New Magnetic Symmetries in $(d+2)$-Dimensional QED

Previous analyses of asymptotic symmetries in QED have shown that the subleading soft photon theorem implies a Ward identity corresponding to a charge generating divergent large gauge transformations on the asymptotic states at null infinity. In this work, we demonstrate that the subleading soft photon theorem is equivalent to a more general Ward identity. The charge corresponding to this Ward identity can be decomposed into an electric piece and a magnetic piece. The electric piece generates the Ward identity that was previously studied, but the magnetic piece is novel, and implies the existence of an additional asymptotic"magnetic"symmetry in QED.


Introduction
In recent years, an intricate relationship between soft theorems and asymptotic symmetries in asymptotically flat spacetimes has been discovered and extensively studied (for a detailed review of this subject, see [1]). It began with the discovery that the leading soft theorems in both four dimensional gauge and gravity theories are equivalent to Ward identities associated to charges generating asymptotic symmetries of the theory [2][3][4][5][6][7]. These results were later extended to all higher dimensions [8][9][10][11].
The relationship between asymptotic symmetries and soft theorems became more intriguing when it was observed that there is also a relationship between the subleading soft theorems and asymptotic charges generating divergent large gauge symmetries in all dimensions [12][13][14][15][16][17][18][19].
However, unlike the case involving the leading soft theorems, the subleading soft theorem is oftentimes a stronger condition than the associated Ward identity. While the subleading soft theorem implies the Ward identity, the reverse is not necessarily true.
Traditionally, one conjectures a 'matching condition' that relates a specific component of a field at past null infinity I − to that at future null infinity I + (see Section 3.1), 1 which can then be massaged into a Ward identity (in the semi-classical theory) for the S-matrix. For example, in gauge theories one imposes such a matching condition for the radial electric field E r , and the corresponding Ward identity is equivalent to the leading soft photon theorem.
Similarly, in gravitational theories the matching condition for the electric part of the Weyl tensor is equivalent to the leading soft graviton theorem. Crucially, each independent matching condition leads to an independent Ward identity or soft theorem. This does not imply a contradiction for the d leading soft photon theorems (one for each polarization), which are not all independent. Rather, they satisfy a trivial identity that results in a single independent leading soft theorem [10]; this is the soft theorem that is equivalent to the matching condition described above. A similar argument holds for the leading soft graviton theorem as well.
However, such an identity does not hold for the subleading soft photon theorem, thereby implying there are indeed d independent subleading soft theorems. It is therefore not possible to demonstrate its equivalence with a Ward identity arising from matching the radial electric field. Rather, the matching condition for the radial electric field leads to a particular linear combination of the subleading soft theorem [19], which we shall henceforth call the subleading electric Ward identity. The origin (from the perspective of asymptotic symmetries or matching conditions) of the remaining d − 1 independent soft theorems is, so far, unknown.
In this paper, we conjecture a matching condition for the d angular components of the magnetic field (a vector matching condition), and then show that it is equivalent to the d independent subleading soft photon theorems. Naturally, our ansatz that there are d matching conditions instead of just one implies that the associated Ward identities must correspond to new symmetries. As it turns out, the subleading electric Ward identity corresponds precisely to one of the d matching conditions. The remaining d−1 matching conditions then give rise to Ward identities corresponding to charges that generate magnetic large gauge transformations, and we shall call these Ward identities the subleading magnetic Ward identities. This suggests that even though there are no global magnetic charges in our theory, there exists finite large gauge symmetries that are generated by asymptotic magnetic charges. This paper is organized as follows. In Section 2, we will summarize all the notations and conventions used throughout the paper. We will also derive the asymptotic expansion of the gauge field near I ± ; because much of the technology used was introduced in [10], we refer the reader there for more details. In Section 3, we will conjecture the set of d matching conditions and derive the corresponding Ward identities. Next, in Section 4, we prove the equivalence between the subleading soft theorems and the Ward identities. Finally, we explain in Section 5 the interpretations of the charges that correspond to these new Ward identities.

Preliminaries
In this section, we introduce the notations employed in this paper (following the conventions of [10]) and review related previous work.

Spacetime Coordinates
We work in flat null coordinates x µ = (u, r, x a ), a = 1, . . . , d, where d ≥ 2. These are related to Cartesian coordinates by and the standard Minkowski metric in flat null coordinates takes the form I ± is located at r → ±∞ while keeping (u, x a ) fixed, and these surfaces have the topology S d × R. The point coordinatized by x a on I + is antipodal to the point with the same coordinate on I − . The boundaries of I + and I − are located at u = ±∞ and are denoted by I + ± and I − ± , respectively.
Momenta Coordinates We will focus on the scattering of massless particles, which satisfy We parameterize such momenta using flat null coordinates so that Massless gauge fields transform under the vector representation of the little group SO(d) and (2.4) One particle in-and out-states with momenta p are created from the vacuum by in (−) and out (+) creation and annihilation operators denoted by O (±) † α (p) and O (±) α (p), where α labels the polarization of the particle. They are canonically normalized, i.e.
Poincaré Algebra The Poincaré algebra is generated by P A and M AB and takes the form (2.7) We define so that the nonzero commutators in the Poincaré algebra are given by (2.9) In addition to the Poincaré transformations, we will also consider scale transformations X A → λX A , which appears as an effective symmetry in the infrared (near the asymptotic regions of spacetime). We denote the generator of scale transformations by S, which satisfies the commutation relations Gauge Theory A U (1) gauge theory is described in terms of a 2-form field strength F µν that satisfies Maxwell's equations, i.e.
where J µ is the matter current. The theory is invariant under the gauge transformations where ε ∼ ε + 2π and Q i ∈ Z is the U (1) charge of the matter field Ψ i . Gauge transformations that vanish at infinity map physically equivalent solutions to each other and are therefore merely redundancies of the theory. We will use this redundancy to impose the gauge condition when carrying out the asymptotic expansion of the radiative field. Note that (2.13) is consistent with the choice of polarization in (2.4).
In flat null coordinates, Maxwell's equations take the form where ∂ 2 ≡ ∂ a ∂ a . The gauge field can be split into two pieces

Radiative Field
In this subsection, we study the radiative gauge field near I ± by expanding it in powers of 1/r. Because we adopt the same strategy and techniques introduced in [10], we refer the reader there for additional details and explanations.
The radiative gauge field satisfies the sourceless Maxwell's equations and hence admits the mode expansion where q 0 = | q | and ε a is the polarization vector defined in (2.4). Switching to the flat null coordinate parametrization of momenta defined in (2.3), we obtain To perform the large |r| expansion, we assume that the creation and annihilation operators admit the Fourier expansion and that the Fourier coefficients in turn admit a soft expansion, i.e. they could be written as 3 Substituting (2.17) and (2.18) into (2.16), and performing the integral over ω, we obtain where K ν is the modified Bessel function of the second kind, k ≡ | k|, z ≡ √ iu √ ir , and ν n = d 2 − 1 + n. Because large |r| corresponds to small z, we can expand the Bessel function about z = 0. This asymptotic expansion for the Bessel function is qualitatively different depending on whether ν n is an integer (d even) or a half-integer (d odd), so we will consider these cases separately.
The full large |r| expansion of the radiative gauge field components in all dimensions is presented in Appendix A for completeness, though for our purposes we only need the large (see Section 3.1), where we have adopted the notation f (±,n) to denote the coefficient of |r| −n near r = ±∞ after expanding the field f in large powers of 1/|r|. Thus, we only need to focus on the terms in the expansion that are 3 In the soft expansion given, we are ignoring potential log ω terms.
O 1/r d and O(u 0 ); the O(u) terms are projected out by 1 − u∂ u , and O(1/u) terms vanish at I ± ∓ . In even dimensions, these terms are where we have simplified the expression by assuming These assumptions are required to cancel logarithmic divergences in the asymptotic expansion in order to render the charge finite, and are discussed in greater detail in [10,19]. In odd dimensions, the relevant terms are (2.22) It follows from (2.20) that in even dimensions, whereas it follows from (2.22) that in odd dimensions, (2.24)

Coulombic Field
Following the approach taken in [10], we know that the Coulombic gauge field A (C) µ has a large |r| expansion given by The conserved current that couples to the gauge field also admits a similar expansion: Substituting these expressions into Maxwell's equations (2.14), we derive various constraint equations order-by-order in large |r|. In particular, we have These equations in turn imply Acting on both sides with ∂ 2 u and using (2.27), (2.28), we find (2.30) 3 Ward Identity

Matching Condition
In [19], it was shown that the Ward identity corresponding to the insertion of ∂ a O (±,1) a is associated with the antipodal matching condition 4 However, we now want to derive a set of d independent Ward identities involving the insertion a , which will ultimately be equivalent to the d subleading soft photon theorems (one for each polarization of the soft photon). To motivate the appropriate matching condition, we begin by noting that Maxwell's equations imply 4 Coordinate x a on I + and I − correspond to antipodal points on the celestial sphere.
It follows that (3.1) is equivalent to matching ∂ a F (±,d) ra across spatial infinity, as the current vanishes on the boundaries of null infinity. Since this matching condition gives rise to a Ward identity corresponding to inserting ∂ a O (±,1) a , it is natural to conjecture that in order to obtain d independent Ward identities corresponding to inserting O (±,1) a , we require the matching so that the charge is classically conserved.

Soft and Hard Charges
In the semiclassical picture, (3.5) implies the following Ward identity for the charge Q Y : Analogous to our decomposition of the gauge field A µ into a radiative field A (R) µ and a Coulombic field A (C) µ , we may decompose the charge into a soft piece and a hard piece, i.e. where (3.8) 5 To verify that this is indeed the charge whose Ward identity implies the subleading soft photon theorem, it must generate appropriate divergent large gauge transformations on the in-and out-states. This is verified in the next subsection.
Thus, the Ward identity becomes The form of the soft charge depends on the spacetime dimension. Using (2.23), the soft charge in even dimensions is and using (2.24), the soft charge in odd dimensions is (3.11) The form of the hard charge, on the other hand, is independent of dimension, and using (2.30) is given by where we have assumed that there are no stable massive particles in the system so that the contribution to Q Y from I ± ± vanishes. We will demonstrate in Section 3.2.1 below that 6 In an S-matrix element, the hard charge acts on multi-particle states as a tensor product of one-particle states. Thus, we can rewrite (3.9) as (3.14)

Action of Hard Charges
In this sub-subsection, we will prove (3.13). For notational simplicity, we do not distinguish between in-and out-states and drop the superscripts (±) on all operators. We will also drop the subscripts on the annilation operators and simply denote them as O(ω, x). 6 Readers who are mainly interested in the final result should feel free to skip Section 3.2.1.
Begin by defining the light-ray operators (LROs) Note that Q(x) is the LRO appearing in the leading hard charge (see [10]), and J a (x) is the LRO appearing in the subleading hard charge: Recall that massless one-particle states are created out of the vacuum by creation and annihilation operators. Since the LROs annihilate the vacuum [22], we have Our objective now is to determine the following commutators We will follow the spirit of the procedure outlined in [23], where the authors determined To utilize this assumption, we note that O(ω, x) is itself a LRO. For instance, the scalar annihilation operator can be constructed out of a scalar field as where Φ (d/2) (u, x) = lim 4. Minimal Coupling: The commutator of an annihilation operator with a LRO is also an annihilation operator (or derivatives thereof ) of the same particle type.
This is required in order to fix the commutators uniquely. As was shown in [24], the subleading soft theorem receives corrections from higher-derivative operators and is therefore not universal. In this paper, we will only focus on the universal part of the subleading soft theorem in minimally coupled theories. 7 Having outlined our assumptions, we now proceed to prove (3. and O(ω, x), J a (x ′ ) .
The Lorentzian separation between the two operators is x − x ′ 2 . They are therefore spacelike separated as long as x = x ′ , which by Assumption 1 implies where · · · represents terms involving additional derivatives of the delta function. Integrating over the transverse directions x ′ , we obtain O(ω, x),Q = L(ω, x). (3.22) Comparing to (3.20), it follows that L(ω, x) = QO(ω, x).
Next, consider L a (ω, x). Utilizing Table 1, we observe that the twist of the left-hand-side of (3.21) is d 2 + d, and the twist of On the other hand, by Assumption 4, L a is locally constructed from O and must therefore have the general form 8 (3.23) All the operators on the right-hand-side have twist d 2 + n. Since n ≥ 0, there are no operators with twist d 2 − 1, 9 thereby implying that L a (ω, x) = 0. Likewise, terms with additional derivatives of the delta function in (3.21) are excluded by the same argument, and so By Assumption 1, the commutator takes the form where we have excluded terms involving higher derivatives on the delta function using the same twist argument as above. By Assumption 4, K (0) and K (1) have the general form p,q ab a 1 ···a n ω p (ω∂ ω ) q ∂ a 1 · · · ∂ a n O(ω, x).

(3.26)
From (3.25), we can deduce that the twists of K (0) and K (1) are d 2 + 1 and d 2 , respectively, and Table 1 further implies that the boost charges of K (0) and K (1) are 0 and −1, respectively. 8 Lorentz invariance forbids explicit factors of x a . 9 The assumption of minimal coupling implies that the Poincaré invariant constants c a 1 ···a n p,q are dimensionless and therefore have vanishing twists.
Matching the twists and boost charges on both sides of (3.26), we find that p = −1 for both K (0) and K (1) , while n = 1 for K (0) and n = 0 for K (1) . There are no constraints on q, so As we shall next show, R ab and B ab can be fixed by checking consistency with the Jacobi where X is a generator of the Poincaré algebra. In particular, choosing X to be P − , P a , where R ′ ab and B ′ ab are operators independent of ω∂ ω .
• P a : We note Using this, (3.29) and (3.32) imply (3.35) Using this, the Jacobi identity implies which is simply the statement that B ′ ab is an SO(d) covariant matrix.
• K a : We note

Connection to the Subleading Soft Theorem
We begin by recalling the subleading soft photon theorem in its standard momentum coordinates. Let A n (p 1 , · · · , p n ) be a scattering amplitude involving n particles with momenta p 1 , . . . , p n , and let A out n+1 ( p γ , ε a ; p 1 , · · · , p n ) be the same amplitude with an additional outgoing photon with momentum p γ and polarization ε a . In a minimally coupled theory, the soft limit (E γ → 0) of the amplitude has the form where the 1 E γ pole is related to the leading soft photon theorem [25,26], and is the subleading soft factor. Here, J i AB is the angular momentum operator, which is the sum of the orbital and spin angular momenta: To write the subleading soft factor in flat null coordinates, we parametrize p A i and p A γ via (2.3) as (ω i , x a i ) and (ω, x a ), respectively. Using (2.4), we have where By the LSZ reduction formula, the left-hand-side of (4.1) corresponds to the insertion of the a (x) in the S-matrix. 10 One can then rewrite the subleading soft photon theorem as

Soft Theorem =⇒ Ward Identity
In this subsection, we want to use the subleading soft photon theorem (4.6) to derive the Ward identity (3.14). For the even dimensional case, we act on both sides of (4.6) with the Applying (3.10) and using the fact that in even dimensions which is precisely (3.14).
For the odd dimensional case, we act on both sides of (4.6) with the operator (4.10) Applying (3.11) and using the fact that in odd dimensions we again obtain (3.14).

Ward Identity =⇒ Soft Theorem
Now, we want to prove the converse, that the Ward identity (3.14) implies the subleading soft theorem (4.6), thereby proving that the two are completely equivalent. First, recall (4.5) and choose Y a (z) = Y a (z) ≡ I a b (x − z)ζ b in (3.14) for a constant ζ, so that where the derivatives are with respect to z. Substituting this choice of Y a into (3.10) and (3.11), and using (4.8) and (4.11), we determine in both even and odd dimensions that the soft charge takes the form 11 Substituting this into (3.14) with Y a (z) = Y a (z) and simplifying yields (4.14) Choosing ζ a = δ a b yields which is precisely the subleading soft photon theorem given in (4.6).

Electric and Magnetic Large Gauge Transformations
In previous sections, we have established the equivalence between the subleading soft theorem and a Ward identity for the charge In this section, we will show that this charge generates divergent electric and magnetic large gauge transformations. We begin by using Hodge decomposition to decompose the vector field We can then write the charge as 12 In proving this, we have integrated by parts in (3.11) and neglected potential boundary terms. To justify this, we can take Y a = Y a in a region encompassing all the points x i in the amplitude and zero outside, in which case all boundary terms are trivially vanishing. 12 We assume λ and K ab fall off sufficiently quickly in |x| so we can integrate by parts and neglect boundary terms.
Using (3.2) and the fact that the current vanishes at I ± ± , we can simplify the first term as ur .

(5.4)
This is precisely the charge studied in [19], where it was shown to generate divergent electric large gauge transformations.
To simplify the second term in (5.3), we use the Bianchi identity It then follows that the second term in (5.3) reduces to which, as we shall now show, generates divergent magnetic large gauge transformations.
In form notation, the charge that generates magnetic gauge transformations on a hyper- where K is a (d − 2)-form. This acts on the dual gauge field viã A →Ã + dK, (5.8) where dÃ = * dA. To study this charge with Σ = I ± , we note where * d is the Hodge dual on R d . Substituting this into (5.7) with Σ = I ± yields To match the charges, we take Recalling that the field strength near I ± ∓ admits the expansion |r| d−1 + · · · , (5.12) 13 In form notation, the charge that generates electric gauge transformations is Q the charge simplifies tõ This charge is formally divergent, and it is in this sense that the symmetry generated by this charge is a divergent magnetic gauge symmetry. However, when this divergent charge is inserted into an S-matrix element, the divergent contribution (the terms that are O(|r|), which are shown explicitly above, as well as terms that are O(u), which are implicitly present in F (±,d−1) ab ) vanishes due to the constraint equation thereby giving rise to a finite Ward identity. Comparing (5.13) with (5.6), we see that the finite part of the charge is exactly (5.6), the second term of (5.3). This concludes our demonstration that the asymptotic symmetry dual to the subleading soft photon theorem is a divergent electric and magnetic large gauge symmetry.

A Asymptotic Expansions
In this appendix, we list the full large |r| expansion of the radiative gauge field components A r and A a near I ± . Recall that (2.19) states where K ν is the modified Bessel function of the second kind, k ≡ | k|, z ≡ Because large |r| is equivalent to small z, we can perform a large |r| expansion of these components by utilizing the known expansions about z = 0 for the Bessel functions and then performing an inverse Fourier transform. For further details, we refer the reader to [10], where this procedure was introduced and more carefully explained.
In even dimensions D = d + 2 > 4, we only need the z = 0 expansion of Bessel functions with nonnegative integer orders. Carrying out the procedure outlined above, we obtain

B Poincaré and Scale Transformations
In this appendix, we determine the Poincaré and scale transformations of the annihilation operators and the LROs (3.15). The goal is to derive Table 1 and the commutators used when verifying the Jacobi identity (3.29). For simplicity, both subscripts labeling the annihilation operators and superscripts ± labeling the currents and LROs will be kept implicit.
We first derive the necessary commutators involving annihilation operators, which transform as