The QED four-photon amplitudes off-shell: part 2

This is the second one of a series of four papers devoted to a first calculation of the scalar and spinor QED four-photon amplitudes completely off-shell. We use the worldline formalism which provides a gauge-invariant decomposition for these amplitudes as well as compact integral representations. It also makes it straightforward to integrate out any given photon leg in the low-energy limit, and in the present sequel we do this with two of the four photons. For the special case where the two unrestricted photon momenta are equal and opposite the information on these amplitudes is also contained in the constant-field vacuum polarisation tensors, which provides a check on our results. Although these amplitudes are finite, for possible use as higher-loop building blocks we evaluate all integrals in dimensional regularisation. As an example, we use them to construct the two-loop vacuum polarisation tensors in the low-energy approximation, rederive from those the two-loop $\beta$-function coefficients and analyse their anatomy with respect to the gauge-invariant decomposition. As an application to an external-field problem, we provide a streamlined calculation of the Delbr\"uck scattering amplitudes in the low-energy limit. All calculations are done in parallel for scalar and spinor QED.


Introduction
The present paper is the second one in a series of four devoted to a first calculation of the scalar and spinor QED one-loop four-photon amplitudes fully off-shell, using the worldline representation of these amplitudes. In part I [1] we already derived these representations, discussed their properties, and calculated them for the simple limiting case where all four photons are taken in the low-energy (or "Euler-Heisenberg" limit). In the present sequel, we calculate them explicitly for the case where only photons 3 and 4 are taken in that limit, but 1 and 2 have arbitrary off-shell momenta, see Fig. 1.  The results are given in compact form in terms of the hypergeometric function 2 F 1 for the dimensionally continued case, and in trigonometric form for D = 4.
Although to our knowledge this special case of the four-photon amplitudes has not been considered by other authors, for k 1 = −k 2 , k 3 = −k 4 it is straightforward to extract them from the well-studied photon polarization tensors in a constant field [2,3,4,5,6,7,8,9], and we use this fact for a check on our calculations.
Off-shell legs can be used for creating internal propagators by sewing, or for connecting them to external fields. As an example for the former, we use our results to construct, by sewing of the legs 1 and 2 that carry full momentum, the two-loop scalar and spinor QED vacuum polarization tensors in the low-energy limit, see Fig. 2.
In this limit the vacuum polarization tensors just reduce to the induced Maxwell terms, from which we can rederive the two-loop β-function coefficients. Apart from providing another check, this also improves on previous work [10] where the worldline formalism had already been applied to the calculation of these coefficients. Although these coefficients were already known for decades [11,12], that work was motivated by the fact that the formalism allows one to obtain parameter-integral representations that unify all the Feynman diagrams contributing to the quenched (single fermion loop) photon propagator at any loop order, see., e.g., Fig. 3 for the three-loop case. The interesting feature of the worldline formalism is that the sewing results in parameter integrals that represent not some particular Feynman diagram, but the whole set of Feynman diagrams shown in Fig. 3.
And it is precisely this type of sums of diagrams that are known for particularly extensive cancellations between diagrams. In 1967 Johnson, Willey and Baker showed [13] that the quenched vacuum polarisation has, at any loop order, only a single overall UV divergence. And the coefficient of this pole, which is essentially the β-function, turned out to be rational up to the four-loop level (see [14] and refs. therein). The role of gauge invariance in these cancellations is not transparent, and remains an object of active investigation even at the two-loop level [15]. The worldline formalism was applied to the recalculation of the two-loop scalar and spinor QED β-functions by M.G. Schmidt and one of the authors in [10] (see also [16]) to achieve significant cancellations already at the integrand level. However, those calculations were done using a different approach based on two-loop worldline Green's functions [17], which provides a shortcut but obscures the role of gauge invariance. Only after that work a refinement of the usual homogeneising Bern-Kosower integration-by-parts procedure was found that led to a decomposition of the four-photon amplitudes into sixteen individually gauge invariant contributions (see [18,19] and part I). This gives us a chance to improve on [10] by asking the following two questions: first, can we identify parts of that decomposition that drop out in the construction of the two-loop β functions? Second, are there cancellations in their computation beyond what is expected from gauge invariance, that is, not only inside each gauge invariant structure, but also between them? Thus we will study here the distribution of the β-function coefficients, as well as the cancellation of the double pole in the 1 expansion, in the gauge-invariant decomposition derived in part I.
As an application to an external-field problem, among the many processes related to the four-photon amplitudes we have chosen here to reanalyse low-energy Delbrück scattering, the deflection of photons in the Coulomb field of nuclei due to the vacuum polarization (partially motivated also by the fact that the worldline formalism has already been applied extensively to calculations involving constant [20,21,7,16,22,23], planewave [24,25] and combinations of the two types of fields [26], but not as yet to Coulomb fields). Delbrück introduced this scattering in 1933 in order to explain the discrepancies Meitner and Kösters had found in their experiment for the Compton scattering on heavy atoms [27]. Later Bethe and Rohrlich computed the angular distribution for small angles and the total cross section for Delbrück scattering [28]. In 1973 DESY reported a first observation of this scattering in the high-energy, small-angle limit [29] which was in agreement with the prediction made in a series of papers by Cheng and Wu [30,31,32,33] and later confirmed in [34,35]. In 1975 the Göttingen group performed another experiment which was the first one where exact predictions based on Feynman diagrams were confirmed with high precision after considering other background phenomena like atomic and nuclear Rayleigh scattering [36]. The most accurate high energy experiment done so far on Delbrück scattering is the one carried out at BINP [37,38]. Here we will use our results for a short and efficient calculation of the Delbrück scattering cross section in the low-energy limit.
This paper is organized as follows. In Section 2 we shortly review the worldline representation of the scalar and spinor QED four-photon amplitudes, and discuss our general computational strategy in this paper. In particular, we provide all the integral formulas required to integrate out low-energy photons without having to split these multi-photon integrals into ordered sectors. The central Section is 3, where we list our explicit final results for the amplitudes, in D as well as in four dimensions, and for both scalar and spinor loops. Section 4 contains the above-mentioned comparison with the amplitudes extracted from the vacuum polarization in a constant field. In Section 5 we recalculate the two-loop β-functions, while Section 6 is devoted to lowenergy Delbrück scattering. Finally, Section 7 gives a summary and outline of future work. Supplementary information and formulas are provided in the appendices.

Worldline representation of the four-photon amplitudes
In part I, we presented in detail the derivation and structure of the worldline representation of the Nphoton amplitudes. Due to the inherent freedom in the integration-by-parts procedure, this representation comes in various slightly different versions. For easy reference, let us summarize here the representation of the four-photon amplitudes that we will actually use in the present sequel 1 : Here we have already done the usual rescaling τ i = T u i such that the exponential part is the bosonic Green's functions are   The extraordinary compactness of this representation is made possible through the introduction of the "Lorentz-cycle" Z n (i 1 i 2 . . . i n ) as is the field strength tensor of photon i, and the "bicycle" It is the "tails" that exist in various versions. For the present computation, we use the one-photon tail T (i) of the original Q-representation and the "short tail" T sh (ij), introduced in part I, as the two-photon tail: The spinor-loop result is obtained by employing the Bern-Kosower replacement rule, i.e. replacing simultaneously every closed (full) cycle asĠ i1i2Ġi2i3 · · ·Ġ ini1 appearing in the integrand of the scalar-loop witḣ where G F ij = sgn(u i − u j ) is the fermionic Green function (here it is understood thatĠ ij = −Ġ ji may have to be used to achieve the cycle form). We write the spinor-loop amplitude as (2.10) Thus, apart from a global factor of −2, the only difference to the scalar QED formula (2.1) is the replacement of Q scal by Q spin according to the rule (2.9). Let us also emphasize once more that equations (2.1), (2.10) are valid off-shell, and that the right-hand sides are manifestly finite term-by-term. The well-known spurious UV-divergences of the four-photon diagrams that usually cancel only in the sum of diagrams would show up here as logarithmic divergences of the T -integration at T = 0, but have been eliminated already at the beginning by the IBP procedure that led from the P-representation to the Q-representation, see part I. To avoid carrying common prefactors we definê (2.11) 2.1. Case distinction for the low energy limit of photons 3 and 4 As has been mentioned above, in the present part II the photons number 3 and 4 are taken in low-energy limit which means that in the expressions Q scal/spin e (·) we only consider those contributions which are linear in both k 3 and k 4 (for more details about the low-energy limit see part I). For the sake of clarity, in this subsection we list the different cases that appear in our calculations: • Case 1: Terms in Q scal which are already linear in both k 3 and k 4 do not need any further factors of k 3,4 from the exponential, so that we can simply replace e (·) → e (·)| k 3 ,k 4 →0 = e T G12k1·k2 . This includes all the "pure cycle" terms, defined by the absence of tails. For instance, the four-cycle (2.14) • Case 2: Terms in Q 3 scal which contain one of the low-energy momenta, say, k 3 , in the three-cycle, but lack k 4 . Those require an expansion of the exponential factor to linear order in k 4 . For instance, the one-tail termĠ(123)T (4) e (·) giveṡ G(123)T (4) e (·) →Ġ(123) Ġ 41 ε 4 · k 1 +Ġ 42 ε 4 · k 2 T G 14 k 1 · k 4 + T G 24 k 2 · k 4 e T G12k1·k2 . (2.15) • Case 3: Terms in Q 3 scal which have two low-energy momenta in the three-cycle, for those we neglect terms in the one-tail that are linear in any of the two low-energy momenta. For instance, the one-tail termĠ(234)T (1) e (·) givesĠ (234)T (1) e (·) →Ġ(234)Ġ 12 ε 1 · k 2 e T G12k1·k2 . (2.16) However, it turns out that the contributions of these particular terms vanish after integration over u 3 and u 4 .
• Case 4: Terms in Q 2 scal which lack both k 3 and k 4 . Such terms occur only in the two-tail terṁ G(12)T sh (34) e (·) . Here the exponential factor must be expanded to linear order in both k 3 and k 4 , • Case 5: Terms in Q 2 scal which have only one of the low-energy momenta in the two-cycle. For instance, the two-tail termĠ(13)T sh (24) e (·) giveṡ where we took the terms linear in k 4 from the exponential.
• Case 6: Terms in Q 2 scal which are already quadratic in at least one of the low-energy momenta can be discarded. For instance, in the two-tail termĠ(34)T sh (12) e (·) we have terms likė In particular, all terms inĠ(34)T sh (12) e (·) can be neglected.
Thus we see that, in the limit of two low-energy photons which is our object of interest in this paper, there are many terms that drop out of the amplitude at the integrand level, and others that drop out after integration. The transition to spinor QED does not lead to any new considerations.

Low-energy limit for two of the photons: computational examples
Next, let us explain our strategy for computing the four-photon amplitude with two low-energy photons, using some sample terms from both Q scal and Q spin .
Example 1: Let us consider the following one-tail term: In order to take the low-energy limit for leg 4, we define Here, for convenience, we set the following convention for low-energy legs: on the left-hand side of expression (2.21) the subscript '(4)' indicates that the leg number 4 with momentum k 4 is to be taken in the low energy limit and integrated out. In Appendix A we give a list of all the occurring integrals (up to permutations). It allows us to perform the integral over u 4 without fixing an ordering for the remaining photon legs (for efficient techniques to derive such formulas see, e.g., [39]). Since in this example we do not have linear terms in k 4 , we take them from the exponential part and we use Eq. (A.5) to integrate (2.24) Combining terms, we can write the result in a manifestly gauge invariant form (2.25) Next we repeat all this with photon number 3. We define Similarly to the above, the subscript '(34)' in equation (2.26) indicates that legs number 3 and 4 are both projected on their low energy limits. Note that some terms in (2.25) are quadratic in k 3 and can be neglected. We use (A.4) to perform the integral over u 3 and, with the aid of the identityĠ 2 ij = 1 − 4G ij , write the result as (2.28) Therefore at this stage the contribution of Q 3 scal (123; 4) to the four-photon amplitude with legs 3 and 4 at low energy is given bŷ This leads us to define The proper-time integral T is elementary and, due to the unbroken translation invariance along the loop, one of the two parameters u 1 , u 2 can be fixed at some arbitrary value, such as u 2 = 0 and u 1 = u. To the remaining u integral we apply a tensor reduction procedure, explained in Appendix A.2, to arrive at the following expression, wherek 12 ≡ k1·k2 m 2 and 2 F 1 (a, b; c; x) is the Gauss hypergeometric function. With these definitions, (2.28) and (2.29) transform intoΓ 3 scal(34) (123; 4) = − (2.32) Proceeding to the spinor QED case, since for the term at hand the integration over u 4 does not interfere with the application of the Bern-Kosower replacement rule (2.9), the term corresponding to (2.25) reads (2.33) It would seem that we now have to extend our table of integrals in Appendix A to integrals involving botḣ G ij and G F ij , but we can avoid this by eliminating the latter in favor of the former. This can be done using the basic identities Proceeding as in the scalar case, one finds that the spinor-QED equivalent of (2.32) becomeŝ as the reader can easily verify using formulas of Appendix A. Example 2: Let us consider another one-tail term, Notice that it has a quadratic term in k 4 and two linear terms in k 4 . Omitting the former, and integration the latter over u 4 using (A.4), we obtain Thus we have a term quadratic in k 3 and a term linear in k 3 , and for the latter the integral over u 3 turns out to vanish. Therefore we get nothing,Γ We leave it to the reader to check that the additional terms generated by applying the Bern-Kosower replacement rule to the (234) cycle vanish, too. Thereforê Example 3: As our final example, we consider the following 4-cycle term in the spinor QED integrand, To reexpress the spin contribution G F 12 G F 23 G F 34 G F 41 , we insert a factor 1 = −G F 13 G F 31 , and once more use the identity Upon integration we find, after some cancellations, Integrating over u 3 we obtain (2.44) Thus with the notation (2.30), the contribution to the amplitude of this term readŝ 3. Results: low-energy limit for two of the photons In this section, we present our results for the complete four-photon amplitudes in both scalar and spinor QED, off-shell but under the restriction that photons 3 and 4 are taken in the low-energy limit i. e., we consider only contributions that are linear in k 3 and k 4 (see part I). These amplitudes represent the main result of the present article. In the worldline formalism, they appear naturally decomposed aŝ Recall that the subscript (34) inΓ i (34) indicates that the integrals over u 3 and u 4 were performed taking the low-energy limit for the corresponding photons. The absolute normalizations of the amplitudes are given by spin (34) .
We give them first in dimensionally regularized form, and then in D = 4. Since the worldline representation of these amplitudes is manifestly finite, taking this limit is trivial and will not involve a cancellation of spurious 1/ poles as would be the case in a Feynman diagram calculation (in our conventions = D − 4).

Dimensionally regularized amplitudes
The integrals appearing in (3.2) are all of the form of (2.30) so that we choose to express them in terms of the functions Y nl that have been given in terms of 2 F 1 in (2.31). We note that this implies some redundancy since, as shown in Appendix A.2, there exist recursion relations between the Y nl functions that would also allow us to rewrite all our results entirely in terms of the Y n0 . However, the resulting representation would be less compact than the one given in the following.

Check: the photon propagator in a constant field
The kinematical regime of the four-photon amplitudes that we are studying in this paper has, to the best of our knowledge, not been treated in the literature before. However, for the special case where k 1 = −k 2 (which then also implies k 3 = −k 4 ) they can be easily extracted from the photon polarization tensor in a constant background field, a quantity that has been intensely studied in spinor [2,4,5,6,7,8,9] and to a lesser degree in scalar QED [3,7]. The field can be treated non-perturbatively using the exact Dirac or Klein-Gordon propagator in the field, indicated by the double line on the left-hand side of Fig. 4 as is customary.
Expanding out the full non-perturbative polarization tensor in powers of the background field one obtains the series depicted on the right-hand side of Fig. 4, containing any even number of interactions with the field. Since a constant field cannot inject energy or momentum into the loop, these interactions are mediated by zero-momentum photons. Thus the second diagram on the right-hand side of Fig. 4 corresponds to our case of two arbitrary and two low-energy photons, the two low-energy legs corresponding to the two photons taken from the background field, however under the restriction k 1 = −k 2 = k imposed by the energy-momentum conservation in the constant field. = + + + . . . We would now like to use this correspondence for a check on our above results. While such a comparison could be made using the results of any of the references cited above, it will be convenient to compare with [7], where the worldline formalism was already used to compute the constant-field photon polarization tensors for both scalar and spinor QED. Although that calculation is still quite different from our present one (mainly because the integrating out of the low-energy photon legs there was effectively done already at the level of the construction of the generalized worldline Green's functions, shown below in (4.22), (4.44)), it is still close enough to our present one to make it possible to verify the equivalence at an early stage, without the need to perform all the integrals. It will be sufficient to work with the intermediate results of our above calculations after integrating out u 3 and u 4 , which we collect in Appendix B. Using suitable integrations-bypart, we identify the resulting set of parameter integrals with the ones obtained by expanding the integral representations given in [7] for the scalar and spinor QED photon propagators in a constant background field to second order in the field strength tensor F µν .

Scalar QED
The above replacement for the momenta (k 1 = −k 2 = k) actually makes some terms in Q scal vanish. The remaining terms in the scalar loop case can be compactly written as:  Note that with the above replacement f µν Rewriting the remaining integrals we have where we have also set u 2 = 0 and u 1 = u so that now G 12 = G(u, 0) = u(1 − u). In order to be able to make the comparison at the integrand level is important to use IBP or equivalently the identity for many of the terms in Q 2 scal (34) . Note that this identity can also be expressed in terms of the Y nl as which will be useful in Section 5. Turning our attention now to the photon polarization tensors in a constant background field, for scalar QED the worldline calculation of [7] yielded the following presentation: where and G Bij is the bosonic worldline Green's function in a constant field background [21], with Z µν = eT F µν . For the exponent of (4.20), we have In order to compare with the results in the present work, we choose F µν = i(f µν 3 + f µν 4 ) and expand the whole polarization tensor to second order in F µν so as to obtain the first two diagrams on the right-hand side of Fig. 4. After some straightforward algebra, we can express the result as Note that the term proportional to (eT ) −2 corresponds to the two-point diagram or the vacuum polarization diagram in the absence of the background field. The remaining terms give the sought after four-photon amplitude, and are found to be in complete agreement with (4.17).

Spinor QED
For the spinor QED case we find, in complete analogy,   (23), (4.40) and the following integral representation As in the scalar case, to be able to compare at the integrand level it is important to make repeated use of (4.18).
The spinor-loop result of the vacuum polarization tensor (from [16,7]) is with the fermionic Green's function in a constant field and its coincidence limit [21]  After the expansion, choosing Z as in the scalar case, the spinor-loop gives which again after removing the term proportional to (eT ) −2 is in perfect agreement with (4.42). Despite of the restriction k 1 = −k 2 , this provides an excellent check on many of the results presented in Section 3 above.

Gauge-invariant decomposition of the two-loop β-functions
In this section, we use the results presented in Section 3 to go to two loops by sewing together the unrestricted photons 1 and 2 (Fig. 2). This yields the two-loop photon propagators in the low-energy limit, that is, the induced Maxwell terms, from whose 1/ -poles we can (up to a contribution from mass renormalization) compute the two-loop β-function coefficients for scalar and spinor QED, recuperating the known results [11,12,40,10,41,16]. As explained in the introduction, besides providing another check this is motivated by the following two questions suggested by the highly structured worldline integrand (2.4): first, are there terms that drop out in the sewing procedure? Second, does the fact that the sixteen terms in this decomposition are (differently from the usual decomposition into Feynman diagrams) individually gauge invariant, have a bearing on the issue of gauge cancellations? Specifically, will the vanishing of the double poles involve only cancellations inside each gauge-invariant structure, or between them?
We use Feynman gauge in the sewing, so that it is done by the replacement with k to be integrated over. The induced Maxwell term in our present set-up will appear as tr(f 3 f 4 ). The sewing will now change the argument of Y nl functions, such that in this section it is understood that Y nl = Y nl (−k 2 , m 2 , D). Going back to the results in D-dimensions in Section 3 for the scalar/spinor-loop, we have the following expressions which will be used in both cases:

(5.2)
Here we have used the Passarino-Veltman type identity where F is an arbitrary scalar function. Note that the three-cycle Z 3 (123) drops out in the sewing, together with its three permutations. Therefore Q 3 scal/spin does not contribute here, and the total amplitudes can be written as (using a similar convention as in (2.11)) (note that a symmetry factor of 1 2 has to be included).

Scalar QED β-function
In this way, we find the following result for the Maxwell term of the two-loop vacuum polarization tensor in scalar QED,

(5.5)
Using the identity (4.19) this can be somewhat simplified, Next, we rewrite this as We then use (A.10) to eliminate the k -integration, which leads to The remaining u-integral can be expressed in terms of the Euler Beta-function as which leads to our final result for the coefficient of the induced Maxwell term, Note the global factor of (D − 4), which makes it clear that, as expected, the 1/ -expansion has only a single pole, After substituting (5.11) into (5.7) one arrives at which is in agreement with previous calculations of the two-loop dimensionally regularized effective action in Scalar QED [10,42] 3 . For the full two-loop renormalization of the photon propagator, one still needs to add an equal contribution ∆Γ (2) scal (f 3 , f 4 ) from mass renormalization [10], This then gives the two-loop photon wave-function renormalization factor and leads to the standard result for the two-loop β-function coefficient in Scalar QED [12,10] β (5.14) More interesting is the distribution of the double and single -pole coefficients a scal and b scal in the decomposition (2.4), shown in Table 1.
Although all terms here are already individually gauge invariant, the vanishing of the double pole still happens through an intricate cancellation between them. Similarly, apart from the vanishing of the threecycle contributions the other terms all contribute to the single pole, without any clear pattern emerging as one might have hoped. γ scal Table 1: Contributions of individual terms in the worldline integrand to the double and single poles of the two-loop Maxwell term in Scalar QED. The a scal sum to zero, the b scal to 2, see (5.11).

Spinor QED β-function
For the more relevant spinor QED case, let us also write down the individual contributions in terms of the Y nl :Γ  Collecting all terms, we obtain Note once more the global factor of (D − 4), which reduces the pole in dimensional regularization to a single one, Adding the appropriate term from mass renormalization [10] ∆Γ (2) we get the total pole of the induced Maxwell term, which finally gives the correct two-loop spinor QED β-function [11,40], Finally, in Table 2 we list the double and single pole contributions of the individual terms in the worldline integrand. The picture that emerges is similar to the scalar QED case above.

Four-dimensional computation of the Spinor QED β-function
Although the calculation of corresponding quantities in Scalar and Spinor QED in the worldline formalism usually proceeds with only minor differences, a surprising finding of [10] had been that, in the case of the two-loop QED β functions, this holds true when employing dimensional regularization but not for strictly four-dimensional regularization schemes such as the use of a proper-time cutoff. In [16], this observation was explained as a consequence of the above-mentioned theorem by Johnson et al. [13] that the total induced effective action can have only a global divergence. Since, unlike Scalar QED, Spinor QED does not possess true quadratic divergences, the required cancellation of subdivergences here leads to two constraint equations. And since, by the structure of the worldline representation of the induced Maxwell term, the integrand before the final integration over u in D = 4 must be of the form (compare (5.18)) one can first conclude from the absence of a quadratic subdivergence that A = 0, and then from the absence of a logarithmic subdivergence that B = 0, leaving only the coefficient C non-vanishing and making the final u-integral trivial (this argument does not work in dimensional regularization due to the principal suppression of quadratic subdivergences by that scheme). Thus one would expect that also the present recalculation should simplify in the Spinor QED case when a four-dimensional scheme is used. And indeed, if we return to (5.16) and put D = 4 there, we find 25) and then using any four-dimensional method for the regularization of the global divergence that is still contained in the global T -integration will lead to a trivial u-integration.

Delbrück scattering at low energies
In this section, we compute the differential cross section of Delbrück scattering in scalar and spinor QED under the assumption that the photon that interacts with the Coulomb field has low energy. For the spinor QED case, this quantity was computed in detail in [43], therefore we will follow their conventions for easy comparison.
Here we use our above results for the four-photon amplitudes with two low-energy photons, and replace the two unrestricted legs 1 and 2 with two Coulomb photons. Furthermore, we now take the two low-energy photons (3 and 4) on-shell.

Summary and outlook
In this second part of our series of papers on the off-shell four-photon amplitudes in scalar and spinor QED we have obtained these amplitudes for the case where two of the photon momenta were taken in the low-energy limit, as defined in part I. We have used the worldline representation of these amplitudes which, as outlined in part I, allows one to treat the scalar and spinor cases in parallel, and to arrive at a permutation and gaugeinvariant decomposition of these amplitudes that is closely related to the one previously obtained by [43]. The coefficient functions in this decomposition are written in terms of Feynman-Schwinger type parameter integrals, and as our main result we have explicitly evaluated them both in four and in D dimensions. Since in our formalism the four-photon amplitudes are manifestly free of UV divergences, the former results are sufficient for one-loop applications. The latter formulas serve the double purpose of making the off-shell four-photon amplitudes useful as input for higher-loop calculations in dimensional regularization, but also giving the correct results for them in other physical dimensions (arbitrary D for scalar QED, even D for spinor QED). After the integrating out of the two low-energy legs these integrals are of two-point type, and can therefore be written in terms of 2 F 1 for general D, and for D = 4 in terms of trigonometric functions.
To the best of our knowledge, for this momentum configuration the four-photon amplitudes have not been obtained before. However, for the special case k 1 +k 2 = k 3 +k 4 = 0 they can be extracted from the low-energy limit of the known vacuum polarization tensors in a constant field, and performing this comparison provided an excellent check on our results for both scalar and spinor QED.
From a technical point of view, probably the most important aspect of our calculation is that it demonstrates how to integrate out a low-energy photon leg without fixing an ordering for the remaining legs. As discussed in part I, when using the light-by-light diagram as a subdiagram in higher-loop calculation this property effectively allows one to unify the calculation of Feynman diagrams of different topologies, and considering the proliferation of diagrams in higher-loop QED calculations it is obviously of great interest to explore what level of simplification can be achieved along these lines.
Off-shell photon legs can be used for creating internal photons by sewing, or for connecting to external fields. As an example for sewing, we have used our results for a construction of the two-loop vacuum polarisation tensors in the low-energy limit, which allowed us to recover the two-loop β-function coefficients for scalar and spinor QED. Although by present-days standards this is easy enough to do using Feynman diagrams, the worldline calculation in the version presented here has the advantage that it replaces the non-gauge invariant decomposition into diagrams by a gauge-invariant decomposition into sixteen substructures. This leads not only to a more compact parameter integral representation, but also has given us an opportunity to probe into the nature of the extensive cancellations that one generally finds in perturbative QED calculations [13,44,14,45]. If, as is sometimes assumed, these were entirely due to gauge invariance, one might have expected the cancellation of the double poles in the scalar and spinor QED β-function calculations to have happened already at the level of each of the gauge-invariant partial structures; instead, we have seen an intricate pattern of cancellations between the various structures. Another interesting result of this calculation has been that four of the sixteen partial structures, namely the one involving a three-cycle, drop out in the sewing. It remains to be seen whether this fact admits a generalization to larger numbers of external photons and multiple sewing.
As an example of an external-field calculation, we have applied our formulas to a calculation of the lowenergy Delbrück scattering cross sections for both scalar and spinor QED. For the spinor QED case this served just as another check of efficiency and correctness, while the scalar QED result is new, to the best of our knowledge. Both the scalar and spinor QED calculations have been presented in a way that would be easy to adapt to other external fields.
The forthcoming third part of this series is devoted to the explicit, D-dimensional calculation of the off-shell four-photon amplitudes with only one leg taken in the low-energy limit and its applications.