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Polarization operator for plane-wave background fields

We derive an alternative representation of the leading-order contribution to the polarization operator in strong-field quantum electrodynamics with a plane-wave electromagnetic background field, which is manifestly symmetric with respect to the external photon momenta. Our derivation is based on a direct evaluation of the corresponding Feynman diagram, using the Volkov representation of the dressed fermion propagator. Furthermore, the validity of the Ward-Takahashi identity is shown for general loop diagrams in an external plane-wave background field.


I. INTRODUCTION
The most precise calculations known so far in physics are provided by quantum electrodynamics (QED). The reason for this is the smallness of the fine-structure constant α = e 2 /(4π 0 c) ≈ 1/137, which allows us to use perturbation theory [1] (e is the electron charge). The most prominent example is probably the electron g-factor, for which experimental and theoretical results have been matched on the record accuracy level of parts per billion [2]. To achieve this outstanding precision, the corresponding theoretical calculation included Feynman diagrams with up to four loops.
A quite different situation is encountered for QED with electromagnetic background fields. A source for strong electromagnetic fields are modern laser systems. If spatial focusing effects are sufficiently small, laser fields can be well approximated by plane-wave fields. For a plane-wave field, we obtain, besides the fine-structure constant, a second gauge and Lorentz-invariant parameter ξ 0 = |e|E 0 /(mcω 0 ), where E 0 and ω 0 are the peak electric field strength and central angular frequency of the plane wave, respectively [m is the electron mass; see also Sec. II C]. If ξ 0 1 the interaction between electron and positrons with the laser field must be taken into account exactly. For optical lasers (photon energy ω 0 ≈ 1 eV), this happens already at intensities of the order of 10 18 W /cm 2 . More precisely, we can generally still treat the interaction of the electrons and the positrons with the quantized radiation field perturbatively as in vacuum QED (QED without background fields), but must include the dependence on ξ 0 to all orders if the threshold ξ 0 ≈ 1 is exceeded.
Another important scale is the so-called critical field E cr = m 2 c 3 /( |e|) = 1.3 × 10 16 V/cm, which corresponds to a peak laser intensity of I cr = 0 cE 2 cr = 4.6 × 10 29 W /cm 2 . A constant and uniform electric field of this strength can, in principle, produce electron-positron pairs from the vacuum [3][4][5].
The current laser intensity record (in the optical * dipiazza@mpi-hd.mpg.de regime) is given by 2 × 10 22 W /cm 2 [6] and future facilities envisage even intensities of the order of 10 24 − 10 25 W /cm 2 [7][8][9]. Thus, the nonperturbative regime (in ξ 0 ) can be entered with presently available laser systems, and even the critical field can be reached in the rest frame of an ultrarelativistic particle (e.g., a ∼ 1 GeV electron [10]). So far only one experiment has been carried out to probe strong-field QED effects using laser fields [11,12]. However, this is expected to change in the near future and, correspondingly, the experimental progress has stimulated many theoretical investigations during the last years . For a more detailed overview, the reader is referred to the review [49].
In contrary to vacuum QED, calculations with a planewave background field have not been carried out beyond the one-loop order (for constant-crossed fields higherorder calculations have been performed, see, e.g., [50][51][52]). This can be attributed to the fact that already diagrams with just a few propagators correspond to quite complicated expressions. It is therefore of general interest to investigate new techniques, which have the potential to make also the calculation of complicated diagrams tractable.
Here we present a new derivation of the first-order contribution to the polarization operator given in Fig. 1 [53,54]. In Ref. [53] an operator approach similar to the one introduced by Schwinger [5] has been used. We show here how the diagram in Fig. 1 can be evaluated directly using the Volkov representation of the dressed propagators. This approach has the appealing feature that it is very similar to the calculation techniques used in vacuum QED. Our final result (which is equivalent to the one in Ref. [53]) has the interesting property that it is manifestly symmetric with respect to the external photon four-momenta q 1 and q 2 (see Fig. 1). Furthermore, we prove the validity of the Ward-Takahashi identity [55,56] for general loop diagrams in a plane-wave background field. The calculation techniques employed here are expected to be useful also for other higher-order calculations with strong plane-wave background fields.
The polarization operator itself is of central importance, since it determines the properties of a photon inside the background field via the Schwinger-Dyson equation for the exact photon propagator [53,57]. As a con- sequence, a plane-wave field acts as an active medium, e.g. the photon obtains a mass and has a nontrivial dispersion relation. Furthermore, due to the unitarity of the S-matrix, the imaginary part of the polarization operator is related to the total photo-production probability for an electron-positron pair (see Fig. 1) [58,59]. The polarization operator is also required for the calculation of radiative corrections to elementary processes like nonlinear Compton scattering or pair production. The significance of the photon polarization tensor can also be inferred from the ongoing effort to calculate it for different field configurations [60,61]. The paper is divided into three parts. In Sec. II the general framework of strong-field QED with plane-wave background fields is described. The actual calculation is then presented in Sec. III. Finally, we show how our results are related to those obtained in Ref. [53] and discuss the special cases of a constant-crossed field, a linearly polarized field in the quasi-classical approximation, and a circularly polarized, monochromatic field in Sec.

IV.
II. STRONG-FIELD QED QED is described by the following Lagrangian density [57,62]: where ψ and A µ are the Dirac spinor field and the electromagnetic four-vector potential, respectively, and F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor [from now on we will use natural units = c = 1 and Heaviside-Lorentz units for the charge, α = e 2 /(4π), see Appendix A for further details]. The equation of motion for the spinor field ψ, which follows from Eq. (1), is the Dirac equation Correspondingly, we obtain for the photon field A µ in Lorentz gauge (∂ µ A µ = 0) the wave equation

A. Vacuum QED
To obtain a quantum theory of electrons, positrons, and photons, both the spinor ψ and the photon field A µ are promoted to operators with canonical (anti)commutation relations (alternatively, the functional integral formalism can be employed). Once derived in either way, the S-matrix element for a given process can be calculated perturbatively using Feynman rules. In vacuum QED, the starting point for the perturbative expansion is a solution of the free Dirac or the free wave equation [Eq. (2) with A µ = 0 and Eq. (3) with J µ = 0, respectively]. An electron with a given four-momentum p µ = ( , p) ( > 0, p 2 = m 2 ) can then be described by the plane-wave solutions [57], and the corresponding propagator is given by For a photon with polarization four-vector µ and fourmomentum k µ = (ω, k) (ω ≥ 0, k 2 = 0), we obtain the following wave-function: and in the Feynman gauge, the photon propagator is given by Finally, the interaction between electrons, positrons, and photons is represented by the interaction vertex where the dots indicate that the vertex is always contracted with two fermion and one photon lines. Thus, one can move the exponential functions from the external lines and propagators to the vertex. After the space-time integrals associated with the vertex are taken, momentum-conserving delta functions are obtained. For a more detailed discussion see e.g. Refs. [57,62].

B. QED with background fields
Vacuum QED has been tested to a very high precision because, due to the smallness of the fine-structure constant, a perturbative treatment is adequate in most situations. However, for very strong external electromagnetic fields, the (conventional) perturbation series breaks down. Modern laser facilities provide a source of such strong external electromagnetic fields. An intense laser field represents a coherent state of the photon field that can be described by a classical four-potential A µ . Since only the highly occupied laser modes can be considered as classical, we separate those modes by writing A µ = A µ rad + A µ in the Lagrangian density. Here, A µ is treated as a classical field, whereas all other modes, described by A µ rad , are properly quantized [63][64][65] (see Appendix B for further details).
To estimate the laser intensity at which we can start to treat the laser as a classical field, we follow Ref. [57], Sec. 5. The energy density of a laser field with intensity I 0 and central angular frequency ω 0 is of the order of I 0 , and the density of modes is of the order of ω 3 0 . Correspondingly, each mode contains N γ photons, where N γ is of the order of I 0 /ω 4 0 . If N γ 1, due to the correspondence principle, we can describe the laser modes by a classical field. Thus, we obtain the following condition for the laser intensity: which is well fulfilled at the relativistic intensities (I 0 10 18 W /cm 2 ) in the optical regime (ω 0 ∼ 1 eV) we are interested in here.
Furthermore, we also have to ensure that the depletion of the laser field is sufficiently low, such that the latter can be treated as a given background field. Now, typical available optical petawatt lasers, which are suitable for the investigation of QED processes in a strong laser field, have an energy of the order of 100 J [49], i.e., a total number of about 10 20 photons. In a typical QED process as nonlinear Compton scattering, at an intensity of the order of 10 22 W /cm 2 , about ξ 3 0 ∼ 10 6 photons are absorbed from the laser by each electron [49] (such an intensity is, in principle, reachable with a petawatt laser). By assuming a number of electrons in the beam of the order of 10 9 , we obtain that about 10 15 photons are expected to be absorbed from the laser field, which remains then practically unaffected (this number of electrons is typical for current laser-plasma-based electron accelerators [10]). In conclusion, we can safely assume here that the background laser field can be treated as a given, classical background field (see also Refs. [66][67][68][69][70]).
Working in the so-called Furry picture [71] the classical background field A µ is taken into account exactly and only the radiation field is treated perturbatively. The starting point for the perturbative expansion of the Smatrix is then the solution of the interacting Dirac equation (2) with the replacement A µ → A µ . Since photons have no self-interactions at tree level, the photon wave functions and propagators are left unchanged by the background field. A more detailed discussion of strongfield QED can be found in Refs. [49,52,63,[72][73][74][75][76] and in the references therein.

C. Plane-wave fields
In this paper we will consider only plane-wave external fields, i.e., we require that the field tensor F µν depends only on the plane-wave phase φ = kx, where k µ is a momentum four-vector which characterizes the plane wave (k 2 = 0). In the absence of charges and currents, the field tensor must obey the homogeneous Maxwell equations [77]: Working in a basis where ka i = a 1 a 2 = 0, with i = 1, 2 (see Appendix C), we can show that the most general (antisymmetric) field tensor obeying Eq. (10) is given by [5] F µν (kx) = f µν where The scalar functions ψ i (kx) are arbitrary, restricted only by the physical requirement that the external field is of finite extent [i.e., ψ i (±∞) = ψ i (±∞) = 0, with ψ i (kx), ψ i (kx) vanishing fast enough at infinity]. Furthermore, we adopt (without restriction) the normalization condition |ψ i (kx)| , |ψ i (kx)| 1. We note that in the Lorentz gauge (∂ µ A µ = 0) the four-potential corresponding to the field tensor in Eq. (11) can be chosen in the form Using the tensors f µν i and an arbitrary fourmomentum vector q µ (most commonly the fourmomentum of an incoming particle), we can define the following quantum nonlinearity parameters [49]: where and where the quantities are the so-called classical intensity parameters. Since both η and χ i are manifestly gauge and Lorentz invariant, also the parameters ξ i are gauge and Lorentz invariant. Due to the normalization condition for the shape functions ψ i (kx), the parameters ξ i characterize the strength of the plane-wave field. It turns out that the plane-wave field must be taken into account exactly in the calculations if ξ i 1 [49]. Modern laser facilities can easily reach this nonperturbative domain, e.g. in Ref. [6] ξ i ∼ 100 was obtained [we point out that the parameter ξ 0 , mentioned in the introduction, can be related to the parameters ξ i by noting that, for a linearly-polarized field with ψ 2 (kx) = 0 and with electric-field amplitude E 0 and central angular frequency ω 0 , the quantity −a 2 1 can be set equal to E 0 /ω 0 ].
It is also convenient to introduce the integrated fieldstrength tensor, which can be written as in the Lorentz gauge [we will use F µν x = F µν (kx) interchangeably to denote the argument]. Both F µν and F µν have the important algebraic property that successive contractions of more than two tensors vanish, and their square is proportional to k µ k ν , e.g., If the background field is a plane-wave field, the Dirac equation [Eq.
(2) with A µ → A µ ] can be solved analytically [78]. The corresponding so-called Volkov solution with the boundary condition Ψ p → ψ p if kx → −∞ [see Eq. (4)] can be written as [50,57,79] where the phase is given by Note that Volkov states, although being an exact solution of the Dirac equation and apart from the spin-terms proportional to / k / A(kx), have a quasiclassical structure ∼ exp[iS p (x)], with S p (x) being the classical action of an electron inside a plane-wave field [77].
The dressed propagator (which is the Green's function of the interacting Dirac equation) is given by whereĒ Thus, in comparison with the vacuum case, the plane waves are replaced by the Ritus E p functions, which depend nontrivially on the plane-wave phase kx. However, they also form an orthogonal and complete set [50]: The E p functions convert the dressed momentum into the free momentum [50]: (these identities hold only if the derivative acts solely on E p,x andĒ p,x , respectively).

D. Dressed vertex
To obtain Feynman rules in momentum space, we can proceed analogously as in the vacuum case and move the E p -functions to the vertex [79]. Correspondingly, we define the dressed vertex by Working in momentum space, the only difference between vacuum QED and strong-field QED is the vertex we have to use [i.e., the free vertex in Eq. (8) is replaced by the dressed vertex in Eq. (27)]. Using the relations given in Appendix D, we can write the dressed vertex as where the phase and the coupling tensors are given by (note that G 1 and G 2 are even in the permutation kp ↔ kp while G 3 is odd). We point out that the expression given in Eq. (28) is manifestly gauge invariant, since it depends on the external field only through the tensor F µν [79].
In position space the dressed propagator in Eq. (23) can be interpreted such that the electron (or positron) continuously interacts with the external field during its propagation. Examined in momentum space, we can also visualize the influence of the external field as a modification of the coupling between the photons of the radiation field and the charged particles. From Eq. (28) we see that, besides the modification of the photon vector current interaction we also obtain a coupling to the axialvector current inside the plane-wave background. This is possible since the external field provides the pseudotensor F * µν .
Since the external field depends only on the plane-wave phase φ = kx, it is useful to use light-cone coordinates, which are discussed in Appendix C. We can then always take the integrals in dx + and dx ⊥ in Eq. (28) and obtain momentum-conserving delta functions in three of four light-cone components, where we used the notation for a general four-vector a µ . Thus, the four-momentum is only conserved up to a four-vector proportional to the plane-wave four-momentum k µ at each vertex.

E. Ward-Takahashi identity
The Ward-Takahashi identity [55,56] is a direct consequence of the gauge invariance of QED, which becomes particularly transparent in the functional integral approach [62,80]. Diagrammatically, it is a functional relation for Feynman diagrams (in momentum space), where the polarization four-vector of an external photon leg is replaced by the corresponding momentum four-vector. In Ref. [81] a perturbative proof of the Ward-Takahashi identity in vacuum QED is given. We will show now how this proof can be extended to electron-positron loops inside a plane-wave background field.
The starting point is the following algebraic identity for the dressed vertex [79] where To verify Eq. (34), we use Eq. (26) and note that the identity holds [79]. Typically, ( / p − m) and ( / p − m) in Eq. (34) cancel an adjacent propagator, and the associated momentumintegral can be taken using the relations which follow from Eq. (25). Using Eqs. (34) and (37), we can simplify diagrams which contain dressed vertices contracted with the corresponding photon four-momenta.
As an example, we consider now a closed electron loop which contains n dressed vertices and electron propagators (see Fig. 2). The ith propagator of such a loop together with its adjacent vertices is given by (the electron four-momenta p i are integrated out). If we insert now a vertex (contracted with its photon fourmomentum) at this propagator, we obtain and, by using Eqs. (34) and (37), we find that this is equivalent to Closed electron loop with n dressed vertices and propagators.
(in the first line, we have changed the name of the integration variable from p to p i ). Thus, the insertion splits the diagram into the sum of twice the original diagram with the additional four-momentum q added once at the adjacent vertex before and after the insertion. If we sum now over all possible insertion points of the loop, we obtain zero since all contributions cancel pairwise (as in the vacuum case [81]).
We point out that the above discussion is shortened, since possible issues arising due to the renormalization of the theory were not addressed (in general, the validity of the Ward-Takahashi identity may be spoiled by anomalies [82]). In this paper, however, we are mainly interested in modifications induced by the background field, which turn out to be finite. Thus, subtleties arising from manipulations of divergent integrals can be addressed as in vacuum QED.

A. General expression
The leading-order contribution to the polarization operator P µν (q 1 , q 2 ) for plane-wave background fields (see Ref. [57], Sec. 104) is determined by the diagram in Fig. 1. This diagram corresponds to the following expression: and T µν = iP µν (see Ref. [57], Sec. 115; [83]). We note that T µν (q 1 , q 2 ) is divergent, but if we write (42) the contribution in square brackets is finite [53], and the regularization of the vacuum contribution is well known [57,62]. In the following, we will focus on the tensor in square brackets which contains only the corrections induced by the external background field.

B. Evaluation of the integrals
Working in light-cone coordinates (see Appendix C) we can take all space-time integrals except of those in dx − and dy − and obtain the momentum-conserving delta Here and in the following, we write q µ if q µ 1 and q µ 2 can be used interchangeably due to the above delta function. Successively, we can take the integrals in dp − and dp ⊥ (for simplicity we will continue writing p and identify p = p + q for the components −, ⊥).
It is now convenient to introduce the two four-vectors: which obey Λ i Λ j = −δ ij , kΛ i = q i Λ j = 0 and They allow us to write the remaining phase as where we defined Due to the appearance of Λ µ i in λ µ , it is more convenient to use modified light-cone coordinates from now on [see Eq. (C11); the calculation so far is independent of this choice]. In modified light-cone coordinates, we obtain the convenient relations which simplify the algebra considerably. If the preexponent would not depend on p + and p + , both integrals could now be taken. We therefore introduce the proper-time representation of the scalar propagators [5,76]: In the following we will drop the pole prescriptions i0 and keep the replacement m 2 → m 2 − i0 in mind. Furthermore, we add the source terms ip µ j µ + ip µ j µ to the phase, which allows us to make the replacement in the trace. Now, the preexponent depends only on p − (through kp and kp ). Taking the derivatives with respect to the sources out of the integrals, we can take the integrals in dp + , dp + , which results in the delta functions, Successively, these delta functions can be used to take also the integrals in dy − and dp − . To this end we rewrite (since s + t ≥ 0) (for simplicity we keep writing y − and p − ). In particular, we obtain the identities which imply for j = j = 0 that where we defined [53] [the motivation for these definitions becomes clear in Eq. (76)]. The remaining part of the phase structure (including the part coming from the propagators and the sources) is now given by Taking the Gaussian integrals in p I and p II , we obtain the prefactor π i(s+t) , and the final phase is given by which reads for zero sources (j = j = 0) where we defined (the prefactor is chosen such that, for j = j = 0 and ψ i = 1, we obtain I i = J i = 1). Finally, we can write the tensor T µν as where the expression for 1 4 tr [. . .] µν is given in Eq. (45) with the replacement in Eq. (54) and where the sources are set to zero after the derivatives are taken.
We point out that the two four-momenta q 1 and q 2 appear asymmetrically in the final expression [see Eq. (62)]. To remove this asymmetry, we shift the x − integration by defining After this shift, the phase contains q 1 q 2 since Furthermore, we obtain (for j = j = 0) symmetric representations for the functions in Eq. (63): since kx = kz − µkq,

C. Tensor structure
In principle, the only remaining task is to evaluate the two derivatives with respect to j and j and then set j = j = 0. Despite being elementary, this is the most tedious part of the calculation, since the sources appear in many places in the final expression. The work is considerably reduced if we expand the polarization operator in a convenient basis [53]. To this end we note that q 1µ T µν (q 1 , q 2 ) = 0, T µν (q 1 , q 2 )q 2ν = 0 (69) due to the Ward-Takahashi identity (see Sec. II E).
Since the four-vectors Λ i appear in the phase [see Eq. (50)] and q i Λ j = 0, it is natural to introduce the two complete sets q 1 , Q 1 , Λ 1 , Λ 2 and q 2 , Q 2 , Λ 1 , Λ 2 , where (Q 2 1 = −q 2 1 , Q 2 2 = −q 2 2 , Q i Λ j = 0, q i Q i = 0). Using the set including q 1 for the index µ and the set including q 2 for the index ν, seven of 16 coefficients vanish due to the Ward-Takahashi identity, and we can decompose T µν (q 1 , q 2 ) as [53] It turns out that also the coefficients c 6 −c 9 vanish. If analyzed perturbatively (with respect to the external field coupling) this can be understood from Furry's theorem [53,54]. Since a closed fermion loop with an odd number of vertices does not contribute to the final amplitude, only diagrams with an even number of external field couplings (eA µ ) contribute to T µν . Due to gauge invariance and the fact that T µν is a tensor, the external field can enter only as F µν (which is linear in A µ ). Since it is not possible to construct a scalar linear in F µν using only the four-vectors q µ 1 , q µ 2 and k µ , the tensor structure cannot involve an odd number of the tensor F µν (note that q 1 Fq 2 = qFq = 0). As a consequence, the coefficients c 6 − c 9 (which are linear in Λ µ i and thus in the external field) should vanish. We will later see that this is indeed the case.
The coefficients c i in Eq. (71) can be determined by contracting T µν (q 1 , q 2 ) with the appropriate four-vectors. Especially, using again the Ward-Takahashi identity, we obtain Thus, effectively, we need to determine the contractions of T µν (q 1 , q 2 ) with the four-vectors k µ and Λ µ i to determine the coefficients c i , i.e. we need to calculate the (−, ⊥)-components of T µν (q 1 , q 2 ) in modified light-cone coordinates. Since k µ has only a +-component, the evaluation of the derivatives is now considerably simplified. Leaving the term proportional to pp aside, we see that all derivatives which act on kj or kj can be ignored. They would result in the replacement of p µ or p µ by k µ . Since k µ F µν = k µ F 2µν = k µ F * µν = 0 and k 2 = kΛ i = 0, we do not need to consider those terms. The derivatives acting on kj or kj are therefore only important to determine the term proportional to pp . However, this is achieved more easily if the calculation presented in Sec. III B is repeated with a scalar source term J pp in the exponent (see Sec. III E).
To calculate the preexponent of the polarization operator, we must now insert the explicit expressions given in Eq. (30) into the trace in Eq. (45). Many terms of the trace, e.g., the terms proportional to F µν , F 2µν , F 2µρ p ρ vanish, as they are contracted with k µ or Λ µ i from each side. Using the relations in Appendix E, we can show that Eq. (45) can be substituted by the following expression: Since the term proportional to pp enters as g µν , it modifies only the diagonal coefficients c 3 and c 4 .

D. Evaluation of the derivatives
Leaving the term proportional to pp aside, we can ignore derivatives acting on kj and kj as discussed above [this implies that the derivatives do not act on kp, kp , and ky; see Eq. (57)]. The remaining nontrivial sourcedependent part of the phase is given by [see Eq. (61) The squared term contributes only if both derivatives act on it, which results in the replacement and the only nonzero contribution arises from the first three terms in Eq. (73). If the derivatives act on the other source terms, we obtain the replacement After these replacements are applied to Eq. (73) and the sources are set to zero, we obtain (effectively) the following expression for Eq. (73): [note that terms proportional to (Fλ) µ , (Fλ) ν can be omitted]. By changing the proper-time integrations from we see that the terms linear in v vanish. Those terms determine the coefficients c 6 − c 9 , which are therefore zero (as already anticipated from Furry's theorem).

E. Scalar term
To determine the term proportional to pp , we add the scalar source term iJ pp to the phase (instead of ip µ j µ + ip µ j µ ) and repeat the calculation presented in Sec. III B. The propagators are represented in the same way [see Eq. (53)], and we replace pp by −i ∂ ∂J . Then, we take the integrals in dx + , dx ⊥ , dy + , dy ⊥ , dp − , dp ⊥ , dp + , and dp + . Instead of Eq. (55), we obtain now The remaining part of the phase (including the part from the propagator) can be written as It is now convenient to shift the proper-time integrations Due to this shift, also the integral boundaries of the proper-time integrations depend on the source. However, if the derivative acts on the integral boundaries, either s or t is set to zero or to infinity. The resulting terms do not depend on the external field since s = 0 or t = 0 implies µ = 0, ky = kx and thus λ µ = 0 and Λ = 0. On the other hand, the terms at s → ∞ or t → ∞ do not contribute because the field-dependent part of the integral is convergent. Since we want to calculate only the field-dependent part of the polarization operator [see Eq. (92)], we will ignore the source dependence of the integral boundaries.
After the shift in Eq. (80), the delta functions read (81) and the phase is given by We can now use the delta functions to take the integrals in dy − and dp − (we keep writing y − and p − for convenience). We then obtain the identities [for J = 0 this agrees with Eq. (57)]. The shift in the proper-time integrals has the advantage that kp and kp are now independent of J . We could have proceeded similarly also in the calculation of the other terms. However, since we ignored sources contracted with k, this was not necessary.
Taking now the Gaussian integrals in dp I , dp II , we obtain the prefactor π i(s+t) , and the final phase is given by where I i and J i are defined in Eq. (63) [for zero sources Eq. (84) agrees with Eq. (62)]. Since pp in the preexponent is only multiplied by g µν [see Eq. (76)], the evaluation of the derivative is not very cumbersome, and we obtain the replacement after J is set to zero (as explained above, we have ignored the source dependence of the proper-time integral boundaries).
To symmetrize the final expression, we change the x −integration by defining [see Eq. (65)] (z − = z − for J = 0), which implies and Finally, we obtain the symmetric replacement (we assume that at x − = ±∞ the external field vanishes, and therefore the derivative does not act on the integral boundaries, which now also depend on the source).

F. Final result
To determine the nonvanishing coefficients c 1 − c 5 of the polarization operator [see Eq. (71)], we combine now Eqs. (64), (65), (76), and (89). Furthermore, we define the following functions: and note that, for j = j = 0, Using these relations, we obtain the following expression for the field-dependent part of the tensor T µν : where the field-independent phase reads [see Eqs. (62) and (66)] [µ = 1 4 τ (1 − v 2 ); see Eq. (59)] and The field-dependent phase is given by [see Eq. (62)] where [see Eq. (67)] and [see Eq. (90)] We note that, using the metric tensor g µν , we can construct the following projection tensor [54]: which obeys and can be decomposed as This shows that the decomposition given in Eq. (92) has the structure claimed in Ref. [54].

A. Comparison with the literature
The expression we obtained for the field-dependent part of T µν in Eq. (92) is manifestly symmetric in q 1 and q 2 . We will now show how the alternative representation found in Ref. [53] can be derived from our calculation. To this end we do not apply the shift in Eqs. (65) and (86), which means that we have to use the replacement given in Eq. (85) [rather than Eq. (89)] for the pp term in Eq. (76). This modifies the coefficients b 3 and b 4 . Furthermore, we introduce the variable which allows us to write [see Eq. (66)] and [see Eq. (57)] (here and in the remaining subsection, we assume that all sources are set to zero). Thus, we obtain the following representation [see Eq. (63)]: where we introduced [53] ∆ i (r) = ψ i (kz − 2kqr) − ψ i (kz ).
Furthermore, it is useful to define [compare with Eq. (90)] which can be written as Finally, we obtain the following alternative representation for the field-dependent part of T µν : and phases This representation coincides with Eq. (2.27) in Ref. [53].

B. Constant-crossed field
The polarization operator for a constant-crossed field was first obtained in Refs. [84,85] (see also Refs. [50,52,86]). We show now how this result can be obtained from the expression in Eq. (92).
A constant-crossed field is characterized by (the latter condition corresponds to ξ 2 = 0, and we will write ξ = ξ 1 in this paragraph). The field tensor and its square are then given by [see Eq. (11)] For a constant-crossed field, we obtain After inserting these expressions into Eq. (92), we can take the integral in dz − and obtain the remaining delta function 2π δ (+) (q 1 − q 2 ), which implies that the polarization tensor for a constant-crossed field is diagonal in the external photon four-momenta. We define therefore the four-vectors [see Eq. (70)] They obey The four-vectors q µ , Q µ , Λ µ 1 , and Λ µ 2 form a complete set, and we obtain the following representation of the metric tensor: From Eq. (92) we obtain now the following representation of the field-dependent part of T µν in a constant-crossed field [see Eq. (112)]: and the phases are given by (in the following, we will make the change of variables τ → t, where τ 3 b = t 3 and ρ = a/ 3 √ b). Here we have introduced the quantum nonlinearity parameter (κ in Refs. [50,84]). We can rewrite now where and obtain [see Eq. (98)] We note the following relations: To obtain the representation given in Refs. [50,52], we pass over to different basis tensors (126) and define the following functions [50,52]: and where Ai and Gi denote the Airy and Scorer functions, respectively [87]. These functions obey the following differential equations: Using the latter, we can replace the function f 1 (x) by f (x) in the following way (if all boundary terms vanish): where G (v) = g(v).
Using the above notation, we can represent the fielddependent part of the tensor T µν for a constant-crossed field given in Eq. (118) by where Since all nonvanishing functions are even in v, we can now apply the following change of variables: which shows that the result in Eq. (133) is equivalent to the one given in Refs. [50,52].
By employing the identity |kq| = m 2 χ/ξ, we can expand all functions depending on µkq (X 12 = X 21 = X 22 = Z 2 = I 2 = J 2 = 0 for linear polarization). Thus, if multiplied by ξ 2 , only the leadingorder terms are independent of ξ, and all others are suppressed. In the limit ξ → ∞, the expressions in Eq. (138) and Eq. (100), we obtain for a linearly polarized planewave field in the quasiclassical approximation the following representation for the field-dependent part of the tensor T µν [see Eq. (92)]: where [see Eq. (134)]
In order to obtain Eq. (155) from Eq. (148), we have used the identity via integration by parts.

V. CONCLUSION
In the present paper, we have proven for the first time the Ward-Takahashi identity for general loop diagrams in a plane-wave background field (see Sec. II E). Moreover, we have presented a new derivation of the leading-order contribution to the polarization operator in a plane-wave background field for arbitrary polarization and dependence on the plane-wave phase (see Sec. III). Our calculation relies on a direct evaluation of the space-time integrals without using Schwinger's operator method [5] that was employed in Ref. [53]. An interesting feature of our final representation is the manifest symmetry with respect to the external photon four-momenta q 1 and q 2 [see Eq. (92)]. ACKNOWLEDGMENTS S.M. is grateful to the Studienstiftung des deutschen Volkes for financial support.

Appendix A: Notation
In this paper we use natural units = c = 1 (in some formulas and c are restored for clarity) and the charge is measured in Heaviside-Lorentz units ( 0 = 1). The electron mass and charge are denoted by m and e < 0, respectively. Thus, the fine-structure constant is given by α = e 2 /(4π) ≈ 1/137. In covariant expressions the space-time metric g µν with signature (1, −1, −1, −1) is used, and ∂ µ = (∂/∂t, ∇) is the four-derivative. This implies ∂ µ x ν = g µν , where x µ = (t, x) denotes the position four-vector. The unit tensor is denoted by δ µ ν = g µρ g ρν = diag(1, 1, 1, 1) (δ µ µ = 4), and space-time indices (lowercase Greek letters) are raised and lowered using the metric a µ = g µν a ν (summation over all types of repeated indices is understood if they do not appear on both sides of an equation). Greek and Latin indices take the values (0,1,2,3) and (1,2,3), respectively. Contractions of four-vectors are denoted by a µ b µ = ab and scalar products of three-vectors by a i b i = ab. We denote the dual of a second-rank tensor T µν by T * µν = 1 2 µνρσ T ρσ , where µνρσ is the totally antisymmetric tensor in four dimensions with 0123 = − 0123 = 1. For contractions of second-rank tensors and vectors, a matrix notation is sometimes used, e.g. aT b = a µ T µν b ν , (T 1 T 2 ) µν = T µ 1ρ T ρν 2 , T 2µν = T µρ T ν ρ , (T a) µ = T µν a ν . All spinors are Dirac spinors (with four components); spinor indices are usual suppressed. The Dirac gamma matrices are denoted by γ µ , / a = a µ γ µ , γ 5 = −iγ 0 γ 1 γ 2 γ 3 and 2σ µν = γ µ γ ν − γ ν γ µ (γ µ γ ν + γ ν γ µ = 2g µν ). For a spinor u, we defineū = u † γ 0 , and for a matrix in spinor space M , correspondinglyM = γ 0 M † γ 0 . A quantization volume V = 1 is assumed for the normalization of the single-particle electron, positron and photon states. The total derivative of a function with respect to its argument is denoted by a prime f (x) = d dx f (x). Integrals without boundaries range from −∞ to +∞. We use i0 as a short notation for i together with the limit lim →0 + . In general, our notation therefore follows Ref. [57] with different units for charge.
Eq. (D1) for both sides of the equation. It is in particular possible to map the gamma matrix algebra to a corresponding tensor algebra once the decomposition of the product of two (arbitrary) gamma matrix expressions is known,