An Addendum to the Heisenberg-Euler effective action beyond one loop

We study the effective interactions of external electromagnetic fields induced by fluctuations of virtual particles in the vacuum of quantum electrodynamics. Our main focus is on these interactions at two-loop order. We discuss in detail the emergence of the renowned Heisenberg-Euler effective action from the underlying microscopic theory of quantum electrodynamics, emphasizing its distinction from a standard one-particle irreducible effective action. In our explicit calculations we limit ourselves to constant and slowly varying external fields, allowing us to adopt a locally constant field approximation. One of our main findings is that at two-loop order there is a finite one-particle reducible contribution to the Heisenberg-Euler effective action in constant fields, which was previously assumed to vanish. In addition to their conceptual significance, our results are relevant for high-precision probes of quantum vacuum nonlinearity in strong electromagnetic fields.


I. INTRODUCTION
One of the striking predictions of quantum field theory (QFT) is that virtual charged particle-antiparticle fluctuations in the quantum vacuum can induce nonlinear interactions among electromagnetic fields [1][2][3]; for reviews emphasizing various theoretical aspects as well as prospects for the experimental detection of such effects, see [4][5][6][7][8][9][10][11][12][13]. Aiming at probing the vacuum of the Standard Model of particle physics with classical electromagnetic fields and low energy photons, the dominant effective interactions are governed by quantum electrodynamics (QED).
For the macroscopic electromagnetic fields presently attainable in the laboratory, the effects of QED vacuum nonlinearities are rather small, making their experimental detection challenging [9,11]. These effective interactions have no tree-level analogue, but are mediated by at least one electron-positron loop. For electromagnetic fields which vary on scales much larger than the Compton wavelength of the electron λ C = 3.86 · 10 −13 m and are weak in comparison to the critical electric field strength E cr ≡ m 2 c 3 e ≃ 1.3 · 10 18 V/m [2,14,15], i.e., fulfill {| E|, c| B|} ≪ E cr , this results in a parametric suppression of the effective interaction by inverse powers of the electron mass. Most of the electromagnetic fields available in the laboratory meet this criterion.
Finally, and perhaps most strikingly, strong electric fields can facilitate the spontaneous formation of real electron-position pairs from the QED vacuum via the Schwinger effect [2,14,15].
It is a fascinating aspect of this plethora of phenomena that they manifest the effective interactions of electromagnetic fields beyond Maxwell's linear theory, which can be summarized elegantly by an effective action that dates back to the early days of quantum field theory [2,3]: the Heisenberg-Euler effective action. Its matured embedding into the modern language of field theory is due to Schwinger [15], who gave a nonperturbative definition of this action by means of the vacuum persistence amplitude, i.e., the Schwinger functional.
Nowadays, QFT is often defined in terms of generating functionals for correlation functions, with the concept of the effective action being identified with the generating functions of one-particle irreducible (1PI) correlators (proper vertices).
In this article, we emphasize that the Heisenberg-Euler effective action is different from -though related to -the 1PI effective action. This fact has, of course, been well known in the specialized literature [4,54] but is sometimes confused in textbooks. We detail the construction of the Heisenberg-Euler effective action from the standard definition of QED in terms of the partition function in the present work. The difference between the two effective actions is manifested by one-particle reducible (1PR) contributions to the Heisenberg-Euler action. In a perturbative loop expansion, such 1PR contributions occur at and beyond the two-loop order. At two-loop order, we find that there is a finite 1PR contribution to the Heisenberg-Euler effective action in constant electromagnetic fields, which was previously believed to vanish.
Using a locally constant field approximation (LCFA), we also study in detail the effective theory of slowly varying classical background fields and low-frequency photon fields in the QED vacuum. The photon polarization tensor derived within the latter contains 1PI, 1PR, as well as disconnected contributions, all of which can be understood as generated by the 1PI effective action.
As higher-loop diagrams are typically suppressed in comparison to the one-loop diagram, a proper inclusion of the previously neglected 1PR diagrams is expected to impact the proposed experimental signatures of quantum vacuum nonlinearities only at subleading order. In fact, most of the theoretical studies listed above exclusively limit themselves to one-loop order.
For instance for vacuum birefringence in weak fields, the two-loop contribution represents only a 1% correction [55].
Our article is organized as follows. Section II is devoted to an in-depth discussion of the Heisenberg-Euler effective action. Here, we elaborate differences and common ground with respect to the standardized 1PI effective action and show how the Heisenberg-Euler effective action emerges from the microscopic theory of QED. Finally, we explicitly sketch its diagrammatic expansion up to two-loop order. In Sec. III we focus on the Heisenberg-Euler effective action in constant fields. Here, we provide the weak-and strong-field asymptotics of the Heisenberg-Euler effective Lagrangian at one-and two-loop order. Their explicit derivation is relegated to Appendix B. Thereafter, in Sec. IV we introduce the LCFA and show how it can be employed to construct an effective theory describing the interactions of slowly varying electromagnetic fields and low-frequency photon fields in the QED vacuum.
Here, we mainly concentrate on fluctuation-induced effective interactions at two-loop order.
Finally, we end with conclusions and an outlook in Sec. V.

A. Partition function and vacuum persistence amplitude in an external field
The Heisenberg-Euler action Γ HE [2] is often viewed as the prototype of an effective action Γ, the latter having become a canonized central object in QFT. Effective actions Γ have a precise meaning as the generating functional of 1PI correlators (proper vertices) and follow from a standardized QFT construction via the Legendre transform of the partition function 1 Here, S denotes the classical action of the theory to be quantized, ϕ summarizes the fluctuation fields such as electrons/positrons and photons in QED, and J is a source that can be used to generate correlation functions from the partition function. The above definition (2) guarantees the 1PI property of Γ, making it a convenient and elegant tool for many purposes of QFT.
As it is of particular relevance for understanding our novel results obtained below, we wish to emphasize that the Heisenberg-Euler action does not fall into the class of effective actions as constructed from Eq. (2). This statement holds both from the perspective of its historical construction as well as from its modern use in strong-field physics.
The physical difference is, for instance, apparent from the fact that the 1PI effective action (2) depends on the so-called classical field φ. The supremum prescription in Eq. (2) relates φ to the expectation value of the fluctuating quantum field, (a relation that can be considered both at J = 0 or for a nonvanishing source). In turn, the field φ is obviously the result of a full quantum averaging process.
By contrast, Heisenberg and Euler [2] as well as Weisskopf [3] have been interested in the response of the quantized electron-positron field to a non-quantized external electromagnetic 1 In order to keep the notation compact, we employ the shorthand forms x ≡ d 4 x and k ≡ d 4 k (2π) 4 for the integrations over position and momentum space, respectively. Besides, we simply use if the integration can be performed in position or momentum space. fieldĀ which is considered to be given from the outside. In absence of quantum fluctuations, this external field would obey an action principle with action S ext [Ā] = − 1 4 F µνF µν , wherē Contrarily, in the presence of quantum fluctuations, the dynamics of the external fieldĀ is governed by the Heisenberg-Euler action, The additional contribution W [Ā] arising from quantum fluctuations has been formalized by Schwinger in terms of the vacuum persistence amplitude [15], parametrizing the probability amplitude for the vacuum to persist in the presence of an external fieldĀ ("the prescribed field" [15]). The Schwinger functional W [Ā] is considered to be a functional of the external field (and not of a source coupled to a quantum field). It can be written as a path integral over fluctuating fields, where we employed the shorthand notation For a proper comparison with the literature, we point out that our phase conventions agree with those of [4] and thus do not include the Maxwell term for theĀ field in W [Ā] (contrary to Schwinger's conventions [15]). 2 In Eq. (6), we have distinguished between the external backgroundĀ and the fluctuating photon field q, the latter being equipped with a kinetic term involving the field strength motion given by whereJ is a classical source for the external field. Upon the inclusion of quantum fluctuations, the dynamics ofĀ is modified such that Γ HE [Ā] governs the dynamics of the external field.
To one-loop order, the photon fluctuations ∼ Dq can be ignored in Eq. (6) and one obtains the historic answer [2,3,15]. At higher loops, starting from two loop on, W [Ā] also contains one-particle reducible diagrams [4,54,56], as is obvious from its definition (6) and will be recalled explicitly below. Hence, Γ HE does not correspond to the standard 1PI effective action.
From a fundamental viewpoint, the concept of a non-quantized external fieldĀ seems somewhat redundant, as the world is fully quantum. Moreover, a separation into internal and external fields might seem purely academic. Nevertheless, this concept is perfectly adjusted to our perception of a real experiment in terms of classically controlled sources and detectors. In the remainder of this section, we detail how this useful concept can be extracted from the full quantum theory.

B. From QED to the Heisenberg-Euler effective action
In order to develop the formalism, it is useful to envisage a typical physical system where the external fieldĀ is generated by suitable sourcesJ. The sources (lasers, magnets, etc.) are macroscopically separated from an interaction region of volume V I (focal volume, interaction cavity, etc.). We consider physical situations where quantum vacuum nonlinearities, i.e., higher-order effective couplings of electromagnetic fields mediated by quantum fluctuations of charged particles, become sizable only within V I .
Then, the physics inside V I can create signals (induced field components, signal photons, etc.) which are ultimately observed in detectors macroscopically separated from the interaction region V I . Due to the smallness of the nonlinear effective couplings among electromagnetic fields induced by quantum fluctuations of virtual charged particles, the signal may often be of quantum nature, as it is, e.g., the case for a single-photon signal to be measured in a single-photon detector. Still, it is useful to think of the signal as a contribution to the external fieldĀ, because it is ultimately measured far away from the region V I .
In order to distinguish between applied fieldsĀ applied (e.g., the fields provided by lasers, or magnets) and the signal photonsĀ signal , one may decompose the external field as As the signalĀ signal is eventually induced byĀ applied , the two components ofĀ will typically exhibit a causal ordering in time. Similarly, it is possible to distinguish between the source partsJ that are responsible for creatingĀ applied and those that interact withĀ signal within the detectors.  With these prerequisites, let us turn to the standard partition function for QED, Concentrating on correlation functions of the electromagnetic field, we only include a source term for the gauge field. Of course, the generalization to sources for the fermions is straightforward. In a next step, we employ the variable substitution in order to rewrite Eq. (11) as We emphasize that -despite its explicit appearance on the right-hand side -this partition function of course does not depend onĀ but is a functional of the source J only. Contrary to the standard QFT treatment where J often plays the role of an auxiliary variable, the source is needed here to sustain the external field. Still, let us not simply reduce J →J, but keep it slightly more general.
As a next step, we classicalize the external fieldĀ: For this, we assume that the fluctuation field q only couples to the electron-positron field, i.e., any direct coupling to the background field should vanish. More precisely, we choose J such that Let us emphasize that for any violation of Eq.
A comparison with Schwinger's vacuum persistence amplitude (6) shows that This suggests introducing the Heisenberg-Euler action by Note that Eq. (17) does not constitute a Legendre transform, since J is subject to the constraint (14).
Since the fieldĀ is ultimately created by the classical sourceJ, we demand for which implies that the correction term in Eq. (10) is given by C µ [Ā] = δW [Ā]/δĀ µ . Hence the correction term can be viewed as a shift in the source term which is needed in Eq. (14) to inhibit that the background as well asJ provide a source for the fluctuation field q. as it also contains one-particle reducible contributions which contribute to the equations of motion of the external field.
We end this section with the remark that once Γ HE is obtained, it can be used for de-terminingĀ both by purely classical means or by describingĀ in terms of a Fock space in a quantum optical setting. Both treatments ofĀ are useful as well as legitimate. In particular, it is natural to treat applied macroscopic fieldsĀ applied classically and the induced weak signal fieldsĀ signal by means of Fock space states, as has been suggested in the vacuum emission picture [52].

C. Diagrammatic expansion of the Heisenberg-Euler effective action
Apart from the classical Maxwell term, the Heisenberg-Euler effective action (17) [4]. Generically, this expansion can be organized in the number of loops, We begin by noting that the fermionic integral in Eq. (7) can be written as a functional determinant, If evaluated at q = 0, this quantity amounts to the one-loop Heisenberg-Euler effective action in the external fieldĀ, i.e., Γ 1-loop ; for a graphical representation, cf. Fig. 1. Since S ψ is a one-loop expression, the two-loop order of the Schwinger functional is already obtained by performing the photonic fluctuation integral ∼ Dq to Gaußian order. For this, we expand S ψ about the external fieldĀ, where we employed the shorthand notation The first-order term corresponds to a one-loop photon current induced by the fieldĀ, and the Hessian is related to the one-loop photon polarization tensor Π µν [Ā] := S evaluated in the external fieldĀ; for completeness note that this definition of the photon polarization tensor differs from that of [57] by an overall minus sign. To Gaußian order, we ignore the terms of O(q 3 ) in the exponent, resulting in In principle, terms of O(q 3 ) in the exponent can, of course, be treated perturbatively to any desired order. The quantity D −1 µν arises from the Maxwell term for the fluctuations and denotes the inverse photon propagator. E.g., in momentum space and accounting for a gauge-fixing term (generalized Lorenz gauge), we have where ξ = 1 in the Feynman gauge. Performing the integration over q in Eq. (24), we arrive at To Gaußian order in the photon fluctuations, we thus obtain for the Schwinger functional The first term on the right-hand side corresponds to the one-loop contribution to the Heisenberg-Euler effective action (called W (1) [Ā] in [4,15]). The other two terms contain the complete two-loop order contribution as well as subclasses of diagrams to arbitrarily high loop order. To make this manifest, we expand the ln det term as follows, where in the last step, we have dropped field-independent constants. The Tr(DΠ) term corresponds exactly to the two-loop contribution to the Heisenberg-Euler action that has first been computed in [54]; see also [4,[58][59][60]. This contribution as well as all higher order terms in Eq. (28) are one-particle irreducible from a diagrammatic viewpoint; see Fig. 2.
The last term in Eq. (26), however, contains In turn, the two-loop Heisenberg-Euler effective action consists of a 1PI and a 1PR diagram and is given by The existence as a matter of principle of the 1PR term in Eq. (30) has been known for a long time. It has, however, been argued that this term vanishes for constant external fields [4,54]. Let us reproduce this argument for reasons of completeness: a crucial building block of the 1PR term is S While this part of the argument holds true in the full analysis, it does not necessarily imply that the 1PR diagram in Fig. 4 vanishes. In fact, the two currents in the 1PR diagram are convoluted with a photon propagator, describing a long-range force with an IR singularity ∼ 1/p 2 in the propagator. Hence, it is a quantitative question as to whether the currents approaching zero are outbalanced by the IR singularity of the photon propagator. In the subsequent sections, we give proof that the result is finite.
Heuristically, it is clear that the above-mentioned argument for the vanishing of the current no longer holds as soon as the external field supports a slightest inhomogeneity somewhere in spacetime. So, j µ [Ā] = 0 strictly relies on ∂ µ F νκ = 0 for all x. On the other hand, the existence of massless long-range fluctuations in QED is independent of the constant-field assumption. Therefore, the 1PR term is expected to be finite for any realistic field.

III. THE HEISENBERG-EULER EFFECTIVE ACTION IN CONSTANT ELECTROMAG-NETIC FIELDS
In the following, we first summarize some of our explicit results for the Heisenberg-Euler effective action, concentrating on fully analytic expressions in asymptotic field-strength limits for spinor QED. This provides for a first glance at the parametric dependence of the various contributions, and elucidates the regime of relevance of the two-loop 1PR term. Details of the calculations are deferred to the subsequent sections.
For constant external electromagnetic fields,F µν = const., Lorentz and gauge invariance constrain Γ HE to depend onĀ only in terms of the two scalar invariants [1,2] Here, ǫ µναβ is the totally antisymmetric tensor (ǫ 0123 = 1), and our metric convention is g µν = diag(−1, +1, +1, +1). In addition, CP invariance of QED dictates Γ HE to be even in the pseudoscalar quantity G, Γ HE [Ā] = Γ HE (F , G 2 ). To keep notations compact, we also introduce the dimensionless quantitiesF = ( e m 2 ) 2 F andG = ( e m 2 ) 2 G. Finally note that the action and the Lagrangian are trivially related in constant fields, differing only by an overall volume factor, i.e., Γ HE = L HE x .
In constant external fields, L 1-loop HE and L 2-loop HE 1PI are known explicitly in terms of parameter integral representations for both spinor [2,54] and scalar [3,56] QED; cf. [60] for a review. For instance, the on-shell renormalized one-loop effective Lagrangian for spinor QED is given by [2,15], where ǫ = √ F 2 + G 2 − F 1/2 and η = √ F 2 + G 2 + F 1/2 are the secular invariants in constant electromagnetic fields. The analogous expression for L 2-loop HE 1PI is given in Eq. (B11) in the appendix for spinor QED. For completeness, we also note that mass renormalization has to be taken into account from two loops on for diagrams involving fermion loops with internal radiative corrections; see, e.g., [58].
As indicated above and determined explicitly below, the 1PR contribution L 2-loop HE 1PR depicted in Fig. 4 is finite also in the constant field limit. Based on the structure of the LCFA, we detail below how the exact expression for L 2-loop HE 1PR in constant fields can be inferred from the constant-field result for L 1-loop HE , yielding In turn, for spinor QED. In the weak field limit, characterized by {F,G} ≪ 1, the well-known literature results read [1][2][3], and [54] where we count O( eF µν m 2 ) ∼ O(ǫ). The terms given explicitly in Eqs. (33) and (34) The contribution given explicitly here stems from the 1PR diagram in Fig. 4 and where In addition to the well-known leading-log terms [4,54,56,61], Eqs. (36) and (37) also account for the strongly suppressed contribution ∼G 2 which is of relevance for the photon polarization tensor (cf. Sec. IV D below). Note that this contribution is suppressed as  Apart from these constant-field results, only a few exact results for Γ 1-loop HE in specific (one-dimensional) field inhomogeneities are known explicitly; cf., e.g., [62][63][64][65], and [60] for a review. Also note that the effective action vanishes identically for the case of a single monochromatic plane wave field [15]. On the three-loop level, first analytical results for the 1PI part of Γ 3-loop HE have been obtained in 1+1 dimensions [66,67]. No further analytical results for Γ l-loop HE with l > 2 as well as for more-dimensional field inhomogeneities are available so far.

A. Locally constant field approximation
In the spirit of the LCFA, the Heisenberg-Euler effective action for constant fields can also be adopted for slowly varying inhomogeneous fields. The LCFA amounts to substitutinḡ F µν →F µν (x) in the constant-field result for the Lagrangian, such that L HE (F , G 2 ) → L HE F (x), G 2 (x) . In turn, the corresponding action becomes a functional of a varying field The deviations of this LCFA result from the corresponding -typically unknown -exact result for Γ HE in the particular inhomogeneous background field profile under consideration are of order O ( υ m ) 2 , where υ delimits the moduli of the frequency and momentum components of the considered inhomogeneous field from above [57,68]. The reasoning to arrive at this conclusion is as follows: As Γ HE is both a Lorentz scalar and a gauge invariant quantity, and the associated Lagrangian should be "almost local" for slowly varying fields, its dependence on the external fieldĀ µ (x) should be expressible in terms ofF µν (x), * F µν (x) and derivatives thereof. Any scalar quantity made up of combinations ofF , * F and ∂ is necessarily even in ∂. Canonical power-counting implies that the occurrence of any derivative ∂ has to be balanced by a dimensionful scale. In QED and for generic laboratory fields, this scale is provided by the electron mass m, leading to the above criterion. This implies that the LCFA constitutes a good approximation for inhomogeneous fields fulfilling υ ≪ m.
In position space this criterion translates to the requirement that the inhomogeneous fields under consideration should only vary on scales much larger than the Compton wavelength λ C and time λ C /c of the electron; cf. Sec. I. Explicit results for higher orders in the derivative expansion show, that the dimensional balancing of derivatives can also be taken over by the field strength itself for strong fields |eF (x)| ≫ m 2 , thereby increasing the validity range of the LCFA in that regime [69].

B. Effective action for low-frequency photons
In a next step, we employ the LCFA result for the 1PI part of Γ HE as an effective action Γ eff , describing the propagation and interactions of dynamical low-frequency photon fields in the quantum vacuum subject to the slowly varying external field. More precisely, we define int . The gray bubbles symbolize the effective couplings of n low-frequency photon fields a(x); generically they are made up of 1PI diagrams of arbitrary loop order. In momentum space they are given by s σ 1 ...σn this effective action as where Γ HE 1PI denotes the 1PI part of the Heisenberg-Euler effective action, and the field strength tensorF is understood to be shifted as follows [18,57,68], After this shift,F (x) describes the slowly varying external field with υ ≪ m, and f µν (x) = It is then convenient to organize Γ int in terms of interactions involving n ∈ N 0 photon fields, i.e., int a(x), F (x), G 2 (x) ∼ a n . For a graphical representation of this expansion, cf. Fig. 5. More specifically, we have which implies that L HE F (x), G 2 (x) 1PI generates effective photon interactions to any order in n [68,70]. As the LCFA results in a local Lagrangian, by construction all these effective interactions are local with respect to f µν (x), and correspondingly in a µ (x). Let us also emphasize that by construction we have to assume that the combination of any given number n of low-frequency photons again gives rise to a low-frequency photon. For the following discussion it is more convenient to turn to momentum space where the derivatives acting on the photon fields contained in f µν (x) translate into multiplicative factors of the associated momenta, i.e., f µν (x) = i p e ipx a σ (p) p µ g νσ − p ν g µσ . This results in with the effective n-photon couplings s σ 1 ...σn (n) (p 1 , . . . , p n ) (1PI proper vertices) given by The latter obviously fulfill the Ward identity (p j ) σ j s The contribution s σ 1 (1) (p 1 ) in Eq. (44) constitutes a photon current [52,71] and s σ 1 σ 2 (2) (p 1 , p 2 ) a photon polarization tensor [57]. In more conventional notations, the quantum corrections to the effective action up to quadratic order in a µ are given by with j σ (p) := s σ (1) (p). The neglected higher-order terms of O(a 3 ) correspond to effective interactions involving three or more photons, giving rise to, e.g., direct light-by-light scattering [1,72], photon splitting [18,43,45] and higher-harmonic generation [50,51,73,74].
Obviously no real (on-shell) photons can be generated from constant external fields, as The physical reason for this is that a constant external field cannot supply momentum to the virtual charged particle-antiparticle fluctuations. Still, the fields a µ can be propagating fields, the free causal propagation of which is described by the usual Feynman propagator (25).
Within the LCFA, we have the additional constraint that the considered momentum transfer is manifestly restricted to the soft momentum regime, i.e., {|p 0 |, | p|} υ ≪ m (cf. above).
Here, we argue that this constraint will be fulfilled automatically in the evaluation of all the Feynman diagrams that can arise as quantum corrections within the effective theory Γ eff of low-frequency photon fields in slowly varying electromagnetic fields. For this, we first stress that Γ eff already incorporates all 1PI proper vertices by definition, such that further quantum corrections to be evaluated within the effective theory of low-frequency photon fields must be 1PR. By construction, the virtual photons in these 1PR diagrams mediate between slowly varying fields only, since the external lines of the 1PI building blocks are either low-frequency photons or slowly varying electromagnetic fields. Hence, the above kinematic constraint is indeed fulfilled automatically.
In a next step, we utilize Γ eff to derive some physically relevant explicit results: as the prime example, we compute the 1PR contribution to the Heisenberg-Euler effective action Γ 2-loops HE 1PR in slowly varying external fields, introduced and discussed already in Secs. II and III above.

C. 1PR contribution to the Heisenberg-Euler effective action
Let us now focus on the effective self-interactions of the external electromagnetic field arising in this theory. At one-loop order these are encoded in Γ , also the 1PR diagram depicted in Fig. 4 (right) contributes. It corresponds to the following expression: where j µ l-loop := s µ (1) l-loop . We emphasize that the integration in Eq. (47), which is formally over all virtual momentum transfers, exclusively receives contributions from the soft momentum regime. This is because the photon currents j µ (p) only induce low-energy modes by construction via the LCFA. The constant-field limit in Eq. (46) provides an obvious example for the underlying mechanism. Inserting the explicit expressions for the currents and the photon propagator (25) in the Feynman gauge, we obtain Here we have defined which fulfills 1 4 g µν G µν (x) = δ(x) and x G µν (x) = g µν . Expressing the derivatives forF in terms of derivatives for F and G (cf. Appendix A), Eq. (48) can be represented as where we employed the identity x ′ , such that the integrations over position space in Eqs. (48) and (50) can be performed right away, resulting in Eq. (32) above.
Let us finally resolve the seeming discrepancy that the constant-field limit of Eqs. (48) and (50)  This implies that no real external photons a σ (p), exhibiting a regular behavior for p σ → 0, can be induced from j σ (p). However, this behavior is clearly not fulfilled by virtual photons.
As the photon propagator (25) scales as ∼ 1 p 2 , the linear momentum dependences of the two individual currents effectively drop out upon combination with two photon currents, leaving us with a finite contribution.
Correspondingly, the two-loop physical effective interaction among generic external elec- ternal electromagnetic field at higher loop orders -even in constant external fields. Besides, they obviously also need to be accounted for in determining the effective interactions between any given number of photons in the quantum vacuum subject to external electromagnetic fields.

D. Low-frequency photon propagation
In this section we study quantum corrections to photon propagation -i.e., photon-photon correlators -in external electromagnetic fields up to order ( α π ) 2 . It is instructive to have a look on the various Feynman diagrams potentially contributing to photon propagation up to this order; see Fig. 5. We organize them into (a) 1PI diagrams, (b) 1PR diagrams, and disconnected contributions which amount to (c) current-current correlators. Note that there are just two diagrams at order α π , namely the first one in Fig. 6(a) and in Fig. 6(c), respectively. All the other diagrams shown are proportional to ( α π ) 2 .
The diagrams depicted in Fig. 6(a) constitute the 1PI part of the photon polarization tensor at one (first line) and two loops (second line). They are contained in Γ (2) int , and are given by cf. Eqs. (43)- (45). Note that s ρσ (2) 1-loop corresponds to the slowly varying field limit of the one-loop polarization tensor Π ρσ ≡ s ρσ (2) 1-loop defined in Sec. II C above. The definition (51) automatically accounts for all the topologically inequivalent 1PI diagrams depicted in the second line of Fig. 6(a). Obviously, we have s ρσ (2) (p, p ′ ) l-loop ∼ ( α π ) l . Expressing the derivatives forF in terms of derivatives for F and G, Eq. (51) can be represented as [57] s ρσ where we have employed the shorthand notations (pF ) µ = p νF νµ , (p * F ) µ = p ν * F νµ , (pp ′ ) = p µ p ′µ , etc. In the constant-field limit,F as well as L HE do not depend on the space-time coordinate and Eq. (52) simplifies significantly: In this limit, the x integration can be performed right away, giving rise to an overall delta function ensuring momentum conservation in constant fields, and thus where we have introduced the tensor structures The associated coefficients c l-loop j (F , G) are given by Note that the tensor structure p ′ µ p α ǫ ρσµα vanishes in constant fields, where p ′ µ = −p µ . Let us now have a closer look on the other Feynman diagrams depicted in Fig. 6. The first diagram in Fig. 6(b) is just an iteration of Π ρσ , and the corresponding expression reads A similar diagram exists in the absence of external fields. Contrarily, all the other diagrams in Fig. 6(b) and those in Fig. 6(c) do not contribute at zero field, because of Furry's theorem.
The last two diagrams in Fig. 6(b) arise from saturating one leg of an effective three-photon coupling with j 1-loop . Both diagrams yield the same result. Their combined contribution is given by Finally, we turn to the disconnected diagrams in Fig. 6. The first three diagrams in Fig. 6(c) The two diagrams depicted in the last line of Fig. 6(c) amount to current-current correlators with one-loop quantum correction to the out-and ingoing photon line, respectively. The left one can be expressed as and the right one corresponds to Eq. (59) with the replacements p ↔ p ′ and ρ ↔ σ.
In constant electromagnetic fields, all diagrams depicted in Fig. 6(c) vanish if at least one of the external photons is real; cf. the discussion in Sec. IV C. Of course, the derivatives of the Lagrangian forF in Eqs. (56)-(59) could again be expressed in terms of derivatives for F and G (cf. Appendix A). While this would allow us to identify the various tensor structures spanning these contributions, the resulting expressions are lengthy so that we do not show them here.
Correspondingly, the photon polarization tensor in the presence of an external field is not only given by 1PI diagrams, but also receives corrections from 1PR and even disconnected diagrams. More precisely, it is made up of all the diagrams that do not correspond to iterations of more elementary diagrams describing quantum corrections to photon propagation.
The full dressed photon propagator in the external field -accounting for quantum corrections to all orders -is then obtained by summing up all possible iterations analogously to a Dyson series of this photon polarization tensor. At l loops, we have where Π ρσ l-loop 1PI := s ρσ (2) l-loop and ∆Π ρσ 2-loop refers to the contributions of 1PR and disconnected diagrams. In a slight abuse of nomenclature, we label contributions to the photon polarization tensor which scale as ∼ ( α π ) l with "l-loop", even though ∆Π ρσ l-loop generically also includes current-current correlators involving higher loop numbers; cf. Fig. 6. The explicit expressions for ∆Π ρσ l-loop at one and two loops are and Equations (61) and (62) account for the five distinct diagrams in Fig. 6(b) and 6(c) that do not correspond to iterations of more elementary quantum corrections to photon propagation.
In inhomogeneous electromagnetic fields, the explicit expressions for Eqs. (61) and (62) written in terms of derivatives of L 1-loop HE for the scalar field invariants F and G are rather lengthy. The main reasons for this are the different space-time arguments of the derivatives of L 1-loop HE . However, in constant external fields these expressions simplify significantly, as all the current-current correlators vanish and the derivatives of L 1-loop HE with respect toF become independent of x. More specifically, in this limit, we obtain ∆Π ρσ 1-loop (p, p ′ ) = 0, and with the tensor structures P ρσ j (p) defined in Eq. (54). Hence, Eq. (63) is spanned by the same tensor structures as Π ρσ 1PI in constant fields as listed in Eq. (54). The coefficients ∆c 2-loop j are given by by differentiation and read The analogous coefficients ∆c 2-loop j can be obtained from Eq. (64) by using the expressions assembled in Eq. (B6), yielding Hence, in the strong-field limit, we read off the scalings ∆c

V. CONCLUSIONS AND OUTLOOK
We have taken a fresh look at the famous Heisenberg-Euler effective action Γ HE , which has played a substantial role in the development of quantum field theory, and its relation to the underlying microscopic theory of QED formulated in terms of the partition function.
We have paid particular attention to the differences and common ground of Γ HE and the nowadays more conventional 1PI effective action. Most noteworthily and distinctively, Γ HE also contains 1PR contributions implying quantitative differences to the 1PI effective action from the two-loop level on.
For an efficient determination of these effective actions, we have constructed an effective theory of low-frequency photons in the QED vacuum subject to slowly varying electromagnetic fields. Apart from discussing various generic features of such an effective theory, our main focus was on the effective interactions generated at O ( α π ) 2 . Here, we in particular studied the effective self-interaction of external electromagnetic fields, and derived the photon polarization tensor in the low-frequency limit. The latter is composed of 1PI, 1PR as well as disconnected diagrams.
One of our main findings is that the coupling of two one-loop vacuum currents via a photon propagator gives rise to a nonvanishing 1PR contribution to Γ 2-loop HE even in the limit of constant electromagnetic fields; cf. Fig. 4 (right). This contribution was previously believed to vanish. To clarify the importance of this newly evaluated 1PR contribution relatively to the well-known 1PI one, we have investigated the limits of perturbatively weak and strong fields. Whereas Γ 2-loop HE 1PR is generically suppressed for weak fields, it can even surpass Γ 2-loop HE 1PI in magnitude for strong fields. Similar results are obtained for the twoloop photon polarization tensor. Also here, the 1PR contributions can surpass the 1PI ones in magnitude for strong fields.
Our results can also be of relevance beyond QED, for instance, for the exotic case of a hypothetical minicharged particle sector beyond the Standard Model of particle physics [75]. Beyond QED, the most essential new feature of the diagram depicted in Fig. 4 (right) as compared to Fig. 4 (left) is that the first one can induce effective interactions mediated by two different fermion species (the two loops in this diagram do not necessarily have to contain the same fermion species), while the latter one features a single fermion loop and thus may only involve one fermion species. This implies a parametrically different dependence of electromagnetic or optical observables on the various coupling and mass parameters starting at two-loop level. As the considerations invoked here can also be adopted for scalar QED, particularly in a combination of scalar and spinor QED, this type of mixed effective interactions can also be generated when one of the loops traces over fermions and the other one over bosons. For 1PI diagrams, such an effective coupling of different species can only happen at three loops or beyond.

Acknowledgments
We acknowledge support by the BMBF under grant No. 05P15SJFAA (FAIR-APPA-SPARC). The authors wish to thank W. Dittrich for helpful conversations and for carefully reading the manuscript. We are particularly grateful to S. Evans for correspondence that lead to the improved version (v4) of this paper. and Appendix B: Strong-field asymptotics for spinor QED In this appendix, we concentrate on constant fields and focus on the limit of |F | = 1 2 ( e B m 2 ) 2 − ( e E m 2 ) 2 ≫ 1 and |G| = e E m 2 · e B m 2 ≪ 1, which is of relevance for either strong electric or magnetic fields.

Two-loop effective Lagrangian in constant external fields
As detailed in the main text, the two-loop Heisenberg-Euler effective Lagrangian consists of a one-particle irreducible and a one-particle reducible contribution, cf. also Appendix B 1.
Closed-form expressions exists for L 2-loop HE 1PI only for the simplified case of self-dual fields [78], but not for the cases of interest here, not even for the special case of G = 0. However, the leading strong-field behavior of L 2-loop HE 1PI /m 4 for |F| ≫ 1 and |G| ≪ 1 is known explicitly [54] (cf. also [4] for an independent verification). ForF > 0, it is given by whereη is one of the secular invariants of the electromagnetic field, defined as η = ( F 2 +G 2 +F) 1/2 andǫ = ( F 2 +G 2 −F) 1/2 .
Note that forF ≫ 1, we havẽ Aiming at determining the leading strong-field asymptotics of Π µν 2-loops (p, p ′ ) const. , we need the complete scaling of the leading contribution ∼G 2 , for which the terms given in Eq. (B9) are not sufficient.