Master formulas for the dressed scalar propagator in a constant field

The worldline formalism has previously been used for deriving compact master formulas for the one-loop N-photon amplitudes in both scalar and spinor QED, and in the vacuum as well as in a constant external field. For scalar QED, there is also an analogous master formula for the propagator dressed with N photons in the vacuum. Here, we extend this master formula to include a constant field. The two-photon case is worked out explicitly, yielding an integral representation for the Compton scattering cross section in the field suitable for numerical integration in the full range of electric and magnetic field strengths.


Introduction
The one-loop effective action in scalar QED has the well-known "worldline" or "Feynman-Schwinger" representation [1], Here m and T denote the mass and proper-time of the loop scalar, and P Dx(τ ) the path integral over closed loops in (euclidean) spacetime with periodicity T in the proper-time.
Strassler in 1992 [2] showed how to convert this path integral into the following master formula for the N-photon amplitudes: Here G B ,Ġ B ,G B are the "bosonic" worldline Green's function and its first and second derivatives, Here a 'dot' always means a derivative with respect to the first variable, and we abbreviate G B (τ i , τ j ) ≡ G Bij etc. G B (τ, τ ) is the Green's function for the second derivative operator d 2 dτ 2 adapted to the periodicity, as well as to the "string-inspired" ('SI') boundary conditions (up to an irrelevant constant that has been omitted). Note thatG B (τ, τ ) contains a delta function that brings together two photon legs; this is how the seagull vertex arises in the worldline formalism. The notation ε 1 ε 2 ···ε N means that the exponential should be expanded, and only the terms linear in each of the polarization vectors be kept. The photons are ingoing and still off-shell, so that these vectors are just bookkeeping devices at this stage.
Originally, the same master formula (2) was derived by Bern and Kosower [3,4] from string theory as a generating expression from which to construct the one-loop on-shell N gluon amplitudes by way of a certain set of rules. It contains the information on the N -photon amplitudes in a form that is not only extremely compact, but also well-organized with respect to gauge invariance, particularly when combined with a certain integration-by-parts procedure [3,4,2,5]. Moreover, it combines into one integral the various Feynman diagrams differing by the ordering of the N photons. This may not seem very relevant at the one-loop level, however when the N -photon amplitudes are used as building blocks for multiloop amplitudes it leads to highly nontrivial representations combining Feynman diagrams of different topologies [6,7] (see also [8]).
In [2] also a generalization to the spinor QED was given (see [7] for generalizations to more general field theories).
Shaisultanov [9] then generalized both the scalar and spinor QED master formulas to the case of QED in a constant external field F µν . For the scalar case, this generalized master formula can be written as [10,7] where we have introduced the abbreviation Z ≡ eF T . This master formula differs from the vacuum one, Eq. (2), only by the additional determinant factor det − 1 2 sin(Z) Z , which represents the dependence of the free (photonless) path integral on the external field, and a change of the worldline Green's function G B to a new one G B that holds information on the external field, This Green's function obeys the same SI boundary conditions as the vacuum one, (4). The master formula (5) and its spinor QED generalization [9,10] are usually more efficient for the calculation of photonic processes in a constant field than the standard method based on Feynman diagrams. Its applications include the vacuum polarization in a constant field [11,12,7], photon splitting in a magnetic field [13,7], and the two-loop Euler-Heisenberg Lagrangian in an electric/magnetic field [10,14,15,7,16] as well as in a self-dual [17] background field. See [18,19,20,21,22,23,24] for extensions to gravity and Einstein-Maxwell theory.
Much less has been done for the analogous amplitudes involving an open line. For scalar QED in the vacuum, already in 1996 Daikouji et al. [25] obtained the following master formula representing the scalar tree-level propagator dressed with N photons (Fig. 1): . . . . . .
Here a different worldline Green's function ∆(τ, τ ) appears, Instead of the SI boundary conditions (4), it is adapted to Dirichlet boundary conditions ('DBC') Contrary to the former, these boundary conditions break the translation invariance in proper-time, so that one now has to distinguish between derivatives with respect to the first and the second argument. A convenient notation is [26] to use left and right dots to indicate derivatives with respect to the first and the second argument, respectively: The Green's functions G B and ∆ are related by [27] (the factor of 2 is conventional) with the inverse relation In [25] the master formula (7) was derived by a comparison with the standard Schwinger parameter integral representations of the corresponding Feynman diagrams. Recently, the same formula has been rederived [28] inside the worldline formalism, starting from the generalization of the path integral representation (1) to the propagator of a scalar particle in the Maxwell background: The master formula (7) so far has been generalized neither to spinor QED, nor to the inclusion of an external field. The purpose of the present paper is to carry out the latter generalization; the extension to the fermonic case (but without an external field yet) will be presented in a companion paper [29]. See [30] for a non-abelian generalization of the dressed scalar propagator. The organization of the paper is as follows. As a warm-up, in section 2 we use the path integral representation to rederive the well-known scalar propagator in a constant field, in configuration as well as in momentum space. In section 3 we obtain our master formulas for the photon-dressed propagator in a constant field in both configuration and momentum space, generalizing the vacuum calculation of [28]. In section 4 we work the momentum space formula out for the N = 2 case, and obtain a compact integral representation for the Compton scattering cross section in a constant field. Section 5 provides a summary and outlook. In appendix Appendix A we give our conventions, while in appendix Appendix B we collect some information on the constant field worldline Green's functions.

The propagator in a constant field
In this section, we use (13) to just rederive the well-known scalar propagator in a constant field, without photons yet.

Configuration space
Choosing Fock-Schwinger gauge, the gauge potential for a constant field can be written as where we have fixed the initial point of the trajectory x as the reference point where the potential will vanish. Further, we decompose the arbitrary trajectory x(τ ) into a straightline part and a fluctuation part q(τ ) obeying Dirichlet boundary conditions, q(0) = q(T ) = 0: Using (14) and (15) in (13), and further defining after some simple manipulations the integrand can be rewritten as The path integral is already of gaussian form, and in the second line we have written it in a form that prepares the formal gaussian integration. Apart from the free path integral normalization, which is (see, e.g., [31]) this involves the determinant and the inverse of the operator − d 2 dτ 2 + 2ieF d dτ . For the case of the SI boundary conditions (4), the relevant formulas have been given already in (5), (6) above. The ratio of the field-dependent and free path integral normalizations are (the 'prime' refers to the elimination of the zero mode which is contained in the path integral for string-inspired boundary conditions). This can be shown by a direct eigenvalue computation [31], and it is easy to see that the spectrum does not change when passing from string-inspired to Dirichlet boundary conditions, so that (19) holds unchanged for the open-line case. The worldline Green's function does change, but still relates to the one for string-inspired boundary conditions in the same way as in the vacuum case, (12): Note that, contrary to the vacuum Green's function (8), it is a non-trivial matrix in the Lorentz space-time indices. Using this Green's function in the usual completing-the-square procedure, we get Here we have now extended the above 'dot' notation to include integration as well as differentiation; a left (right) 'open circle' on ∆ (τ, τ ) denotes an which brings us to the well-known proper-time representation of the constantfield propagator (see, e.g., [32]),

Momentum space
We Fourier transform (23), and changing the integration variables to we get As the final result becomes

The dressed propagator in a constant field
We now wish to dress the propagator with N photons in addition to the constant field. As before, we start in configuration space.

Configuration space
For this purpose, the potential in (13) has to be chosen as where A ext is the same as in (14), and A phot represents a sum of plane waves: Each photon then effectively gets represented by a vertex operator integrated along the scalar line. This leads to the following path integral representation of the constant-field propagator dressed with N photons: For the evaluation of the path integral, it will be convenient to rewrite the photon vertex operator (31) as Applying the path decomposition (15) we get the following generalization of (17), The path integral is already in a form suitable for gaussian integration. "Completing the square" using the Green's function (20), and using (22), we get the following x -space master formula: For the special case of a purely magnetic field, this x -space master formula was obtained already in 1994 by McKeon and Sherry [33].

Momentum space
The transition to momentum space is quite analogous to the photonless case. We Fourier transform according to (24), and change the variables to (25). The x + integral produces the global delta function for energymomentum conservation, and the x − integral is gaussian. Performing it we get our momentum space master formula: Here we have defined The master formula (36) describes the same set of Feynman diagrams depicted in Fig. 1, only that now all the scalar propagators are the "full" ones in the external field (usually indicated by a double line). When applying it to the calculation of physical processes, one has to take into account that it describes the untruncated dressed propagator, i.e. the final propagators on each end of the scalar line in Fig. 1 are included. To obtain the matrix element T , we have to cancel these final propagators using (28): Moreover, it will be convenient to Wick rotate from euclidean to Minkowski space; the rules for the Wick rotation are given in appendix Appendix A together with our conventions. In appendix Appendix B we collect the formulas necessary to write the integrand in explicit form. We use (20) to write the Green's function ∆ (τ, τ ) in terms of G B (τ, τ ), which is translation invariant and obeys (4) which will be very useful here. We then explain how to write G B (τ, τ ) explicitly for a generic constant field. Finally, let us remark that eventual poles in the global proper-time integral due to the determinant factor in (36) are spurious, because when cos Z = 0 the factor e −T b( tanZ Z )b will vanish too (differently from the corresponding one-loop amplitudes, where such poles lead to an imaginary part related to pair creation).

Compton scattering in a constant field
We will now work out the N = 2 case, i.e. Compton scattering in a constant field. Expanding out the exponentials in (36) and projecting to the terms linear in both polarization vectors, we find (omitting now the global factor for energy-momentum conservation): and Squaring, and performing the sum over the photon polarizations via we get the following for the Compton cross section: After writing the integrand explicitly with the help of the formulas of appendix Appendix B, this expression is suitable for numerical integration.

Summary and Outlook
Using the worldline path integral formalism, we have derived a Bern-Kosower type master formula for the scalar propagator in QED, in a constant field and dressed by an arbitrary number of photons. The x -space version of this formula generalizes the one obtained by McKeon and Sherry for the purely magnetic case [33]; the p -space version generalizes the vacuum master formula of Daikouji et al. [25] on one hand, the closed-loop master formula of Shaisultanov [9] on the other. Our master formula is valid off-shell, and combines the various orderings of the N photons along the scalar line. It can thus be used as a convenient starting point for the construction of higherloop scalar QED processes in a constant field. On-shell, it yields parameter integral representations for linear and nonlinear Compton scattering in the field, as well as the various processes related to it by crossing.
To make this paper self-contained, we have also provided all the machinery necessary for writing the integrands in explicit form. We have worked out the integrand for the linear Compton scattering case explicitly, arriving at a compact representation suitable for numerical integration. The results of such a numerical computation will be presented in a forthcoming publication. Compton scattering in magnetic fields is a process of potential relevance for astrophysics, but, to the best of our knowledge, so far has been studied only in the strong-field limit [35].
The advantages of G B (τ, τ ) are that it is translation invariant, so that we do not have to distinguish between right and left derivatives, and that it fulfills the same nonlocal boundary conditions as the vacuum Green's function (4), The latter property will be very useful for the 'circled' Green's functions. Moreover, the Lorentz matrix structure of G B (τ, τ ) has already been worked out for the various types of constant fields [12,7].
Using (B.1), the various derivatives and integrals of ∆ (τ, τ ) appearing in the master formulas become

Appendix B.2. General properties of the Green's function G B
Here we cite a few general properties of the Green's function G B and its derivatives; for derivations and more details, see [12,7,23]. We can write these functions as power series in the matrix Z ≡ eF T as follows [10]: By absorbing the dependence on τ, τ in terms of the derivative of the vacuum Green's function,Ġ B (τ, τ ), one avoids having to make an explicit case distinction between τ 1 > τ 2 and τ 1 < τ 2 that would become necessary otherwise [9]. Let us note also the coincidence limits of G B ,Ġ B : Note that they are independent of τ . As Lorentz matrices, G B and its derivatives have the following symmetry properties: For weak background fields, it is often justified to approximate the Green's function by the first few terms of its expansion in F µν . To order F 2 , one finds Finally, a matrix decomposition of G B will be necessary. This can be achieved in a Lorentz invariant way [12], but from a practical point of view it is simpler to work in a Lorentz frame well-adapted to the external field. Here, we will be satisfied with treating (i) the case of a generic field and (ii) the purely magnetic field case; see [12] for the more special 'crossed field' and 'self-dual' cases. In all cases it will be useful to decompose G B as where S B is the even part of G B as a function of F , and A B the odd one. For S B , the following trigonometric rewriting is often useful: where we have rescaled τ = T u, τ = T u .
Appendix B.3.1. The generic case For a generic constant field, both Maxwell invariants B 2 − E 2 and E · B are nonzero. By Lorentz invariance there then exists a Lorentz frame where the electric and magnetic field vectors both point along the z -axis, and by parity invariance we can assume that they both point along the positive zaxis. In euclidean conventions, we then have Using this Lorentz frame and base, and defining z ⊥ ≡ eBT, z ≡ ieET , (B.11) the matrix functions S B and A B can be decomposed as [12,7] S µν B12 = − T 2 α=⊥,