On the low-energy limit of the QED N-photon amplitudes: part 2
James P. Edwards Adolfo Huet Christian Schubert
 




a]James P. Edwardsc1
jedwards@ifm.umich.mx

b]Adolfo Huet
adolfo.huet@uaq.mx




a]Christian Schubert
schubert@ifm.umich.mx

[a]Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de
Hidalgo, Edificio C-3, Ciudad Universitaria.  Francisco J. Mújica s/n. Col. Felícitas del Río.  58040 Morelia, Michoacán, México.

[b]Facultad de Ingeniería, Universidad Autónoma de Querétaro, Cerro de las Campanas s/n, Colonia Las Campanas, Centro Universitario, 76010, Querétaro, Querétaro, México. 

[c1]Corresponding author


	


Photon amplitudes Quantum electrodynamics Euler-Heisenberg Lagrangian tadpole diagram  





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.  


















ε












































[ε_1^+,k_1;…;_K^+,k_K;_K+1^-,k_K+1;…;_N^-,k_N]

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cosech
sech
coth
tanhtanh
tantan

12
13
14
15
16
17
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19
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c.c.
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1-4mu l 1-4mu l
1-4.5mu l 1-5mu l
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Z-0.4em Z
Z-0.3em Z
Z-0.2em Z
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        [#1, #2]



 

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                § INTRODUCTION: THE QED PHOTON AMPLITUDES
                
                
                Despite the remarkable progress that has been achieved in recent years in the calculation of 
                on-shell amplitudes, particularly in the massless and/or SUSY cases (see, for example, 
                <cit.>),
                presently explicit calculations of loop amplitudes in gauge theory are, except for special helicity configurations,
                still confined to a small number of particles. Even the prototypical QED one-loop N-photon amplitudes are currently
                known only up to the six-point level <cit.>. 
                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 <cit.> 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 <cit.>).
                
                At one loop, the QED effective Lagrangian for the constant field strength
                case is just the well-known Euler-Heisenberg
                Lagrangian <cit.> (see <cit.> for a review), whose
                weak field expansion is known in closed form.
                In <cit.> (called “part one” in the following) this expansion was used together
                with the spinor helicity technique <cit.> 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 <cit.>. 
                
                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
                <cit.>
                is sufficiently explicit to obtain corresponding all - N formulas
                at the two-loop level. Nevertheless, the formulas given in <cit.>
                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 <cit.>
                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 <cit.>.
                In <cit.>, 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. <ref> (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. <ref>. 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., <cit.>). 
                However, Gies and Karbstein <cit.> 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:
                
                 L^(2) 1PR   =    L^(1)F^ L^(1)F_  .
                
                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 <cit.> and spinor QED <cit.>, as well as to the two-loop
                photon vacuum polarization <cit.>. 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 <ref> we give our results for the effect of the reducible diagram and tabulate updated coefficients taking these new contributions into account. 
                
                
                
                
                § 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
                _1,… ,_N, and defines for every leg the
                field strength tensor
                
                
                F_i^μν   ≡    k_i^μ_i^ν -  k_i^ν_i^μ  .
                
                Then define the sum
                
                
                F_ tot   ≡   ∑_i=1^N F_i  .
                
                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:
                
                Γ
                [k_1,ε_1;…;k_N,ε_N]
                   =    L(iF_ tot)|_F_1⋯ F_N  .
                
                For the four-photon case this is a standard textbook exercise <cit.>. 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 Γ^(EH)
                with K positive-helivity photons and L=N-K negative-helicity photons, the following steps should be taken:
                
                
                  * 
                Replace F by iF in the effective Lagrangian L(F). 
                
                
                  * 
                Rewrite the effective Lagrangian in terms of the invariants a,  b, that are the invariants of the Maxwell field, defined by (as usual _μν := 1/2ϵ_μναβF^αβ)
                
                
                a^2    =   √((F_μνF^μν)^2
                +(F_μνF̃^μν)^2)
                + F_μνF^μν  ,
                
                
                b^2    =   √((F_μνF^μν)^2
                +(F_μνF̃^μν)^2)
                - F_μνF^μν  ,
                
                
                such that a^2-b^2=B^2-E^2, (ab)^2 = ( E· B)^2.
                The charge e will often be set to unity in the following.
                
                
                  * 
                Change variables from a,b to χ_± via
                
                
                a    =   √(χ_+) + √(χ_-)  ,
                
                b    =    -i(√(χ_+)-√(χ_-))   . 
                
                  * 
                Expand the effective Lagrangian in powers of χ_+,χ_-. 
                
                
                  * 
                Retain only the terms involving χ_+^K 2χ_-^L 2. This selects the contribution to the particle loop dressed by K (L) low energy photons of helicity + (-) from the constant background.
                
                
                  * 
                In those, effect the replacement
                
                χ_+^K 2→χ_K^+    ≡   
                
                
                (K/2)!
                 2^K 2{
                [12]^2[34]^2⋯ [(K-1)K]^2 +    all    permutations}  ,
                
                χ_-^L 2→χ_L^-    ≡   
                
                
                (L/2)!
                 2^L 2{⟨ (K+1)(K+2)⟩^2⟨ (K+3)(K+4)⟩^2⋯⟨ (N-1)N⟩^2 +    all    perm.}  ,
                
                
                where [ij] and ⟨ ij⟩ are spinor products (our spinor helicity conventions follow <cit.>). 
                
                
                
                 §.§ 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 <cit.> for spinor QED, and to Weisskopf <cit.>
                for scalar QED:
                
                 L_ spin^(1)   =   
                -
                1 8π^2∫_0^∞dT T ^-m^2Te^2abtanh(eaT)tan(ebT)
                -e^2 3(a^2-b^2)
                -1 T^2  ,
                
                 L_ scal^(1)   =   1 16π^2∫_0^∞dT T ^-m^2Te^2absinh(eaT)sin(ebT)
                +e^2 6 (a^2-b^2) -1 T^2]   .
                
                Here T denotes the proper-time of the
                loop fermion.
                Using the Taylor series,
                
                x/ tan  x   =   ∑_n=0^∞(-1)^n2^2n B_2n (2n)!x^2n  ,
                
                x/sin x   =   
                -∑_n=0^∞
                (-1)^n(2^2n-2) B_2n (2n)!x^2n
                (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
                
                    ℒ^(1)_spin(i F)    = -m^4/8π^2∑_N = 4^∞(2e/m^2)^N∑_K = 0^Nc^(1)_spin(K/2, N-K/2)χ_+^K/2χ_-^N - K/2  , 
                    ℒ^(1)_scal(i F)    =   m^4/16π^2∑_N = 4^∞(2e/m^2)^N∑_K = 0^Nc^(1)_scal(K/2, N-K/2)χ_+^K/2χ_-^N - K/2  ,
                
                where both sums are over even numbers and the coefficients are given by
                
                
                c_ spin^(1)(K 2,N-K/2)
                   =   
                (-1)^N 2(N-3)!
                ∑_k=0^K∑_l=0^N-K
                (-1)^N-K-l B_k+l B_N-k-l
                k!l!(K-k)!(N-K-l)!  ,
                
                
                c_ scal^(1)(K 2,N-K/2)
                   =   
                (-1)^N 2(N-3)!
                ∑_k=0^K∑_l=0^N-K
                (-1)^N-K-l(1-2^1-k-l))
                (1-2^1-N+k+l)
                 B_k+l B_N-k-l
                k!l!(K-k)!(N-K-l)!  .
                
                
                The remaining steps then yield the N-photon low energy scattering amplitudes (with K positive helicities and L negative helicities)
                
                Γ_ spin^(1)(EH)   =   
                -m^4 8π^2(2e m^2)^N
                 c_ spin^(1)(K 2,N-K 2)
                χ_K^+χ_N-K^-   ,
                
                
                Γ_ scal^(1)(EH)   =   m^4 16π^2(2e m^2)^N
                 c_ scal^(1)(K 2,N-K 2)
                χ_K^+χ_N-K^-
                
                (for N≥ 4). 
                We note that the coeffcients c_ spin,scal^(1)(K 2,L/2) 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 <cit.> the integral representations given in <cit.> 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 
                
                Γ_ spin^(2)(EH)
                [ε_1^+;…_K^+;_K+1^-;… ;_N^-]
                   =   
                -απ m^4 8π^2(2e m^2)^N
                 c_ spin^(2)(K 2,N-K 2)
                χ_K^+χ_N-K^-   ,
                
                Γ_ scal^(2)(EH)
                [ε_1^+;…_K^+;_K+1^-;… ;_N^-]
                   =   απ m^4 16π^2(2e m^2)^N
                 c_ scal^(2)(K 2,N-K 2)
                χ_K^+χ_N-K^-   ,
                
                with coefficients c_ spin^(2) and c_ scal^(2) given in Table 1 of <cit.> and α = e^2/4π the usual fine structure constant. For the “all +” helicity case the following closed-form expressions can be obtained <cit.>:
                
                
                c^(2)_ spin(n,0)    =   1 (2π)^2{2n-3/2n-2  B_2n-2
                +3∑_k=1^n-1 B_2k 2k B_2n-2k (2n-2k)}  ,
                
                
                c^(2)_ scal(n,0)   =   1 (2π)^2{2n-3/2n-2  B_2n-2
                +3/2∑_k=1^n-1 B_2k 2k B_2n-2k (2n-2k)}  ,
                
                
                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 (<ref>), inverting these expressions to write the field strength tensor and its dual as
                
                    F^2 = 2(a^2 - b^2) ,      (F )^2 = (4 a b)^2   .
                
                Moreover, one notes that 2(a^2 + b^2) = √((F^2)^2 + (F )^2). With the further results
                
                
                    ∂ a^2/∂ F_μν   = 1/2(F^μν + 1/2F^2F^μν + F ^μν/a^2 + b^2)   , 
                    ∂ b^2/∂ F_μν   = 1/2(-F^μν + 1/2F^2F^μν + F ^μν/a^2 + b^2)
                
                we can express the covariant formula for the reducible contribution as (the cross terms vanish)
                
                
                    16∂ℒ^(1)/∂ F_μν∂ℒ^(1)/∂ F^μν   = 1/a^2[F^2 + (F^2)^2 + (F )^2/a^2 + b^2 + 1/4F^2( (F^2)^2 + (F)^2)/(a^2 + b^2)^2] (∂ℒ^(1)/∂ a)^2, 
                       + 1/b^2[F^2 - (F^2)^2 + (F )^2/a^2 + b^2 + 1/4F^2( (F^2)^2 + (F)^2)/(a^2 + b^2)^2](∂ℒ^(1)/∂ b)^2 .
                
                Rewriting the field strengths in terms of a and b leads to a re-writing of (<ref>) in the form
                
                
                    ∂ℒ^(1)/∂ F_μν∂ℒ^(1)/∂ F^μν = 1/2[ (∂ℒ^(1)/∂ a)^2 - (∂ℒ^(1)/∂ b)^2].
                
                Note that this result is valid for either the spinor or scalar Lagrangian. To proceed we use the explicit formula for  and  in (<ref>) and (<ref>).
                For spinor QED the derivatives of  with respect to a and b are
                
                
                    ∂/∂ a   = -e^2/8 π^2∫_0^∞dT/T ^-m^2 T[b/tanh(e a T) tan(e b T) - 2 e a b T/sinh(2 e a T)tanh(e a T) tan(e b T)   - 2a/3]   ,
                    ∂/∂ b   = -e^2/8 π^2∫_0^∞dT/T ^-m^2 T[a/tanh(e a T) tan(e b T) - 2 e a b T/sin(2 e b T)tanh(e a T) tan(e b T)   + 2b/3]   ,
                
                so that squaring these and taking their difference yields
                
                
                    ∂/∂ F_μν∂/∂ F^μν = 
                    e^4/128π^4∫_0^∞dT/T∫_0^∞dS/S ^-m^2(T + S){   b^2(1 - 2 e a T/sinh(2 e a T))(1 - 2 e a S/sinh(2 e a S))- a^2(1 - 2 e b T/sin(2 e b T))(1 - 2 e b S/sin(2 e b S))/tanh(e a T)tanh(e a S)tan(e b T) tan(e b S)
                    
                    -   4 a b/3 1 -   e a T/sinh(2 e a T) -  e b T/sin(2 e b T)/tanh(e a T)tan(e b T)	
                    +
                    (T→ S )
                    
                    +4 a^2 - b^2/9}.
                
                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 (<ref>) and (<ref>) that express the one-loop Lagrangian in terms of χ_±. As above, we want to change variables in the covariant formula (<ref>), 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± i F derivatives with respect to the field strength can be converted to derivatives with respect to χ_±,
                
                
                    ∂/∂ F_μν = 1/4(F^μν + i ^μν)∂/∂ + 1/4(F^μν - i ^μν)∂/∂.
                
                From here it is straightforward to derive
                
                
                    ∂/∂ F_μν(^k/2^N - k/2) = k/8(F^μν + i ^μν)^k-2/2^N - k/2 + N-k/8(F^μν - i ^μν)^k/2^N - k-2/2  .
                
                Squaring this and rewriting F^2 and F in terms of χ_± provides (again the cross terms vanish)
                
                
                    ∂/∂ F_μν(^k_1/2^N_1 - k_1/2)
                    ∂/∂ F^μν(^k_2/2^N_2 - k_2/2)
                       =   
                          1/4[k_1k_2 ^k_1 + k_2 - 2/2^N_1 + N_2 -(k_1 + k_2)/2 + (N_1 - k_1)(N_2 - k_2) ^k_1 + k_2/2^N_1 + N_2 - (k_1 + k_2) - 2/2].
                
                Beginning with spinor QED, (<ref>) enters the summand of the reducible contribution by applying it to the power series representation of the one-loop Lagrangian in (<ref>) (recall the sums over N_1, N_2 and k_1, k_2 are over even integers only):
                
                
                    ∂/∂ (iF_μν)∂/∂ (iF^μν) =   
                    - m^8/256 π^4∑_N_1 = 4^∞∑_N_2 = 4^∞(2 e/m^2)^N_1 + N_2∑_k_1 = 0^N_1∑_k_2 = 0^N_2 c^(1)_spin(k_1/2, N_1 - k_1/2) c^(1)_spin(k_2/2, N_2 - k_2/2) 
                       ×[k_1 k_2^k_1 + k_2 - 2/2^N_1 + N_2 - (k_1 + k_2)/2 + (N_1 - k_1)(N_2 - k_2)^k_1 + k_2/2^N_1 + N_2 - (k_1 + k_2) - 2/2].
                
                Here the one-loop coefficients for spinor QED are given in (<ref>). 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
                
                
                    ∂/∂ (i F_μν)∂/∂ (iF^μν) =   
                    - m^8/256 π^4∑_N_1 = 4^∞∑_N_2 = 4^∞(2 e/m^2)^N_1 + N_2∑_k_1 = 0^N_1∑_k_2 = 0^N_2 
                     k_1 c^(1)_spin(k_1/2, N_1 - k_1/2) k_2 c^(1)_spin(k_2/2, N_2 - k_2/2) 
                    ×   [ ^k_1 + k_2 - 2/2^N_1 + N_2 - (k_1 + k_2)/2 +
                     ( ↔) ].
                
                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 ^K/2^L/2. For the first term in (<ref>) 
                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 (<ref>) we find a contribution to the two loop coefficient, c^(2)_spin, equal to 
                
                
                    c̃^(2,red)_spin(K/2 , L/2)   =   1/2 π^2∑_N_1 = 4^K+L-2∑_k_1 = max(N_1 - L,  0)^min(N_1, K+2)  c^(1)_spin(k_1/2, N_1 - k_1/2) c^(1)_spin(K+2 - k_1/2, L - N_1 + k_1/2) 
                    
                          × k_1(K+2-k_1),
                
                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 (<ref>), amounting to a symmetrization in K and L:
                
                
                c^(2,red) _spin(K/2 , L/2) = c̃^(2,red) _spin(K/2 , L/2) 
                + c̃^(2,red) _spin(L/2 , K/2)    .
                
                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 (<ref>), (<ref>), where the one-loop coefficients are now given by  (<ref>).
                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 (<ref>), (<ref>)). 
                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. (<ref>) we must, in addition to changing the one-loop coefficients, also replace the global prefactor 1/2 π^2
                by -1/4 π^2. 
                
                
                
                Finally, tables (<ref>) and (<ref>) show explicit numerical values for the two-loop coefficients up to order F^10. These correct the corresponding tables presented in <cit.> 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 <cit.>. 
                
                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 (<ref>) contributes in this case, and that (<ref>) simplifies to 
                
                
                
                
                
                
                    c̃^(2,red) _spin(K/2 , 0)	
                    =  1/2π^2∑_N_1 = 4^K-2[N_1   (K + 2 - N_1) c^(1)_spin(K+2 - N_1/2, 0)c^(1)_spin(N_1/2, 0) ]   .
                
                Now using the fact that according to our normalisations in (<ref>) and (<ref>) the all plus / all minus coefficients for scalar and spinor QED coincide as
                
                
                    c^(1)_spin(n, 0) = c^(1)_scal(n, 0) = (-1)^n+1 B_2n/2n(2n-2)
                
                with n = N/2 as in (<ref>), we can give the above results in a form similar to (<ref>):
                
                
                    c^(2,  red)_spin(n, 0)    = (-1)^n+1/2π^2∑_m=2^n-1 B_2m B_2(n-m+1)/(2m - 2)(2(n-m+1)-2)  ,
                    
                    	c^(2,  red)_scal(n, 0)    = -c^(2, red)_spin(n, 0)/2  .
                
                These results are in agreement with the entries reported in the tables above for L = 0. Analogous results can be found for coefficients with two minus photons by setting L = 2 and repeating the above analysis. 
                
                
                
                § SUMMARY AND OUTLOOK
                
                We have worked out here the changes to the results of <cit.> on the two-loop N-photon amplitudes in the low energy limit, necessary to take into account the 
                recently discovered non-vanishing of the reducible contribution to the two-loop QED effective Lagrangian in a constant field. Contrary to the irreducible contributions,
                the “new” reducible ones can be given in closed form for all helicity assignments, written in terms of the known one-loop coefficients. 
                
                The formulas for the reducible contributions are written in a form advantageous for carrying out an asymptotic analysis. An analysis of special cases like "all-plus-helicities" or equal numbers of plus and minus helicities is also feasible. Consequences for the imaginary part of the effective Lagrangian and Schwinger 
                pair creation will be given elsewhere. Furthermore, these reducible contributions will be present at all higher loop orders too, so that future work on such calculations must take them into consideration.
                
                
                
                 §.§ Acknowledgements
                
                
                C.S. thanks CONACyT for financial support through grant Ciencias Basicas 2014 No. 242461. J.P.E receives funding from PRODEP. Adolfo Huet acknowledges support from CONACyT.
                
                
                
                
                
                
                
                
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