27th ANNUAL INTERNATIONAL LASER PHYSICS WORKSHOP, Nottingham, July 16-20, 2018
1PR contribution to the one-loop electron propagator in a constant field Naser Ahmadiniaz (
[email protected]) Institute for Basic Science (IBS) Center for Relativistic Laser Science (CoReLS), Gwangju, Korea In collaboration with F. Bastianelli, O. Corradini, A. Huet, J. P. Edwards and C. Schubert
2
Outline
Introduction to the worldline formalism One-loop master formula in a constant field Tree-level master formula in a constant field 1PR contribution to the one-loop electron propagator in a constant field
3
History and introduction In 1948, Feynman developed the path integral approach to non-relativistic quantum mechanics (based on earlier work by Wentzel and Dirac). Two years later, he started his famous series of papers that laid the foundations of relativistic quantum field theory (essentially quantum electrodynamics at the time) and introduced Feynman diagrams. However, at the same time he also developed a representation of the QED S-matrix in terms of relativistic particle path integrals. Why worldline formalism? No need to compute momentum integrals and Dirac traces. Worldline formalism works well for massive particles (on- and off-shell) not even at tree-level but at loop order too. The difference between open line and loop (purely bosonic): Dirichlet boundary conditions (topology of a line)
hx|e
−HT
0 |x i =
Z x(T )=x x(0)=x 0
Dx(τ ) e
−S[x,G ]
Periodic boundary conditions (topology of a closed line)
Z Dx(τ )e x(0)=x(T )
−S[x,G ]
4
Free-propagator: Free scalar propagator that is the Green’s function for the Klien-Gordon equation:
D
x,x 0 0 = h0|T φ(x)φ(x )|0i = hx| 0
1 − + m2
4 ∂2 X 0 |x i , = 2 i=1 ∂xi
We exponentiate the denominator using a Schwinger proper-time parameter T . This gives xx 0 D0 =
Z ∞
dT e
−m2 T
hx| exp
0 − T (−) |x i =
Z ∞
dT e
−m2 T
0
0
Z x(T )=x x(0)=x 0
Dx e
R − 0T dτ 1 x˙ 2 4
This is the worldline path integral representation of the relativistic propagator of a scalar particle in euclidean spacetime from x 0 to x. xx 0 D0 =
Z ∞
dT e
−m2 T
e
−
(x−x 0 )2 Z 4T
0
Dq(τ ) e DBC
⇒ Fourier transform ⇒ familiar momentum space representation pp 0 = D 0
Z d
=
D
xe
ip·x
Z d
D 0 ip 0 ·x 0 xx 0 x e D0
D D 0 (2π) δ (p + p )
1 p 2 + m2
R − 0T dτ 1 q˙ 2 4
5
Coupling to electromagnetic field To get the the ”full” or ”complete” propagator for a scalar particle, that interacts with background field A(x) continuously while propagating from x 0 to x
D
xx 0
[A] =
Z ∞
dT e
−m2 T
Z x(T )=x x(0)=x 0
0
Dx(τ ) e
R − 0T dτ 1 x˙ 2 +ie x·A(x(τ ˙ )) 4
Effective action: The effective action encodes the nonlinear properties of a system due to quantum fluctuations, analogously to how the thermodynamic partition function encodes the effects of thermal fluctuations.
Γscal [A] =
Z ∞ dT 0
T
e
−m2 T
Z Dx(τ ) e
R − 0T dτ 1 x˙ 2 +ie x·A(x(τ ˙ )) 4
x(0)=x(T )
Note that we now have a dT /T , and that the path integration is over closed loops; those trajectories can therefore belong only to virtual particles, not to real ones. The effective action contains the quantum effects caused by the presence of such particles in the vacuum for the background field. In particular, it causes electrodynamics to become a nonlinear theory at the one-loop level, where photons can interact with each other in an indirect fashion.
6 After the following decomposition (to fix the average position of the loop) x
µ
(τ )
=
Dx(τ )
=
µ µ + y (τ ) 0 Z Z D d x0 Dy (τ )
x
Z
,
x
1 Z T µ µ dτ x (τ ) ≡ 0 T 0
(1)
The remaining y (τ ) path integral is performed using the Wick contraction rule
hy
µ
ν µν (τi )y (τj )i = −δ GB (τi , τj )
,
GB (τi , τj ) ≡ GBij = |τi − τj | −
(τi − τj )2
(2)
T
The free Gaussian path integral gives Z
R − 0T dτ 1 y˙ 2 −D 4 De = (4πT ) 2
,
D = spacetime dimension
Now if we specialize the background A(x), which so far was an arbitrary Maxwell field, to a sum of N plane waves, N X µ ik ·x µ A (x) = ε e i i i=1 After expanding the interaction term we get
Γscal [A] = (−ie)
N
Z ∞ dT 0
e
−m2 T
T ⇒ V
E −D D γ γ (4πT ) 2 V [k , ε ] · · · V [k , εN ] scal 1 1 scal N
γ [k, ε] ≡ scal
Z T 0
dτ ε · x˙ e
ik·x
(3)
7
At this stage the zero-mode integration can be performed Z d
D
x0
N Y
N ik ·x D D X e i 0 = (2π) δ ( ki ) i=1 i=1
After some mathematical manipulations one get the following master formula which is known as Bern-Kosower master formula for external photons
Γscal [k1 , ε1 ; · · · ; kN , εN ]
=
×
N D D X (−ie) (2π) δ ( ki )
exp
X N h1 i,j=1
2
Z ∞ dT 0
T
N Z T − D −m2 T Y (4π) 2 e dτi i=1 0
GBij (pi · pj ) − i G˙Bij (εi · pj ) +
1 2
i G¨Bij (εi · εj ) lin(ε1 ···εN )
where G˙Bij
=
G¨Bij
=
2(τi − τj ) = sign(τi − τj ) − dτi T d 2 GBij 2 = 2δ(τi − τj ) − dτ 2 T i dGBij
(4)
8
One-loop master formula in a constant field
The presence of an additional constant external field, taken in the Fock-Schwinger gauge centered at x0 changes the path integral Lagrangian only by a term quadratic in the fields 1 ∆L =
iey 2
µ
Fµν y˙
ν
(5)
The external field then can be absorbed by a change of the free worldline propagators, replacing GBij , G˙Bij , G¨Bij by
GBij =
T 2Z 2
Z sin Z
−iZ G˙Bij e + iZ G˙Bij − 1
,
G˙ Bij = · · · ,
¨ G Bij = · · ·
(6)
where Z = eFT and the change in the free path integral due to the external field: −D −D − 1 h sin Z i (4πT ) 2 → (4πT ) 2 det 2 Z Retracing our above calculation of the N-photon path integral with the external induced we arrive at the following generalization of the one-loop Bern-Kosower master formula representing the scalar QED N-photon scattering amplitude in a constant field
(7)
9
Z ∞ dT − D −m2 T − 1 h sin Z i N D D X (−ie) (2π) δ ( ki ) (4π) 2 e det 2 0 T Z X N h1 N Z T i Y 1 ¨ dτi exp ki · GBij · kj − iεi · G˙ Bij · kj + εi · G · × Bij εj lin(ε1 ···εN ) 0 2 i,j=1 2 i=1
Γscal [k1 , ε1 ; · · · ; kN , εN ]
=
which represents the following set of diagrams
+
+···
+
! !
!
=
+
+
+···
10
Tree-level master formula in a constant field Now, let’s go back to the worldline formula for scalar propagator: 0 D[x ; x]
=
Z ∞
dTe
−m2 T
Z x(T )=x x(0)=x 0
0
Dx(τ ) e
−1 4
RT 2 ˙ ))] 0 dτ [x˙ +ie x·A(x(τ
After completing the square in the exponential, we obtain the following tree-level “Bern-Kosower-type formula” in configuration space 0 N D[x ; x; k1 , ε1 ; · · · ; kN , εN ] = (−ie)
×
Z T Y N 0
dτi e
Z ∞
dTe
0 2 1 − D −m2 T − 4T (x−x ) 2 4πT e
0
PN PN τ (x−x 0 ) ∆ij pi ·pj −2i •∆ij εi ·kj −•∆•ij εi ·εj +iki ·(x−x 0 ) i +iki ·x 0 i=1 εi · T T e i,j=1
i=1
where we used the following derivatives of the Green function • ∆(τ1 , τ2 )
=
• ∆ (τ1 , τ2 )
=
• • ∆ (τ1 , τ2 )
=
τ2
1
1 sign(τ1 − τ2 ) − 2 2 1 1 − sign(τ1 − τ2 ) − T 2 2 1 − δ(τ1 − τ2 ) T +
T τ1
where left and right dots indicate derivatives with respect to τ1 and τ2 respectively.
lin(ε1 ε2 ···εN )
.
11
k1 k2 k3
kN
···
p
k2 k1 k3 p′
+
p
k2 k3
kN ···
p + .. .
···
p′
+
···
p′
+
···
+
+ k1
kN
k2 k1 k3 p′
+
kN ···
p + .. .
12
Now, we Fourier transform the master formula
0 D[p; p ; k1 , ε1 ; · · · ; kN , εN ] =
Z d
D
Z x
d
D 0 ix 0 ·p 0 +ix·p 0 x e D[x ; x; k1 , ε1 ; · · · ; kN , εN ]
After some algebra, we get the momentum-space version of our master formula (off-shell) X Z ∞ −T (m2 +p 2 ) 0 N D D dT e p+p+ ki D[p; p ; k1 , ε1 ; · · · ; kN , εN ] = (−ie) (2π) δ 0 i h i |τi −τj | τi +τj PN Z T Y N −2ki ·pτi +2iεi ·p+ − pi ·pj −i sign(τi −τj )−1 εi ·pj +δ(τi −τj )εi ·εj 2 2 × dτi e i,j=1 lin(ε1 ε2 ···εN ) 0 i=1 It includes all the possible Feynman diagrams which one needs to calculate for any order of external photons.
N. A, A. Bashir and C. Schubert, PRD 93, 045023 (2016)
13
The propagator in a constant field The propagator of a scalar particle in the Maxwell background:
D
xx 0
[A]
Z ∞
=
0
−m2 T dT e
Z x(T )=x x(0)=x 0
R − 0T dτ 1 x˙ 2 +ie x·A(x) ˙ 4 Dx e . (8)
where A = Aext + Aphot
(9)
Choosing Fock-Schwinger gauge, the gauge potential for a constant field can be written as 1 µν µ 0 ν A (y ) = − F (y − x ) , 2
(10)
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:
0 ∆ (τ, τ ) ≡ hτ | ^
d2 dτ 2
− 2ieF
d −1 dτ
1 0 0 0 GB (τ, τ ) − GB (τ, 0) − GB (0, τ ) + GB (0, 0) . | τ iDBC = 2
(11)
The dressed propagator in momentum space:
D
pp 0
(F )
=
D 0 (2π) δ(p + p )
Z ∞ 0
−Tp( tanZ )p Z −m2 T e dT e . 1 det 2 [cosZ]
(E.S. Fradkin, D.M. Gitman, S.M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum, Springer 199)
(12)
14
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. For this purpose, the potential in (8) has to be chosen as
A = Aext + Aphot ,
(13)
where Aext is the same as in (10), and Aphot represents a sum of plane waves:
N X µ µ ik ·x A (x) = ε e i . i phot i=1
(14)
Each photon then effectively gets represented by a vertex operator
A V [k, ε] =
Z T 0
ik·x(τ ) dτ ε· x(τ ˙ )e ,
(15)
integrated along the scalar line. This leads to the following path integral representation of the constant-field propagator dressed with N photons:
D
xx 0
(F |k1 , ε1 ; · · · ; kN , εN )
=
(−ie)
N
Z ∞ 0
−m2 T dT e
R ˙ ext (x)] − 0T dτ [ 1 x˙ 2 +ie x·A 4 Dx e
Z P
×V [k1 , ε1 ]V [k2 , ε2 ] · · · V [kN , εN ] . (16)
15
Applying the path decomposition and some nontrivial calculations we arrive to the following x-space representation of the dressed scalar propagator in a constant background field:
in x-space:
D
xx 0
1 h Z i − 1 x Z cot Zx −D −m2 T − dT e (4πT ) 2 det 2 e 4T − sin Z x x τ P Z T Z T N εi · − +iki · − i +iki ·x 0 T T dτN e i=1 dτ1 · · · × 0 0 X N N X ◦ 2e • • • ◦ • ×exp ki ∆ ij kj − 2iεi ∆ ij kj − εi ∆ ij εj + F ∆ i ki − iF ∆ i εi . x− ^ ^ ^ ^ ^ ε1 ε2 ···εN T i,j=1 i=1
(F | k1 , ε1 ; · · · ; kN , εN )
=
N (−ie)
Z ∞ 0
(17) R R left (right) ‘open circle’ on ∆ (τ, τ 0 ) denotes an integral 0T dτ ( 0T dτ 0 ). ^
For the special case of a purely magnetic field, this x - space master formula was obtained already in 1994 by McKeon and Sherry (Mod. Phys. Lett. A9 (1994) 2167).
16 which describes the following Feynman diagrams:
k1 k2 k3
kN
···
p
k2 k1 k3 p′
+
p
k2 k3
kN ···
p + .. .
···
p′
+
···
p′
+
···
+
+ k1
kN
k2 k1 k3 p′
+
kN ···
p + .. .
17
and in p-space
D
pp 0
(F |k1 , ε1 ; · · · ; kN , εN )
×
Z T 0
dτ1 · · ·
Z T 0
=
N Z ∞ X 1 −m2 T 0 N D D dT e p+p + ki (−ie) (2π) δ i 1h 0 i=1 det 2 cosZ
PN ∆ ij kj −2iεi • ∆ ij kj −εi • ∆• ij εj −Tb( tanZ )b i,j=1 ki ^ ^ ^ Z e
dτN e
ε1 ε2 ···εN
. (18)
Here we have defined
Zµν
=
eTFµν
b
≡
p+
N h 1 X T i=1
τi − 2ieF
i ◦ ◦ • ∆ k − i 1 − 2ieF ∆ i εi . ^i i ^
A. Ahmad, N. A, O. Corradini, S. P. Kim and C. Schubert, NPB, 919 (2017) 9.
(19)
18
1PR contribution to the propagator in a constant background field
The Euler-Heisenberg Lagrangian (EHL), one of the first serious calculations in QED describes the one-loop amplitude involving a spinor loop interacting non-perturbatively with a constant background electromagnetic field which is given by
L
1 Z ∞ dT −m2 T e 2 ab (1) (a, b) = − e spin 8π 2 0 T tanh(ebT )tan(ebT )
(20)
where a and b are related to the two invariants of the Maxwell field by 2 2 2 2 a −b = B −E
,
2 2 (ab) = (E · B)
The EHL contains the information on nonlinear QED effects such as photon-photon scattering (Heisenberg and Euler, Z. Phys. 98 (1936) 714). photon dispersion (Adler, Ann. Phys. 67 (1971) 599). photon splitting (Adler, Ann. Phys. 67 (1971) 599; Adler and Schubert, PRL 77 (1996) 1695).
(21)
19 x0
x
The first radiative corrections to these Lagrangians (scalar and spinor), describibg the effect of an additional photon exchange in the loop, were (2) in terms of certain two-parameter integrals which are obtained in the seventies by Ritus (Sov. Phys. JETP 42 (1975) 774). He obtained L scal,spin intracable analytically, closed-form expressions have obtained for their weak-field expansions for the purely electric or magnetic cases (Dunne and Schubert, NPB 564( 2000) 59).
In all these calculations it was assumed that the only diagram contributing to the EHL at the two-loop level is the one particle irreducible (1PI) one:
At the same loop order, there is also the one-particle reducible (1PR) diagram
2
which was generally believed not to contribute!!
20
Gies-Karbstein discovery
Recently, Gies and Karbstein (JHEP 1703 (2017) 108) made the stunning discovery that actually this diagram does give a finite contribution, if one takes into account the divergence of the connecting photon propagator in the zero-momentum limit which leads to the the following simple formula
L
(1) (1) (1) (1) h ∂L(1) 2 ∂L(1) 2 i ∂L ∂L ∂L ∂L (2)1PR EH EH EH EH = F EH EH = − + 2G EH µν ∂F ∂Fµν ∂F ∂G ∂F ∂G
(22)
with the following renormalized one-loop EH Lagrangian for spinor QED
L
o (eT )(eηT ) 2 1 Z ∞ dT −m2 T n (1) 2 e − (eT ) F − 1 = − EH 8π 2 0 T3 tan(eT )tanh(eηT) 3
and two invariants in constant field q 1/2 = ( F 2 + G2 − F )
,
q 1/2 η = ( F 2 + G2 + F )
(23)
21 Right after this paper appeared two of my colleagues (Edwards and Schubert NBP 923 (2017) 339) pointed out that a similar 1PR addendum exists also for the QED scalar propagator already at the one-loop level:
x
x!
but there is also a finite contribution from the 1PR:
× x0 kµ x"
x
By sewing the one-loop with one off-shell photon to the scalar propagator with one off-shell photon using the photon propagator in the Feynman gauge and Z d
D
k D k δ (k)
µkν k2
η µν =
(24) D
one obtains: L
(1)1PR scal
=
e
2
Z ∞ 0
×
Z ∞ 0
where Zµν = eTFµν and
GB =
− D −m2 T − 1 h sin Z i dT (4πT ) 2 e det 2 Z 0 D − 1 h sin Z i 1 0 0 − −m2 T 0 dT (4πT ) 2 e det 2 tr(G˙ B · G˙ 0 B ) Z0 D
T 2Z 2
Z · cot Z − 1
,
G˙ B = i
2Z T
GB
(25)
22
Worldline derivations To linear order in the photon momentum, the vertex operator integrand takes the form i [ε · q(τ ˙ ) k · q(τ ) + 2ε · ψ k · ψ] which when inserted under the integrand is equivalent to acting with a derivative with respect to F :
Γ1 k = −2iε ·
∂
Z ·k
d
∂F
D
x0 e
R 2 Z Z − 0T dτ 1 q· − d 2 +2ieF d ·q 4 dτ −m2 T q(T )=0 ik·x0 ∞ dτ Dq e dT e q(0)=0 0 h i R I − 0T dτ 1 ψ· d −2ieF ψ 2 dτ × Dψ e .
The remaining path integral is just the one-loop (zero-photon) amplitude in the worldline representation. The x0 integral provides the expected momentum conserving δ function " # ∂ D D (1) 3 Γ1 = −2ie(2π) δ (k) ε · ·kL + O(k ) . ∂F In Feynman gauge, the sewing of two diagrams by an intermediate photon is achieved by replacing products of polarisations by a metric tensor:
1PR Γ2 =
Z
dD k (2π)D
Γ1 [k, ε1 ]Γ1 [−k, ε2 ] δµν . ε1µ ε2ν → 2 k
Only the piece linear in k on both sides survives and provides Z d
D
∂L(1) k µ k ν ∂L(1) ∂L(1) ∂L(1) D k δ (k) −→ ∂Fρµ k2 ∂Fν ρ ∂Fµν ∂F µν
(D = 4).
23 We extended these results to the spinor case and found the one-particle reducible contribution to the electron propagator in a constant electromagnetic background field (Ahmadinaiz et. al, NPB, 924 (2017) 377).
This time the vertex operator, linear in momentum, takes the form
V
xx 0
[k, ε] = i
Z T
dτ k ·
0
τ 0 + q(τ ) ε · x + (x − x) T
x0 − x
! + q(τ ˙ )
T
that can also be produced by differentiation of the zero-photon propagator: x0x ie 0 x 0 x ∂D x0x + x D x · k + ε · L · k D1 k = −2iε · ∂Fµν 2 where Lµν is an unimportant symmetric tensor.
Now we can sew this to the loop in the same way as before:
Σ
x 0 x 1PR
Z =
dD k x0x Γ1 [k, ε1 ]D1 [−k, ε2 ] δµν . (2π)D ε1µ ε2ν → 2 k
The result in configuration space is Σ
x 0 x 1PR
0 ∂D x x ∂L(1) =
∂Fµν ∂F µν
ie +
D 2
(1) x 0 x 0µ ∂L ν x x ∂F µν
and in momentum space only the first term survives: ∂D(p) ∂L(1) 1PR Σ(p) = . ∂Fµν ∂F µν
.
24
The explicit expression for the momentum space version of the reducible contribution takes the form
S
(1)1PR
(p)
=
−T (m2 +p· tan Z ·p) Z dTT e
D − 1 h tan Zp i 0 0 − −m2 T 0 dT (4πT ) 2 e det 2 Zp h i Z − sin Z · cos Z ∂ 0 0 × m − γ · (1l + itan Z) · p − Tp · · Ξ · p + Ξµν Z 2 · cos2 Z ∂Zµν i η·tan Z·η 2 0 −1 , e4 −iγ · sec Z · Ξ · p symb e
2
Z ∞ 0
Z ∞ 0
(26)
where
Ξ ≡
1 Z
−
1
"
d =
sin Z · cos Z
tan Z
tr ln dZ
# ,
(27)
Z
and 0 Ξµν
∂ symb ∂Zµν
−1
n
i η·tan Z·η o i η·tan Z·η −1 i 2 0 e4 = symb η · sec Z · Ξ · η e 4 4 i 2 0 µ ν µναβ 2 0 = − sec Z · Ξ [γ , γ ] − sec Z · Ξ (tan Z)αβ γ5 . µν µν 4 (28)
25
Reducible Tadpole Contributions in Constant Field QED James P. Edwards ifm.umich.mx/∼jedwards
LPHYS’18 July 2018
In collaboration with Naser Ahmadiniaz (IBS, South Korea), Fiorenzo Bastianelli (INFN, Italia), Olindo Corradini (INFN, Italia), Adolfo Huet (UAQ, M´ exico) and Christian Schubert (UMSNH, M´ exico). Ongoing work with Anton Ilderton (Plymouth, UK), Idrish Huet (UAC, M´ exico) and Naser Ahmadiniaz James P. Edwards Reducible Tadpole Contributions in Constant Field QED
26
Introduction As is gradually being recognised, there is a great deal of work to do to incorporate reducible contributions to processes in constant field QED. The main ingredient is the constant field tadpole diagram,
that is sewn with a photon propagator to another part of a Feynman diagram, kµ
R
µ ν
d D k δ D (k) k k k2
=
η µν D
to ensure that the correct theoretical predictions are recorded. A natural question in light of these new considerations is ¿How significant are the reducible contributions? James P. Edwards Reducible Tadpole Contributions in Constant Field QED
27
Introduction As is gradually being recognised, there is a great deal of work to do to incorporate reducible contributions to processes in constant field QED. The main ingredient is the constant field tadpole diagram, kµ
∼ δ D (k)k µ
that is sewn with a photon propagator to another part of a Feynman diagram, kµ
R
µ ν
d D k δ D (k) k k k2
=
η µν D
to ensure that the correct theoretical predictions are recorded. A natural question in light of these new considerations is ¿How significant are the reducible contributions? James P. Edwards Reducible Tadpole Contributions in Constant Field QED
28
Outline
1 Weak Fields Pair Creation
2 Explicit Frames Crossed Field Constant magnetic Field
3 Low Energy Scattering
4 Higher Order Calculus
James P. Edwards Reducible Tadpole Contributions in Constant Field QED
29
Quantum electrodynamics in a background electromagnetic field Electromagnetic backgrounds greatly effect the properties of quantum fields seen in vacuum. Quantum electrodynamics presents effects that are non-linear such as Vacuum birefringence and other polarisation dependent structure. Mass shifts associated to the matter fields. Instability to particle (Schwinger) pair creation, predicted as early as 1931. Crucial tools for their study include the propagator dressed by the background field and the one-loop Euler-Heisenberg[1] (Weisskopf[2] ) effective Lagrangians for spinor (scalar) QED in a constant background:
k1 P∞
P∞
N=0
k1 1 2
N=0
...
. kN
Z. Phys. 98, (1936), 714 Kong. Dans. Vid. Selsk. Math-fys. 6, (1936)
James P. Edwards Reducible Tadpole Contributions in Constant Field QED
. .
kN
30
Light by light scattering and pair creation The one-loop effective actions allow for the examination of the photon propagator and investigation of light by light scattering.
Through the optical theorem, they also contain information on the pair creation rate. For a constant electric field, E , expanding the worldline representation and carrying out the “proper time” integral leads to the Schwinger summation ∞ eE 2 X (+1)n − nπm2 m4 (1) = Lspin = e eE 8π 3 m2 n2 n=1 ∞ eE 2 X (−1)n − nπm2 m4 (1) = Lscal = − e eE 16π 3 m2 n2 n=1
... James P. Edwards Reducible Tadpole Contributions in Constant Field QED
31
Weak field expansion For a weak constant electric field the two-loop reducible formula reduces to ∂L(1) ∂L(1) = . ∂E ∂E with well known series representations (spinor QED). (2)
Lred = 2
