Strong-field QED in Furry-picture momentum-space formulation: Ward identities and Feynman diagrams

The impact of a strong electromagnetic background field on otherwise perturbative QED processes is studied in the momentum-space formulation. The univariate background field is assumed to have finite support in time, thus being suitable to provide a model for a strong laser pulse in plane-wave approximation. The usually employed Furry picture in position space must be equipped with some non-obvious terms to ensure the Ward identity. In contrast, the momentum space formulation allows for an easy and systematic account of these terms, both globally and order-by-order in the weak-field expansion. In the limit of an infinitely long-acting (monochromatic) background field, these terms become gradually suppressed, and the standard perturbative QED Feynman diagrams are recovered in the leading-order weak-field limit. A few examples of three- and four-point amplitudes are considered to demonstrate the application of our Feynman rules which employ free Dirac spinors, the free photon propagator, and the free Fermion propagator, while the external field impact is solely encoded in the Fermion-Fermion-photon vertex function. The appearance of on-/off-shell contributions, singular structures, and Oleinik resonances is pointed out.


I. INTRODUCTION
With the advent of permanently increasing laser intensities by advanced technologies [1] one gets access towards the strong-field regime of QED [2] in a hitherto uncharted region of parameter space.Among the famous examples are the "vacuum break-down" by the Sauter-Schwinger effect (cf. [3][4][5]7] for recent activities and entries to extensive citations) and the unsettled implications [6,8,9] of the Ritus-Narozhny conjecture (cf.[10] for an introduction).While QED delivers unprecedentedly accurate results in certain regions of the parameter space, at high energies of the involved particles and for processes in strong background fields there is still room for testing the theory for "physics beyond the Standard Model" or verifying long-standing predictions.For instance, the soft-photon theorems -at the heart of the IR structure of QED -seem to fail when many hadrons are involved in the final state of a high-energy strong-interaction collisions.This issue triggered activities for new detector concepts and plans of experimental investigations at the LHC [11].In fact, Strominger's IR triangle [12] finds recently much interest culminating in "new symmetries of QED" [13], see e.g.[14].
The current standard approach to calculations of QED processes in strong background fields deploys the Furry picture in position space.Fermions (electrons and positrons) are dressed by accounting exactly for the (quasi-classical) interaction with the electromagnetic background field, while the interaction of Fermions with photons is dealt with in order-α perturbative expansion, see [15][16][17] for reviews.A convenient model of the laser field is accomplished in the plane-wave approximation.The external field, taken as given background (for back-reaction dynamics, cf.[18,19]), depends then only on one variable, the invariant phase φ := k • x = ωt − with light-front time t − = x 0 − x in a coordinate system where k e z .Due to the high symmetry, the Dirac equation can be solved exactly, delivering the Volkov solution, which depends trivially on three components of space-time and, generally, highly nontrivial on φ or t − via the background.For a laser pulse, the background has finite support in the variables φ and t − , and the details of the temporal structure shape the final phase-space distribution in a distinct manner.The limiting case of an infinitely long-acting external field is called a monochromatic field.It is to be contrasted with the sandwich field, where -like in the situation of a passing gravitational wave -an observer sees the vacuum, followed by the pulse, and is left afterward in the vacuum again, irrespective of memory effects imparted on test particles.
It is often taken for guaranteed that the weak-field limit of the background field (which is a classical field) and the monochromatic limit (i.e.infinite support at constant strength) yields the standard perturbative QED results obtained via Feynman diagrams in momentum space.The role of loops is less obvious in that correspondence.A particular situation is when considering a short weak (classical background) field pulse: A straightforward treatment by perturbative QED via Feynman diagrams does not catch the features caused by the higher/lower Fourier components of the weak field, see [20].In particular, [21] points out that, to preserve Ward identities, one has to add to the Furry-picture position-space Feynman diagrams some terms "by hand".It happens that a concise formulation in momentum space 1provides, in a clear manner, these necessary terms, while their origin and relation to gauge invariance are obscured in position space.
It is the goal of the present paper to dilate on the strong-field QED Furry-picture in the momentum space.Our approach has been outlined in [23].The key is to employ Ritus matrices [24] and to accumulate all dependencies on the external classical field in the dressed vertex [25], while keeping vacuum photon and Fermion lines for propagators and in/out states.Our presentation enables easy access to the systematic expansion of amplitudes and probabilities/cross sections in powers of the classical laser intensity parameter a 0 .In the lowest order of small a 0 we obtain "pulsed perturbative QED" which accounts for temporal pulse shape effects [20,23]; the very special case of a monochromatic external classical field recovers standard perturbative QED.
Our paper is organized as follows.In Section II we present the formal development of the momentum-space Feynman rules with emphasis on implementing gauge invariance, definition of the Fermion-Fermion-photon vertex and the graphical representation.The monochromatic limit is considered too and some remarks on soft factors are supplied as well.
Examples of applications are introduced in Section III.The explication of the three-point amplitude, that is for the one-vertex processes nonlinear Compton/Breit-Wheeler/one-photon annihilation, is spelt out in Section IV to demonstrate the path from our rules towards the basic formulation of already exhaustively analyzed phenomena.The two two-vertex processes and related four-point amplitudes are dealt with in Section V with some details w.r.t.non-linear two-photon Compton and nonlinear Møller scatterings and crossing channels.Special aspects of Oleinik resonances, singularity structures, and the monochromatic limit are uncovered here.We conclude in Section VI.Appendix A deals in some detail with the regularization to ensure gauge invariance.

A. Background field description
The following considerations apply to Lorenzian null fields, i.e. the classical background field has the structure where, in units with c = = 1, m and e denote the electron's rest mass and electric charge, µ 1,2 refer to the laser's polarization vector (e.g.µ 1 = (0, 1, 0, 0) and µ 2 = (0, 0, 1, 0) in a reference frame where k µ = ω(1, 0, 0, 1)), and the carrier envelope phase reads φ CEP .Side conditions specify further this model of a laser pulse in plane-wave approximation: or elliptic polarization for other values of ξ. g denotes the pulse shape function or envelope for a shortcut with ∆φ as the pulse width parameter.Scalar products of four-vectors are henceforth noted as dot products.

B. Dressed vertex decomposition
The dressed vertex is defined by [25] ∆ where p (p ) is the in-(out-) going Fermion four-momentum, and the outgoing photon fourvector is denoted by k .By inserting the Ritus matrices, e.g.
the second line follows, defining the dressed vertex function: Note (i) the crucial "photon number parameter" as Fourier conjugate of the phase φ and (ii) the decomposition into elementary vertices {γ µ , Γ µν 1 , Γ µ 2 }, depending on {p, p , k}, and phase integrals {B 0 , B µ 1 , B 2 }, depending on as well The function G reads where φ 0 → −∞ is a useful choice for pulses and The phase integral (11) in Eq. ( 7) needs a regularization which takes care of the Ward identity k • Γ = 0.This is accomplished by (see Appendix A) where the instruction P means taking Cauchy's principle value and Eventually, the dressed vertex function (6,7) is decomposed as We have suppressed the pertinent arguments p, p , k in all functions, but highlighted the dependence.The term labeled by "div" could be named "gauge invariance restoration term", since it emerges just from that requirement.
The decomposition of the vertex function implies the following modification of standard Feynman rules in momentum space: use d 2π (−ie)Γ µ ( ) for the Fermion-Fermion-photon vertex, instead of −ieγ µ , and integrate over internal momenta.While a N -vertex Furrypicture diagram in position space involves N space-time integrals (cf.[15] of how to process the expressions), in momentum space one meets N integrations over the respective vertexattached "photon number parameter" .
In a follow-up paper, we show in more detail that, in leading order of a series expansion in powers of a 0 , the standard momentum Feynman rules are recovered and explicate the NLO terms.

C. Expansion in powers of a 0
One benefit of our momentum space formulation is the possibility of a straightforward series expansion of the amplitude in powers of a 0 .The temporal pulse shape is imprinted transparently in the weak-field limit.To begin with, we introduce for the bookkeeping of powers of a 0 the tilde notation: every quantity with a tilde is free of any dependence on a 0 , e.g.A µ (a 0 , φ) = a 0 Ãµ (φ), which implies when using the abbreviations Ã1 := α µ Analogously, the phase integrals (12,13) become sums of Fourier transforms: Proceeding in such a manner we represent the vertex functions (19,20) as where the Heaviside distribution ensures to take only N > 1 contributions in the last term.
We note (i) Γµ div N =0 = 2πγ µ δ( ) due to G± N =0 = 1; δ( ) enforces = 0 meaning that this vertex part does not exchange energy-momentum with the external field, and (ii) the N = 1 contribution to Γ µ reg is solely related to Bµ 1 = ∞ −∞ dφ e i φ Ãµ (φ), i.e. the Fourier transform of the background field; it is the only linear contribution (in a 0 and the background field), while contributions ∝ a N 0 with N > 1 are nonlinear in the background field.

D. Graphical representations
The above vertex function decomposition facilitates the following graphical representation with the a 0 expansion in the bottom lines: .
Each expression must be supplemented by a factor of −ieδ (4) (p+ k−p −k ).The pertinent arguments , p, p , k are not displayed here.Note that the expanded vertex functions in (28) are ∝ a N 0 , thus allowing for easy bookkeeping.
E. Monochromatic limit: ∆φ → ∞ The above formalism is devised to include arbitrary shape functions g(φ, ∆φ).For specified (explicitly given) temporal pulse structures, G(φ) can be explicated and the phase integrals B 0 , B µ 1 and B 2 can be executed to arrive at special functions (cf.[26] for such an example w.r.t.nonlinear Compton).A very special case is the monochromatic planewave background field.Formally, ∆φ → ∞ and the envelope function in Eq. (1) obeys g(φ, ∆φ) = 1 for all φ ∈ R. Setting φ 0 = 0 is a suitable choice for monochromatic planewave backgrounds, since then the initial condition is known due to A µ (φ 0 ) = a µ 1 cos ξ and the nonlinear phase defined in ( 14) is finite for finite values of φ.The distinction of Γ µ div and Γ µ reg is no longer suitable, since also B 0 , B µ 1 and B 2 in Γ µ reg contribute to the so-called δ-comb, which refers to individual harmonics.The carrier-envelope phase φ CEP is irrelevant and can be skipped.Under these prepositions, Eq. ( 21) yields where , and a ≡ a 0 m/|e|, β := 1 2 a 2 α 2 .Insertion into Eqs.(11,12,13) results in and allows the representation of the fully dressed vertex as where we use the abbreviation Γ µ ± := −,ν Γ µν 1 ; cf.Eqs.(50,51) below for the definition of µ ± and the decomposition of B µ 1 into B ± .To make some intermediate steps more obvious, we note Using the Jacobi-Anger expansion of the non-Fourier part one gets where J n stand for Bessel functions of the first kind.The phase integral (36) reads then and the index shift n → n ± 1 yields Eq. ( 31).

F. Recovery of Feynman rules of perturbative QED
The relation to standard perturbative QED Feynman rules has two facets.
(i) Get the Fermion-Fermion-photon vertex from the dressed vertex by inspecting Eq. ( 28) and noting that, at a 0 → 0 and for off-shell legs, G → 2π and the leading order term in the (ii) However, in the case of a monochromatic background and for on-shell legs (e.g. for Compton), the leading-order terms in the a 0 expansion stem from the vertex ¡ = ∞ N =1 a N 0 ¡ N since (for on-shell amplitude) ¡ = 0.The powers of a 0 are ex- pelled off the regularized vertex, which becomes ¡ N → ¡ . . .with N incoming laser photon lines ¡ in all permutations.
On the level of cross-section, dσ/dΩ dω ), where, upon executing the spin and polarization sums, , the Klein-Nishina cross section is recovered; the energy-momentum balance via the delta distribution arises from phase space integration, dΦ , and only for ∆φ → ∞.
G. Soft photons: k → 0 Lowest-order soft theorems 2 allow for factorizing amplitudes as M = M hard × S, where S is the soft factor accounting for the emission of a soft photon and M hard is the amplitude of hard interaction.Focusing first on the soft-photon emission off an external leg, e.g. an incoming electron, the corresponding matrix element reads where M hard is part of the matrix element which emerges from the interaction of the electron with other non-soft particles and µ (k ) denotes the polarization of the emitted soft photon.
Since k is small compared with the electron momenta, one has the out-going momentum p appears only in the product p • k in Γ µ ( , p, p ). Inserting the decomposition (18) in the matrix element (42), the infrared behavior of k must be examined for two parts: (i) The gauge-restoration part Γ µ div : The corresponding matrix element becomes where the integration is executed by employing the δ-distribution leading to = 0. Considering the nonlinear Volkov phase G defined in ( 14  Similarly to Section II G, the soft interaction with the background field refers to the limit k → 0, which is equivalent to ω → 0, where we set k µ = ωn µ with the normalized reference momentum n µ .Consequently, this soft limit needs to be examined separately for the two parts of the dressed vertex. (i) Regularized part Γ µ reg -Considering the elementary vertices defined in Eqs. ( 9) and ( 10), we find Γ µν nn µ 2 n•p n•p , which are both finite in the limit ω → 0. On the contrary, the kinematic factors α µ 1 , α 2 defined in (15) have the typical form of Weinberg's soft factors [28], hence a divergence in the soft limit.Therefore, for ω → 0, we have Γ µν 1 γ µ α ν 1 and Γ µ 2 γ µ α 2 for all finite .This leads to the soft limit of the finite part of the dressed vertex (20): lim ω→0 Γ µ reg ( ) = γ µ S sof t reg ( ), where the regularized soft factor reads Linearizing the phase integrals in a 0 , i.e. considering only Bµ 1,1 ( ) given in (23) and dropping B 2 ∼ a 2 0 , this soft-factor results in , which recovers the softfactor elaborated in [21] for the case of Bhabha scattering.Furthermore, considering linear polarisation A µ = a µ g(φ, ∆φ) cos φ and performing the simultaneous limit a 0 → 0 and ∆φ → ∞, the soft-factor reads S sof t reg ( ) → −aα µ 1 = ae p k•p − p k•p , which indeed is, up to a constant normalization, Weinberg's well-known soft-factor.
(ii) Gauge restoration part Γ µ div -First we note that the integrals appearing in the prefactor of the gauge restoration part G defined in Eq. ( 17), ±∞ φ 0 A µ (φ ) dφ and ±∞ φ 0 A 2 (φ ) dφ , are finite in the soft limit ω → 0, whereas the exponents G ± of G diverge due to the presence of the factors α µ 1 , α 2 .This means, the factor G itself acts like a soft factor, which highly oscillates in the soft limit.However, considering the linearization of this soft factor in a 0 , we dφ , which again recovers the form known from Weinberg's soft factors, but this time with the integrated fields acting as polarization vectors.
In summary, it can be stated that in both cases, soft photon emission and soft interaction with the background field, generalized versions of typical soft factors appear, which can be, in a suitable limit, connected to soft factors well-known from monochromatic QED.However, the cancellation of the soft factors shown here with higher-order vertex corrections, i.e. finding a generalized version of the Bloch-Nordsieck theorem, deserves separate work.

III. EXAMPLES & FUTURE APPLICATIONS
The above formalism is ready for direct numerical applications.Elements are Dirac spinors, Dirac matrices, the metric tensor, momentum, and polarization four-vectors; Fermion and photons propagators are as in free-field and could be defined as stand-alone objects; most importantly, the nonlinear phase integrals encode solely the external field and require some care and numerical optimization.For a given exclusive reaction, these elements are to be connected by scalar and matrix products, thus delivering a few partial amplitudes (e.g.direct and exchange terms or the multitude of diagrams with the same out-state) to be summed up to one complex number -the amplitude M. Its mod-square, |M| 2 , is to be garnished to arrive eventually at probability or cross-section which depend on spins, polarizations, and invariants referring to the initial state including the background field and final phase space.Partial or complete integration over the final phase space variables need often specially adapted procedures, while spin/polarization summations, if required, are straightforward.Handling of the δ distributions is analog to position space formulation: All dependence is integrated out before squaring the amplitude, and finally use We refrain here from such specific numerologies but instead stress the need for an in-depth understanding of the essential dependencies and singular structures of M, as the the core of |M| 2 , prior to numerical evaluations.
To sketch applications of the presented formalism we (re)consider one-and two-vertex processes related to three-and four-point amplitudes of nonlinear (one-and two-photon) Compton and Møller scattering processes.The detailed application to nonlinear trident is relegated to an accompanying paper.The following Section IV recalls the one-vertex processes by demonstrating how the above rules lead to the known matrix elements, in particular for nonlinear Compton with a few supplementing remarks on nonlinear Breit-Wheeler.The next-to-one Section V considers the two two-vertex processes with emphasis on nonlinear two-photon Compton and nonlinear Møller scattering.

IV. ONE-VERTEX PROCESSES/THREE-POINT AMPLITUDE
The matrix element of one-vertex processes has, symbolically, the structure M ∼ J µ L E µ (k) with current J µ L ∼ ūL (p )Γ µ u L (p), where the label "L" is a reminder of the laser dressing of charged Fermions with wave functions u and its adjoints ū, and E µ (k) stands for the photon L ⇒ e ∓ L applies.

A. Nonlinear Compton
The matrix element for nonlinear Compton (nlC) scattering reads where * µ (k ) stands for the polarization four-vector of the outgoing photon, and we mark tied Fermion lines by [• • • ].The vertex decomposition (18) facilitates two contributions, M div nlC and M reg nlC .The one related to Γ µ div (19) refers to the "gauge restoration part": which vanishes since the δ( ) term, upon integration, enforces the balance equation p−p −k for on-shell momenta, leaving no phase space.The non-zero term, related to Γ µ reg (20), becomes where in the last two lines the light cone coordinates are employed: δ lf (q) := 1 2 δ(q − )δ (2) (q ⊥ ).The photon number parameter is 0 = (p+k ) 2 −m 2 2k•p .The matrix element (48) is the starting point for many investigations of one-photon nonlinear Compton in a pulsed plane-wave background.To make this relation explicit we rewrite Eq. ( 14) by means of (1) and a := where the phase integrals (cf.(12,13)) in ( 48) can be cast in the form yielding eventually Eq. (3.26) in [29] with many accompanying and subsequent works.The one-photon nonlinear Compton process based on the one-vertex diagram seems to be exhaustively analyzed (cf.[15] for a recent review).The special setup of multi-color laser background field, e.g. the superposition of aligned optical and XFEL beams, i.e. x-ray scattering at an electron moving in the laser field [30,31], offer further interesting facets, up to polarization gating to produce a mono-energetic γ beam [32,33]).Furthermore, nonlinear Compton has non-perturbative contributions (analog to nonlinear Breit-Wheeler), and [34] shows how to isolate them.
With respect to power counting of e and a 0 , the assignments are displayed in Eq. ( 48), where (• • • ) stands for the series expansion of the exp{iG} term.

B. Nonlinear Breit-Wheeler
The matrix elements of nonlinear Breit-Wheeler (nlBW) refer to and read, analog to the nonlinear Compton as crossing channel Eqs. ( 45 -48), where M div nlBW → 0 is a result of combining the δ distributions δ( 0 ) δ lf (k − p p − p e ) → δ (4) (k − p e − p e ) which can not be satisfied by on-shell momenta.
While, in nonlinear Compton, the initial electron momentum may be zero, p = 0, due to the action of the external classical field, the electron can emit a real photon(s), whatever the external-field central-frequency ω > 0 is.One can imagine this as shaking off photons due to the quiver motion in the external field.The crossing channel, i.e. nonlinear Breit-Wheeler as one-photon decay, γ → e + L e − L with matrix element ∝ k µ [v p Γ µ ūp ]), faces a severe threshold, making nonlinear Compton and nonlinear Breit-Wheeler quite distinctive, even the amplitudes are related by crossing symmetry.The balance equations in the monochromatic case read with quasi-momenta q = p + a 2 0 m 2 /(2p • k) and q = p + a 2 0 m 2 /(2p • k) which facilitate Explication for nonlinear Compton (head-on laser-electron collisions) reads p = (m cosh y, −m sinh y, 0, 0), ( 62) where E = m cosh y and | p | = m sinh y relates energy E and momentum p, E 2 + p 2 = m 2 , with rapidity y, and express ω and E as a function of cos Θ .Forward (backward) scattering is defined by cos Θ = 1 (−1).The out-electron angle is determined by sin θ = −ω sin Θ /|p |.
In nonlinear Breit-Wheeler, the quasi-momenta q and q symmetrically enter the corresponding kinematic equations.The threshold energy, for the monochromatic case, is de- However, the sub-threshold pair production is enabled in short pulses, as emphasized in [46][47][48][49][50]. Temporal double pulses or bichromatic pulses enhance further the pair rate, as suggested in [51][52][53].
While Compton has a classical analog (shaking off the e.m. field accompanying an accelerated charge in the form of asymptotically outgoing waves), Breit-Wheeler is said to be a quantum process, i.e. "converting light into matter".A particularly interesting aspect is the relation to vacuum birefringence, see [54], which is experimentally searched for in dedicated and highly specialized & optimized set-ups, e.g.pursued by HIBEF [55][56][57][58].

V. TWO-VERTEX PROCESSES/FOUR-POINT AMPLITUDE
The two two-vertex diagrams have the symbolic matrix elements (i) M ∼ with S F and D µν as Fermion and photon propagators.They have one (i) and two (ii) tied Fermion lines.Again, depending on the orientation of the four-momenta, several processes related by crossing symmetry are conceivable: L + e + L (nonlinear trident) and several crossing channels as well (e.g.nonlinear Bhabha scattering with s and t channel diagrams).Also, the involvement of two different lepton species is conceivable, e.g.electrons and muons.
We explicate now our momentum space Furry-picture Feynman-rules for nonlinear twophoton Compton (subsection V A) and nonlinear Møller (subsection V B).
A. Two-photon nonlinear Compton

Diagrams and matrix element
The two-photon nonlinear Compton (2nlC) e − (p) + laser → γ(k 1 ) + γ(k 2 ) + e − (p ) + laser as a two-vertex tree-level diagram requires a somewhat more intricate treatment, see [29,[35][36][37][38], despite the simple matrix element (direct term, i.e. left diagram; the exchange term, i.e. right diagram, is to be processed analogously) where is Feynman's free Fermion propagator.Collinear divergence and IR behavior as well as the on/off-shell behavior of the Fermion propagator provide some challenges.In addition, the soft-photon theorem (cf.[39] for contemporary reasoning) might be explicated here.
We consider now only the structure of the direct matrix element (68), where p and p denote the momenta of the in-and out-going electrons, and k 1,2 are momenta of the outgoing photons with polarization four-vectors µ,ν * (k 1,2 ); Q refers to the intermediate electron.
Using one of the δ-distributions, the integral over the intermediate-electron momentum Q can be solved analytically: where Q is now related to the external momenta and the photon number parameters via The remaining δ-distribution in (69) can be used to solve one of the photon-number parameter integrals by applying light-cone coordinates to the involved momenta: where δ lf(q) = 1 2 δ (2) (q ⊥ ) δ(q − ).The second δ-distribution in (71) can be used to solve one of the integrals over the photon number parameter, e.g. the r-integral, which leads to where the two photon number parameters are no longer independent, but related by r := 0 − with

Singularity structures
One obvious source of singularities is the vanishing denominator of the field-free electron propagator S F (Q). Therefore, the general resonance condition is given if the intermediate electron goes on-shell, i.e.Q 2 = m 2 .One convenient way to keep control of the propagator singularity is to define the virtuality of the intermediate electron: Eq. ( 70), the virtuality is a function of either one of the two photon-number parameters: where we have ν ≡ ν r = 0 − .The resonance condition ν := Q 2 − m 2 is then equivalent to ν = ν r = 0. Written with the virtuality ν , the denominator of the field-free fermion propagator reads where we employ the replacement 2k•(p−k 1 ) → and we use the abbreviation Analogously, written with the virtuality ν r , one gets Consequently, the singularity structure of the electron propagator in the matrix element (72) is directly related to the values of the photon number parameters at the respective vertex, i.e. ν = 0 ⇔ = on or equivalently ν r = 0 ⇔ r = r on .
As we will show in the sequel, these types of singularities are the only ones, which may appear in the matrix element (72).

Asymptotically vanishing field case
As illustrated in Section II B in the case of plane-wave pulses, i.e. asymptotically vanishing fields, the manifestly gauge-invariant dressed vertex function Γ µ decomposes into a finite Γ µ reg and a gauge-restoration Γ µ div part.Consequently, inserting the decomposition by Eqs.(18 -20) into Eq.( 72), the matrix element of strong-field two-photon-Compton scattering becomes where the vertex structure of the particular matrix element introduces constraints on the respective photon-number parameter.We consider term by term: (i) No energy-momentum transfer -The first part of (79) corresponds to the case, where both vertices are given as the divergent part of the dressed vertex, which implies that for this diagram, there is no energy-momentum transfer w.r.t. the background field on neither of the vertices.This follows also directly from the corresponding part of the 2nlC matrix element, which reads where the δ-distributions from the gauge-restoration parts are solved by = 0 = r 0 ≡ 0 .
The remaining δ-distributions only depend on the external particles.Therefore, using ( 71) and (73) leads to However, there is no physical phase space solving p − k 1 − k 2 − p = 0 with all on-shell momenta.Furthermore, for the first part of (79), the virtualities read Therefore, there is no singularity to cancel the vanishing phase space, thus there is no contribution of S 2nlC to the matrix element.(ii) Contribution from the left vertex -The second term in the decomposition (79) represents the case, where at the left vertex, the photon-number parameter does not vanish, i.e. = 0, whereas, on the right vertex, the photon-number parameter is identically zero: r = 0 − = 0.The corresponding part of the strong-field 2nlC matrix element is given by = −e 2 π (2π where 0 is given by Eq. ( 73) and Γ µ reg is the finite part of the dressed vertex defined in Eq. (20).The Cauchy principal value operator in Γ µ reg ensures l 0 = 0.For this part of the matrix element, the virtualities (74) and (75) read thus, there is no singularity in S 2nlC .(iii) Contribution from the right vertex -The third term in the decomposition (79) represents the case, where at the right vertex, the photon-number parameter does not vanish, i.e. r l = 0, whereas, at the left vertex, the parameter is identically zero: = 0.The corresponding part of the strong-field 2nlC matrix element is given as where r 0 ≡ 0 is given by Eq. ( 73) and Γ µ reg is again the regularized finite part of the dressed vertex defined in Eq. ( 20) and the Cauchy principal value operator in Γ µ reg ensures r 0 = 0.For this part of the matrix element, the virtualities (74) and (75) read thus, there is no singularity in S 2nlC .(iv) On-shell-and off-shell contributions -The fourth term in the decomposition (79) represents the case, where on both vertices the photon-number parameters are non-zero, i.e. = 0 and r = 0.The corresponding part of the strong-field 2nlC matrix element is given as where Γ µ reg is the finite part of the dressed vertex defined in Eq. (20).The Cauchy principal value operator in Γ µ reg ensures = 0 = r .However, according to Eq. ( 76), the condition = on , with on defined in (77), is equivalent to Q 2 − m 2 = 0 inducing the propagator m 2 +i to diverge in the limit → 0. This divergence can be handled by applying the Sokhotski-Plemelj theorem: where, on the r.h.s., in the first term the denominator of the propagator is removed and Q is set on-shell, i.e.Q 2 != m 2 , or equivalently != on , where on is given in Eq. ( 77).In the second term of the r.h.s. of equation ( 94), the Cauchy principal value operator ensures that the denominator of the propagator never vanishes, which removes the singularity caused by In the language of virtualities introduced in Section V A 2, this means for the first term of the r.h.s. of (94), the virtualities read ν = ν r = 0 , i.e. the intermediate electron goes on-shell.However, since for this term, the denominator of the propagator is removed, the onshell intermediate electron does not induce a pole.The second term of the r.h.s. of Eq. ( 94) does not induce a pole neither, because the Cauchy principal value operator protects the denominator of the electron propagator from vanishing, i.e.P 1 Q 2 −m 2 = P 1 ν , which excludes the value ν = 0 from the integration region.Therefore, there is no singularity in S (2) 2nlC .In summary, it can be said, therefore, that in the case of asymptotically vanishing background fields, ∆φ < ∞, the matrix element (72) of two-photon Compton scattering has no singularities except for a single light-front δ-distribution, which ensures the conservation of the transverse and the minus components of the external momenta.

Infinitely extended plane wave/Oleinik resonances
The special case of a monochromatic plane-wave background field that is infinitely extended (see Eq. ( 1) with ∆φ → ∞ or g = 1) should be considered separately.Now, the two-photon Compton matrix element (72) reads where the mode-wise dressed vertex function Γ µ IPW,n is given in Eq. (34).We mention that the functions β(p, Q) and β(Q, p ) do not depend on the photon-number parameter , since β(p, Q) = β(p, p − k 1 ) and β(Q, p ) = β(p + k 2 , p ), respectively.Therefore, exchanging the integration and the two summations, one can solve the integral by using one of the δ-distributions, e.g. the first one, which leads to = reso n = β(p, p − k 1 ) + n.Then, the respective virtuality (74) results in where we use the abbreviation β p = a 2 e 2 2 1 k•p .Finally, if one employs the resonance condition ν = reso n = 0, for the emitted photon with four-momentum k µ 1 = ω 1 n µ 1 , one finds for the resonance energy4 Singularities of this type are called Oleinik resonances [40,41], which were already identified for the nonlinear two-photon Compton process in [30,37,38].For further investigations of the diagrams (98) w.r.t.Oleinik resonances we refer the interested reader to [63][64][65][66] and further citations therein, where one of the photon lines is attributed to a "field photon" of a nucleus.
The multi-photon nonlinear Compton with more than two vertices, e.g.[42,43], perpetuates this line of arguments and offers a test bed of gluing techniques, such as developed in [44,45].

B. Nonlinear Møller
As a further application of the momentum-space Furry-picture Feynman rules to twovertex processes we consider nonlinear Møller scattering (nlM), i.e. e − L (p 1 ) + e − L (p 2 ) → e − L (p 1 ) + e − L (p 2 ) in the laser background field (1).Here, the Oleinik resonances are attributed to the on-shell contributions of the photon propagator, cf.[67].The leading-order tree level two-vertex diagrams are The direct term (first diagram) corresponds to the matrix element where D µν is the free photon propagator.Executing the Q integration with one of the δ (4)   distributions yields where The intermediate photon's virtuality is defined by ν := Q 2 , which is related to the photon number parameters with δp n := p n − p n , n = 1, 2. Thus, the photon number parameters and r are intervened.
With aid of light cone variables, the four-momentum balance can be rewritten as to execute the integral in Eq. (100) with the result where r := r 0 − r with r 0 = (p The decomposition (18) facilitates four contributions to the direct term: Clearly, w.r.t.Eq. ( 19), the black-bullet vertices do dependent on the background field since , meaning that the first diagram, S 0 , links to the Møller scattering in vacuum with momentum balance p 1 + p 2 = p 1 + p 2 , thus reproducing standard perturbative QED.The background field, in particular, its temporal shape encoded in g(φ), enters the other three diagrams.A suggestive interpretation is proposed in [21]-Fig.2 by attributing a temporal ordering to the diagrams.
For instance, the second diagram, S 11 , would refer to virtual Compton under the influence of the background field (corresponding to the hatched vertex Γ µ reg ), while the subsequent virtual-photon absorption in the black vertex Γ µ div would proceed after the impact of the external field.Such an interpretation would ascribe the first diagram to proceeding before or after the action of the external field, while the last diagram would refer to both sub-processes within the action of the field; the third diagram, S 12 , would be accordingly interpreted as virtual Compton prior to the external impact.Independent of such an interpretation, the fourth diagram, S 2 , facilitates on-and off-shell contributions.
Analog to the sequence of steps in elaborating the matrix elements in Section IV A one can easily explicate the above diagrams to obtain the decomposition of the four-point amplitude corresponding to Eq. (2.23) in [21].As pointed out in Section II H, in the case of soft interactions with the background field, the finite (Γ µ reg ) and gauge restoration (Γ µ div ) parts of the dressed vertices factorize into a hard scattering part and a generalized soft factor.
Consequently, this soft/hard factorization also sets in for each diagram in the decomposition (105).Therefore, in the simultaneous limit a 0 → 0 and ∆φ → ∞, the corresponding soft versions of the five-point functions in perturbative monochromatic QED appear, where soft photons couple to each Fermion line.
Considering again monochromatic plane-wave background fields and inserting (34)  Via crossing symmetry, the trident amplitude ¡ rk lk has a similar decomposition as given above by the four direct-term diagrams, to be supplemented by the exchange terms.
The changed in-and out-phase space, however, modifies the treatment/interpretation of individual contributions, to be dealt with in a follow-up paper.The nonlinear Møller scattering is laser-assisted, while the nonlinear trident is laser-enabled.Oleinik resonances show up in both channels [68].

VI. SUMMARY
Following [20,23] we present the comprehensive momentum-space Furry-picture Feynman rules for QED in an external classical background field.Our emphasis is on formal aspects of gauge invariance and Ward identity, thus providing a general framework well-suited for n-point amplitudes.The special case of four-point amplitudes, dealt with in [21] within a somewhat different formulation, emerges naturally.Three-point amplitudes are considered exhaustively in the past and are uncovered as well.The benefit of our formalism is a systematic approach to the weak-field approximation, i.e. the series expansion of the S matrix element in powers of the laser intensity parameter a 0 .The leading-order term represents "pulsed perturbative QED" which accounts, in contrast to standard perturbative QED, for the temporal structure of the external classical field, where the Fourier transform of that field enters decisively.The limiting case of a monochromatic external classical field recovers the standard perturbative QED in terms of Feynman graphs and their rules of are not independent.The relation (A2) is a well-known formula, which appears in several investigations of specific processes in strong-field QED, e.g.[29] in the case of Compton scattering or for the trident process [61].However, the connection to the Ward identity and, therefore, to gauge invariance was not stressed there.
The Ward identity (A1) must be solved for one of the phase integrals, e.g.B 0 , in a distributional manner.This can be formulated as follows.Let be b 0 ( ) a solution of Eq. (A2), i.e. 0 = b 0 ( ) + α µ 1 B 1µ ( ) + α 2 B 2 ( ), then b0 ( ) := b 0 ( ) + Gδ( ) is a solution as well, where G is an arbitrary, but finite, function of the momenta.This seems trivial, because δ( ) = 0, but it turns out that this term leads to non-negligible contributions.However, the Ward identity (A2) does not determine the delta distribution's prefactor G. Instead, it can be derived by regulating the integral in the definition of B 0 , Eq. (11).Adapting the procedure in [62] for Compton scattering in a generic case by inserting e − |φ| with > 0 in the integral in Eq. ( 11) we get B 0 ( ) = lim 2 + 2 e iG(0) = 2πδ( )e iG(0) .(A5) in A µ , i.e. there is no A µ -independent term in an expansion of Γ µ w.r.t. the background field.

( 43 )
Therefore, we have a special case of a sub-leading Low theorem, where one part of the matrix element, M (1) reg , has an infrared pole and the other term, M (2) reg , does not.The elaboration of the general version of M (1,2) reg is relegated to separate work.H. Soft background field: k → 0