N-photon amplitudes in a plane-wave background

We use the worldline formalism to derive master formulas for the one-loop N-photon amplitudes in a plane-wave background, for both scalar and spinor QED. This generalises previous work by Ilderton and Torgrimsson for the vacuum polarisation case, although with some change in methodology since, instead of evaluating the path integral on the semi-classical trajectory, we use the special kinematics of the plane-wave background to uncover the crypto-gaussian character of this type of worldline path integral.


Introduction
In strong-field QED, there are two external field configurations that play a special rôle: the constant field, and the plane-wave one. Both are not only important from the physics point of view, but also special mathematically, since they allow for an exact solution of the Dirac equation in the field, which makes it possible to perform non-perturbative calculations in such fields (see, e.g., [1,2]). Nevertheless, beyond the simplest special cases such calculations tend to be extremely lengthy and tedious [3,4,5,6,7,8,9]. For example, the one-loop QED vertex in a plane-wave field has been calculated only very recently [10].
For the constant-field case there exists an alternative approach, based on Feynman's worldline path integral formulation of QED [11,12] and concepts originally borrowed from string theory [13,14], that has been shown to offer various technical advantages for closed-loop photonic processes [15,16,17,18] and recently also for amplitudes involving open scalar [19] and fermion [20] lines -for reviews of this formalism see [21,22].
The plane-wave case has attracted much attention in recent years because of its relevance for laser physics [23,24,25]. However, the application of the worldline formalism to this case has turned out to be less straightforward. A calculation of the scalar and spinor QED vacuum polarisations along these lines was achieved by A. Ilderton and G. Torgrimsson [26], but it is not obvious how to extend their approach to the general N-photon amplitudes. Here we will use a slightly different approach, based on a direct rewriting of the worldline path integral as a gaussian one, to construct compact master formulas for the scalar and spinor QED N-photon amplitudes in a plane-wave background.
We start in the following section with a short summary of the worldline representation of the N-photon amplitudes in vacuum (for details see [21]). The following two sections are devoted to the derivation of master formulas for the N-photon amplitudes in a plane-wave background, first for scalar and then for spinor QED. As a check, in section 5 we work out the N = 2 cases and recover the results of [26]. In the final section we shortly summarize our results and point out possible generalisations.

N -photon amplitudes in the worldline formalism
The starting point for the calculation of the scalar QED N-photon amplitudes in the worldline formalism is Feynman's [11] worldline representation of the one-loop effective action Γ scal [A]: The path integral runs over all closed trajectories in spacetime obeying the periodicity condition x(T ) = x(0) in proper time. The N-photon amplitudes are obtained from this by expanding the "interaction exponential" and Fourier transformation, which leads to the "vertex operator representation" of the N-photon amplitude: Here each photon is represented by the following photon vertex operator, integrated along the trajectory: After a formal exponentiation ε i ·ẋ i e ik i ·x i = e ε i ·ẋ i +ik i ·x i | ε i the path integral can be done by gaussian integration using the basic correlator This results in the following "Bern-Kosower representation" of the N-photon amplitude [27,13,14], Here a 'dot' denotes a derivative acting on the first variable, and we abbreviate G ij ≡ G(τ i , τ j ) etc. The factor (4πT ) − D 2 represents the free Gaussian path integral determinant factor, and the (2π) D δ( k i ) factor is produced by the integration over the zero mode x µ 0 ≡ 1 T T 0 dτ x µ (τ ) of the path integral. The exponential must still be expanded and only the terms be retained that contain each polarisation vector ε i linearly: with certain polynomials P N . For spinor QED, a generalization of (1) suitable for analytical calculations is given by the Feynman-Fradkin representation [12,28] Here ψ µ (τ ) is a Lorentz vector whose components are Grassmann functions, {ψ µ (τ ), ψ ν (τ ′ )} = 0, and the path integral Dψ has to be taken over antiperiodic such functions, ψ µ (T ) = −ψ µ (0). Note that it is already gaussian as it stands.
Applying the same procedure as for the scalar case above, one obtains the following generalization of (2) to the spinor QED case: The photon vertex operator for spinor QED V γ spin differs from the scalar one (3) by a second term representing the interaction of the fermion spin with the photon, with f µν ≡ k µ ε ν − ε µ k ν the photon field-strength tensor. Thus the N-photon amplitude is naturally obtained in terms of a spin-orbit decomposition where S denotes the number of spin interactions, and the sum {i 1 i 2 ...i S } runs over all choices of S out of the N photons as the ones assigned to those interactions. It is then straightforward to arrive at the following master formula for Γ Here the polynomials P are now defined by (compare with (7)) where the notation on the left-hand side means that one first sets the polarisation vectors ε i 1 , . . . , ε i S equal to zero, and then selects all the terms linear in the surviving polarisation vectors. In particular, one has the extremal cases P For the Wick-contraction of the spin interaction terms we have introduced the notation to be evaluated with the basic correlator This object possesses the following closed-form description. Define a "Lorentz cycle of length n" Z n by and a "fermionic bi-cycle of length n" by Then we can write Here the sum runs over all inequivalent possibilities to distribute the indices 1, . . . , S among the arguments of any number cy of bi-cycles, and n k denotes the length of the bi-cycle k. Working out (18) up to S = 4, we find 3. N -photon amplitude in a plane-wave background (scalar QED) In general, a plane-wave field can be defined by a vector potential A(x) of the form where n µ is a null vector, and, as is usual, we will further impose the light-front gauge condition Note that we absorb the charge e in the definition of a µ . Repeating the procedure of the previous section with the addition of the potential a µ to the worldline Lagrangian, we straightforwardly get a representation of the Nphoton amplitude in the plane-wave background that generalizes the vacuum formula (2), Fixing the zero-mode problem as usual by separating off the average position x µ 0 of the trajectory, x µ (τ ) = x µ 0 + q µ (τ ), we note that, differently from the vacuum case, it now appears not only in the exponents of the vertex operators, but also in the argument of a µ (n·x). Thus it will now be convenient to introduce (euclidean) light-cone coordinates adapted to the null vector n µ . Thus we set n µ ≡ 1 √ 2 (0, 0, 1, i), and define x + ≡ n · x = 1 √ 2 (x 3 + ix 4 ) ("lightfront time") and x − ≡ 1 √ 2 (−x 3 + ix 4 ). We will further denote x ⊥ ≡ (x 1 , x 2 ). Defining also k ± ≡ 1 √ 2 (±k 3 + ik 4 ), and using the decomposition allows us to integrate out x µ 0 but for its x + 0 component: The calculation of the functional integral at first sight looks like an intractable problem, since the integration variable q(τ ) appears in the argument of the unknown function a µ . In [26] this problem was solved for the two-point case using the fact that the plane-wave path integral possesses the gaussian property that its semiclassical approximation is exact. For the N-point generalization, we find it more convenient to exhibit the crypto-gaussian nature of the path integral using the relations (21) and (22). In principle, we could do the functional integral by expanding and then Taylor-expanding (Note that we use a 'prime' for the derivative of a function with respect to its argument, while the 'dot' will be used for the total derivative with respect to proper time.) Now, we observe that a factor n · q m can neither be Wick contracted with another such factor because of (21), nor with a factor oḟ q n · a (k) (x + 0 ) because of (22). Thus each n · q m has to be contracted with the exponential e N i=1 (ik i ·q i +ε i ·q i ) , and this will convert it into We can then resum (27) into and subsequently also (26), where we can now, with some abuse of notation, replace Thus we have removed the functional integration variable from the argument of a µ , and converted the functional integral (25) into a gaussian one. Now the usual "completing-the-square" procedure can be applied, and yields The first term in the exponent on the right-hand-side can, introducing the worldline average and using (6) be rewritten as Similarly, we can rewrite For the integral involvingĠ(τ, τ i ), we introduce the periodic integral function Integrating by parts, we get Putting the pieces together, we get the following master formula for the scalar QED N-photon amplitude in a plane-wave background 1 Note that the appearance of the polarization vectors in the argument of a µ makes it still messy to extract the terms linear in all of them. Further substantive simplification can be achieved by choosing the ε i such as to obey which is possible for generic momenta by a gauge transformation, and will be assumed for the rest of this paper. This will reduce (31) to The master formula (38) can then be written more explicitly as where the polynomials P N are defined by (compare (7)) 1 The reader familiar with the worldline formalism may wonder why we could drop the additive constant T 6 from this Green's function, which is customary but relies on momentum conservation. It is easy to verify that here, in light-cone coordinates, the removal of the constant requires only that

N -photon amplitude in a plane-wave background (spinor QED)
Proceeding to the spinor QED case, the same argument that we applied above to eA µ = a µ can be used to convert also the argument of the eF µν = n µ a ′ ν − a ′ µ n ν appearing in the spin part of the worldline Lagrangian in (8) in the same way as in (29). Thus the only new element is that the fermionic Wick-contraction rule (14) now has to be calculated with a generalised worldline Green's function inverting the field-dependent operator The appropriate generalisation of (15) is where (45) and we have further defined Note that the modified Green's function has a non-zero coincidence limit, and satisfies the anti-symmetry relation G F (τ ′ , τ ) = −G ⊺ F (τ, τ ′ ). Proceeding as in the vacuum case, we get a spin-orbit decomposition as in (11) with The polynomials P N S now are defined by and W(k i 1 , ε i 1 ; . . . ; k i S , ε i S ) denotes the correlator (14) evaluated with the modified fermionic Wick contraction (44). For the calculation of this correlator we can still use the cycle decomposition formula (18), only that the fermionic bicycle (17), now must be replaced by Note that, differently from the case of a constant external field [16,18], the fermionic path-integral determinant factor is not affected by the presence of the plane-wave field, and remains at its free value 2 D 2 .

The case N = 2
As a check, let us show that the above master formulas correctly reproduce the results of [26] for the N = 2 case. We first give the general off-shell results before specialising to the on-shell helicity flip process studied there. We work throughout with the gauge choice (39) for convenience and note that momentum conservation in the + direction gives k + 2 = −k + 1 .

Off-shell
Using the notation introduced in (11), for Γ 20 we find from (50) which is sufficient to produce the scalar QED result when substituted into (12). Furthermore, for the spinor case we also require These lead to a spin-orbit decomposition where we have omitted the momentum conserving δ-functions in the + and ⊥ directions. The term in the exponent 2 i=1 k i · I(τ i )− I at the two-point level could be removed by imposing on a µ , instead of (22), the stronger gauge condition of full transversality. To see this, it is easiest to return to (37). Using the conservation of momentum along the + and transversal directions together with (40), and choosing a function b µ (x) such that b ′ µ = a µ , we have

On-shell
In the on-shell case and for N = 2 photons we gain additional simplifications due to the mass shell condition which, by conservation of momentum in the + and ⊥ directions, implies the additional condition k − 2 = −k − 1 so that k 1 = k = −k 2 with k 2 = 0. This removes the exponent e G 12 k 1 ·k 2 from (59). Further imposing the transversality conditions ε 1 · k = 0 = k · ε 2 , the components of the spin orbit decomposition reduce to
This indeed correctly reproduces the parameter integrals determined in [26] as can be easily seen using the integral representations (66)

Summary and Outlook
We have used the worldline formalism to construct a master formula for the N-photon amplitudes in a general plane-wave background field, for both scalar and spinor QED. We believe that it compares favourably with other available techniques for calculations in plane-wave backgrounds. As usual in applications of the worldline formalism to QED, it unifies the scalar and spinor cases in the sense that any spinor QED calculation yields the corresponding scalar QED quantity as a side result. In particular, with the formalism developed here a calculation of the photon-photon scattering amplitudes in a plane-wave background should be quite feasible. In a more extensive publication, we will give a more detailed derivation, including an alternative approach using the worldline super formalism for the spinor QED case, and explore the N = 3 and N = 4 cases. It should also be interesting to generalize the mapping of worldline averages to spacetime averages, introduced in [26], to the N-point case. Also under consideration is the extension of the formalism to the open-line case, i.e. the photon-dressed scalar and spinor propagators in a plane-wave background.