Chiral fermion anomaly as a memory eﬀect

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I. INTRODUCTION
Not all symmetries of a classical theory remain exact after quantization.When this occurs, i.e., when a symmetry of the action is broken by quantum effects, one speaks about anomalies [1].Anomalies were first discovered in the late 1960s, in the seminal works by Adler, Bell and Jackiw, as an attempt of solving the pion decay puzzle [2,3].They found that the chiral symmetry of the action of a massless Dirac field Ψ(x) that interacts with an electromagnetic background is broken in quantum field theory.Mathematically, this outstanding result is beautifully encoded in the non-conservation equation of the fermionic chiral current j a A (x) = Ψ(x)γ a γ 5 Ψ(x), which on a 3+1 dimensional Minkowski spacetime takes the form where F ab is the field strength of the background electromagnetic field, ⋆ F ab its Hodge dual, and q the charge of the fermion.This is the celebrated chiral or axial anomaly.Besides electromagnetic fields, gravitational backgrounds have also the ability of triggering a chiral anomaly, as it was soon after found in [4][5][6].Mathematically, this contribution generalizes the previous equation by adding a new term proportional to the pseudo-scalar curvature invariant R abcd ⋆ R abcd , where R abcd is the Riemann tensor and * R abcd its Hodge dual with respect to the first two indices: The following years experienced an outbreak of fascinating results involving anomalies, both regarding physics and mathematics.Examples include, besides the prediction of the neutral pion decay rate to two photons, applications to the matter-antimatter asymmetry of the universe, the U(1) and strong CP problems in QCD, implications for anomaly cancellation in the Standard Model (see [7] for a nice summary of all these applications), and connections with the index theorems in geometric analysis [8][9][10].The notion of chiral anomalies has also been extended to other fields, including integer spin fields [11][12][13][14].
In this article, we investigate yet another aspect of chiral anomalies, related to global properties of both the fermionic and the background fields.These global properties appear when discussing the Noether charge associated with the chiral current, namely Q A = Σ dΣ j 0 A , where the integral is computed on any constant-time Cauchy hypersurface Σ.For the sake of clarity, let us focus on electromagnetic backgrounds, although similar arguments apply also for gravitational backgrounds -except for some important differences that we unravel in this article.Classically, the chiral charge Q A of a Dirac field measures the difference in the amplitude of the two helical components of the field.Quantum mechanically, this quantity translates to the difference in the number of positive and negative helicity particles, together with possible contributions from "vacuum polarization".The charge Q A is strictly conserved in the classical theory, but it is not quantum mechanically due to non-conservation of the current (2).This can be easily shown by considering any two Cauchy hypersurfaces, Σ in and Σ out , and noticing that the change in the vacuum expectation value of Q A between Σ in and Σ out is equal to the integral of ∇ a j a A in the four dimensional region R bounded by Σ in and Σ out : where, in the second equality, the second term vanishes due to Stoke's theorem and standard fall-off conditions of the fields at spatial infinity.Hence, a non-zero value of the integral R d 4 x ∇ a j a A implies the non conservation of the chiral charge of the quantum fermionic field.
Our goal is to understand what characteristics the electromagnetic backgrounds must have to produce a non-zero value of this integral.It is easy to check that the pseudo-scalar F ab ⋆ F ab appearing in (1) can be written as the divergence of a vector, j a CS = 2A b ⋆ F ab , where A a is the electromagnetic potential -CS stands for Chern-Simons.Repeating the steps used to produce Eqn.(3), Eqns.( 1) and (3) automatically imply that the fermionic chiral charge Q A fails to be conserved if and only if the scalar Q CS = Σ dΣ j 0 CS associated with the vector j a CS changes between Σ in and Σ out : This simple observation provides an interesting strategy to classify electromagnetic backgrounds that are able to trigger an anomalous non-conservation of the chiral charge of fermionic quantum fields propagating thereon. 1 This strategy was initiated in [15,16] for both electromagnetic and gravitational backgrounds, where some aspects of the scalar Q CS were analyzed for asymptotically flat spacetimes, in which the hypersurfaces Σ in and Σ out can be chosen to be past (I − ) and future null infinity (I + ), respectively.This is a natural choice when studying massless quantum fields.It was shown that, at these limiting surfaces, the scalar Q CS receives a contribution from the net helicity of the radiative content of the electromagnetic and gravitational fields.This implies that, if there are sources in the bulk emitting helical or circularly polarized radiation -in gravity, this happens, for instance, in the coalescence of a large family of binary black hole mergers [15,17]-there is a net change of Q CS between I − and I + , which induces a change in the chiral charge Q A of fermionic quantum fields propagating thereon.This is a profound relation between the radiative content of the background field and the chiral charge of quantum fields.We emphasize that this is a quantum effect; classically, Q A is strictly conserved for massless fields, regardless of the properties of the electromagnetic and gravitational backgrounds.This article unravels another contribution to Q CS -which, consequently, also acts as a source of fermionic helicity Q A -originated in the existence of certain electromagnetic infrared, or "soft" charges.Infrared charges have received a good deal of attention in the recent past, due to their theoretical importance in the study of the S-matrix in quantum electrodynamics and quantum gravity, and due to their connection with soft theorems (see the reviews [18,19] and references therein).On the other hand, non-zero infrared charges indicate the generation of "memory effects" in physical systems.To give an example, test charged particles can experience a permanent change in their velocity (a "kick") after the passage of electromagnetic waves [20].In electrodynamics, this was the first example of memory reported in the literature.Other memory effects have been identified in recent years (see for instance [21,22] for an effect related with the helicity of radiation, and references therein).Therefore, the results of this article can be interpreted as another type of memory effect produced by infrared charges, now on test quantum fields rather than on classical test particles.Quite interestingly, we find that this new manifestation of electromagnetic memory effect does not occur for the gravitationally-induced chiral anomaly.
The rest of this article is organized as follows.Section II introduces a simple example of pedagogical value: a massless Dirac field in 1+1-dimensional flat spacetime coupled to an electromagnetic background.Section III contains a brief summary of the asymptotic properties of the electromagnetic field at past and future null infinity, including the notion of soft charges and memory effect in this framework; readers already familiar with the notation can skip this section.Section IV contains the main analysis of this article, where the contribution of soft electromagnetic charges for the Adler-Bell-Jackiw anomaly is derived.This section also includes a simple example of an electromagnetic configuration for which the relevant infrared charges are different from zero.The gravitational case is discussed in section V, and section VI closes the paper with a few conclusions and remarks.
Throughout this paper, we use geometric units in which G = c = 1, and we keep explicit in our equations to emphasize quantum effects.The metric signature is chosen to be (−, +, +, +); ∇ a represents the Levi-Civita connection; the Riemann tensor is defined by 2∇ [a ∇ b] v c =: R abc d v d for any covector v d .Unless otherwise stated, all tensors will be assumed to be smooth.

II. CHIRAL ANOMALY IN TWO DIMENSIONS
This section discusses a massless, charged Dirac field in a 1+1-dimensional flat spacetime, propagating on a homogeneous electric background with finite support in time.To make our arguments simpler, we assume that the spacetime manifold is R × S 1 , i.e., the spatial dimension has been compactified to the circle.The electric field is assumed to be strong enough to make the backreaction of the quantum field negligibly small.This setup has great pedagogical value to illustrate some of the main messages of this paper, particularly the relation between the chiral fermion anomaly and the memory effect.We also discuss the relation of these concepts with spontaneous quantum particle-pair creation.
In a 1+1-dimensional spacetime, F ab has only one independent component -the electric field-F ab = E ǫ ab , where ǫ ab is the totally anti-symmetric tensor.Among the two Maxwell equations, dF = 0 is a trivial identity in 1+1dimensions, since it involves anti-symmetrizing three indices which can only take two different values.The other set of Maxwell equations, d ⋆ F = ⋆ j, lead to ǫ ab ∂ b E = −j a .These equations imply that the electric field cannot vary out of the support of the sources j, neither in space nor in time.Hence, in 1+1-dimensions there are neither magnetic fields nor electric waves.
Let us consider a fixed, time dependent electric field that is uniform in space.This can be generated by a timedependent current of the form j a = (0, j(t)).We further assume that the electric field is different from zero only during a finite interval, E(t) = 0 for t in < t < t out .Despite the fact that dF = 0 is an identity and there is no magnetic field, it is still useful to introduce a vector potential, F = dA, in terms of which the electric field reads E = ∂ t A θ − ∂ θ A t .A gauge transformation changes A a → A a + ∂ a α, with α a continuous function in the spacetime manifold.We can always use this freedom to make A t = 0, so that, under this gauge choice, E = ∂ t A θ .
The right hand side can be written as q 2π ∇ a j a CS , where j a CS = 2 ǫ ab A b .Although this vector is not gauge invariant, its divergence, as well as the scalar Q CS (t) ≡ t dxj 0 CS (t, x), are both gauge invariant. 2Following the argument described in the introduction, equation (5) implies that the change of the chiral charge Q A (t) = L 0 dθ j 0 A (t, θ) from t in to t out can be written, for any quantum state, as where As mentioned above, this quantity is manifestly gauge invariant.Recall also that this scalar is purely electric, i.e., it does not know anything abut the Dirac field.From this, we have where in the last equality we have used that E = ∂ t A θ , that the electric field is homogeneous, and that L is the length of the spatial sections.Hence, Eqn.(6) tells us that the anomalous non-conservation of the chiral charge Q A is dictated by the value of the time integral of the electric field This result shows that the vacuum expectation value Q A "keeps memory" of the past history of the electric field.
In particular, the effect of switching on an electric field for some period of time t in < t < t out can leave a residual, permanent helicity contribution on the vacuum state of the quantum field (quantified by the value of Q A at late times), even after switching off completely the external field.One can think about this residual helicity as the way the quantum field retains information about the past influence of the electric background.The integral on the RHS above also features in the memory effect found for classical particles [20].A test charged particle in our background would suffer a permanent change in its velocity after the passage of this electromagnetic pulse if and only if tout tin dtE(t) = 0. From this point of view, the permanent change in the vacuum expectation value Q A found above can be thought of as another manifestation of the electromagnetic memory effect, but now on quantum fields.
It may be surprising that, despite the fact that the external electric field vanishes, the quantum system does not return to its original configuration.To better understand this effect, one has to resort to the electromagnetic potential, which makes manifest that the memory actually originates from transitions between inequivalent vacua of the electric field (hence, the background does not really return to the same exact configuration either).To see this, recall that our electric background evolves from a vacuum configuration, E(t in ) = 0, to another vacuum configuration E(t out ) = 0. Classically, the two electric vacuum states are equivalent, but quantum mechanically they may not be. 3.In particular, note that the change in potential from t in to t out , A a (t out ) − A a (t in ), is non-trivial.This can be seen from the fact that the loop integral S 1 dℓ a (A a (t out ) − A a (t in )), which is gauge invariant, is different from zero if and only if tout tin dt E(t) = 0.But since the electric field vanishes at early and late times, A a (t out ) can only differ from the initial potential A a (t in ) by a residual gauge transformation, left by the dynamical evolution of the electric field.A straightforward calculation shows that this gauge transformation is given by A a (t out ) = A a (t in ) + ∂ a α with α(θ) = θ tout tin dtE(t) + α 0 , for some constant α 0 .However, this is not an ordinary gauge transformation because α(θ) is not a continuous function on S 1 (because α(L) = α(0)).Instead, it belongs to the family of so-called "large" gauge transformations [23], which carry physical implications, and which can be used to label inequivalent notions of vacuum states of the quantum electromagnetic theory. 4 In summary, the passage of an electric pulse with tout tin dtE(t) = 0 induces a large gauge transformation in the vector potential, which produces a memory effect, not only on classical charged particles, but also on the states of quantum fermion fields.As will be discussed below, in 3+1 dimensions there is another contribution to the chiral anomaly coming from the radiative content of the electromagnetic field; such contribution does not arise in 1+1 due to the absence of electromagnetic radiation.
The permanent change in the vacuum expectation value of the chiral charge Q A , described on the LHS of (8), can also be understood in terms of the standard notion of electromagnetic memory for particles.Heuristically, virtual charged particles populating the quantum vacuum would suffer a permanent change in its velocity after switching on this electric pulse, provided the integral tout tin dtE(t) does not vanish.In 1+1 dimensions these charges can only propagate in two directions, left or right.Therefore, positive charges suffer a "kick" in the direction of the electric field, while negative charges are kicked in the opposite direction.Both particles in the pair have the same helicity.If the kick is strong enough, it will turn virtual charges into physical excitations out of the quantum vacuum.This results in a net creation of helicity, which explains the permanent change of the quantum state or of the chiral charge This heuristic picture can be made rigorous through a calculation of particle-pair creation using Bogoliubov coefficients.We finish this section with a brief allusion to this.If the electric field is non-zero only during a finite interval, t in < t < t out , we can define natural "in" and "out" notions of vacua and particles.The question of interest is: if the field is prepared in the "in" vacuum before t in , and evolved until a time after t out , what is the number of "out" quanta in the final state?
This question can be answered without much difficulty in the case in which the electric field is uniform at all times (Appendix A contains a detailed derivation of this and of the general case of an electric field that varies both in space and time).As before, let us work in a gauge in which the vector potential is purely spatial, A t = 0. Without loss of generality, we can also consider A θ (t) = 0 for t < t in .Let A 0 = tout tin E(t) dt denote then the value of A θ at late times, after t out .In short, a non-zero value of A 0 induces a permanent frequency shift between the "in" and "out" basis of 3 The most prominent example of this is the Ahranov-Bohm effect [24]. 4 More precisely, when the potential Aa is viewed as a gauge connection on a U (1) principle bundle over R × S 1 , we can speak of infinitesimal gauge transformations, as well as of global or finite gauge transformations.In the temporal gauge fixing At = 0, a finite gauge transformation, Aa → Aa − ig −1 ∇ag, is determined by a continuous map g : S 1 → S 1 , given in a local coordinate system by g(θ) = e iα(θ) .Continuous maps on S 1 can be divided in different (homotopy) classes, where two elements of the same class can be deformed continuously into each other.The classification of continuous maps is determined by the first homotopy group, Π(S 1 ) ≃ Z, which shows that each class of gauge functions is labeled by an integer.This is easy to infer from the requirement that g is continuous, because it demands α(L) − α(0) = 2πn, for n ∈ Z. Two gauge functions g, g ′ belonging to different classes cannot be deformed continuously into each other.An ordinary gauge transformation is a gauge transformation g that belongs to the trivial class or n = 0, while gauge functions with n = 0 lead to "large" gauge transformations [23].
solutions of the field equations, which define the "in" and "out" vacua, respectively.Namely, for modes with spatial dependence e ikθ , with k ∈ (2π/L) n and n ∈ Z, the "in" modes oscillate with frequency ω in = k, while "out" modes oscillate with frequency ω out = k + q A 0 .Because k is discretized (due to the compactness of the spatial sections) there is a finite number of modes within the frequency interval (0, q |A 0 |).Namely, there are q |A 0 | L 2π modes within this interval, where the square brackets denote integer part.
This shift in frequency automatically implies that the evolution creates a number q |A 0 | L 2π of fermion-anti fermion pairs.Because of linear momentum conservation (note that the background is homogeneous), anti-fermions (positive charges) move in the direction of E(t), while fermions (negative charges) move in the opposite direction.However, helicity is not conserved in this process.Fermions moving to the right (left), and anti-fermions moving to the left (right), both have negative (positive) helicity.As a result, both members of the pair have positive (negative − ) helicity if A 0 > 0 (A 0 < 0).The total helicity carried by the excited pairs is 2 q A 0 L 2π = 2 q tout tin E(t) dt L 2π .This agrees, except for the non-integer part, with the prediction for Q A (t out ) − Q A (t in ) given in Eqn.(8).The difference is due to the "vacuum polarization", i.e. the helicity leftover in the "out" vacuum, which did not reach the threshold to excite another pair.
Although, for the sake of pedagogy, in this section we have restricted to uniform, background electric fields, all the arguments generalize to arbitrary functions E(t, θ).Appendix (A) contains information about this generalization and further details which have been omitted in this section.

III. ASYMPTOTIC STRUCTURE OF THE ELECTROMAGNETIC FIELD AND INFRARED CHARGES: A BRIEF REVIEW
The rest of this paper will focus on the chiral anomaly in asymptotically flat spacetimes in 3+1 dimensions.The presence of electromagnetic radiation, not present in 1+1 dimensions, makes it more convenient to use past and future null infinity for the initial and final Cauchy hypersurface of zero-rest mass fields.This section contains a brief summary of tools concerning the asymptotic structure of the electromagnetic field at null infinity and infrared charges.These tools are well-known [18,[25][26][27], and the reader familiar with them can skip this section.

A. Review on the asymptotic structure of the electromagnetic field
The electromagnetic radiation generated by charges and currents can be rigorously studied within the framework of asymptotically flat spacetimes [25][26][27].This framework makes use of the notions of conformally compactified spacetimes introduced by Penrose in the 1960s [28].
Let (R 4 , ηab ) represent the physical, Minkowski spacetime, and let (M , η ab ) denote an, extended (unphysical) spacetime obtained from (R 4 , ηab ) by an ordinary conformal compactification, i.e. by the addition of "points at infinity". 5More precisely, the new metric is obtained from the physical one by a conformal transformation η ab = Ω 2 (x)η ab , while the new manifold is constructed by attaching smoothly a null boundary J to the physical manifold, M = R 4 ∪ J .Locally, J corresponds to the hypersurface Ω = 0, and has null normal η ab ∇ b Ω = 0. From a physical viewpoint, the elements of J represent the "points of (null) infinity", i.e. the points that can be asymptotically reached by following radial, null geodesics in the physical spacetime.The boundary J is made of two portions, past (J − ) and future (J + ) null infinity.In the following, we will focus on J + .The construction is similar for J − .
For example, in a Bondi-Sachs coordinate system {u, r, θ, φ}, where u = t − r is the standard retarded time, the Minkowski metric reads dŝ 2 = −du 2 + 2dudr + r 2 dω 2 and one uses Ω = 1/r to obtain ds 2 = −Ω 2 du 2 + 2dudΩ + dω 2 after the conformal transformation mentioned above.The restriction of this line element to the Ω = 0 hypersurface gives a well-defined (although degenerate) metric.The limit r → ∞ keeping u, θ, φ constant follows the geodesics of outgoing radiation propagating to future null infinity, getting to Ω = 0 in finite time as measured by the unphysical metric.The extended manifold is obtained then by including all these limiting points {u, Ω = 0, θ, φ} to the original manifold, and future null infinity is described then by the submanifold R × S 2 .
This framework makes it possible to study the behavior of the electromagnetic field in a neighborhood of infinity (which in this case is simply a boundary of the spacetime manifold) using standard techniques in differential geometry.To see this, let us first note that the electromagnetic field tensors are conformal invariant, Fab = F ab , Âa = A a .These tensors are well-defined in the entire extended spacetime, including at the boundary J + .The electromagnetic field F ab has six independent components.In a Newman-Penrose basis {n a , ℓ a , m a , ma } [29], where typically one takes ℓ a = −∇ a u as the vector tangent to outgoing null geodesics, the 6 electric and magnetic components of F ab can be captured in the following 3 complex scalars If we assume smooth fields, the Peeling theorem guarantees that these scalars admit the following Taylor expansion in Ω in a neighborhood of future null infinity [30]: where we denote Φ 0 2 (u, θ, φ) ≡ Φ 2 (u, Ω = 0, θ, φ) and similarly for Φ 0 1 (u, θ, φ) and Φ 0 0 (u, θ, φ).These fields encode all the information about the electromagnetic field at J + .They are, however, not independent.Using Maxwell's equations, one finds where ∂ u f = n a ∇ a f for any function f (this is a consequence of the Newman-Penrose normalization n a ℓ a = −1), and where ð is a spin-weighted derivative operator [30], defined by . .T bc... ), for arbitrary tensors V a i and T bc... .These equations determine the evolution of the scalars Φ 0 0 and Φ 0 1 along the retarded time u in J + , upon giving initial conditions, and also some input for Φ 0 2 .In contrast, the dynamics of Φ 0 2 along u is not determined by Maxwell equations.This scalar serves as the free data for a characteristic value formulation of Maxwell theory at J + .
By switching back to the original physical spacetime, with the appropriate conformal rescaling of the Newman-Penrose vectors, one can see that Φ2 ∼ O(r −1 ), and Φ1 ∼ O(r −2 ).From this, one identifies the scalar Φ 0 2 as describing the two radiative degrees of freedom of the electromagnetic field, while Φ 0 1 represents the Coulombic part of the field.In fact, the total energy flux radiated to J + is given by , where T ab is the energy-momentum tensor, and therefore is entirely determined from Φ 0 2 .Similarly, the electric charge of sources in the bulk can be determined from J + using Gauss' Law as Q = 1 2π dS 2 Re Φ 0 1 (u, θ, φ), and is completely determined from Φ 0 1 .Using Maxwell equations in J + , it is straightforward to check that ∂ u Q = 0, reflecting the conservation of the electric charge.Note also that the requirement of finite energy flux at J + , F < ∞, requires Φ 0 2 (u, θ, φ) → 0 as u → ±∞.
In terms of an electromagnetic potential, one introduces the scalars A 2 := A a ma , A 1 = A a n a , A 0 = A a ℓ a .The fall-off conditions of the potential for large r is not given beforehand from the theory and requires some input.Physical considerations require that these components admit an asymptotic series with leading behavior O(r −1 ) [31,32].Using F = dA, one can obtain the following formulas, valid at future null infinity J + : where dot denotes derivative with respect to retarded time u.Furthermore, by integrating Maxwell equations at J + , one can further obtain: where G(u 0 , θ, φ) arises as an integration factor.From (9) we see that the two electromagnetic radiative degrees of freedom are distributed between A 0 2 and A 0 1 .But this is 3 real-valued scalars, so there is, as expected, some gauge redundancy in the description.A useful gauge fixing is A 0 1 ≡ A a n a = 0.With this gauge choice, the real and imaginary parts of A 0 2 represent the two radiative degrees of freedom, electric and magnetic respectively.The Coulombic aspects of the field are all encoded in G(u 0 , θ, φ), in particular Q = 1 2π S 2 G(u 0 , θ, φ).

B. Electromagnetic soft charges and the memory effect
The phenomenon of memory effect is well-known in Maxwell's theory [20].The most prominent example is a charged point-like particle of initial velocity v 1 that suffers a "kick", and changes its direction of propagation to v 2 after the passage of an electromagnetic pulse.This is an example of electric-type memory.In the intermediate process, the charged particle emits radiation by Bremsstrahlung; the properties of the emitted radiation carry information about this effect.
At future null infinity, the phenomenon of electromagnetic memory is encoded in the following quantities: where α is a smooth real function on the sphere S 2 .The complex numbers q α are called soft charges, and they measure permanent changes in the multipolar structure of the Coulombic part of the electromagnetic field after some process.From (15), one infers that q α = 0 only if Φ 2 (u, θ, φ) = 0, i.e. only, if there is a flux of electromagnetic radiation reaching infinity.When this happens, one says that the electromagnetic field keeps memory on the radiation flux emitted to infinity in the past.The relation of the charges q α and the radiation reaching J + can be explicitly shown by using Maxwell equations to replace Φ 0 1 by Φ 0 2 in (20): Expanding in spin-weighted spherical harmonics, this can be further simplified as The problem is reduced to study the basis q ℓm = +∞ −∞ duΦ 0 2ℓm (u) of charges.Note that, for ℓ = 0, i.e. when α(θ, φ) = const, the soft charge is identically zero.This is consistent with the fact that the monopole of the electromagnetic field (the electric charge) is conserved and cannot be radiated away.In contrast, dipolar and higher order structure (ℓ ≥ 1) can be radiated away.That phenomena is encoded in q ℓm = 0 for ℓ > 0.
As argued in the previous subsection, finiteness of energy fluxes require that Φ 2ℓm (u) belongs to L 2 (R, C), which implies that it admits a Fourier transform on J + : This automatically implies that the charges q ℓm are simply the "zero mode" of Φ0 Therefore, only the zero-frequency modes of the emitted electromagnetic radiation leave a memory on the multipolar structure of the field.This is the reason why these charges are called "soft", as they are associated with "soft photons" [19,33].These charges are intrinsically associated to asymptotic symmetries of Maxwell theory.One way of looking into this is by considering the phase space Γ of the electromagnetic degrees of freedom at J + .This phase space is made of pairs of canonically conjugate fields 6 ma7 .Γ can be endowed with the structure of an infinite-dimensional Banach manifold.The usual symplectic structure for Maxwell theory can be written on future null infinity as Ω( (A (1)  a , E (1)a ), (A (2)  a , Together with suitable fall-off conditions at u → ±∞ required to make this integral well-defined, the pair (Γ, Ω) defines the phase space for the radiative degrees of freedom of Maxwell theory.Now, consider the (restricted) family of gauge transformations A a → A a + D a α, with α = α(θ, φ).This transformation is generated in phase space by the quantity where in the second equality we have used Maxwell equations (15) to write D a E a ∝ Re Φ0 1 − Re ðΦ 0 2 = 0. Recalling that ∂ u = n a D a , the RHS of this equation happens to be equal to the real part of the soft charges defined in (20).(A similar analysis using a "dual" potential Z a produces the imaginary part of the soft charges).
This observation tells us that soft charges q α can be identified with the generators of gauge transformations in the radiative phase space.Since soft charges can be different from zero, one concludes that transformations A a → A a +D a α in J + are actual symmetries of our phase space (Γ, Ω), rather than mere gauge transformations.Therefore, they have physical significance (which is, precisely, the electromagnetic memory).From the viewpoint of the bulk, these are gauge transformations that do not vanish at infinity.To distinguish them from ordinary gauge transformations, they are called "large" gauge transformations.The set of all large gauge transformations constitutes the infinitedimensional, asymptotic symmetry group of Maxwell theory.

IV. THE CHIRAL ANOMALY INDUCED BY AN ELECTROMAGNETIC BACKGROUND
This section contains the main results of this article.We consider a quantum, massless Dirac field propagating in Minkowski spacetime in 3+1 dimensions with metric η ab , coupled to an electromagnetic field F ab .The spin 1/2 field is treated as a test field, i.e. we neglect its back-reaction on the electromagnetic and spacetime backgrounds.This external electromagnetic field is assumed to be generated by some distribution of electric charges and currents, that are smooth and confined in space, but otherwise arbitrary.To keep the parallelism with the 1+1-dimensional chiral anomaly discussed in the previous section, the sources will be "switched on" only for a finite amount of time, in the sense that they become stationary at sufficiently late and early times.All possible electromagnetic waves are radiated during a finite period of time.
As discussed above, the electromagnetic field can induce a change in the helicity of the fermionic field due to the chiral anomaly.Our starting point is expression (3) for the change of the chiral charge of the quantum field If we integrate over the entire spacetime manifold, R = M ≃ R 4 , then Σ in and Σ out correspond to past and future null infinity, respectively.This choice makes it possible to use the machinery summarized in the previous section to disentangle the properties of the electromagnetic field that can make the RHS different from zero.This problem was worked out in [16], where it was found that, assuming no incoming electromagnetic radiation from past null infinity, the RHS of ( 26) can be written in terms of boundary data on future null infinity as Here, α 0 is a smooth real-valued function on the sphere and ðα 0 is a pure gauge potential (i.e. it produces no electromagnetic field, Φ 0 2 = 0).This expression was derived in the gauge A 0 1 = 0. Notice that a non-zero value is obtained in the integral (27) because of the weak decay behavior of the radiative solutions of Maxwell equations in a neighboorhood of future null infinity: A 2 ∼ 1/r, Φ 2 ∼ 1/r (recall the discussion of page 6).These two radial factors compensate the r 2 factor in the integral measure. 8.
To analyze the physical interpretation of the RHS of the previous equation it is convenient to work with a compactified retarded coordinate u.We will consider u ∈ [−L/2, L/2] and let L → ∞ at the end of the calculation.As explained above, if Φ 0 2 is the electromagnetic radiation field, the requirement that the energy flux across J + is finite implies Φ 0 2 (•, θ, φ) ∈ L 2 (R) for all (θ, φ) ∈ S 2 , and in particular Φ 0 2 → 0 as u → ±∞.Therefore, we will consider functions Φ 0 2 (•, θ, φ) ∈ L 2 ((−L/2, L/2)), for all (θ, φ) ∈ S 2 , with boundary conditions given by Φ 0 2 (± L 2 , θ, φ) = 0.This will guarantee that Φ 0 2 → 0 as u → ±∞ at the end of the calculation.Since the functions Φ 0 2 (•, θ, φ) happen to be periodic with period L, an orthonormal basis for L 2 ((−L/2, L/2)) is given by { e −iωnL/2 √ L e −iωnu } n∈Z , where ω n = 2π L n, so one can expand in Fourier series: The inverse Fourier series is The basis modes are orthonormal with respect to the L 2 norm: The continuous limit will be recovered using the formula lim L→∞ In order to disentangle the potential contribution of IR charges to equation (27) we will make an explicit distinction between the zero frequency mode Φ0 2 (0, θ, φ) = 0 and the rest of modes.Let us then write the field and potential as where the second line is derived using Φ 0 2 = Ȧ0 2 , which is valid in the gauge A 0 1 = A a n a = 0. Next, we substitute this expansion in (27) and keep track of the contribution of the zero mode.The calculation is tedious, and is written in detail in Appendix B. We focus here in the result and its physical meaning: The first term in the RHS of this equation contains the contribution from electromagnetic radiation with non-zero frequencies reaching J + ; the subtraction of Φ0 2 (0, θ, φ) removes the zero mode from the integral.This, in turn, makes the integrand finite in the limit ω → 0 (the integral is well-defined for ω → ±∞ by Plancherel Theorem).This term can be further expressed as [16]: where Φ R (ω, θ, φ) := Φ0 2 (ω, θ, φ) − Φ0 2 (0, θ, φ) defined for ω > 0, and ΦL (ω, θ, φ) := Φ0 2 (−ω, θ, φ) − Φ0 2 (0, θ, φ) defined for ω < 0, describe right-and left-handed circularly polarized radiation, respectively.Expression (34) has a neat physical interpretation: it measures the net electromagnetic helicity radiated to J + .
The second and third terms in (33) come entirely from the zero mode of the electromagnetic field, and correspond to two infrared charges of magnetic and electric type, Imq α and Req β , respectively: Re In these equations the real-valued functions α and β are defined from the longitudinal and transverse part of the electromagnetic potential at future timelike infinity, as follows.In the gauge we are using, in which A 0 1 = A a n a = 0, the 1-form A 0 a lives on the cotangent space of each cross section S 2 of J + .Therefore, it can be expressed as the sum of a gradient and a curl: , where D a is the covariant derivative on S 2 .This equation defines α(u, θ, φ) and β(u, θ, φ).Using ( 18)-( 19), they can be solved from Re Φ 0 1 (u, θ, φ) = ∆α(u, θ, φ) + G(θ, φ) and Im Φ 0 1 (u, θ, φ) = ∆β(u, θ, φ), where ∆ denotes the 2-dimensional Laplacian.Since the soft charges Im q α and Re q β are specified from functions α and β, which depend on the electromagnetic potential, these are field-dependent soft charges.Since α, β originate from a gradient and a curl, respectively, the function α can be thought of as the electric degree of freedom of the emitted waves, while β is the magnetic one.
In summary, the change of the chiral charge of a quantum, massless, Dirac field between past to future null infinity, resulting from its coupling to an electromagnetic background, yields This is the main result of this paper.It shows that the anomalous non-conservation of fermionic helicity receives two types of contributions from an external electromagnetic field.Namely, Q A can change in time if (1) a distribution of electric currents and charges in the bulk are able to radiate chiral electromagnetic waves, and (2) there is a change in the infrared sector of the external electromagnetic field, such that the two soft charges ( 35)-( 36) are different from zero.The presence of emphasizes that this is a quantum effect with no classical analog; it originates from the chiral anomaly.
We finish this section with a few remarks.Remark 1.We have assumed no incoming radiation from J − .If the electromagnetic field is not trivial at past null infinity, we just need to replace quantities at J + above with differences between J + and J − .
Remark 2. The contribution from soft charges bears some similarity with the rationale behind the theory of instantons [34,35], in which quantum-mechanical transitions between "topologically inequivalent" vacuum states of the Hilbert space underlying a non-abelian gauge theory induces an anomaly.In the quantum theory of the electromagnetic field, for each non-trivial IR sector one has a representation of the canonical commutation relations which is unitarily inequivalent to the usual Fock representation.So, just like with the interpretation of the instantons, we can say here that tunneling transitions between the different IR vacuum states of the electromagnetic field induces the fermionic chiral anomaly 9 .In contrast, in this approach there is no need to work with Euclidean field equations.In fact, by working with solutions of the Lorentzian Maxwell equations we also get a radiative contribution, in addition to the contribution from soft charges.This radiative contribution is not predicted in the Euclidean case, where everything is "instantaneous".

A. Examples
We discuss now examples of electromagnetic sources that are able to trigger the chiral anomaly obtained in (37).A physical configuration of electric charges and currents that can radiate circularly polarized electromagnetic waves was described in [16], namely, an electric-magnetic oscillating dipole.In this subsection we focus on examples that produce non-zero values of the infrared charges (36) and (35) Soft charges of electric-type are determined by the Coulombic contribution (i.e.∼ 1/r 2 ) of the radial component of the electric field: The canonical example where these soft charges are not zero is a charged particle with some initial velocity that interacts with an external source (a nucleus, another charged particle, etc.) and changes its velocity [20].In this process, the charged particle emits Bremstrahlung, which is known to possess zero-frequency photons.We shall review this example here for completeness.The electromagnetic field generated by a moving charged particle can be obtained in closed form from the Lienard-Wiechart potentials [36]: 9 Notice that this was precisely the origin of the chiral anomaly in 1+1 dimensions discussed in section II.
where r s (t) is the location of the charge, v s = ˙ r s its velocity, n s = r− rs(t) || r− rs(t)|| , and t r = t − || r − r s (t r )|| is the retarded time (which is a function of (t, r)).For a particle with constant velocity, ˙ v s = 0, and the electric field above can be rewritten as [36] Changing to Bondi-Sachs coordinates {u, r, θ, φ}, and taking the limit r → ∞ keeping {u, θ, φ} constant, one obtains the following expression for the radial component of the electric field where in the last equality we have chosen ẑ in the direction of v s .From this expression and ( 38) one readily obtains For a particle that always moves with the same constant velocity, Re q ℓm = 0 for all ℓ.This is easy to see if we choose the reference system comoving with the particle, so that v s = 0 and Re Φ 0 1 (u, θ, φ) = q 4π .However, if the particle interacts with some external potential and changes its velocity, then we can no longer choose an inertial reference system attached to the particle at all times.While at early times we may have Re Φ 0 1 (u → −∞, θ, φ) = q 4π , at late times we will have Re Φ The electric soft charges can be now computed: Re In particular, Re q 00 = 0 due to electric charge conservation, as expected.By taking different values of ℓ one can check that this expression is indeed different from zero.In summary, soft charges of electric type can be generated by Lorentz boosting an electric charge.Soft charges of magnetic-type are determined by the Coulombic contribution (i.e.∼ 1/r 2 ) of the radial component of the magnetic field: Because magnetic charges have not been observed, we do not have, in principle, a magnetic analog of a boosted charge.To the best of our knowledge, there are no examples of magnetic memory reported in the literature.We discuss here one such example which, although it could be challenging to materialize physically [37], it certainly contains pedagogical value.
To think in potential situations that exhibit magnetic memory it may be useful to rewrite the radial component of the magnetic field of a moving particle (40) as where a s = ˙ v s is the acceleration of the charge, and L r is the radial component of the particle's angular momentum L = m r s × v s .Equivalently, these terms are related to the magnetic dipole moment of the moving charge: Equation (46) shows, in particular, that for a charged particle moving with constant velocity, ImΦ 0 1 = 0, and all soft charges of magnetic type are zero.
Only the first two terms in (46) may lead to ImΦ 0 1 = 0, as the third term decays as O(r −3 ).Furthermore, a priori one would expect that physically reasonable situations demand a s (u → ±∞) → 0. So, for sources consisting of a moving charged particle, to get magnetic memory one needs a situation in which the particle acquires a permanent rate of change for the radial angular momentum at infinity, Lr = 0.

V. THE CHIRAL ANOMALY INDUCED BY A GRAVITATIONAL BACKGROUND
As remarked in the Introduction, if instead of an electromagnetic field we consider an external gravitational background, described by a curved spacetime (M , g ab ), Dirac fields (as well as the electromagnetic field itself [11][12][13][14]) experience a gravitationally-induced chiral anomaly (see Eqn. ( 2)).Similarly to what we did for electromagnetic backgrounds, we explore here global properties of this chiral anomaly by studying the change of the chiral charge Q A .
As usual in General Relativity, we restrict to globally hyperbolic spacetimes to ensure the well-posedness of the Cauchy problem.This allows us to foliate the manifold in the form M ≃ R × Σ.We will further assume that the spatial slices are Σ ≃ R 3 .Performing a similar analysis as in (3), the permanent change in the chiral charge predicted by the chiral anomaly is dictated now by the Chern-Pontryagin integral where {x a } is a global coordinate system for M ≃ R 4 .The RHS of this equation was investigated in [15,16].Although it may appear intractable from an analytical viewpoint, it is actually possible to rewrite it in a form that allows us to extract information of physical value without having to resort to numerical techniques.More precisely, assuming no incoming gravitational waves from past null infinity J − , it is possible to rewrite it as an integral over future null infinity only: In this expression, Ψ 4 (u, θ, φ) = − lim r→∞ rC abcd ma n b mc n d is a complex scalar constructed from the Weyl tensor C abcd , which carries the two radiative degrees of freedom of gravitational waves; it is the gravitational analog of the complex scalar Φ 0 2 (u, θ, φ) in electrodynamics (compare with equations ( 9) and ( 12)).On the other hand, N (u, θ, φ) = N ab (u, θ, φ)m a m b is the relevant component of the Bondi News tensor N ab [25], which measures the time evolution of the asymptotic shear of outgoing null geodesics at J + .It is a symmetric, transverse (N ab n b = 0) and traceless tensor on J + that, just like Ψ 4 , captures the two gravitational degrees of freedom at future null infinity.The two quantities are related by Ψ 4 = − 1 2 Ṅ , so N can be thought of as the gravitational analog of the electromagnetic potential A 0 2 (compare with equation ( 17) with the gauge choice A 0 1 = 0).The total amount of energy carried away by the gravitational waves across J + is proportional to Because of this, the Bondi News indicates unambiguously if a system is radiating gravitational waves.If N = 0 then the sources do not emit radiation, while N = 0 indicates the presence of radiation.Finiteness of this energy flux requires N (•, θ, φ) ∈ L 2 (R, C) for all (θ, φ) ∈ S 2 , and in particular N → 0 as u → ±∞.These properties carry over to Ψ 4 .
In view of the results found in Sec.IV, it is natural to ask if gravitational soft charges, or gravitational memory, may also contribute to the fermion chiral anomaly (58).The gravitational memory effect [38][39][40][41][42][43][44] consists in the permanent relative displacement that a set of free test masses may experience after the passage of a gravitational wave burst.The deformation of a congruence of free observers or curves is controlled by the shear.If σ(u, θ, φ) denotes the asymptotic shear of outgoing null geodesics at future null infinity, a flux of gravitational radiation will make σ(u, θ, φ) evolve with time u, while it remains constant otherwise.As commented above, this effect is captured precisely in the Bondi News, which is related to the shear via the equation N = 2 σ.Because N → 0 as u → ±∞, σ(u, θ, φ) reaches constant values at early and late times.However, σ(−∞, θ, φ) = σ(∞, θ, φ) in general, and there can remain a permanent distortion in the shear.The amount of gravitational memory encoded in free test masses is quantified then by the overall change in the asymptotic shear σ(u, θ, φ) of outgoing null geodesics between early and late times: where α is an arbitrary real-valued function on the sphere.These quantities are called gravitational infrared charges [27] (compare this definition with the electromagnetic analog (20)).Following the analogy with the electromagnetic case, it can also be proven that these charges can be identified with the Hamiltonian generating BMS supertranslations in the radiative phase space of General Relativity [45,46].From the point of view of the bulk, supertranslations are diffeomorphisms (the gauge transformations in General Relativity) that do not vanish at infinity, as a result of which they are called "large".Notice that the physical manifestations of the gravitational and electromagnetic memory effects are qualitatively different.An electromagnetic field does not generate a permanent, relative displacement of electrically charged particles; instead, it generates a permanent, relative velocity between the charges.
Using the relation between the shear and the Bondi news, we can formulate the gravitational infrared charges in terms of the radiative degrees of freedom, Expanding in a basis of spin-weighted spherical harmonics, this expression reduces to for real-valued coefficients α ℓm .In the second equality, we have defined the parameters q ℓm = +∞ −∞ duN ℓm (u).Now, because N (•, θ, φ) ∈ L 2 (R, C), each of its harmonic modes admits a Fourier transform on J + Therefore, just like in the electromagnetic case, we conclude that the infrared charges are determined by the zerofrequency mode of the gravitational radiation as described by the Bondi News N .Namely, Notice, however, that, in sharp contrast with the electromagnetic case (20), the infrared charges are determined by the zero modes of the "potential" N (u, θ, φ) and not by the zero modes of the "field" Ψ 4 (u, θ, φ).While this may look an irrelevant comment, it is an important point in our analysis.The calculation of the RHS of ( 58) is formally equal to the electromagnetic case (27) if we identify A 0 2 with N , and Φ 0 2 with Ψ 4 .In the previous section we found that the electromagnetic infrared charges contribute to the chiral anomaly through the zero-modes of the electromagnetic field Φ 0 2 (u, θ, φ).Similarly, in the gravitational case, (58) only receives contributions from the zero modes of Ψ 0 4 (u, θ, φ), while the zero mode of N (u, θ, φ) never appears.However, Ψ 0 4 (u, θ, φ) has no zero mode: where in the second equality we made use of Ψ 4 = − 1 2 Ṅ , and the last equality follows from N (±∞, θ, φ) = 0.This is in sharp contrast with electrodynamics, where Φ 0 2 (u, θ, φ) -the electromagnetic analog of Ψ 4 -does have a zero mode.As a consequence, only the radiative part of the gravitational field contributes to the chiral fermion anomaly in (58).There is no gravitational memory contributing to the change of the chiral charge Q A , and the total change from J − to J + is determined by the helicity carried away by gravitational waves generated in the bulk [15,16].

VI. CONCLUSIONS
Chiral fermion anomalies have been extensively studied in the literature for several decades and from multiple viewpoints.Despite that, this topic is sufficiently rich to allow for yet another intriguing insight.We have found one such new aspect by studying global aspects of the chiral anomaly, related to the failure of the chiral charge Q A of a massless Dirac field to be conserved.This charge is strictly conserved classically, as well as in quantum field theory for free Dirac fields.However, the presence of background fields, either electromagnetic or gravitational, may induce a local non-conservation of the chiral current j a A by quantum fluctuations, which can potentially produce a time evolution in the vacuum expectation value Q A .
The identification of external fields that can or cannot trigger a change of Q A is a non-trivial problem.For non-abelian gauge fields, a traditional approach is to look for instanton solutions in an euclidean spacetime, which display a complex topological/global structure.To address this question, we have evaluated instead the change in Q A between past and future null infinity using familiar, global techniques within the framework of asymptotically flat spacetimes.For an external electromagnetic field, our results are neatly summarized in equation (37).This equation tells us that Q A can change between past and future null infinity if (i) electromagnetic sources in the bulk emit circularly polarized electromagnetic waves (i.e.radiation with net helicity) and/or (ii) if electromagnetic sources in the bulk produce transitions between certain infrared sectors of Maxwell theory.The relevant transitions are determined by a concrete pair of infrared charges of electric and magnetic type, respectively, written in equation (35) and (36).To gain physical intuition, we have devised an academic example where the required soft charges are different from zero.
Physically, non-zero infrared charges are known to produce memory effects on physical systems.This is how the transitions between the infrared quantum vacua can leave observable imprints.To the best of our knowledge, the only electromagnetic memory effects known to date involve classical systems.Here, we have shown that quantum states of a field theory can also keep memory of the past influence of electromagnetic radiation, by storing a certain amount of helicity.
The connection of electromagnetic memory and the change of Q A has also been worked out in 1+1 dimensions, which is cleaner because there are no electromagnetic waves.This example also allowed us to interpret this new memory effect in terms of "kicks" of virtual charges and excitation of particle pairs out of the quantum vacuum.
Overall, the results in this paper, together with our previous analysis [15,16], open up an unforeseen connection between chiral anomalies, the radiative content of the electromagnetic field, infrared charges and the memory effect.
Although our approach is qualitatively different, the contribution from soft charges to the chiral anomaly bears some similarity with the rationale underlying instantons in Euclidean gauge-field theories.According to the usual interpretation [34,35], instantons mediate quantum-mechanical transitions between inequivalent vacuum states of the Hilbert space of the background (non-abelian) gauge field.These transitions, which are labeled by the instanton charge, are able to induce the chiral anomaly [47].On the other hand, the quantization of the electromagnetic field at future null infinity leads naturally to a Hilbert space that can be divided in different, disjoint infrared sectors [26,48], which represent inequivalent notions of quantum vacua.The infrared charges label transitions between the different infrared sectors, and therefore play the same role of the instanton charge.We have shown in this article that these transitions contribute to the chiral anomaly in a specific manner.
To finish, we have also checked that, quite interestingly, gravitational infrared charges do not contribute to the fermion chiral anomaly.
However, this operator is not well-defined in our Fock space, its expectation values produce divergent sums.This is because Q 5 (t) is quadratic in the quantum fields, and, consequently, the evaluation of expectation values requires renormalization.The quantity of interest is in|Q 5 (t)|in ren , whose time evolution tells us whether there exists an anomaly or not.To obtain this result one can apply renormalization directly.However, there is an alternative, indirect procedure which, as we shall see, provides useful insights on the physical interpretation of in|Q 5 (t)|in ren .Let us introduce the following fiducial ("normal-ordered") operator: : This operator, which is given in terms of particle number operators of the out vacuum state, is now well-defined on the Fock space.In particular, the expectation value in| : Q The first contribution on the RHS depends only on the Bogoliubov coefficients and does not require renormalization.Then, it can be understood in terms of particle pairs created with net helicity by the external, electric background.
The second term on the RHS requires renormalization, and it represents a vacuum polarization effect.To the quantum anomaly, both effects contribute. 10 As a side remark, it is interesting to note the similarity of this result with the Hawking effect for black holes.In the formation of a black hole by gravitational collapse, one can compute the expectation value of the particle number operator using Bogoliubov transformations between past and future null infinity (with in and out states, respectively), as Hawking originally did.This calculation is well-defined and doesn't require renormalization.In our problem, this would be analogous to the calculation of in| : Q 5 : (t out )|in , which is related to the particle number operators.On the other hand, one can also study the Hawking effect by computing the non-diagonal, flux component of the expectation value of the stress-energy tensor across future null infinity.This calculation, based only on the in state, does require renormalization.This is because, apart from the particle pair creation, there is yet another contribution coming from vacuum polarization effects.In our case, since Q 5 is a quadratic operator, the evaluation of in|Q 5 (t out )|in ren is analogous to the calculation of the Hawking effect using the stress-energy tensor and not via the particle number operator.
The evaluation of in| : Q 5 : (t out )|in in (A32) is straightforward from the expressions (A26)-(A29) above and the canonical commutation relations.It produces: Note that each sum is convergent because each summand decays in both indices as n → ∞.Using (A20)-(A21), (A24)-(A25) one can write: 10 Mathematically, : Q 5 : (and not Q 5 ) is the relevant operator that is related with the Index Theorem in geometric analysis [53], and, because of this, one may be tempted to identify it with the chiral anomaly.Historically, chiral anomalies were studied on compact manifolds without boundary, that arise naturally using Euclidean techniques, and in these cases the chiral fermion anomaly was found to match the predictions from the Atiyah-Singer Index Theorem.As a result, the statement that chiral anomalies are predicted by Index Theorems became a standard lore.However, this is not true in more general cases.In particular, for manifolds with boundary, extra contributions arise in the index theorem, like the APS eta index η AP S [53], and the agreement with the anomaly fails.Physically, these extra boundary terms are represented by the vacuum polarization effects out|Q 5 (tout)|out ren pointed out in this appendix. .Combining all three terms above, and for u 0 = −L/2, which allows to further simplify some terms, we end up with .The second term above can also be greatly simplified and has a nice physical interpretation.To see this, note first that, in the gauge A 0 1 = A a n a = 0, the 1-form A 0 a lives on the tangent space of S 2 , so it can be expressed as the sum of a gradient and a curl: A 0 a = D a α + ǫ b a D b β, where D a is the connection on S 2 .Thus, A 0 2 = ð(α + iβ), with α, β ∈ C ∞ (R × S 2 , R).There still exists a residual gauge freedom represented by A 0 a → A 0 a + D a Λ, with Λ = 0.Under this transformation β remains invariant, and although α doesn't remain invariant, the combination α − α 0 does.With this new terminology we can reexpress the second term on the RHS above as where in the last step we made use of Maxwell equations (15) and the definition of infrared charges (20).