Axions and Superfluidity in Weyl Semimetals

An effective field theory (EFT) for dynamical axions in Weyl semimetals (WSMs) is presented. A pseudoscalar axion excitation is predicted in WSMs at sufficiently low temperatures, independently of the strength of the Weyl fermion self-coupling. For strong fermion self-coupling the axion is the gapless Goldstone boson of chiral $U(1)^{\text{ch}}$ spontaneous symmetry breaking. For weak fermion self-coupling an axion is also generated at non-zero chiral density for Weyl nodes displaced in energy, as a gapless collective mode of correlated fermion pair excitations of the Fermi surface. This is an explicit example of the extension of Goldstone's theorem to symmetry breaking by the axial anomaly itself. In both cases the axion is a chiral density wave or phason mode of the superfluid state of the WSM, and the Weyl fermions form a chiral condensate $\langle\bar{\psi}\psi\rangle$ at low temperatures. In the presence of an applied magnetic field the axion mode becomes gapped, in analogy to the Anderson-Higgs mechanism in a superconductor. 't Hooft anomaly matching from ultraviolet to infrared scales is directly verified in the EFT approach. WSMs thus provide an interesting quantum system in which superfluid, non-Fermi liquid behavior, and a dynamical axion are predicted to follow directly from the axial anomaly in a consistent EFT that may be tested experimentally.

could shed on the CP 'naturalness' problem of the Standard Model. For this, it is important to identify the requirements and physical mechanism(s) by which an axion can emerge as a collective mode in a realistic many-body system that can be studied in a laboratory environment, where the parameters of the system can be varied and controlled. This is just what WSMs provide.
From a condensed matter perspective, a possible axion excitation in WSMs has been discussed primarily through the introduction of four-fermion interactions [13] of the Nambu-Jona-Lasinio (NJL) type [33,34], which induce the spontaneous symmetry breaking (SSB) of U (1) ch chiral symmetry. The axion is then the Nambu-Goldstone boson generated by this SSB of a global symmetry. As a result of this SSB, typically described by the introduction of a scalar field order parameter developing a non-zero expectation value in its ground state, the Weyl fermions acquire a mass gap. A preliminary claim of detection of an axionic mode in a WSM has been made in [20], although this awaits confirmation and study of the excitation spectrum and its detailed properties.
At the same time the axial or chiral anomaly breaks U (1) ch symmetry explicitly, violating the Ward-Takahashi (WT) identities of U (1) ch invariance. Thus it is not clear how this explicit breaking of a global symmetry by the anomaly is to be reconciled with the usual formulation of Goldstone's theorem, where the symmetry is assumed to be exact at the microscopic level, and only the ground state of the theory spontaneously breaks the symmetry. The usual statement of Goldstone's theorem and massless Goldstone boson depend upon the existence of a bosonic order parameter and degenerate states of the same energy, related by the symmetry. Neither a scalar order parameter nor the requisite U (1) ch symmetry are immediately apparent in a fermionic theory possessing an anomaly, which explicitly violates this very symmetry.
Despite these apparent differences with the usually considered requirements of Goldstone's theorem, it was shown in a previous paper that a gapless pseudoscalar chiral density wave (CDW) in the anomalous fermion theory follows from the axial anomaly itself [35]. The CDW may be understood as a collective excitation of the Fermi surface and a result of fermion-antifermion (particle-hole) pairing, similar to Cooper pairing in a superfluid or superconductor. The gapless collective mode is made manifest as a 1/k 2 massless pole of the axial anomaly triangle amplitude [36]. Thus it appears that the Goldstone phenomenon applies also the case of anomaly symmetry breaking (ASB), despite-or rather as a direct consequence of-the axial anomaly itself.
Clarifying this state of affairs and applying it to WSMs where the pseudoscalar CDW is an axion is the first principal aim of this paper. We will show that the familiar description of SSB by a scalar order parameter and proof of Goldstone's theorem can be extended to the case of anomalous symmetry breaking (ASB). Indeed both can be derived from the same UV complete QFT in different limits and, remarkably, both lead to a essentially the same low energy effective theory of a relativistic superfluid with an axion excitation.
An interesting related issue is that of decoupling of heavy (gapped) degrees of freedom from the low energy spectrum, and the status of anomaly matching between the SSB phase, where the fermions become massive, and the ASB phase, where they were assumed (at least at first) to remain massless. The expectation of 't Hooft anomaly matching [37] is that the form and magnitude of the chiral anomaly should be renormalization group invariant, and hence the same across phases and across widely different distance scales, even if the fermions develop a mass gap.
Naively a large fermion mass gap might lead one to expect that the fermions and the entire axial anomaly decouple entirely at low energies or large distances, which would appear to violate 't Hooft anomaly matching across scales. In the Standard Model this anomaly matching from the short distance scales of perturbative QCD to the low energy pion dynamics of the confined theory is critical to accounting for the experimentally verified low energy decay rate π 0 → 2γ, in terms of the quantum numbers of the constituent colored quarks [38]. Historically this success played an important role in the establishing of QCD as the theory of the strong interactions. However, a proof of anomaly matching between the short distance, high energy quark degrees of freedom of QCD and the long distance, low energy theory of mesons in which quarks are confined and do not appear at all in the low energy spectrum of QCD is necessarily indirect in full QCD [37], because a complete understanding of the mechanism of quark confinement is still lacking.
In the effective field theory (EFT) motivated for application to WSMs of this paper, it turns out that anomaly matching across energy scales and phases can be checked and verified directly by perturbative methods. Consequently, this EFT provides a consistent framework from UV to IR in which the fundamental question of decoupling-or its opposite, 't Hooft anomaly matching-can be addressed systematically. Providing an explicit example of derivation of low energy superfluid behavior emerging from an EFT, itself obtained from a UV complete QFT, in which the axial anomaly remains intact is the second principal goal of this paper.
In order to avoid possible misunderstanding across different sub-fields, let us remark at the outset that there are three different notions of EFTs in common use in the literature: (i) The full one-particle irreducible (1PI) quantum effective action Γ[ϕ] obtained by computing quantum corrections to a classical action S cl [φ], independently of scale; (ii) The low energy expansion of this non-local 1PI quantum effective action in inverse powers of the heavy mass M of some quantum field(s) which are completely 'integrated out,' to obtain a local (Wilson) effective action S eff,M applicable to energy scales lower than M ; (iii) The hydrodynamic or Fluid effective action S Fluid defined only by symmetries, conservation laws and the equation of state of the system in local thermodynamic equilibrium, applicable only at macroscopic distance scales, with no reliance upon any specific microscopic theory.  [39,40], evaluated at J ϕ (x) ≡ J[ϕ(x)] obtained by solution of the implicit equation which must be inverted to find J ϕ (x) and inserted into (1.2).
The 1PI effective action Γ[ϕ] so defined is the generating functional of proper vertices, and the {ϕ i (x)} are the background or mean fields of the background field method [41]. If the original S cl is renormalizable and UV complete, its 1PI effective action Γ[ϕ] and the equations of motion following from it apply at any scale without restriction. However, in general Γ[ϕ] formally defined by (1.2) is a non-local functional of ϕ, that cannot be calculated exactly. Restrictions then inevitably arise in the approximations necessary to calculate it. As a familiar example, the oneloop approximation to Γ[ϕ] for the ϕ i spacetime constants (not necessarily fundamental fields) are the order parameters of the effective potential V eff (ϕ) at finite temperature or density that can be used to diagnose SSB and the restoration of symmetry [42], as well as to prove the Goldstone theorem [43,44]. The 1PI effective potential is therefore closely related to the Landau theory of phase transitions, with parameters fit to data in condensed matter applications rather than calculated from first principles of microphysics [45].
If some of the quantum fields in (1.1) are massive with large mass gap M , one expects that their effects can be neglected at low energies E ≪ M , or equivalently distances much larger than 1/M , as a consequence of decoupling [46]. In that case, such heavy fields may be 'integrated out' completely in (1.1), with no sources or mean fields specified for them. This procedure is also usually difficult to carry out explicitly, especially when the 'fundamental' or UV complete theory is itself unknown. In that case one relies on the symmetries of the low energy theory and scaling dimensions to expand the effective action in a power series of local operators of the remaining light fields in ascending powers of 1/M . A familiar example of this EFT approach is Chiral Perturbation Theory [47,48]. Since the separation of scales specified by M is closely related to the Wilsonian classification of infrared (IR) relevant and irrelevant operators in statistical and condensed matter physics [49][50][51], the EFT (ii) organized this way in terms of a heavy mass scale is referred to here as the Wilson effective action [52].
Finally, at the lowest energy scales when almost no information is available about the underlying microscopic theory or its fundamental interactions, one can consider a hydrodynamic or Fluid EFT (iii) which is based only upon symmetries and conservation laws, and a specified equilibrium equation of state of the system. Assuming local equilibrium on microscopic scales (i.e. on order of the mean free path and smaller) is maintained, small deviations from the equilibrium relations at the longest wavelength macroscopic scales such as sound waves can be studied [53].
Generally, each of these three meanings of 'effective field theory' is different from the others and the relationship between them is non-trivial. The remarkable fact following from explicit anomaly matching in WSMs is that the chiral anomaly admits a consistent description from fundamental principles of QFT connecting these three approaches, starting from a single renormalizable theory of Weyl fermions coupled to scalars and electromagnetism in both the SSB and ASB cases, as we shall show, at least at zero temperature and in a pure sample without defects.
The organization of this paper is as follows. We begin in the next section 2 with a review of the axial or chiral anomaly of fermionic QED 4 , and the gapless collective boson excitation of the

The Axial Anomaly in QED 4
The Lagrangian of Dirac fermions ψ coupled to electromagnetic potential A µ with charge strength e is ,ψ = ψ † γ 0 , and we have allowed both for a non-zero fermion mass m and coupling to an external axial vector potential b µ ≡ A 5 µ with unit strength. The Dirac γmatrices satisfy the anti-commutation property {γ µ , γ ν } = −2 η µν with η µν = diag(−1, 1, 1, 1), such that γ 0 = (γ 0 ) † is Hermitian while the γ i = −(γ i ) † are anti-Hermitian. The chirality matrix is also Hermitian, with eigenvalues ±1 corresponding to right-handed or left-handed chiralities respectively. The for any m. When m = 0 (2.1) is also invariant, at the classical level, under the additional U (1) ch local chiral phase transformation in which the right and left chiralities transform oppositely. By Noether's theorem these two classical invariances of the action S f = d 4 x L f at m = 0 imply the conservation of both the electric and axial currents, as follows from the Dirac equation obtained by variation of (2.1). The m = 0 case thus has an apparent U (1) EM ⊗ U (1) ch symmetry.
As is well-known, this apparent larger classical symmetry at m = 0 does not survive in the quantum theory, since it turns out to be impossible to simultaneously satisfy the requirements of Fig. 1: The one-loop axial anomaly J µ 5 J α J β triangle diagram, which is Γ µαβ 5 (p, q) of (A.4) in momentum space. The solid lines represent the propagators of Dirac fermions and the wavy lines external photon legs, which carry off four-momenta p and q from the electromagnetic current vertices J α and J β . The incoming four-momentum at the axial current vertex J µ 5 is k = p + q by momentum conservation.
Lorentz invariance, U (1) EM gauge invariance and U (1) ch chiral invariance at the quantum manyparticle level [8,54,55]. This conflict of symmetries first appears at the one-loop level of the triangle diagram of Fig. 1.
The triangle diagram with three fermion propagators and one internal loop momentum integration is naively ultraviolet (UV) linearly divergent at large loop momenta or short distances.
Hence it is a priori undefined and in need of some further prescription. However the linear UV divergence is proportional to an arbitrary four-vector (not equal to k, p or q) which, if present, would imply that the vacuum or ground state is not Lorentz invariant. Conversely the imposition of Lorentz invariance automatically sets this possible linear divergence to zero identically. Indeed With these four requirements Γ µαβ 5 (p, q) is completely well-defined and finite, but does not obey axial or chiral invariance (2.5) in the massless fermion limit m → 0.
In fact, contraction of Γ µαβ 5 (p, q) defined by requirements (a)-(d) with the external momentum k µ = p µ + q µ entering at the axial vector vertex gives the well-defined result where υ αβ (p, q) ≡ ǫ αβρσ p ρ q σ = υ βα (q, p) (2.7) and A is most conveniently presented as an integral over Feynman parameters x, y in the form where the denominator D in (2. in position space, in the presence of external electric E and magnetic B fields. Here F µν ≡ 1 2 ǫ αβµν F αβ is the dual of F µν , and α = e 2 /4π ≃ 1/137.036 is the fine structure constant. One should recognize that conservation at the axial vector vertex J µ 5 could be required, by defining Γ µαβ 5 (p, q) to violate the charge conservation condition (b) instead. The anomaly is fundamentally a conflict between symmetries, forcing a choice between the classical conservation equations (2.5), which can only be decided by additional physical input. Since charge conservation is well-established, and it is known that the m → 0 limit is not equivalent to the m = 0 theory when e = 0, which limit is beset with infrared (IR) divergences, cf. [57][58][59] and Appendix A, the choice is made to preserve U (1) EM and discard U (1) ch as a fundamental symmetry.
The determination of Γ µαβ 5 and (2.8) solely by the conditions (a)-(d) above shows that these symmetry requirements are all that are necessary and sufficient to determine the anomaly (2.10), which is finite and independent of any UV divergences or renormalization. Thus exactly the same anomaly is obtained by any method which satisfies these symmetry requirements, among them Pauli-Villars regularization [60], dimensional regularization [55], heat kernel methods [61], or Fujikawa's regularization of the fermion functional integral for the axial current [62,63]. The fact that these various methods were developed in QFT to regularize its short distance behavior, as the first step to removing UV divergences in a renormalization procedure, led to some obscuring of the physical basis of the axial anomaly as an essentially IR phenomenon. The only role these more sophisticated regularization methods play is simply to require Γ µαβ 5 to satisfy conditions (a)-(d), which are properties of the vacuum or ground state of the theory. Thus the axial anomaly may be viewed as a low energy or macroscopic feature of QFT, which is why it is relevant for EFT treatments of both particle physics and condensed matter systems such as WSMs.
The next important observation is that the triangle amplitude Γ µαβ 5 (p, q) for massless fermions may be decomposed as where Γ µαβ 5 ⊥ is transverse, k µ Γ µαβ 5 ⊥ (p, q) = 0, and hence non-anomalous, while the first term in (2.11) which is responsible for the anomaly contains a massless 1/k 2 pole. Whereas the transverse part may (and does) receive all manner of radiative corrections from higher order processes, the longitudinal anomalous pole contribution in (2.11) is protected from such corrections by the Adler-Bardeen theorem [64]. This remarkable fact can be understood to be a result of the topological character of the anomaly equation (2.10) [65,66], and another indication of its long distance properties. The topologically protected massless pole at k 2 = 0 in the anomaly term describes a two-particle fermion/anti-fermion intermediate state in the triangle diagram, with this fermionic pair propagating coherently and co-linearly at the speed of light [67], cf. Fig. 2. This fermion pair state of opposite helicities is just that of a massless boson [36]. This gapless bosonic mode is a collective excitation of the Fermi-Dirac sea, revealed first in one spatial dimension in the Schwinger model of massless QED 2 [68], and in fermionic condensed matter systems as Luttinger liquids [69][70][71]. Similar Luttinger liquid-like behavior has been conjectured in higher dimensional fermion systems at low temperatures [72][73][74]. It is the 1/k 2 anomaly pole in (2.11) that provides the theoretical basis for this conjecture to be physically realized in WSMs, with the Fermi velocity v F replacing the speed of light c, and the fermion pairing to be analogous to the Cooper pairs of a superfluid condensate [35].
The bosonic excitation arising directly from the axial anomaly itself provides a new realization of Goldstone's theorem, anomaly symmetry breaking (ASB), discussed in Sec. 9, without (so far) any explicit reference to a bosonic order parameter expectation value violating the U (1) ch symmetry, as in the usual case of SSB. We shall see that despite the fact that the axial anomaly modifies the WT identities of the theory whereas SSB preserves them, the two apparently distinct situations of ASB and SSB are very closely related and can be described by the same low energy EFT, as a superfluid with a Goldstone sound mode that is an axion excitation in a WSM.

Anomaly Effective Action and Superfluid EFT of Weyl Nodes Displaced in Energy
Since the axial current J µ 5 is the variation of the action with respect to the external axial potential b µ , the longitudinal projection of (2.11) is the variation of the non-local 1PI effective action (i) [36,75,76] in position space, where −1 xy = 1 4π 2 (x − y) −2 denotes the massless scalar propagator inverse of the wave operator = ∂ µ ∂ µ = −∂ 2 t + ∇ 2 in 3 + 1 spacetime dimensions and A 4 the axial anomaly given by (2.10). The effective action (3.1) is non-local, although its variation with respect to b µ which gives J µ 5 , and subsequent acting with ∂/∂x µ , reproduces the local result (2.10). The non-local 1PI effective action (3.1) can also be expressed in a local form upon the introduction of the local pseudoscalar potential η. The variation of (3.2) with respect to η reproduces the anomaly (2.10), while the variation of (3.2) with respect to the hydrodynamic current J µ 5 produces the constraint ∂ µ η+b µ = 0. Solving this constraint for η gives η = − −1 ∂ µ b µ , and substituting this back into (3.2) reproduces the non-local action (3.1) with its massless pole, after integration by parts. This variational method with respect to a current in which no reference to the underlying QFT variables ψ is made, is employed for effective actions of fluid hydrodynamics (iii), in which context η is called a Clebsch potential [77].
Note that apart from the anomaly term ηA 4 , (3.2) depends upon the linear combination This is a suggestive remnant of the local U (1) ch symmetry (2.3), although the fermions themselves do not appear explicitly in the anomaly effective action (3.2). The fluid form has only bosonic variables, and a fluid variational principle in which only the full current J µ 5 is varied, not its fermionic constituents. Thus it is not immediately apparent from (3.2) to what complex bosonic variable Φ the phase η corresponds, or how its magnitude |Φ| is related to the underlying fermion QFT.
On the constraint ∂ µ η + b µ = 0, the ηA 4 term alone in (3.2) is the form of the effective action of the axial anomaly that can be shown to be responsible for the anomalous Hall conductance, as well as the chiral magnetic and chiral separation effects [12,35]. Further, if 2b µ = (2b 0 , 2b i ) is the difference in the energies and momenta of the two Weyl nodes of a WSM respectively, illustrated in Fig. 3, then η = −x µ b µ = tb 0 − x · b is the chiral phase obtained by a fixed chiral rotation on the fermions through Fujikawa's method [62,63], and (3.2), giving the effective action of the axial anomaly for such WSMs in [1,12,14].
In the WSM literature, the phase θ = −2η multiplying the axial anomaly A 4 in (3.2) is called an 'axion.' However, in QFT the axion of [27,28] is a propagating pseudoscalar field, not simply a Such kinetic terms may just be assumed to be present in the EFT argument, as IR relevant terms in the Wilsonian sense (ii), e.g. [78,13,79]. However the origin of these kinetic terms is not specified in that case and their coefficients must be treated as unknown free parameters that can only be fixed by experiment in such an approach.
On the other hand, continuing with the Fluid description (iii) of free massless fermions with displaced Fermi levels for left-and right-handed Weyl fermions automatically provides just such kinetic terms for small deviations from equilibrium, with coefficients fixed by the equilibrium equation of state, once the energy density ε(n 5 ) of the fermions as a general function of the chiral number density n 5 is taken into account. Adding this non-anomalous −ε(n 5 ) term to the local anomaly effective action (3.2) results in the fluid effective action for the massless fermion system at finite n 5 . The variation of this effective action with respect to J µ 5 is now non-trivial and leads to instead of the constraint ∂ µ η + b µ = 0, and where the Lorentz frame-invariant definition of n 2 5 = −J µ 5 J 5 µ has been used. The quantity µ 5 , likewise invariantly defined by carries the interpretation of the axial chemical potential of the chiral effective fluid. Using these definitions, (3.3) can be expressed in the form where P = µ 5 n 5 − ε is the equilibrium pressure of the fluid at zero temperature.
For a WSM whose two Weyl nodes of opposite chirality are displaced in energy and momentum space by 2b 0 and 2b i respectively, as in Fig. 3, in the rest frame of the material in equilibrium all time derivatives vanish, whereas J 0 5 = n 5 is non-vanishing. Thus when the non-zero ε(n 5 ) and µ 5 of the fluid in (3.3) are taken into account, we havē and the electric current and axial currentJ 5 = 0 vanish in equilibrium from (3.4) and (3.7a), including also in the presence of a background magnetic field B. In equilibrium with E = 0, the anomaly A 4 also vanishes, and ∂ µJ µ 5 = 0. Now since η is a dynamical field in the effective fluid description and J µ 5 depends on η through (3.4), small fluctuations δη(t, x) from equilibrium can be considered and these will generate small fluctuations in µ 5 and J µ 5 , according to for the chiral current components and their departure from equilibrium. To first order in these departures from equilibrium the variation in (3.9b) may be replaced by its value from the equilibrium equation of state, where we drop overbars on the equilibrium values of µ 5 , n 5 , which are independent of t, x. Now using the definition of the sound speed in the fluid the axial anomaly equation ∂ µ (δJ µ 5 ) = δA 4 for the linear variations in (3.9) may be expressed in the form This follows also from (3.3), expanded to quadratic order in the departures from equilibrium, i.e.
If the anomaly term δA 4 is assumed to be of the same order of small variation δη from equilibrium, the coupling of δη to electromagnetism would need to be considered. However since is quadratic in the EM fields, this will only be the case if there is a non-zero background field of either E or B. The latter is considered in the next section. that couples linearly to F F and therefore may be identified as an axion for a WSM arising from displaced Weyl nodes, as in Fig. 3, that fluctuate from their equilibrium value. This axionic CDW arises from the axial anomaly itself, independently of direct fermion self-interactions. For the specific example of free Weyl fermions we have the zero temperature equilibrium relations and the speed of the CDW, v s in this specific case of free fermions with Weyl nodes displaced in energy and momentum, is given by

The Axion for WSMs in a Constant Uniform Magnetic Field: Dimensional Reduction
As a second example of the application of the hydrodynamic EFT to a WSM, consider a WSM, for free or very weakly self-interacting fermions, and with no offset in energy of the Weyl nodes, i.e. b 0 = 0, but placed in a constant, uniform magnetic field B = Bx. In this case the triangle diagram and vertex Γ µαβ 5 reduces to its longitudinal component only and the effective action restricted to excitations along B becomes exact [35]. The details of this dimensional reduction in a constant uniform magnetic field with the fermions in their lowest Landau level (LLL) are reviewed in Appendix B for the convenience of the reader.
The energy density of chiral fermions in their LLL in 3 + 1 spacetime dimensions ε(n 5 ) can be related to the energy density of fermions in 1 + 1 spacetime dimensions ε 2 by is the 3 + 1 chiral number density of fermions in the LLL, in terms of the chiral density and energy density of 1 + 1 dimensional fermions restricted to moving in thex direction along B. The 2D chiral current and number density in (4.3) are denoted by a andñ respectively.
Factoring out a common factor of the electron number density per unit area eB/2π in the LLL leads to the 2D effective action of a 2D chiral superfluid, where the two dimensional axial anomaly is where the a, b indices range over 0, 1 only and the electric field E = F 10 is also in thex direction along B. The 2D axial chemical potential is in fact the same as that evaluated in 4D. Note that in 2D the axial anomaly (4.5) is linear in the electric field, so that variations in A 2 will be of the same order as those of η in 2D, or in 4D as a result of dimensional reduction in a classical background B field.
Apart from the overall factor of eB 2π d 2 x ⊥ , the 2D effective action in (4.4) for excitations along the B direction is thus [35] in terms of a local pseudoscalar boson field χ, related to the chiral phase η by where the arbitrary constant θ phase angle associated with the anomaly is set to zero here. The 2D effective fluid action (4.7) for excitations parallel to the B field is then recognized as the bosonic effective action of the fermionic sector of the Schwinger model, i.e. QED 2 of massless fermions [68,76,35] in which the gauge potential in ǫ ab F ab /2 = ǫ ab ∂ a A b of the 2D model has been replaced by eA a + ǫ a c b c of the dimensionally reduced 4D theory.
The variations express the bosonization rules for the 2D chiral and electric currents. The extremization recovers the 2D axial anomaly (B.8), and the 2D wave operator 2 shows that the χ (or η) boson is a true propagating axion degree of freedom as a result of the anomaly itself. The Green's function of the 2D wave operator −1 2 xy = − 1 π ln(x − y) 2 in (4.11) is the massless scalar propagator in d = 2 spacetime dimensions, describing propagation along the B direction. The non-local form of (4.7) is assuming the electric field E = E(t, x)x is also along the B direction and independent of the transverse coordinates y.
By the Gauss law ∂ x E = ρ, an electric field parallel to the B field direction is induced. Finally, variation of (4.4) with respect to η reproduces the axial anomaly back in 3 + 1 dimensions which is equivalent to the linear wave equation with the mass gap M 2 = 2αeB/π. The energy-momentum offset of the Weyl modes may be set to zero, b a = 0, at this point. It has been retained in (4.14) only to illustrate that such an offset which is varying in time or space acts as a source for the axion phase mode aligned with the magnetic field B.
The wave equation (4.14) describes chiral density waves (CDWs) and chiral magnetic waves

Fermion Self-Interactions and SSB of Chiral Symmetry
In the condensed matter literature on WSMs, it is usually supposed that the fermion excitations at the Weyl nodes may be subject to a generic four-fermion interaction of the Nambu-Jona-Lasino (NJL) type [13,33,34] of unspecified origin in the material. This NJL Lagrangian is invariant under the global U (1) EM ⊗ U (1) ch chiral symmetry ψ → e iα+iβγ 5 ψ just as the massless fermion Lagrangian (2.1) is at the classical level. For G > 0 the interaction (5.1) is attractive. Since this fermion self-coupling G has mass dimension −2, the NJL theory is non-renormalizable in the UV and must come equipped with a cutoff at some high energy scale Λ, corresponding to a short distance cutoff 1/Λ.
The fermion self-coupling G is most conveniently handled by introducing a two-component charge neutral scalar field in the Dirac matrix space, with both scalar (φ 1 ) and pseudoscalar (φ 2 ) components, coupled to the fermions through the Yukawa interaction i.e. Φ is electrically neutral but carries a chiral charge that is opposite in sign to that of ψ and twice as large. The interaction Lagrangian is equivalent to the original four-fermion NJL interaction Lagrangian (5.1) after a Hubbard-Stratonovich transformation, obtained by extremizing L int with respect to φ i , yielding and substituting the result into L int , whereupon L NJL is recovered. Thus, the fermion bilinears of the NJL model are replaced by the Φ boson field in this description.
In the NJL model the U (1) ch symmetry is spontaneously broken, ψ ψ = 0, and the fermion acquires a mass gap m if the self-coupling G exceeds a certain critical value G c . This follows from the solution of the minimization of the effective potential for ψ ψ , which leads to condition in terms of the ultraviolet cutoff Λ ≫ m. If G < G c , the only solution of (5.7) for real m is m = 0, the U (1) ch symmetry remains unbroken and the fermion remains massless. On the other hand, if G > G c a second solution appears with m > 0, given by so that the Φ field develops an expectation value φ 1 ≡σ = 0 in the ground state that spontaneously breaks the U (1) ch symmetry if κ < 0. Comparing the quadratic term in (5.9) to that of the fermion loop in (C.29) shows that and the condition for SSB in the scalar theory κ < 0 coincides with the condition G > G c in the fermionic NJL model. Upon identifying also from the quartic term induced by the fermion loop in (C.29), we then find that and that the kinetic terms for the A µ and b µ = A 5 µ potentials involve only their respective field strength tensors, namely F µν F µν and F 5 µν F 5 µν should also be included for a UV complete theory in general. A mixed F αβ F 5 αβ term is disallowed by the discrete symmetries of charge conjugation or parity spatial reflection.
These terms are generated by fermion loop integrations, with a logarithmic dependence upon the UV cutoff Λ, as shown in Appendix C, just as one would be expect by power counting and the U (1) EM ⊗ U (1) ch classical symmetry. A UV renormalizable effective theory must contain these terms from the very beginning. In such a UV completion of the EFT the couplings κ, λ and g and the fermion mass gap m will all be be finite renormalized parameters independent of the UV cutoff Λ, to be fixed by experiment, and the scalar potential (5.9) can be treated at tree level, as in the classical Landau theory of phase transitions [45]. This is the approach we take in the next section.

A Renormalizable Theory for WSMs Encompassing both ASB and SSB
The effective Lagrangian in the two cases of free or weakly interacting fermions and strongly self-interacting fermions with SSB can be derived as different limits of one and the same UV renormalizable classical theory with Lagrangian with the bosonic Lagrangian where we keep all possible relevant terms up to dimension four, fully invariant under the U (1) ⊗ U (1) ch and Lorentz symmetries. The couplings g, κ, λ will now be arbitrary renormalized (i.e. UV cutoff independent) parameters that ultimately would have to be fixed by experiment in any given WSM. The fermions are taken to be massless initially, with any mass generated only through the Yukawa interaction, and the scalar Φ developing a vacuum expectation value Φ = φ 1 = 0.
As already discussed, one should also allow for kinetic terms for the axial vector potential, such as F µν 5 F 5µν /f 2 5 for the transverse part of b µ , with an independent normalization and chiral coupling f 2 5 analogous to the electromagnetic coupling e of L cl . Anticipating that the U (1) ch symmetry will be broken by the axial anomaly, a kinetic term (∂ µ b µ ) 2 for the longitudinal component with another independent coupling as well as a mass term b µ b µ could also be allowed in (6.1). However, once the U (1) ch symmetry is explicitly broken by the axial anomaly, there is nothing preventing this mass term from being large, of the order of the cutoff scale Λ, in which case all the components of the axial vector potential will be gapped and play no role in the low energy EFT at long wavelength scales. Thus we omit from the effective Lagrangian L cl of (6.1) all such kinetic and mass terms involving the axial potential b µ from the outset.
The U (1) EM ⊗ U (1) ch symmetry of the classical action S cl = d 4 x L cl , with (6.1), (6.2) results in the electromagnetic and axial currents 3) being conserved by Noether's theorem, independently of SSB or a non-zero fermion mass, upon use of the classical equations of motion following from S cl , i.e. before any consideration of the axial anomaly. Defining the polar representation for Φ, the bosonic part of the effective action (6.2) becomes 6) and the axial current has both the fermionic and bosonic contributions where is the bosonic part of the axial current.
Note that the bosonic terms in both (6.6) and (6.7) depend on the chiral phase ζ of (6.5) only through the combination ∂ µ ζ + b µ , while the U (1) ⊗ U (1) ch invariant quartic potential (5.9) depends only upon σ = |Φ|. In the presence of a non-zero b µ , the chiral chemical potential is identified as the invariant combination [80] in the fluid description, analogous also to (3.5), and thus the effective potential to be minimized with respect to σ when µ 5 = 0 is showing that a non-zero chiral chemical potential enters V eff with a negative sign, always tending to destabilize the U (1) ch symmetric state at σ = 0. Indeed the condition with λ > 0 admits the two possible solutions In the SSB case (b), (6.8) together with (6.9) informs us that connecting the EFT fluid description (iii) in the equilibrium ground state with the scalar field description. By standard arguments [80] this SSB state defines a relativistic superfluid with the polar angle field ζ the Goldstone mode of U (1) ch SSB that dominates the long distance/low energy spectrum and can be identified with the axion. Equation (6.13) may be viewed as a definition of n 5 , but the essential scale of SSB is of courseσ.
In the U (1) ch symmetric case (a) the bosonic fields (φ 1 , φ 2 ) are a gapped doublet with equal mass squared κ − 4µ 2 5 > 0. Thus they play no role at large distances and can be dropped entirely in the low energy EFT, while with gσ = 0 the fermions remain massless. In the limit of non-selfinteracting and massless fermions the axial anomaly must be taken into account by adding the anomaly effective action (3.1) or (3.2) to the classical action of (6.1). In other words in (a), taking the quantum fermion loop and the axial anomaly into account means making the replacement of the classical (2.1) coupled to electromagnetism by the 1PI effective action in which theψγ µ γ 5 ψ term of (2.1) is included in (3.2) and no fermion self-interactions appear explicitly. Since the anomaly leads to the fermion pairing as in Fig. 2, single fermonic excitations do not appear in the spectrum at zero temperature and the fermionic term in (6.14) can also be dropped, so that we obtain the low energy effective action in the ASB case in the T = 0 vacuum, with S anom given by (3.2). This is the low energy effective action that applies for a WSM for Weyl nodes displaced in energy, when supplemented by the non-vacuum −ε(n 5 ), as in Sec. 3, or for a WSM placed in a strong magnetic field, with excitations aligned with the B field in the dimensional reduction of Sec. 4. In both cases η describes an axionic mode. The axion remains gapless if EM interactions can be treated as higher order, as in Sec. 3, or becomes gapped if they are of the same order, as in the effectively 2D case of Sec. 4.
We shall next demonstrate that the low energy effective action in the SSB case (b) leads to essentially the same low energy effective action as (6.15), with the polar angle ζ of the scalar Φ field replacing the chiral angle and Clebsch potential η of S anom in the ASB case (a).

Perturbations from Equilibrium: Superfluidity and the Goldstone Sound Mode
In terms of the parameters of the potential V whereσ is given by (6.12b). The absence of any axial current in equilibrium requires J 5 = 0 so that ∇ζ = −b , (7.2) whileζ = 0 andJ 0 5 =n 5 ,μ 2 5 = (b 0 ) 2 in applying this EFT to WSMs in the case of Weyl nodes displaced in energy and momentum as in Fig. 3. From (6.13) and the minimization condition (6.11) at σ =σ, we also have in the equilibrium ground state of U (1) ch SSB.
These equilibrium relations may also be used for small deviations away from equilibrium in the long wavelength limit of the EFT in the effective boson description. Expressing the polar field variables as their equilibrium values plus small time and space dependent perturbations, i.e.
σ =σ + δσ , with b 0 = µ 5 , and expanding the action (6.6) to the second order in perturbations around the SSB ground state, we find (cf. [80]) where The last term of (7.5) shows that there is a mixing between the gapless Goldstone mode δζ and gapped δσ mode, which affects the low energy Goldstone mode. Substituting the complex Fourier decomposition δζ ∼ e −iωt+ik·x we find the 2 × 2 Hermitian matrix operating on the two-component vector (δσ, 2σδζ) of perturbations. Setting the determinant of this matrix to zero gives a quadratic equation for ω 2 , which yields the spectrum, consisting of one solution for ω 2 that is gapped at the scale M 2 σ , and a second solution at which is a gapless acoustic Goldstone mode with speed of sound that differs from the speed of 'light' c = 1 of the Weyl node Fermi velocity. This sound speed is given instead by which agrees with that obtained from a hydrodynamic approach to a sound mode.
Thus although v s = 1 due to the spontaneous breaking also of Lorentz symmetry by the condensate rest frame where J 0 5 = n 5 but J 5 = 0, the Goldstone mode remains gapless. This is an axionic acoustic sound mode with speed (7.9). Note also that this agrees with the v 2 s = 1/3 and (7.1) agrees with the equation of state p = ε/3 of free massless fermions if and only if κ = 0, in which case the EFT is conformal. Any κ = 0 corresponds to non-vanishing self-interactions of the fermions in the NJL description which breaks conformal invariance.
The eigenmode of (7.7) propagating with the sound speed v s of (7.9) is the linear combination to lowest order in k 2 /M 2 σ for long wavelength acoustic excitations of the superfluid. Thus for this linear combination the second action to second order of the perturbations (7.5) becomes for the Goldstone sound mode at long wavelengths. In obtaining this last relation (3.11) and (7.9) has been used. Utilizing (6.8), (7.4) and (7.10), the linear perturbations in the axial current are Thus, the axial anomaly equation becomes if the anomaly source is also varied to linear order. Comparing (7.14) for the perturbations in the phase field δζ in the case of SSB in the effective potential V eff of (6.10) with the perturbations in the Clebsch potential δη of the anomaly effective action (3.12), we see that they coincide, and with the same sound speed v s when κ = 0, corresponding to the case of free fermions considered in Sec. 3. This shows the case of ASB and more familiar SSB in fact describe the same low energy hydrodynamic effective theory of a chiral superfluid in the sense of (iii) of the Introduction. This result, perhaps at first sight surprising, may be understood from the following considerations.
In the SSB case (b) the fermions acquire a mass gap m = gσ from the Yukawa interaction in (6.1) and no longer appear at distances greater than 1/m. Thus they can be integrated out entirely at low energies as in Appendix C. Since the EFT of (6.1) is renormalizable, the effect of integrating out the fermions is to renormalize the parameters of the remaining bosonic effective theory, and to add the finite anomaly action (C. 19) in the bosonic sector. That the axial anomaly survives intact even when the fermions become massive due to SSB is seen by the explicit computation of the effective action, and in particular the term (C. 19) of Appendix C, which coincides with the last term of (3.2) with η = ζ. Thus in this case (b) of SSB the effective bosonic action of the WSM becomes with L Φ given by (6.6). However in L Φ there remain both the angular phase mode ζ which is gapless and the radial mode δσ with mass gap M σ of (7.5)-(7.6). The latter decouples at energy scales less than M σ , so that the effective low energy action for small fluctuations from equilibrium is only (7.11), but this is exactly the same result following from the fluid effective action (3.6) in the ASB case for two Weyl nodes displaced in energy, with ζ taking the place of η. Thus in the SSB case (b) the effective action for low energy gapless excitations only is which is the same as (6.15) after the addition of the −ε(n 5 ) term and replacement of η by ζ. To lowest order in fluctuations from equilibrium the effective action in the SSB case becomes and its variation gives (7.14), with v 2 s given by (7.9). The reason for identification of the Clebsch potential η with the bosonic polar phase angle ζ of the SSB phase is also not difficult to understand. Inspection of the Yukawa interaction in the polar representationψ Φψ = σψ exp 2iζγ 5 ψ (7.18) shows that ζ of the scalar Φ can be shifted to the phase of the fermion fields by a U (1) ch transformation (2.3) by (5.4), with β = ζ. Then e 2iζγ 5 is the complex phase factor of the fermion condensate matrix ψ ψ . The fact that η appears in the EFT of ASB in (6.15) in exactly the same way as ζ does in the EFT of SSB in (7.16) implies that there must be a fermion condensate in the ASB with this chiral phase, induced by the axial anomaly alone, even it is not immediately apparent from the introduction of η in (3.2). As discussed further in Sec. 9, the cases of ASB and SSB, which appear at first sight to be quite different, are both associated with formation of a fermion chiral condensate which spontaneously breaks the U (1) ch symmetry, and leads to the same low energy collective axion excitation, although by a different route.

't Hooft Anomaly Matching and Non-Decoupling of the Axial Anomaly
The effective action of the axial anomaly (3.2) is based on massless fermions, m = 0. As soon as the condition (6.12b) for SSB is satisfied,σ = 0, and the fermion becomes massive, then for any finite fermion mass the fermionic contribution to the axial anomaly A is reduced according to (2.8). This m dependent reduction in A is the non-anomalous contribution to the divergence of the chiral current in position space, where P =ψγ 5 ψ. Using the definition of the canonically normalized Goldstone boson propagator and going over to momentum space, the bosonic contribution to the amplitude (A.1) represented by the diagram of Fig. 4 is where Λ αβ 5 (p, q) is given by (A.10) and m = gσ for the fermion mass has been used. The contraction of (8.3) with k µ therefore gives −2mΛ αβ 5 (p, q) which is equal and opposite in sign to the mass-dependent contribution of the fermion vertex J µ 5 [ψ] given by (A.10) [9]. Its contraction with k µ gives a contribution to the axial anomaly which is equal and opposite to the mass-dependent contribution of the fermion vertex J µ 5 [ψ] of (6.7) given by (A.10).
Since the bosonic contribution to the non-anomalous divergence of the axial current (6.3) cancels the fermionic mass contribution, the axial anomaly

The Goldstone Theorem for Anomalous Symmetry Breaking
The Noether current (6.4) of the classically U (1) EM ⊗ U (1) ch invariant effective action S cl = d 4 x L cl , given by (6.1) and (6.6), satisfies the identity so that the axial current is classically conserved on-shell by use of the equations of motion following from variation of the action S cl . In the quantum theory this relation becomes a WT identity of ψ andψ, and the corresponding terms in the anomalous axial WT identity may be dropped.
In the familiar situation of SSB with no anomaly, the WT identity (9.2) with A 4 = 0 is the basis upon which Goldstone's theorem is established. In that case (9.2) is first integrated over the spacetime volume d 4 x, selecting the Fourier component at zero four-momentum. Since the first term on the left-hand side is a total divergence and the 1PI effective action Γ of (1.2) by definition does not contain one-particle singularities at p µ = 0, this total divergence term gives zero contribution. Then the result is functionally differentiated once more with respect to φ 2 (x ′ ) to obtain If this is evaluated at the extremum of Γ, at which the first variation at right vanishes, and if this extremum occurs for a non-zero constant φ 1 =σ, we obtain in momentum space, since it follows from the definition of Γ in (1.2) that the second variation of Γ at left is (minus) the inverse propagator G −1 22 of the φ 2 field. Since at zero four-momentum G −1 22 (0) is the φ 2 mass gap, (9.4) shows that the φ 2 scalar is a massless Goldstone boson in the SSB state whereσ = 0. Now we may ask how this standard result is affected by the presence of the anomaly A 4 in the anomalous WT identity (9.2). Note first that the anti-symmetric variation in (9.2) becomes in the polar representation (6.5). Thus repeating the step above of integrating over the spacetime volume d 4 x, setting to zero the integral of the total divergence, but now functionally differentiating with respect to the polar angle ζ(x ′ ) gives Since the expression at left is proportional to the ζ field inverse propagator at p µ = 0, the anomalous term A 4 does not spoil Goldstone's theorem or give rise to a mass gap of the polar ζ mode, provided that the anomaly A 4 itself is independent of ζ, at least to first order at the extremum of the effective action Γ.
For the scalar EFT of Sec. 6, (9.6) in the polar representation is clearly equivalent to the standard form of (9.4) by the change to polar field variables (6.5). However, the results of Secs. 3 and 7, in particular comparison of (3.12) and (7.14) shows that they are identical also upon substitution of η for ζ. One can check that this general theorem holds in the EFT of Secs. 3 and 7, with ζ is replaced by η in the former case. It fails in the 2D dimensional reduction of Sec. 4 because the 2D anomaly itself depends on the chiral rotation angle η = ζ at linear order, by (4.5) and (4.12), so that the right side of (9.6) is non-vanishing after dimensional reduction.
This dependence generates an effective mass gap for the would-be Goldstone mode for dynamical electric fields by the Anderson-Higgs mechanism.
Note that (9.4) and (9.6) hold and the ζ angular mode is gapless even if Lorentz invariance is broken and the p 2 terms in G −1 22 (p 0 , p) or G −1 ζ (p 0 , p) of (9.6) have unequal coefficients and the velocity of propagation differs from c (or v F ). This is clearly relevant to the case of Weyl nodes displaced in energy considered in Secs. 3 and 7, where a preferred rest frame is defined by the non-zero µ 5 . In this case the speed of propagation of the Goldstone mode is not v F (the surrogate for the relativistic speed of light in a WSM), but the sound speed (3.15) or (7.9) instead, determined by the equilibrium equation of state and dp/dε. Although the CDW propagates at a different sound speed, it remains gapless, and the Goldstone theorem (9.6) holds in the case of ASB as well as the more familiar SSB case, resolving the apparent paradox of ASB giving rise to a Goldstone mode despite the anomaly explicitly violating the U (1) ch symmetry. It also supports by explicit examples in WSMs the more abstract argument that a gapless Goldstone mode should be expected also in the case of anomalous continuous symmetries, based on the construction of non-invertible defect boundary operators [81,82].
The alert reader will have noticed that (9.4) is a relation for a canonically normalized scalar field φ 2 and is multiplied by the expectation valueσ in the SSB case. In the U (1) ch symmetric phase (6.12a)σ = 0 and (9.4) becomes null: there is no Goldstone boson. On the other hand (9.6) is expressed entirely in terms of the polar phase angle field ζ, which is dimensionless, and not a canonically normalized field (except in d = 2). Referring to the explicit examples of Secs. 3 or 7, we see that the missing factor with the correct dimensions is supplied by dn 5 /dµ 5 in the case of Weyl nodes displaced in energy. Thus this quantity proportional toσ 2 must be non-vanishing for (9.6), or the corresponding relation in terms of η, and the U (1) ch symmetry must be spontaneously broken for (9.6) to be non-null and a Goldstone boson to exist.
Thatσ must be non-zero for a Goldstone boson to exist leads to the conclusion that a fermion condensate ψ ψ = 0 must be present also in the ASB case of Weyl nodes separated by energy, at non-zero µ 5 , even though it does not appear explicitly in (3.12) and (3.13). This is also indicated in the bosonic description by the fact that any non-zero chiral chemical potential with even free massless fermions in Sec. 3, corresponding to the critical value of κ = 0 in V (σ), destabilizes the symmetric ground state to the broken symmetry state. This is the chiral analog of recent studies of non-zero chemical potentials and charge densities being connected with SSB of U (1) symmetry and superfluid behavior as well [83].
Since no direct four-fermion self-interaction was postulated in the ASB case for a system of apparently 'free' fermions, one may ask what the mechanism of symmetry breaking is that can produce a superfluid state and Goldstone mode. Since the fermion pairing in the ASB case is due to the axial anomaly itself, one must recognize that the Weyl fermions are not actually 'free,' but still subject to the electromagnetic interaction that leads to the anomaly. We are thus led to the conclusion that it is the same electromagnetic interactions of the fermions responsible for the axial anomaly that are also responsible for the formation of non-zero ψ ψ = 0 condensate, spontaneously breaking the U (1) ch symmetry in addition.
In fact it has been known for some time that long range magnetic interactions which are not Debye screened in a plasma of finite charge density and chemical potential µ lead to logarithmic divergences of the fermion self-energy Σ ∼ ln(E−µ) for excitations E close to the Fermi surface [84,85], an effect which is enhanced in degenerate relativistic plasmas [86], and which leads to non-Fermi liquid behavior of the plasma [87]. The corresponding calculations have not been performed for a degenerate chiral plasma of massless fermions at non-zero µ 5 to our knowledge. It is natural to conjecture however that the same unscreened long range magnetic interactions provide an attractive channel for fermion pair interactions of Fig. 5, which generates an effective four-fermion interaction even if none were present initially. In that case one would also expect that evaluation of the fermion self-energy in Fig. 6 in the chiral plasma will lead to a non-trivial solution to the fermion gap equation with a chiral condensate ψ ψ = 0 that spontaneously breaks U (1) ch symmetry in the case of non-zero µ 5 as well. ψ ψ Fig. 6: The one-loop fermion self-energy Σ to be calculated for non-zero chiral potential µ 5 needed for the gap equation in the ASB case. The photon propagator is the same as in Fig. 5.
If that is indeed the case, the superfluid axion CDW mode in the ASB case would be accounted for as the bound state of the Cooper pairs produced by the anomaly itself as in Fig. 2, purely by their electromagnetic interactions for arbitrarily small coupling e, consistent with the existence of the axial anomaly due to those same interactions with electromagnetism. This same calculation at finite temperature T and µ 5 should also determine the critical temperature T c ∼ µ 5 at which the condensate first vanishes and with it the Goldstone mode and superfluid behavior. The fermion anomaly diagram of Fig. 1 itself should also be calculated at non-zero µ 5 to directly verify the existence of the gapless pole with the CDW sound speed v 2 s = 1 3 v 2 F of (3.15), and complete the picture in the ASB case.

Summary and Discussion
Exploiting the formal identity of gapless Weyl nodes in WSMs to massless Weyl fermions in relativistic QFT, we have proposed in this paper a UV renormalizable effective theory in Sec. 6 that encompasses both the case of SSB by four-fermion interactions of the NJL kind as in Sec. 5, and the case when these interactions between the Weyl nodes are weak and below the SSB threshold of G G c of κ 0. This provides a consistent QFT framework and basis for deriving controlled low energy EFT limits for WSMs in both cases. There is of course no claim that (6.1) represents the 'true' microscopic degrees of freedom of a WSM, which depend upon the electronic valence and conduction bands of the material. The point is rather that by expressing the EFT in terms of a small number of finite parameters that are insensitive to short distance physics, which can be determined by measurements in each WSM, the Landau paradigm of EFTs of types (i) and (iii) can be found for WSMs, and their consequences reliably deduced independently of the microscopic physics. Integrating out the Weyl fermions when they become gapped can also be studied, as in EFTs of type (ii) in Appendix C.
That the axial anomaly is a consequence purely of the symmetries of the ground state, independently of high energy or short distance physics has been emphasized in Appendix A, to elucidate why it plays an important role in the low energy EFT of a WSM. The massless 1/k 2 pole and finite sum rule associated with the anomaly are low energy features of the axial anomaly, which leads to a propagating collective axion mode that should be present in WSMs even for cases in which direct fermion-fermion interactions are weak or absent. In the framework proposed by starting with (6.1), the axial anomaly and the axionic collective mode to which it leads can be derived from first principles of QFT in WSMs, which directly gives a superfluid EFT of type (iii).
This axion excitation is a collective mode of correlated fermion pairs at the Fermi surface, and arises in a zero temperature WSM, whether or not there are strong four-fermion interactions of the Weyl modes. In the case of strong four-fermion interactions the axion is a Goldstone mode of the SSB of U (1) ch symmetry breaking with a non-zero scalar order parameterσ in its ground state.
Eq. (9.6) shows that despite the axial anomaly explicitly contributing an anomalous term to the U (1) ch WT identity in (9.2), it does not spoil Goldstone's theorem as long as A 4 itself is independent of U (1) ch rotations to first order, when evaluated at the ground state extremum of the effective action Γ. Thus the effective action of the axial anomaly itself (3.2) predicts the existence of a propagating gapless CDW which is linearly coupled to the E · B anomaly, which can be identified as an axion even in WSMs with weakly interacting Weyl modes, where such a collective boson mode might not have been expected. This extends Goldstone's theorem to the ASB case, supporting more formal arguments in [81,82] with an example of laboratory realizable system of anomalous symmetry breaking.
In both the SSB and ASB cases the axion is a collective Goldstone excitation of a low temperature superfluid phase of a WSM, as in both cases the low energy EFT is of the fluid form (iii) of a relativistic superfluid with the Fermi velocity v F of the Weyl nodes replacing the speed of light c of relativistic QFT. In both cases the low-energy EFT of axions η = ζ in WSMs takes the form is the axial vector potential, whose equilibrium value is one-half the separation of Weyl nodes in energy and spatial momentum respectively, cf. This dimensionally reduced effective action is equivalent to massless QED 2 , i.e. the Schwinger model [68], in which case the axionic excitation acquires the squared mass gap M 2 = 2αeB/π, in analogy with the Anderson-Higgs mechanism in a superconductor.
The η variable was introduced in (3.2) as a Clebsch potential in an effective fluid description, with no apparent scalar order parameter of spontaneous U (1) ch symmetry breaking to which it is attached. Yet the fact that the axionic mode in this case obeys exactly the same equation of motion (3.12) as the phase mode ζ (7.14) of the bosonic order parameter is a clear indication that the two cases of ASB and SSB are in fact very closely related. In particular a chiral condensate ψ ψ scalar order parameter that spontaneously breaks the U (1) ch symmetry (even with the anomaly present) should exist for which exp(2iη) is its complex phase. Verifying this will require finding a non-trivial solution of the gap equation resulting from the self-energy of Fig. 6 with ψ ψ = 0.
In that case the spontaneous breaking of the classical U (1) ch symmetry would be due to fermion pairing through the attractive channel of Fig. 5 and the axion mode of Fig. 2  This EFT approach to WSMs provides an interesting application of the 't Hooft anomaly matching condition across scales from the UV renormalizable (6.1)-(6.2) to the macroscopic superfluid effective theory of emergent axion excitations of (6.15) or (7.16). An intriguing possibility is that additional insights into the operation of the axial anomaly, anomaly matching and possible emergent axionic mode in the Standard Model of particle physics could result from detailed study of the properties and excitations of WSMs, to which the EFT of this paper can be applied. Defining the psuedo-tensor υ αβ (p, q) by (2.7), which satisfies

Acknowledgments
it is easily shown that there are exactly six tensors meeting all of the symmetry requirements of τ µαβ i (p, q), which are listed in Table 1 [36]. Thus, general conservation and invariance principles prescribe that the triangle amplitude (A.1) must be expressible as where the form factors f i = f i (k 2 ; p 2 , q 2 ) are Lorentz scalar functions of the three independent Lorentz invariants p 2 , q 2 and k 2 = (p + q) 2 . Since each of the six tensors satisfying (A.2) in Table   1 are homogeneous of degree 3 in the external momenta, 3 powers of momenta have been extracted, the remaining scalar amplitude functions f i are of degree −2 and therefore may be expressed as Feynman integrals that are quadratically convergent, i.e. independent of any high energy or short distance cutoff. The full amplitude (A.4) must also be Bose symmetric, under simultaneous interchange of p, q and α, β.
Owing to the Bose symmetry and overcompleteness of the six tensors in Table 1, finally only two of the scalar coefficient functions, f 1 and f 2 say, need to be independently computed. The finite scalar coefficient functions are given in the literature [89,8], and most conveniently expressed and the denominator D of (A.6) is which defines the quantity S(x, y; p 2 , q 2 ; m 2 ) for arbitrary finite fermion mass m. Both D and S are strictly positive for spacelike momenta, k 2 , p 2 , q 2 > 0.
Computing the contraction with the momentum k µ = (p + q) µ entering at the axial vector vertex results in the well-defined finite result (2.6) with by (A.6), (A.8) and Table 1 in the same Feynman parameter representation, with P =ψγ 5 ψ the pseudoscalar density [56]. If the fermion mass m = 0, this last term of (A.9) proportional to m 2 vanishes, and (2.9) is the finite and non-zero axial current anomaly. On the other hand, in the limit m 2 → ∞, the denominator D of (A.8) can be replaced by m 2 , and the entire A of (A.9) is cancelled since 2 1 0 dx 1−x 0 dy = 1, hence A = 0, so that the fermion decouples entirely and there is no anomaly pole in this limit.
This derivation of the chiral anomaly and anomaly pole relies only upon the low energy symmetries of the theory. To emphasize its independence of extreme high energy or short distance physics, and relation to fermion pairing, one may also derive (2.6), (2.8) by a dispersion approach. For this one observes that the absorptive part of the triangle amplitude (A.4) is determined by the on-shell matrix elements n 0|J µ 5 |n n|J α J β |0 for timelike k µ , with |n a complete set of two-particle intermediate fermion/anti-fermion states. This on-shell absorptive part is well-defined and finite, and in no need of regularization. Then the full amplitude is obtained from the Kramers-Kronig independently of m (and also of p 2 , q 2 0), which can be verified directly from (A.14).
Thus the axial anomaly in the real or dispersive part of the triangle amplitude at m = 0, (2.9) is associated with the UV finite spectral sum rule (A.16) which holds for all p 2 , q 2 , m 2 > 0, and also in the limiting case where p 2 = q 2 = 0 and m 2 → 0 + . This is possible despite the fact that ρ 1 (s; p 2 , q 2 ; m 2 ) vanishes pointwise as p 2 , q 2 , m 2 → 0 + (in any order) for any non-zero s, because becomes a singular Dirac δ-function in this limit, as may also be verified by inspection of (A.14).
The imaginary, absorptive part of the axial current divergence A of (2.6) or (A.13) vanishes for m = 0, in accordance with expectations from the classical theory, since (k 2 + s)ρ 1 (s) s=−k 2 = 0.
However computing the real, dispersive part of k µ Γ µαβ 5 requires dividing (A.13) by k 2 + s (the relativistic analog of the energy denominator of first order perturbation theory) which cancels the first factor, leaving ρ 1 (s) itself. By direct calculation the spectral function ρ 1 (s) does not vanish, even in the m → 0 limit. The reason for this can be traced to the fact that even in the massless limit, a charged Dirac fermion can make a transition to a virtual state of the opposite helicity by the emission of a photon [57][58][59], that is to say, massless Dirac fermions do not become two decoupled Weyl fermions in the presence of electromagnetism. This results in the matrix elements n|J α J β |0 of the two electromagnetic current vertices to the fermion pair intermediate state in the imaginary part of the triangle diagram Fig. 1 remaining non-vanishing in the massless fermion limit, and contributing positively to ρ 1 (s; p 2 , q 2 ; m 2 = 0) as in (A.14). That its integral over s, as in proportional to a δ-function at q µ = 0. The vectors k µ = p µ + q µ = p µ of Fig. 1 are then equal to each other. If their spatial components are taken to lie also in thex direction, parallel to B, then the τ 2 tensor in Table 1 is the only one contributing toΓ µαγ 5 (p, q)Ã γ (q), which for γ = 2, 3 is non-zero only for µ = a, α = b ranging over the 0, 1 components. We then obtain for the constant, uniform B field, in terms of the anti-symmetric tensor ǫ ab ≡ ǫ ab23 of the remaining two a, b = 0, 1 spacetime indices. Next noting from (A.8) that D = p 2 x(1 − x) = k 2 x(1 − x) for m = 0 and q = 0, the function f 2 evaluates simply to explicitly showing the 1/k 2 pole in this case. Thus the full anomaly vertex Γ abγ 5 (p, q) is given entirely by its longitudinal projection in this case of constant, uniform B.
In fact this result for the (three-point) triangle amplitude is closely related to the (two-point) polarization tensor of massless fermions in 1+ 1 dimensional QED 2 . This 2D polarization tensor (B.4) satisfies the usual vector WT identity k a Π ab 2 (k) = 0 of electric current conservation. On the other hand the 2D chiral current is which is dual to the electric current j b . Thus the chiral current polarization satisfies the anomalous WT identity k a Π ab 2 (k) = − k a ǫ ab π (B.7) which is the axial anomaly in 1+1 dimensions, illustrated in Fig. 7. Here E = F 10 the electric field strength in thex direction parallel to B. dimensions becomes proportional to the two-point polarization tensor Π ab 2 the 4D axial anomaly becomes proportional to the 2D axial anomaly, i.e.
after rescaling the electric field E → eE parallel to B, in order to accord with standard (Heaviside-Lorentz) conventions in 4D electromagnetism. In the last replacement of (B.10) we have allowed for the fact that in 2D the chiral potential b a = ǫ a c A c is also dual to the electric potential, i.e. b 0 = A 1 , b 1 = A 0 , and ǫ ab ∂ a A b → ǫ ab A b + b a , so that A 2 is given by (B.8) when both the electric and axial potentials are non-vanishing.
A consequence of this dimensional reduction from 4D to 2D is that the gapless collective boson composed of fermion/anti-fermion pairs propagating along the B direction, described by the 1/k 2 anomaly pole in 4D, becomes in this case for k ⊥ = 0, precisely the 1/k 2 propagator pole of the effective boson of the 2D Schwinger model for vanishing 2D electric coupling constant [68,76,35].
The dimensional reduction (B.9) from 4D to 2D for semimetals in a constant, uniform magnetic field amounts to the LLL approximation for the 3 + 1 dimensional axial polarization tensor in terms of the 1 + 1 dimensional one (B.4), along the magnetic field direction [91][92][93][94][95]. Integrating over the transverse y direction, the exponential factor in (B.11) is set to unity, so that the Fourier transform in t, x gives (B.12) by (B.9). Thus the axial anomaly of the 4D triangle diagram coincides with the 2D chiral polarization of gapless fermions in a constant, uniform magnetic field in the LLL approximation. The 4D axial current along the magnetic field direction J a 5 = (eB/2π)j a 5 , a = 0, 1 in terms of the 2D chiral current j a 5 since eB/2π is just the electron number density per unit area in the LLL. The relation (B.12) may also be understood as an expression of the low energy or infrared nature of the axial anomaly. Since the higher Landau levels are effectively gapped by eB, only the LLL's gapless excitations contribute to the excitations parallel to the B direction at distance scales much larger than 1/ √ eB. If an external electric field is turned on adiabatically, only fermions in the LLL can be excited, and the d = 4 axial anomaly factorizes into its d = 2 counterpart with a transverse density proportional to the uniform magnetic field strength B [5,96]. The contribution of the axial anomaly to the effective action of massless fermions becomes exact in this limit. Thus the 4D triangle anomaly provides a relevant interaction in the low energy, long distance EFT of gapless fermionic modes in Weyl and Dirac semimetals placed in a constant, uniform B field.
There are no terms linear in φ 2 and b with no derivatives because the corresponding trace is zero, but with one derivative in addition we have the sum of diagrams The first three diagrams are equal, as are the last three. A straightforward calculation gives Further terms of relevance to our discussion arise at fifth order. More specifically, we are interested in terms with one power of φ 2 and four derivatives: Since [D µ , D ν ] = ieF µν , we find Collecting the contributions (C.14), (C. 16) and (C.19), we have Further terms involving derivatives of b µ should be included, e.g. an F 5 µν F 5µν term with F 5 µν = ∂ µ b ν − ∂ ν b µ (see [106]), but we omit them here.
Terms quadratic in A and terms quadratic in b are given respectively by k µ k ν (k 2 + m 2 ) 2 , (C. 22) and To preserve gauge invariance we need to demand L D 2 = 0, or J µν 1 = 1 2 η µν I 1 . With this choice we still get a mass for the axial vector. Adding this to (C.20) we get An infinitesimal chiral rotation of the original fermion with x-dependent parameter β = β(x), namely ψ → (1 + iβγ 5 )ψ ,ψ →ψ(1 + iβγ 5 ) , (C. 25) which results in the appearance of an anomaly term d 4 x e 2 8π 2 βF µν F µν in the effective action, leaves the theory (C.24) invariant if gφ 2 → gφ 2 − 2mβ and b µ → b µ + ∂ µ β. This guarantees that (C.17) is the only source of the anomaly in this picture.
With the polar representation (6.5), φ 2 1 + φ 2 2 = σ 2 and the field ζ is dimensionless. Using (6.5) in our action, given by (2.1) with zero bare mass along with (5.5), we get To compute the effective action associated with integrating out ψ we define a chirally rotated fermion ψ ′ = e iζγ 5 ψ. Then, 32) and the fermion ψ ′ can again be integrated out as above. It is important to note that the chiral rotation of the fermion results in a contribution to the effective action associated with the transformation of the path-integral measure, namely the last term in (C.32). This contribution has been discussed in [107] using covariant derivative expansion methods. The linear dependence on ζ in (C.32) follows from the Adler-Bardeen theorem [64], together with Bardeen's general results for the axial anomaly [108].