization of the axion couplings to the SM gauge fields.We assume that we have a single QCD axion whose mass m a comes from the coupling to QCD.We focus on the effective axion-photon coupling g aγγ defined as Here F EM,µν is the field strength for the canonically normalized EM gauge field (2.5).The effective coupling g aγγ receives contribution from the bare coupling E as well as the mixing with π 0 [69]: where m a is the axion mass and α = e 2 /4π is the fine structure constant.We have written the above expression in the standard notations of E, N .Recall that K 3 = N DW = 2N is the number of axion domain walls.
The quantization condition in Section 2 implies that the value of g aγγ /m a cannot be an arbitrary real number, but is subject to some rationality constraint from the ratio E/N .While any given real number can be approximated arbitrarily well by a rational number, an accurate approximation requires a large denominator N , which is related to the number of axion domain walls N DW = 2N .In the absence of other mechanisms, stable domain walls formed after inflation will survive till today and are inconsistent with the current observations.Therefore, to avoid the axion domain wall problem in cosmology, the value of N DW is preferred to be small in realistic, post-inflationary axion models.For any given upper bound on N DW , there is a strict lower bound on |g aγγ |/m a from the quantization conditions on E and N .
Another cosmological constraint on the UV completion of the axion model is the stable relic problem [70,71].There are typically new heavy fermions in UV axion models.These heavy particles have to be able to decay into the SM particles, otherwise they will form stable relics.This is possible if the UV particles carry trivial gauge charge under the Z 6 ⊂ SU (3) × SU (2) × U (1) Y subgroup, as all SM particles do.While particles with Z 6 gauge charge can potentially pair annihilate into the SM particles, such scattering processes are suppressed as the universe expands, and some relic density remains.Therefore, the global form of the SM gauge group is preferred to be [SU (3) × SU (2) × U (1) Y ]/Z q=6 in any axion model without the stable relics in cosmology.This is also the gauge group that is compatible with the various GUT models such as SU (5), Spin( 10), E 6 .See, for example, [18] and references therein for an extensive discussion on the phenomenological criteria for axion models.For this reason, we focus on the q = 6 case below.
The quantization conditions (2.22) for E and N in the q = 6 case are If we want to avoid the domain wall problem, i.e., N DW = 2N = 1, then g aγγ in (3.2) cannot be arbitrarily small because of the quantization condition (2.22).This leads to the following model-independent lower bound: The lower bound is saturated by which is the closest rational number E/N to 1.92 subject to the condition (2.22) and N DW = 1.This ratio is famously realized by one of the classic DFSZ models [48,49].(It is worthwhile noting that the standard DFSZ models have N DW = 3 or 6, while N DW = 1 can be achieved by relaxing PQ charge universality such as in [72].)It is also the ratio realized by the SU (5), Spin( 10), E 6 GUT models [50,73].In other words, the region to the right of the E/N = 8/3 line on the |g aγγ |-m a plot must necessarily face the domain wall problem in a post-inflationary scenario (see Figure 1, modified from [74]).This provides an invariant meaning to the ratio E/N = 8/3 in the landscape of axion models.
There are many proposed solutions to the axion domain wall problems in the literature, so models with N DW > 1 are still potentially phenomenologically viable.Here we merely point out that those QCD axion models violating the lower bound (3.3) must necessarily have N DW > 1.
It should be emphasized that this lower bound relies on the assumption that the axion mass m a is dominated by the contribution from QCD. 15 It is also worth pointing out that in the post-inflationary scenario with N DW = 1, there is an upper bound on f to avoid overproducing axions as dark matter from axion string simulations.The precise bound is a matter of some debate [75,76,77], but conservatively corresponds to f ≲ 10 11 GeV.This would greatly restrict the bottom left region of the E/N = 8/3 line in Figure 1. 16 0 As we increase the allowed tolerance of N DW , eventually the lower bound will be smaller than the uncertainty in (3.2) and cease to be meaningful.For instance, the lower bound with N DW ≤ 2 is |gaγγ | ma ≥ 0.05(1) GeV −2 , which is saturated by E/N = 5/3.17For other values of q = 1, 2, 3 (which are not phenomenologically preferred because of the stable relic problem), the quantization conditions for E, N are different, and the lower bounds on |g aγγ |/m a are saturated by E/N = 35/18, 2, 13/6, respectively.However, these lower bounds with q ̸ = 6 are of the same order as the uncertainty in (3.2).
Finally, it is worth mentioning that in the simplest version of the KSVZ model [42,43], the heavy fermion transforms as (3, 1, 0) under the su(3) × su(2) × u(1) Y gauge algebra.Therefore, the global form of the SM gauge group must correspond to q = 1 or q = 2.To realize the q = or q = 6 cases, the heavy fermion must to have a nonzero hypercharge [21].To be more specific, the condition (2.15) requires 6Q Y = 1 mod 3 for such a fermion.Relatedly, the ratio E/N = (which is commonly labeled as the "KSVZ" line in the |g aγγ |-m a plot) cannot be realized for q = and q = 6 by a single heavy fermion transforming under the fundamental of SU (3), because such a fermion must carry a nontrivial electric charge and therefore E ̸ = 0.

Higher symmetries of the Standard Model coupled to an axion
In this section we show that the SM coupled to an axion enjoys a myriad of generalized global symmetries, and discuss their implications on the possible UV completions of the theory [40,32].

Higher-group and non-invertible global symmetries
We focus on two kinds of generalized global symmetries of the SM coupled to a single axion.One is the higher-group symmetry which involves the center 1-form symmetry and the winding 2-form symmetry.We sometimes also refer to the center 1-form symmetry as the electric 1-form symmetry, since it acts on the Wilson lines.The winding 2-form symmetry comes from the conserved 3-form current J winding = 1 2π ⋆ dθ, where the superscript denotes the form degree.The other generalized symmetry that we consider is the non-invertible 1-form symmetry discussed in [32,33].The non-invertible 1-form symmetry arises from the part of the electric U (1) (1) 1-form symmetry of the free Maxwell theory, which appears to be explicitly broken in the presence of an axion.It was realized in [32,33] that this U (1) (1) symmetry is not entirely broken by the coupling to the axion; rather, it is resurrected as a non-invertible 1-form symmetry labeled by Q/Z (which is dubbed as the "non-invertible Gauss law" in [32]).The non-invertible 1-form symmetry acts invertibly on the Wilson lines, but non-invertibly on the monopoles and axion strings.
These two generalized symmetries are insensitive to the details of the axion potential and mass, unlike the axion shift symmetry.Moreover, they are not broken by any higher-dimensional local operators, which is a general feature of higher-form global symmetries [63].To break them, one needs to introduce additional dynamical degrees of freedom such as new matter fields that transform nontrivially under the center of the gauge group, dynamical monopoles and/or axion strings.The existence of these two generalized symmetries is solely determined by the values of K i 's, which is summarized in Table 2. 18 We also write the results in terms of the couplings N and E below the EWSB scale.The derivation of these generalized global symmetries largely follows the formalism in [40,32], which we review in Appendix A. q Higher-Group Non-Invertible Table 2: Summary of the higher-group for the center 1-form and winding 2-form symmetries, and non-invertible 1-form symmetries of the SM coupled to an axion, written both in terms of (K 3 , K 2 , K 1 ) and (N, E).Whenever an entry in one of the columns is nonzero there exists the corresponding symmetry.Here K ≡ gcd(6, K 1 ) = gcd(6, 36E).Note that K 1 and K are always integer multiples of q due to the quantization condition in Table 1.The q = 6 case does not have these generalized global symmetries.

Bounds on string tension and monopole mass
Whenever the values of the axion couplings admit a higher-group or non-invertible symmetry, then there must be a hierarchy to the energy scales at which the different symmetries are broken.These constraints have been considered for the higher-group symmetries of axion-Yang-Mills, axion-QCD, and axion-Maxwell theory [40], as well as for the non-invertible symmetries of axion-Maxwell theory [32].In our case the symmetry structure is more complicated but the overall logic remains the same, which we now discuss.
The generalized global symmetries we discuss are typically emergent in the IR, and they are therefore broken as we go to higher energies.However, as a consequence of the higher-group and non-invertible symmetries, they are not all on the same footing.Sometimes a "child" symmetry G C is subordinate to a "parent" symmetry G P , in the sense that the former symmetry cannot exist without the latter.For instance, the child non-invertible symmetry G C is typically tied to a parent invertible symmetry G P , and the algebra of symmetry elements of G C contains those of G P . 19This For generalized global symmetries, it is common that one "child" symmetry G C cannot exist on its own without a "parent" symmetry G P .In axion physics, G C , G P are emergent in the IR, and this hierarchical structure constrains the energy scales E C , E P these symmetries are broken.
hierarchical structure constrains the renormalization group flows, since it is not possible to have an effective field theory at an intermediate scale where the child symmetry is preserved while the parent one is broken.This gives universal inequalities on the energy scales where the symmetries become emergent.These inequalities take the form E C ≲ E P , where E C and E P are the energy scales above which the child and parent symmetries are broken, respectively.Since the symmetries are emergent, the inequalities we derive are approximate.See Figure 2.
Let us first consider the case where there is a nontrivial higher-group symmetry.There are two invertible symmetries of interest: the U (1) (2) winding 2-form symmetry, and the Z (1) K/q electric 1-form symmetry, where as usual superscripts denote the form degrees of the global symmetries and K ≡ gcd(6, K 1 ).We denote E winding and E center as the energy scales below which U (1) (2) and K/q become emergent, respectively.Now suppose we flow to a scale below E center .Due to the higher-group symmetry structure explained in Appendix A, turning on the background gauge field for the Z (1) K/q electric symmetry automatically activates the background gauge field for the U (1) (2)  winding symmetry as well, which only makes sense if we are also below the scale E winding [83].In other words, the Z (1) K/q electric symmetry is the "child" symmetry G C which cannot exist without the "parent" symmetry G P = U (1) (2) .We conclude that E center ≲ E winding . (4.1) Its algebra D × D = 1 + η involves an invertible parent Z 2 symmetry generated by η (i.e., η 2 = 1), and therefore the child non-invertible symmetry D cannot exist without its parent η.
When we have a non-invertible symmetry, we obtain stronger bounds.We denote the emergence scale for the non-invertible 1-form symmetry again as, by an abuse of notation, E center , since it acts on the Wilson lines in the same way as the ordinary center 1-form symmetry.In addition, we also associate to the U (1) (1) magnetic 1-form symmetry an emergence energy scale E magnetic .The magnetic 1-form symmetry is generated by the 2-form current J (2) magnetic = 1 2 ⋆ (qF 1 ), where F 1 is the field strength for U (1) Y .The fusion algebra of the non-invertible 1-form symmetry operator explained in Appendix A shows that it cannot exist without the winding 2-form symmetry and the magnetic 1-form symmetry.In other words, the non-invertible center 1-form symmetry is the "child" symmetry G C , and both the magnetic 1-form symmetry and winding 2-form symmetries are "parent" symmetries G P .We therefore find the constraint So far, we have been agnostic to the physical interpretation of the emergence scales.Let us now explore the implications for physical quantities of interest.E winding physically corresponds to the energy scale at which axion strings become dynamical and may unwind.This is naturally associated with the string tension, E winding ∼ √ T , although it may be quite a bit smaller than √ T depending on the UV theory.For example, in the KSVZ model considered in [40], it was found to be associated with E winding ∼ (m ρ f ) 1/2 , where m ρ is the mass of the radial mode of the scalar field responsible for PQ symmetry breaking, and f its vacuum expectation value.At weak couplings, E magnetic is most naturally associated with the mass of the lightest hypercharge monopole, E magnetic ∼ m monopole , although depending on the UV theory, it may again be significantly smaller.For example, if the dynamical monopole is an 't Hooft-Polyakov monopole associated with a UV non-abelian gauge group G, then the magnetic symmetry breaking scale is instead the Higgsing scale v at which G is broken to U (1) Y .The mass of the 't Hooft-Polyakov monopole is m monopole ∼ v/g, which is much larger than v for a weakly coupled theory.
Similarly, E center can be associated with the mass of the lightest particle charged under the center of the gauge group, which we denote as E center ∼ m center .To be more precise, for the case of higher-group symmetry, m center is the mass of the lightest particle charged under the Z K/q subgroup of the center of the gauge group.For the case of non-invertible symmetry, m center is the mass of the lightest particle charged under the Z 6/q center of the gauge group.In both cases, we refer to m center loosely as the mass of the lightest "Z 6 -charged particle."We emphasize that such a particle can either be a new fundamental field with Z 6 gauge charge, or a solitonic excitation of the monopole and axion string dictated by the anomaly inflow.
Combining these interpretations, we conclude that our inequalities imply whenever the higher-group symmetry exists, and whenever the non-invertible 1-form symmetry exists.In short, the tension of axion strings and the mass of hypercharge monopoles are generically bounded from below by the mass of the lightest particle charged under the center of the SM gauge group.
The higher-group symmetry inequality is easy to understand for UV completions based on the PQ mechanism.For example, consider an axion with couplings K i obeying the quantization conditions of q = 1 in Table 1 but not q = 6.Any PQ UV completion would then require at least one heavy fermion charged under the Z K/q subgroup of the gauge group to be integrated out near the scale f . 20Therefore, m center ≲ √ T ∼ f is automatically satisfied.On the other hand, if the true SM gauge group had q = 1 but the axion couplings were to satisfy the q = 6 quantization conditions, then the higher-group symmetry would be trivial from Table 2, and we would not have any inequality.In terms of a PQ UV completion, this means that the heavy fermion(s) integrated out near the scale f may have trivial charge under the Z K/q subgroup of the gauge group, and so there is no constraint on m center .This also provides an intuitive explanation for why the highergroup symmetry conditions in Table 2 are so similar to the quantization conditions in Table 1, at least in the context of PQ UV completions.We should emphasize, however, that our inequalities apply universally to any UV completion, not restricted to the PQ mechanism.
The non-invertible symmetry implies two inequalities, m center ≲ √ T as well as m center ≲ m monopole .The former can be intuitively understood in PQ UV completions, similar to the highergroup case.On the other hand, m center ≲ m monopole can be verified in models where U (1) Y is embedded in a UV non-abelian gauge group G.In this case, m center ∼ gv is the mass of the additional massive gauge bosons, whereas m monopole ∼ v/g.The inequality m center ≲ m monopole is then satisfied at weak couplings.More generally, these inequalities can also be satisfied by the solitonic excitations on the monopoles and the axion strings from anomaly inflow (see Section 7.1 of [32]).

Conclusions
The ambiguity in the global form of the SM gauge group generally has few experimentally observable effects, and is not often discussed as a result.However, for an axion, we have shown that 20 If the global form of the SM gauge group corresponds to q = 1 and K 1 ∈ 6Z, so that we have the invertible 3 1-form symmetry, it is possible that only the Z 3 ) subgroup participates in the highergroup symmetry, depending on the values of K i 's.In this case, for a heavy fermion that is neutral under the Z 3 ) subgroup of the original Z (1) 6 , its mass would not be subject to our inequality.this global structure is of central importance in dictating the allowed values of its quantized couplings to the SM gauge fields, resulting in the conditions in Table 1.These quantization conditions have immediate phenomenological implications.In particular, the original KSVZ axion model is incompatible with both the q = 3 and q = 6 quantization conditions unless the heavy fermion is given a hypercharge.For the case of q = 6 -the phenomenologically preferred value from the perspective of GUTs and the non-observation of cosmologically stable exotic relics -the quantization yields a highly non-trivial correlation between E and N .We have used this correlation to show that the smallest allowed effective coupling to photons |g aγγ | for a post-inflationary QCD axion with domain wall number N DW = 1 is realized by the ratio E/N = 8/3.
Of course, if an axion were discovered, other consequences would become immediately relevant.Unambiguously measuring all of an axion's couplings may be extremely challenging depending on the region of parameter space it populates, and would require several different experimental observables.Nevertheless, by measuring the couplings E and N separately and testing the quantization conditions, one could potentially falsify q = 6, regardless of whether the axion is of pre-inflationary or post-inflationary type and without ever discovering a Z 6 -charged particle.Such a discovery would greatly restrict viable UV completions of the SM gauge groups.Additionally, generalizing [40,32], we have shown that there is a higher-group symmetry structure between the electric 1-form center symmetry and the U (1) (2) winding 2-form symmetry of the axion whenever the couplings satisfy the conditions in Table 2. Similarly, the electric 1-form symmetry also becomes non-invertible depending on the value of E. These symmetries result in model independent constraints between the masses of Z 6 -charged particles, the masses of hypercharge magnetic monopoles, and the axion string tension.
There are several avenues for further study.We have considered the simplest case of a single axion coupled only to the SM gauge group.However, in recent years, there have been a number of axion models developed going beyond these minimal assumptions [18].A clear next step would be to generalize these results to theories of multiple axions or extended gauge groups, which may have more complicated symmetry structures and quantization conditions.We have also neglected discussing the 0-form shift symmetry of the axion, for good reason, since it is generically broken by the potential generated by QCD or UV instantons.However, it is known that the shift symmetry can also lead to a higher-group symmetry [38,39,40,84,85] or become non-invertible [30,31,32], which would then lead to additional inequalities.These may be relevant for axions that do not couple to QCD or for theories with multiple domain walls.[86,87] for coordinating submission.Shortly after our paper appeared on arXiv, another paper [88] was also posted, which discusses the quantization of the axion-gauge couplings too.The work of HTL was supported in part by a Croucher fellowship from the Croucher Foundation, the Packard Foundation and the Center for Theoretical Physics at MIT.The work of SHS was supported in part by NSF grant PHY-2210182.This work was performed in part at Aspen Center for Physics during the workshop "Traversing the Particle Physics Peaks: Phenomenology to Formal," which is supported by National Science Foundation grant PHY-2210452.The authors of this paper were ordered alphabetically.
A More on the higher-group and non-invertible symmetries In this Appendix, we provide derivations of the generalized global symmetries of the SM coupled to a single axion through the coupling (2.1) (or equivalently, (2.6)), which were used in the main text to obtain lower bounds on the tension of axion strings and the mass of hypercharge monopoles.The results are summarized in Table 2.We work in the Euclidean signature for convenience.

A.1 Higher-group symmetries
We first derive the higher-group symmetries of the theory, following [36,37,38,39,40].We will start from the q = 1 case.Let us first consider an SU (3) × SU (2) × U (1) Y gauge theory with an axion and no other matter fields, after which we will add the SM fermions.There are two invertible symmetries of relevance: |K 1 | electric 1-form symmetry , U (1) (2) winding 2-form symmetry .
The electric 1-form symmetry corresponds to the product of the center symmetries of SU (3) and SU (2) with the Z ), where α ∼ α + 2π is a U (1) group parameter.It measures the winding number of the axion field θ(x) around a closed loop γ, and the charged objects are two-dimensional worldsheets of probe axion strings.There are also a 0-form shift symmetry and a magnetic 1-form symmetry that we will not consider, although they may lead to a higher-group symmetry as well [40,38,39,84,85].
To find whether the electric 1-form symmetry and the winding 2-form symmetry combine into a higher-group symmetry or not, one may couple the theory to the background gauge fields for The surface operator D(Σ) is topological, which can be heuristically understood from its relation to (A.28).For a more rigorous proof, see [32].Moreover, it obeys a non-invertible fusion algebra, It satisfies a non-invertible fusion algebra similar to (A.30), which shows that the "child" noninvertible 1-form symmetry cannot exist without the "parents" magnetic 1-form symmetry and the winding 2-form symmetry.
The analysis is the same for the q = 3 case and we will only mention the result.For q = 3, we have K 1 ∈ 3Z, and the non-invertible 1-form symmetry exists when where now U (Σ) ≡ η 2 (Σ)U π (Σ).The "child" non-invertible 1-form symmetry again cannot exist without the two "parent" invertible higher-form symmetries.
For q = 6, there is no interesting non-invertible 1-form symmetry operator, since we do not have a center 1-form symmetry in the SM to begin with even in the absence of an axion.

B Fractional axion-gauge couplings
Throughout the paper we assume that there are no other degrees of freedom in addition to the SM fields and a single axion field.However, certain topological degrees of freedom can result in a fractional axion-gauge coupling [53], which we review below.It would be interesting to further explore phenomenological consequences of this mechanism.
We demonstrate this phenomenon for a dynamical axion field θ(x) coupled to an SU (N ) gauge field A with coupling K 8π 2 θ TrF ∧F with K ∈ Z.We introduce two new dynamical fields, a periodic scalar field θ(x) ∼ θ(x) + 2π, and a 3-form gauge field c.We consider the following couplings: Substituting this back into (B.1),we find an fractional axion-gauge coupling K/N for θ.Globally, however, we cannot integrate out c and relate two compact scalar fields of the same period 2π as in (B.2) (and hence the quotation marks), and (B.1) is the precise, gauge-invariant description of the model.This is similar to the construction in [96,30,97,32].

Figure 1 :
Figure1: Constraints on the effective axion-photon coupling g aγγ versus the axion mass m a , modified from[74].The model-independent quantization conditions (2.22) imply that any QCD axion model in the gray region below the E/N = 8/3 line necessarily faces the axion domain wall problem in a post-inflationary scenario, i.e., N DW > 1.

Table 1 :
Quantization conditions of the axion couplings to the gauge fields for the possible global forms of the SM gauge group.