Master formulas for N -photon tree level amplitudes 3 in plane wave backgrounds

The presence of strong electromagnetic fields adds huge complexity to QED Feynman diagrams, such

Only recently has much attention been paid to worldline master formulas for processes with external matter lines, or processes at tree level [51][52][53][54][55][56].Furthermore, while external photon lines typically appear in the worldline formalism already Lehmann-Symanzik-Zimmermann (LSZ) amputated, matter lines do not, and it has not yet been fully established how one should perform the required LSZ amputation which turns correlation functions into amplitudes.
We fill in some missing pieces of this puzzle in this paper, which is organized as follows.In Sec.II we construct worldline master formulas for all tree level (N þ 2)-point correlation functions describing the emission of N photons from a massive particle in a background plane wave, in both scalar and spinor QED.In Sec.III we turn to the LSZ amputation of the master formula, converting it into an allmultiplicity formula for the corresponding N-photon emission/absorption amplitudes from a massive particle in a plane wave background.Example calculations in which we compare with known literature results at low multiplicity are presented in Sec.IV.We conclude in Sec.V.The Appendix contains additional checks on our results.
Conventions.We set ℏ ¼ c ¼ 1.We work throughout in Minkowski space with light front coordinates, so that ds 2 ¼ dx þ dx − − dx ⊥ dx ⊥ where x ⊥ ¼ ðx 1 ; x 2 Þ are the "transverse" directions.We introduce a null vector n μ which projects onto the "light front time" direction, that The goal of this section is to write down and evaluate the 120 worldline path integral master formulas for tree level 121 correlation functions of N photons and two charged particles 122 in the presence of a plane wave background, valid for 123 arbitrary N. We will do this in both scalar and spinor QED.124 Our plane wave background may be described by the tion of light front time x þ .We may always choose 127 a ⊥ ð−∞Þ ¼ 0, but then a ⊥ ð∞Þ ≕ a ∞ ⊥ is in general nonzero 128 (and carries an electromagnetic memory effect [57][58][59]). 129 The corresponding field strength is , where a prime denotes an x þ derivative.
F1:1 FIG. 1.We consider tree level scattering amplitudes of two massive charges and N photons, as illustrated on the right (for scalar QED). F1:2 The double line represents the presence of a plane wave background, the coupling to which is treated exactly.Amplitudes are obtained F1:3 by LSZ reduction of the corresponding correlation functions.In the worldline approach, a natural starting pointing is the partially F1:4 amputated correlator, or "dressed propagator," in which the photons are already reduced out, but the matter fields are not.This is F1:5 illustrated on the left.Thus LSZ reduction is still required for the external matter lines.
DxðτÞ e iS WL ½xðτÞ;A : ð2Þ Note that A μ is not integrated over, rather it appears as a 162 given field-it is well known (see, for example [62]) that 163 correlation functions with N external photons in vacuum 164 can be extracted from (2) by fixing A μ to be a sum over 165 asymptotic photon wave functions with momenta k i and 166 polarizations ε i : Inserting this into (2) and expanding to multilinear order, 177 the path integral to be performed is in which the weight is now given by the reduced action while the N external photons appear (following the 182 expansion to multilinear order) through the vertex functions 183 184 [We leave implicit a causal and IR convergence factor 185 expð−ϵTÞ under the dT integral in (5).] 186 Our task is to evaluate the integrals in (5).Let us first 187 consider the x μ integrals, and in particular the Dirichlet 188 boundary conditions (BCs).To deal with these we follow 189 the standard procedure used for the evaluation of such integrals in vacuum, and expand x μ ðτÞ into a straight line trajectory and a fluctuation qðτÞ according to The fluctuation must satisfy the homogeneous Dirichlet BCs Gaussian in q μ and can thus be computed analytically. 1Here, however, the fluctuation appears inside the background field , and this has an arbitrary functional form.At first glance this seems to destroy the Gaussianity of the path integral, and prohibit its evaluation.
However, it has been shown for one-loop photon-scattering processes (meaning no external matter lines, and a path integral with periodic rather than Dirichlet BCs) that the properties of the plane wave background mean the integral is still effectively Gaussian [33,37].It is thus crucial to demonstrate that the hidden Gaussianity of the path integral is also present here.
To do so we follow the approach of [34], introducing a Lagrange multiplier χðτÞ and auxiliary field ξðτÞ into the path integral through the equality These auxiliary integrals render that over qðτÞ to be Gaussian.The crucial point, as we show below, is that after evaluating the q integral, the remaining integrals over ξ and χ can still be evaluated, for a plane wave background.
We now compute the fluctuation integral.As is usual in this "string-inspired" approach, it is convenient to manipulate the vertex operators as follows.We exponentiate the polarization-dependent factor, so that it appears linearly in an exponent in the operator, with the understanding that the result should later be expanded to linear order in (each of) the ε i , so we write The result of this is that all dependence on the particle trajectory xðτÞ, or rather the fluctuation qðτÞ to be integrated out, now appears linearly under the path integral.
The integrals to be evaluated are now 1 This is also the case for a constant background in Fock-Schwinger gauge [54].
in which J μ ðτÞ is an effective (operator valued) source Since the fluctuation integral is now Gaussian, it is easily computed in terms of the worldline Green function BCs, which is found to be It is easily checked that Dirichlet BCs hold: Δð0; With this, the fluctuation integral becomes With this, we write out the exponent of (13), using that the 252 background is transverse and on-shell (n • a ¼ 0 and n 2 ¼ 0) 253 to simplify.We find, writing The trivial dependence on χ means that this field can now be integrated out, yielding a δ-functional: This δ-functional has the effect of shifting the argument of the background field, such that from here on we have The dynamical fluctuation is thus replaced by a coupling of the plane wave to the N scattering photons [33,37].This is particular to plane wave backgrounds because (a) for n 2 ≠ 0 Eq. ( 16) picks up a contribution quadratic in χ, while (b) for n • a ≠ 0 there is an additional term linear in χ that depends on the background; instead of (18) one would have obtained via (17) only an implicit equation for a μ .
All remaining background-dependent terms in (17) may be expressed in terms of just two worldline respectively.These would have to be computed for a given background once the functional form of a μ has been fixed.computed.Gathering everything together we obtain our 277 master formulas for the N-photon dressed propagator Kibble "mass" [63] which typically appears in pulsed plane waves [64], while Px 0 x is defined by Observe that in this case changing variables to u ¼ τ T , the worldline average becomes T-independent and can be taken outside the T integral.It may be written as a spacetime average (see [37]), and as such M 2 ðaÞ ¼ m 2 − ha 2 i þ hai 2 now corresponds exactly to the Kibble mass.
Equation ( 22) is equivalent to the standard momentumintegral representation of the Volkov propagator, and offers a concise version of the position-space propagator in [65,66].For N ¼ 1 we recover the (two-scalar one-photon) three-point function, and so on.Since the correlators themselves are not of immediate interest, we will present these checks later, implicitly, as part of our checks on the corresponding formula for scattering amplitudes.
The actual computation of the dressed propagator (and, later, the amplitudes) is greatly simplified by observing that we can choose the gauge n This removes the polarization vectors from the argument of a μ , and thus extraction of the multilinear piece of (24) reduces to the expansion of Pðε 1 ; …ε N Þ alone.We adopt this gauge from here on in order to present the simplest possible expressions and also match to the strong-field QED literature, where this gauge is common.Doing so, then, we can write the master formula in this gauge as 315 316 where the polynomial Px 0 x N is defined by the expansion of the polarization-dependent terms to multilinear order: 319 320 These polynomials generalize those defined for closed worldlines in vacuum (P N ) in [49], for open lines in vacuum ( PN ) in 321 [31], and for the closed loop in a background field (P N ) in [33] (in position space for the time being).For convenience let us 322 write out the first few terms: We now turn to the computation of the analogous N-photon dressed propagators in spinor QED, denoting these by S x 0 x N .
Due to the spin degrees of freedom this is a Dirac matrix-valued function, but we suppress the corresponding indices for brevity.Referring the reader to [51,67] for details, we begin by writing down the analog of the "propagator" (2) in an arbitrary background, but now accounting for the spin of the fermion: The kernel K x 0 x contains an integral over relativistic particle trajectories, as for the scalar case, and also a path integral over Grassmann-valued fields ψðτÞ, obeying antiperiodic (A/P) BCs ψð0Þ ¼ −ψðTÞ.These represent the spin degrees of freedom of the fermion and are minimally coupled to A through its field strength FðxðτÞÞ appearing in the action SWL .
An additional Grassmann variable η also appears; the Dirac-matrix structure of the propagator is produced by acting on this variable by the (inverse of the) symbolic map, defined by This map converts between antisymmetric combinations of Dirac matrices (a combinatorial factor of 1=n! factor is assumed) and products of Grassmann variables η.Use of the symbol map avoids lengthy Dirac-matrix algebra as it automatically produces the kernel in the (even subalgebra of the) Clifford basis of the Dirac algebra.Note that all η-dependence in ( 30) and (31) or any of our expressions vanishes after evaluation of the inverse map; it is therefore pragmatic to state once and for all the results relevant to us in (3 þ 1) dimensions as Now, taking A as in (3) to introduce both our background plane wave and the N external photons, we expand (29) to multilinear order in the photon polarizations to obtain the N-photon dressed propagator where SB ½ψðτÞ; xðτÞ; a is given by replacing eFðxðτÞÞ in SWL ½ψðτÞ; xðτÞ; A with fðxðτÞÞ.In the "N-photon kernel" , the proper time and bosonic integrals are the same as in the scalar case-these represent the orbital degrees of freedom which remain unchanged.In the so-called subleading term involving K x 0 x N−1 , for each term in the sum in = A γ ðx 0 Þ we remove the corresponding photon from the kernel to maintain the projection onto the multilinear sector.Finally, writing fiμν ¼ k iμ ε iν − k iν ε iμ for the linearized field strength associated with the ith photon, the vertex operator is now given by 398 399 and the corresponding spin factor is produced through To compute the integral in (39) we require the (spinor) 402 worldline propagator in the field, G μν ðτ; τ 0 Þ.This will define 403 the fundamental contraction between the Grassmann fields, From the quadratic part of the operator appearing in the path 406 integral action, G must obey as well as antiperiodic boundary conditions Gð0; τ 0 Þ ¼ −GðT; τ 0 Þ and Gðτ; 0Þ ¼ −Gðτ; TÞ.Observe that G has the antisymmetric property G μν ðτ; τ 0 Þ ¼ −G νμ ðτ 0 ; τÞ.The general homogeneous solution of (42) for arbitrary fðτÞ is written conveniently in terms of an auxiliary function Oðτ; τ 0 Þ, which takes care of the ordering of τ and τ 0 , defined by where Θ is the Heaviside step function, However, there are notable simplifications in our particular case that f is a plane wave because, as is well known, the field strength is then nilpotent of order 3. Further, f evaluated at different τ commute.The Green function thus reduces to 2 Equipped with the Green function, we compute the integral in (39) by completing the square, using the shift ψðτÞ ¼ (this should be contrasted with the constant field case, where the normalization picks up a nontrivial field dependence [27,29]).
2 This is an alternative way of writing the Green function given in Eq. ( 45) of [33], with the advantage of being manifestly gauge invariant.There G μν was written in terms of periodic integrals of the derivative of aðτÞ which made its antiperiodicity easier to see.
Gathering all of the above together, the Grassmann integral as defined in (39) becomes The Grassmann path integral is therefore formally computed.In particular, where G μν ðτ i Þ ≔ η μν − R T 0 dτ½Gðτ i ; τÞ • fðτÞ μν and T denotes the transpose in Lorentz indices-in particular we have Putting all of this together, the N-photon dressed propagator can be written in a "spin-orbit decomposition" by summing over assignation of the N external photons to either the spin or bosonic part of the vertex [33], as follows: The sum on the second line runs over the allocation of S, out of the N, photons to the spin part of the vertex operator, V x 0 x η , which subsequently appear in Spinð fi 1 ; …; fi S Þ. Then the remaining N − S photons appear in the polynomial Pfi 1 ∶ i S gx 0 x NS , defined by where the notation on the far right means that the polarization vectors ε i 1 to ε i S should be put to zero before the remaining expression is expanded to multilinear order in the ε i Sþ1 to ε i N .These polynomials generalize those introduced in vacuum ( Pfi 1 ;i S g

NS
) in [51] and satisfy Again, these are position-space expressions, but below we shall transform to momentum space for the purpose of propagator that differ by permutation of the external photons.Obtaining such a formula from the standard formalism (Furry picture, say) of strong-field QED would be a significantly more complicated task.
For completeness, we note that the N ¼ 0 case provides a worldline representation of the well-known Volkov propagator as a one-parameter integral where we used Spinð0Þ computed the integral in the average explicitly, and reexponentiated using n 2 ¼ 0. This is again equivalent to other representations of the Volkov propagator [8,65,66].

III. LSZ FOR SCATTERING AMPLITUDES
The objective of this section is to take the master formulas for the dressed propagators D x 0 x N and S x 0 x N above and produce from them equivalent master formulas for (2-scalar) N-photon scattering amplitudes (for N ≥ 1).To do so we must perform LSZ reduction on the two massive, external legs of the dressed propagators.
In previous worldline literature, amputation was often done "by hand," by obtaining the N-point correlation functions in momentum space and then-once the proper-time integral had been computed-removing external legs with the appropriate inverse matter propagators [51,52].Only then could the external particles be taken onshell-the proper-time integral produces the pole structure of the correlation functions with respect to external matter legs and so is divergent in the on-shell limit.This is a notable example where the Feynman diagram prescription to omit external propagators had appeared less trivial from a worldline perspective.Recently, however, [68,69] showed how amputation can be achieved under the proper-time integral for scalar matter legs, with the inverse propagators simply modifying the bounds on the proper-time and parameter integrals.This exposes the on-shell residue of the correlation functions without the need to carry out amputation by hand.We will here generalize this approach to spinor theories, and also show it is unspoiled by the plane wave background.
To perform LSZ we draw the external legs out to asymptotic times and Fourier transform.Alternatively, we can Fourier transform to momentum space and find the residues of the dressed propagator as the momenta are taken onto the mass-shell.Starting with scalar QED, the amplitude takes the form where in the second line we defined p0 ¼ p þ a ∞ and 530 introduced the momentum-space propagator D p 0 p N , defined 531 by The expression ( 57) is (almost) textbook-standard LSZ in The integrals over x 0⊥;− and x ⊥;− generate, 3 as in the vacuum 555 case, four δ-functions, explicitly TÞ, where we write K ¼ To evaluate similar integrals in the existing literature it was found to be convenient to change variables to end-point center of mass and relative separation (z).However, for our later LSZ amputation of the external legs it is more useful to integrate separately with respect to the end-point coordinates.
δ-function allows us to trivially perform, e.g., the x 0þ integral, so that we can replace in what remains; in particular, the classical trajectory on which the gauge field depends throughout D x 0 x N , as in (18), is modified to, where g ≡ gðfτ i gÞ ≔ 565 566 Thus we can do all but one of the Fourier integrals, which 567 eventually yield aðτÞ ≡ aðx þ cl ðτÞÞ.Note that in the vacuum limit a μ → 0 we can carry out the xþ integral to complete the conservation of 4-momentum and so recover one version of the master formula given in [27,51].
To convert (63) into a master formula for the amplitudes, we have to perform LSZ on each massive scalar leg (these are produced by the parameter and proper-time integrals).
To do so we observe that (58) has, using (63), the following form, writing down only the relevant structures: The on-shell limit p 2 → m 2 − i0 þ therefore returns the residue of the mass-shell pole of the function defined by the integral.To isolate this pole we proceed as in [68][69][70] where LSZ was considered for, e.g., the N-graviton-dressed propagator in vacuum. 4We integrate by parts (off-shell) in order to expose the residue, as so: We can now take p 02 → m 2 and 0 þ → 0 (in either order), 589 upon which the integral becomes exact, and we have lim vacuum, which we comment on further after performing the 596 second amputation, below).We thus find lim We note that all terms with worldline averages have ultimately been replaced with (convergent) integrals over R þ .This was the advantage of having computed the The reason for this change of variable is that it allows us to reexpress (67) in a form which renders the second LSZ amputation immediate.To achieve this, we first rewrite the proper-time integrals appearing in (67) in terms of the new variables as [note the factor of 1 N in the δ-function is missing in (3.18) of [69] ] We also make a change of variable for the x þ -integration, gÞ, and it is convenient to change variables in all dτ integrals from τ to τ ≔ τ − τ 0 , such that the background gauge field now appears as In terms of the shifted variables fx þ ; τ 0 ; τi g, the once-623 amputated propagator (67) takes the form In the previous section we carried out the Fourier transform 702 of D x 0 x N literally, to obtain D p 0 p N .Expression (75) shows a 703 more "direct" approach to deriving the master formula in 704 (63), through a modification of the boundary conditions on the path integral.This fits in more naturally with the "worldline philosophy" of incorporating all information into the worldline path integral.Note that evaluation of (74) requires a worldline propagator with different boundary conditions.Indeed, this helps explain a puzzle arising in [26] (Section 3, footnote 3), where a version of the momentum space master formula was given that involves a Green function with mixed boundary conditions: by expanding about a suitable reference trajectory, ( 75) can be cast into a path integral for the fluctuation variable that must satisfy the mixed boundary conditions This discussion prompts us to study the propagator D xp N with mixed boundary conditions which, examining (75), is given by the integral To see the significance of the mixed propagator, consider the case N ¼ 0, that is the tree level two-point function for the scalar field, with mixed boundary conditions.In To confirm this, we first compute the path integral in (76) for N ¼ 0 (we drop the product of vertex operators).We do not dwell on this step; the entire integral turns out, unsurprisingly given the nature of the Volkov solutions and hidden Gaussianity of the worldline path integral, to be equal to its semiclassical value exp½iS cl ðTÞ, i.e. the exponential of the classical action evaluated on the classical path obeying the mixed boundary conditions, which is The final step is to take p 2 → m 2 and identify the on-shell residue via lim Of course it is clear from the preceding calculations how to proceed; we perform the same manipulations as for the master formula, in particular taking the T → ∞ limit, immediately finding COPINGER, EDWARDS, ILDERTON, and RAJEEV PHYS.REV.D XX, 000000 (XXXX) The right-hand side is precisely the incoming scalar Volkov  60), writing S x 0 x N in terms of the kernels appearing in 766 (51) and evaluating the ∂ x 0 , ∂ x derivatives (using integration 767 by parts) in ( 60) to find Next, following [52] we use the on-shell relation 770 ūs 0 ðp 0 Þð= p 0 þ mÞ −1 ¼ ūs 0 ðp 0 Þð2mÞ −1 , (which is allowed 771 since it does not remove the associated pole, or affect the 772 final expression), and likewise for ð= p þ mÞ −1 u s ðpÞ to find Due to the worldline approach being based on the 775 second-order formalism of QED, the exponent under the proper-time integral of the spinor amplitude contains the same terms as for the scalar amplitude-in particular the parameter and proper-time integrals produce (free) scalar propagators.Hence it suffices to revise the scalar case for this argument.The difference lies in the spin factor of the kernel, the subleading contibutions (those proportional to K N−1 ), and the δaðx þ0 Þ factor from the covariant derivative.
However the differences do not impede processing the T, and later τ 0 , proper time integrals as for scalars.The result is that the LSZ amputation is realized in precisely the same way, by taking T; τ 0 → ∞ as in Eqs. ( 64)- (69).Moreover, after taking the Fourier transform, the conservation of momenta The LSZ truncation projects onto asymptotic late time, taking aðx 0þ Þ → a ∞ when T → ∞, canceling the fielddependent term in square brackets of (81).One may then express (81) in terms of the momentum-space kernel Now we address the subleading terms.These are seen to have poles not in the required mass-shell p 02 − m 2 , but Contributions involving these shifted poles hence vanish after taking the on-shell limit of This is a remarkable generalization of the vacuum case [52].We can be more precise with how this cancellation comes about.In the kernel of the subleading terms, K , one must first remove an ε i and k i , and then replace a ∞ with a ∞ þ k i in (73).This operation leaves p0 þ K invariant, but it does affect the term R ∞ 0 dτp 0 • δaðτÞ, which was convergent as τ → ∞, but now produces a rapidly oscillating phase; noting that the proper-time integral calculates the Laplace transform of the function FðTÞ in (64), the Abelian final value theorem can be invoked to confirm that the subleading contributions must vanish.
Since the manipulations are similar to the scalar case, let us simply record the spinor amplitude in its final form as where φ in p is the incoming scalar Volkov wave function 857 of (79) while φ out p 0 is the outgoing wave function, proper-time integral, we expect this to provide some advantage over the standard formalism, at least in various physical limits of interest.This will be discussed elsewhere.

B. N = 1, nonlinear Compton scattering in spinor QED
Let us now confirm the N ¼ 1 case for spinor QED, which requires expanding the master formula (84) to linear order in ε 1 .Since the field dependence of the exponent in for spinor QED contains that of scalar QED one may write the resulting amplitude using the scalar Volkov wave functions, (91), as requiring only the evaluation of the spin factor (we have again used the Fourier representation of the δ-functions).
Before embarking upon the comparison to the standard formalism, we should emphasize that the approach outlined here, namely writing in terms of spacetime averages with steps to follow, is necessary to make the connection to the perturbative Furry picture with Volkov wave functions.
and therefore the factor without photon 896 insertion is readily determined to be Next, we express the photon momentum, k 1 , in terms of the electron momenta and asymptotic value of the background field.For the þ; ⊥ components we can use momentum conservation, carry out an integration by parts with respect to x þ .We illustrate this step, to be applied to the various k 1 terms in (94), with the following manipulation: In fact, if additional factors of aðx þ Þ appear under the above integral, in turns out that the additional derivatives produced by integrating by parts always contract away.
Therefore (95) can be used throughout (94).Moreover, applying the above procedure to k 1 in the γ 5 term of (94), one can see that in effect k μ 1 → p μ − p0μ , since the two n μ contract to zero against the Levi-Civita tensor.In fact the only term in which the n μ part of (95) survives after these replacements is the first term on the RHS of (94).

926
Last, since we are taking the on-shell limit we may Using the above steps to replace k μ 1 in the remaining terms of (94), after some algebra one may gather terms to 937 find that 938 where we have used the spinor Volkov wave functions, 942 which read This successfully verifies that the worldline approach 947 reproduces the known amplitude for the N ¼ 1 process.

V. CONCLUSIONS
We have presented worldline master formulas for allmultiplicity tree level scattering amplitudes of two massive charged particles and N photons, in a plane wave background, in both scalar and spinor QED.The background field may have arbitrary strength and functional profile, and is treated without approximation throughout.This is particularly relevant as the target application of our results is to laser-matter interactions in the high intensity regime where the field is characterized by a dimensionless strength (the coupling to matter) larger than unity, and hence must be treated without recourse to perturbation theory.
Our master formulas have been derived using the worldline approach to quantum field theory.While several previous publications have derived wordline master formulas for various correlation functions in vacuum, or even at higher loop level in background fields, our focus here has been on scattering amplitudes involving external matter.As such it was necessary to identify the worldline description of LSZ reduction in a plane wave background.We found this to be a fairly direct generalization of the known worldline prescription for LSZ amplitudes in vacuum [68,69].A second notable generalization from known results in vacuum holds for the spinor case: namely that in the second-order formalism, which implies a split into "leading" and "subleading" terms, only the former survives the on-shell limit once the LSZ prescription is imposed.
Furthermore, the background-field-dependent part of this leading term also drops out in the asymptotic limit.This allows for a large number of terms to be discarded (and in the vacuum case allowed for the gauge invariance of the amplitudes to be manifest).
We have checked our results against the existing liter- It is fair to say that the master formulas for amplitudes we have derived here still require, for a chosen number of photons N, some processing in order to extract all their physical content.In future work we will pursue methods of evaluating the remaining proper-time integrals in an efficient manner, or in an approximate manner relevant to interesting physical regimes.Here, benefit should be gained by not breaking the parameter integrals into ordered sectors corresponding to photon permutations, which will maximally exploit the calculational efficiency.Constructing observables from our amplitudes at N > 2 (which are lacking in the literature) will help to benchmark numerical codes which approximate multiphoton processes using sequential single photon emissions.It would be revealing to compare our expressions with the compact all-multiplicity results of [74,75].We also plan to generalize our results to higher-loop orders, in order to pursue the Ritus-Narozhny conjecture on the behavior of loop corrections at very high intensity, see [8,14] for reviews.

APPENDIX: MASTER FORMULA CHECK FOR N = 2
In this appendix we confirm that the master formula (73) correctly reproduces, at N ¼ 2, the amplitude for "double nonlinear Compton scattering" [76,77] in scalar QED, that is the emission of two photons from a particle in a plane wave background.(By crossing symmetry this is directly related to the amplitude for the Compton effect in the background.)Recall that in scalar QED, the standard approach would require evaluation of three separate The gauge field at the interaction points AEτ 1 (indicating the 1065 insertion point of photon with momentum k 1 ) takes the 1066 values This motivates us to make the change of variable , such that the field-1072 independent terms (A1) transform to 1073 1074 where we have defined q ¼ p − k 2 and used the fact the 1075 momenta are on-shell to simplify.We shall shortly need the 1076 last term −ið2p 0 þ a ∞ Þa ∞ τ 1 to simplify some of the field-1077 dependent terms.Before going into that, we return to the 1078 exponent of ( 73) and note that the following field-depen-1079 dent term is already sufficiently simplified: The rest of the field-dependent terms combine with 1082 −ið2p 0 þ a ∞ Þa ∞ τ 1 from (A4) to yield MASTER FORMULAS FOR N-PHOTON TREE LEVEL … PHYS.REV.D XX, 000000 (XXXX) dτ½2p 0 • δaðτÞ − δa 2 ðτÞ: ðA6Þ We now use the dependence of a μ ðx cl ðτÞ on the classical solution to transform the proper-time integrals into spacetime integrals and simplify the above terms as where we have used momentum conservation to replace p⊥ þ K ⊥ with p ⊥ , and p⊥ þ k 1⊥ with q ⊥ .The contribution A p 0 p 2þ to the amplitude from τ 1 > 0 can then be written as We are now going to show that the right-hand side of the above expression is equivalent to one of the three Feynman where D denotes the background-covariant derivative and Gðx 0 ; xÞ ¼ D x 0 x 0 is the scalar particle propagator in the plane wave background (the double arrow indicates the right-left alternating derivative).We then observe that this is equivalent to Taking this expression, we start by using the Fourier representation of Gðx 0 ; xÞ to rewrite it as We can easily evaluate the x 0−;⊥ ; x −;⊥ , and r −;⊥ integrals and rewrite the propagator denominator using a standard Schwinger proper-time integral to obtain COPINGER, EDWARDS, ILDERTON, and RAJEEV PHYS.REV.D XX, 000000 (XXXX) The r þ integral can now be evaluated to give 2πδðx þ − x 0þ þ 8q − τ 1 Þ.The remaining x 0þ integral is therefore trivialized and effects the replacement Taking the multilinear limit, one recovers precisely the right-hand side of (A9) as promised.
116 II.MASTER FORMULAS FOR (2 + N)-POINT TREE 117 LEVEL CORRELATORS IN PLANE WAVE 118 BACKGROUNDS 119 formulas we derive in this section, the N 133 external photons will be LSZ-amputated, but the matter 134 lines not, and thus our correlation functions carry spacetime 135 indices x and x 0 , as well as a dependence on the N-photon 136 momenta fk i g and polarizations fε i g.We hide the latter 137 dependencies, denoting the partially reduced correlators, or 138 dressed propagators as they are called in the worldline 139 literature, by D x 0 x N ; see Fig. 1.We take all photons to be 140 outgoing; other configurations are trivially obtained by 141 sending k → −k.142 The worldline representation of such correlation func-143 tions is given in terms of a path integral over relativistic 144 point particle trajectories, denoted x μ ðτÞ with τ the proper 145 time of the trajectory.The trajectories obey Dirichlet 146 boundary conditions x μ ðTÞ ¼ x 0μ , x μ ð0Þ ¼ x μ , correspond-147 ing to the spacetime dependence of the dressed propagator.148 The trajectories have length T, which is ultimately also 149 integrated out, respecting reparametrization invariance of 150 the path integral [60,61].To write down this path integral, 151 we start from the worldline action that minimally couples a 152 relativistic point particle to an arbitrary gauge field A μ , μ ðτ 0 Þ − ⟪a μ ⟫;ð19Þ272 273 second term represents the spin coupling of the 367 external photons to the particle trajectories.368 Despite the obvious added complexity from the spin 369 coupling to the photon fields, we stress that the same 370 hidden Gaussianity is present here as in the scalar case.371 Consider again the path integral over x μ ; we treat it as we 372 did above, introducing auxiliary fields to yield a Gaussian COPINGER, EDWARDS, ILDERTON, and RAJEEV PHYS.REV.D XX, 000000 (XXXX) path integral in the fluctuation q μ .While there is now an 374 additional dependence on the background f μν introduced 375 by the spin factor, this behaves in the same way as above 376 when integrating out the auxiliary fields, i.e. f in the spin 377 factor is ultimately evaluated at a shifted argument, a μ earlier (recall we have gauged ε þ i ¼ 0 for 380 convenience).In short, and as is natural, the only real 381 difference compared to the scalar case lies in the evaluation 382 of the Grassmann path integral, which is the focus of the 383 remainder of this section.384 Observe that the vertex operators (35) introduce factors 385 of ψ η ðτÞ ≡ ðψðτÞ þ ηÞ under the Grassmann integral.This 386 motivates us to introduce the following functions, The integral over ψ then generates the determinant Detð 1 2 d dτ þ fÞ (for antiperiodic boundary conditions) which because of the nilpotency of f simply gives a factor of 2 D 2 , being the number of degrees of freedom of the fermion in D (even) spacetime dimensions 471 evaluating scattering amplitudes.Although this master 472 formula appears lengthy, it is important to emphasize that 473 it represents a formal evaluation of the path integral for an 474 arbitrary number of photons inserted along the background-475 dressed propagator, conveniently split into contributions 476 from the vertex function representing orbital interactions 477 (in Pfi 1 ∶ i S gx 0 x NS ) and spin interactions [in Spinð fi 1 ; …; fi S Þ]. 478 All of these insertions are integrated along the particle 479 trajectories, so that the master formula represents a sum 480 over all Feynman diagrams contributing to the dressed COPINGER, EDWARDS, ILDERTON, and RAJEEV PHYS.REV.D XX, 000000 (XXXX) 534position space but to compensate for the fact that our 535 potential becomes pure gauge in the far future, the on-shell, 536 outgoing momentum p 0 in the Fourier kernel is shifted to537 p0 ¼ p 0 þ a ∞ [57,63].The expression (58) makes it clear 538 that the amplitude A p 0 p N is the residue of D p0 p N at on-shell 539 momenta.In our conventions A p 0 p N describes N-photon 540 emission from a particle traversing the plane wave.541 Absorption and pair-production/annihilation amplitudes 542 are of course obtained by crossing.543 Similarly for the spinor case, starting from the master 544 formula for the dressed propagator (51), we can extract the 545 spin-polarized amplitude M p 0 p Ns 0 s as to compactify notation.The first three δ-functions 558 describe the (expected) conservation of light front three-559 momentum in the plane wave background.The final 3 , performing the first amputation on (63) is 592 equivalent to dropping the integral over proper time T and 593 its accompanying mass-shell exponent, and taking the limit 594 T → ∞ of what remains (this is the same argument as in 595

Fourier integrals with respect
to the individual end points 603 as discussed above.Equation (67) is the one-side ampu-604 tated propagator.605 Turning to the amputation with respect to p, at this stage 606 it is advantageous to introduce the mean and deviation 607 proper-time variables as follows:

Feynman
diagram language, this is just an external leg, Fourier transformed at one end.Taking the momentum at this end onto the mass-shell, i.e. performing LSZ reduction, we must recover the scalar Volkov wave functions.These are solutions of the Klein-Gordon equation in a plane wave background which reduce to e AEip:x in the asymptotic past/ future and thus represent incoming and outgoing particles in scattering amplitudes.

750 wave function φ in p ðxÞ which
reduces to e −ip•x in the 751 asymptotic past.A similar amputation of the propagator 752 D px 0 (where the boundary conditions are swapped) yields 753 the outgoing Volkov wave functions, i.e. those which 754 reduce to e þi p0 •x in the asymptotic future.Of course the 755 same procedure can be applied to the spinor propagator, 756 wherein the path integral with mixed boundary conditions 757 produces the spinor Volkov wave functions.Worldline path 758 integrals analogous to (76), with mixed boundary con-759 ditions, have also been used before, in a similar context, to 760 recover the exact solutions of the Klein-Gordon equation in 761 a constant external electromagnetic field [73].For numeri-762 cal studies of open line instantons see [41].763 C. Spinor QED 764 Turning to LSZ reduction in spinor QED, we proceed 765 from ( is precisely the expected result for non-860 linear Compton scattering in scalar QED, providing a 861 positive check on our master formula.862 We stress that the method we employed above to process 863 the worldline integrals was meant only to allow direct 864 comparison with existing results.It is not the approach we 865 wish to take in future work; instead, we will use the 866 worldline representation to deal directly with the τ inte-867 grals.Since the major advantages of the worldline approach 868 include that (a) one does not have to split amplitudes into 869 sectors according to permutations of external legs, and 870 (b) internal momentum integrals are recast in terms of the 871 are determined using (48) and (49) 892 under the LSZ reduction (86) and the inverse symbol 893 map, (33).Because of the nilpotency of f one has, 894 under the inverse symbol map already transformed the parameter integral 899 to a spacetime average and computed its value.This is 900 simply the Dirac-matrix structure necessary to construct the 901 spinor Volkov wave functions.902 Let us next treat the single photon spin factor, Spinð f1 Þ. 903 Beginning with the Grassmann integral with one photon 904 insertion, provided in (49) we apply the inverse symbolic 905 map in (33) and realize the LSZ reduction according 906 to (86).The various worldline averages are then trans-907 formed into their corresponding spacetime averages as was 908 done in the N ¼ 1 scalar case, to find Spinð 927 use the Dirac equation for the sandwiching spinors so as 928 to send their corresponding = p and = p 0 to m, anticommu-929 tating where necessary.Again, illustrating this step with 930 the γ 5 term in (94) we rewrite γ 5 in terms of products of 931 four matrices using (33).After acting on the spinor 932 solutions at most three matrices will remain.After this 933 process, the γ 5 term, as it appears in the amplitude (92), 934 becomes MASTER FORMULAS FOR N-PHOTON TREE LEVEL … PHYS.REV.D XX, 000000 (XXXX)

948C.τ 1
photon-scalar vertex) first appears.We will describe the ature, which contains only low-multiplicity amplitudes derived using Feynman rules.Explicitly, these are the cases N ¼ 1 and N ¼ 2, or single and double nonlinear Compton scattering.Moving beyond scattering amplitudes, we have also seen how to recover off-shell quantities, in particular the scalar and spinor correlation functions dressed by the background and the Volkov wave functions, from worldline path integrals.The latter is a particularly interesting case as it exposes the relevance of mixed boundary conditions; the relevant path integrals carry Dirichlet conditions at one limit, representing the local spacetime argument of the wave function, and Robin boundary conditions at the other limit, encoding the asymptotic momentum characterizing the Volkov solution.