On the low-energy limit of the QED N-photon amplitudes: part 2

In recent work, Gies and Karbstein have discovered that the two-loop Euler-Heisenberg Lagrangians for scalar and spinor QED have non-vanishing reducible contributions in addition to the well-studied irreducible ones. This invalidates previous applications of those Lagrangians to the computation of the two-loop $N$-photon amplitudes in the low energy limit. Here we compute the corrections to those amplitudes due to the reducible contributions.

of particles. Even the prototypical QED one-loop N -photon amplitudes are currently known only up to the six-point level [13,14,15,16]. For the massless case, there is also a vanishing theorem by Mahlon for the amplitudes with N ≥ 6 and all or all but one helicities equal [17] but less is known for the massive case.
Things are very different if one wants only the low-energy limit of these amplitudes, i.e. for photon momenta such that all kinematic invariants k i · k j are small compared to m 2 . In this limit the information on the photon amplitudes is fully contained in the effective Lagrangian L(F ) for a background field with a constant field strength tensor F µν . The extraction of the low-energy amplitudes from the effective Lagrangian is straightforward in principle, and for the four-point case can be found in textbooks (for example see [18]).
At one loop, the QED effective Lagrangian for the constant field strength case is just the well-known Euler-Heisenberg Lagrangian [19] (see [20] for a review), whose weak field expansion is known in closed form. In [21] (called "part one" in the following) this expansion was used together with the spinor helicity technique [22,23,24] to arrive at a closed-form expression for the one-loop N -photon amplitudes for any number of photons and any helicity distributions. This was also done in parallel for the scalar QED case, where the corresponding effective Lagrangian is due to Weisskopf [25].
In part one, this program was also carried to the two-loop level. Here none of the known representations of the two-loop effective Lagrangians in a constant field [26,27,28,29,30,31,32] is sufficiently explicit to obtain corresponding all -N formulas at the two-loop level. Nevertheless, the formulas given in [26,27,28] were good enough to obtain the weak-field expansions of these two-loop effective Lagrangians up to the order F 10 , which allowed the explicit calculation of the two-loop N -photon amplitudes up to the ten-point level with arbitrary helicities, in this low-energy limit.
However, something special happens again for the "all equal helicity" amplitudes. For the effective action, those correspond to the special case of a self-dual field [33,34,35,36,37] and for such a (constant) background it is possible to compute the effective action explicitly even at the two loop level, for both scalar and spinor QED [38,39]. In [39], this fact was used to derive simple closed-form expressions for these "all +" amplitudes even at the two-loop level.
A qualitative result of part 1 was the following "double Furry theorem:" while the N -photon amplitudes corresponding to K (L) helicity + (−) photons with full energies restrict only the sum K + L = N to be even, in the low-energy limit both K and L have to be even, i.e. the amplitudes with K or L odd vanish in this limit; thus the Euler-Heisenberg Lagrangian holds no information on them. This follows from a lack of non-vanishing invariants, and thus must hold at any loop order.
A crucial point is that in these calculations to date it was assumed that the only diagram contributing to the EHL at the two-loop level is the one particle irreducible ('1PI') one shown in Fig. 1 (the double line denotes the full electron propagator in a constant field). At the same loop order, there is also the one-particle reducible ('1PR') diagram shown in Fig. 2. However, since the one-photon amplitude in a constant field formally vanishes on account of gauge invariance and momentum conservation, this 1PR diagram previously was generally discarded in the literature (see, e.g., [29,40]). However, Gies and Karbstein [41] recently showed that this diagram actually gives a finite contribution, if one takes into account the divergence of the connecting photon propagator in the zero-momentum limit. A careful analysis of that limit led them to the following simple covariant formula that expresses this contribution to the two-loop Lagrangian in terms of derivatives of the one-loop Lagrangian: This discovery has many consequences for constant-field QED, of which some have already been worked out, namely the tadpole contributions to the one-loop propagators in scalar [42] and spinor QED [43], as well as to the two-loop photon vacuum polarization [44]. In particular it renders incomplete all the results obtained in part 1 for the two-loop N -photon amplitudes, starting at the six-point level. The purpose of the present paper is to work out the changes to those results implied by the non-vanishing of the reducible diagram.
In the next section, we will shortly summarize what was previously known about the (scalar and spinor) QED N -photon amplitudes in the low energy limit. To avoid undue repetition, here we will refer the reader to part 1 for some of the details. In section 3 we give our results for the effect of the reducible diagram and tabulate updated coefficients taking these new contributions into account.
2. The N -photon amplitudes in the low-energy limit: summary of known results Since in the abelian case the ordering of the legs does not matter we assume that photons 1, . . . , K carry the helicity '+' and the remaining L photons the helicity '-'. Furthermore due to the double-Furry theorem mentioned above we can take both K and L to be even and we denote their sum by K + L = N .

Low-energy photon amplitudes from Euler-Heisenberg Lagrangians: general procedure
The extraction of the N -photon amplitudes from the effective action proceeds as follows: One chooses photon momenta k 1 , . . . , k N and polarisation vectors ε 1 , . . . , ε N , and defines for every leg the field strength tensor Then define the sum The low-energy amplitude is obtained by inserting F tot into the effective Lagrangian, expanded to the appropriate order, and selecting the terms involving each F 1 , . . . , F N once: For the four-photon case this is a standard textbook exercise [18]. To carry it out in the general N -photon case, it is convenient to use a helicity basis for the polarisations and apply the spinor helicity technique. An efficient method was developed in part 1: to obtain the amplitude with K positive-helivity photons and L = N − K negative-helicity photons, the following steps should be taken: Rewrite the effective Lagrangian in terms of the invariants a, b, that are the invariants of the Maxwell field, defined by (as usual F µν := 1 2 µναβ F αβ ) The charge e will often be set to unity in the following. 3. Change variables from a, b to χ ± via 4. Expand the effective Lagrangian in powers of χ + , χ − .

Retain only the terms involving
− . This selects the contribution to the particle loop dressed by K (L) low energy photons of helicity + (−) from the constant background. 6. In those, effect the replacement where [ij] and ij are spinor products (our spinor helicity conventions follow [3]).

One-loop N -photon amplitudes
To summarise the existing results at one-loop level, we use the well-known integral representations of the effective Lagrangians due to Euler and Heisenberg [19] for spinor QED, and to Weisskopf [25] for scalar QED: Here T denotes the proper-time of the loop fermion. Using the Taylor series, x (the B 2n are Bernoulli numbers) steps 1-4 of the above procedure yield a power series expansion for the one-loop Euler-Heisenberg and Weisskopf Lagrangians where both sums are over even numbers and the coefficients are given by The remaining steps then yield the N -photon low energy scattering amplitudes (with K positive helicities and L negative helicities) (for N ≥ 4). We note that the coeffcients c are symmetric in their arguments, as is required by the CP invariance of the QED photon amplitudes.

Two-loop N -photon amplitudes
At the two-loop level, in [21] the integral representations given in [26,27] were used to compute the weak-field expansion of the irreducible contributions up to the order (F 10 ). This yielded the low-energy photon amplitudes up to ten-point order in the form with coefficients c (2) spin and c (2) scal given in Table 1 of [21] and α = e 2 4π the usual fine structure constant. For the "all +" helicity case the following closed-form expressions can be obtained [39]: where n = N/2. In the following we update these results by including the one particle reducible contribution to the two-loop Euler-Heisenberg and Weisskopf Lagrangians.

Two-loop: one-particle reducible contributions
Here our aim is to express the reducible contribution in terms of kinematic invariants and to relate the two-loop, reducible contributions to the N -photon amplitudes to the one-loop coefficients reviewed above.

The reducible Lagrangian
To write the reducible contribution in a manifestly Lorentz invariant way we use the standard invariants a and b defined in (5), inverting these expressions to write the field strength tensor and its dual as Moreover, one notes that 2(a 2 + b 2 ) = (F 2 ) 2 + (F F ) 2 . With the further results we can express the covariant formula for the reducible contribution as (the cross terms vanish) 16 Rewriting the field strengths in terms of a and b leads to a re-writing of (1) in the form Note that this result is valid for either the spinor or scalar Lagrangian. To proceed we use the explicit formula for L (1) scal and L (1) spin in (9) and (8). For spinor QED the derivatives of L (1) spin with respect to a and b are so that squaring these and taking their difference yields This formula can be expanded or evaluated numerically. The same procedure can be used for scalar QED, leading to very similar formulas. However, rather than finding explicit formulae for the two-loop coefficients from these proper time representations, in the next subsection we instead express them in terms of the one-loop coefficients.

Reducible coefficients in terms of one-loop coefficients
To relate the two-loop coefficients to their one-loop counterparts we make use of (16) and (17) that express the one-loop Lagrangian in terms of χ ± . As above, we want to change variables in the covariant formula (1), this time to χ ± which will allow us to manipulate the series representations of the one-loop Lagrangians directly. So we note that with 8χ ± = F 2 ± iF F derivatives with respect to the field strength can be converted to derivatives with respect to χ ± , From here it is straightforward to derive Squaring this and rewriting F 2 and F F in terms of χ ± provides (again the cross terms vanish) Beginning with spinor QED, (31) enters the summand of the reducible contribution by applying it to the power series representation of the one-loop Lagrangian in (12) (recall the sums over N 1 , N 2 and k 1 , k 2 are over even integers only): Here the one-loop coefficients for spinor QED are given in (14). Note that by changing variables k i → N i −k i for i = 1, 2 and using the symmetry of those coefficients with respect to interchange of their arguments, we can write this more concisely as This form has the further advantage of explicitly displaying the CP invariance of these reducible two-loop contributions.
For the two-loop coefficient appropriate to the scattering of K (L) low energy helicity plus (minus) photons we look for the terms proportional to χ − . For the first term in (33) we can fix N 2 = K + L + 2 − N 1 as long as N 1 K + L − 2, and set k 2 = K + 2 − k 1 subject to K + 2 k 1 and k 1 N 1 − L. Then with the same normalisation as in (18) we find a contribution to the two loop coefficient, c (2) spin , equal tõ recalling that for the k 1 summation one takes the first even integer satisfying the lower condition. The full reducible coefficient is obtained by the addition of the second term in (33), amounting to a symmetrization in K and L: Thus knowledge of the one-loop coefficients is sufficient to determine the reducible contribution to the two-loop coefficients. For scalar QED, the process is the same, and leads to the same formulas (34), (35), where the one-loop coefficients are now given by (15). However, care must be taken in the overall normalisation of the amplitudes. For the irreducible contribution we had a single scalar/spinor loop, and we chose to leave the corresponding factor of −2 accounting for the difference in statistics and degrees of freedom as a global factor, rather than absorbing it into the coefficients c (2,irr) (see (18), (19)). For the reducible contribution we have two such factors of −2, and one of them must now be absorbed into the coefficients for consistency. Thus to obtain the scalar QED equivalent of eq. (34) we must, in addition to changing the one-loop coefficients, also replace the global prefactor 1 2π 2 by − 1 4π 2 . Finally, tables (1) and (2) show explicit numerical values for the two-loop coefficients up to order F 10 . These correct the corresponding tables presented in [21] that included only the irreducible contributions, making explicit the contribution of each of the two diagrams (irreducible and reducible) and the new total values for the coefficients.
Let us note two properties of the new contributions from the reducible diagrams. First, they start contributing only from the six-photon level. This is because the renormalized one-loop    Lagrangians start only at the four-photon level, thus their square only at the eight-photon level, and taking two derivatives lowers the starting point to the six-photon level. Second, the "all +" coefficients c 2,red (N/2, 0) come in a fixed ratio of −2 between the spinor and scalar cases. The reason is that those involve only the coefficients of the one-loop "all +" Lagrangians, and those come from the self-dual scalar and spinor Lagrangians that coincide at one loop (after renormalization and up to the global normalization, but the latter difference has been eliminated by our convention for the coefficients). This is due to the fact that the Dirac operator in a self-dual field has a supersymmetry [33,45,46,47].
In this special case that all photons have equal helicities our formulas above simplify considerably. Setting L = 0, we find that only the first term in the decomposition (35) contributes in this case, and that (34) simplifies tõ c (2,red) spin