1PR contribution to the one-loop electron propagator in a constant - - PowerPoint PPT Presentation

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 (ahmadiniaz@ibs.re.kr)

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) x|e−HT |x′ = x(T)=x x(0)=x′ Dx(τ) e−S[x,G] Periodic boundary conditions (topology of a closed line)

• x(0)=x(T)

Dx(τ)e−S[x,G]

4

Free-propagator: Free scalar propagator that is the Green’s function for the Klien-Gordon equation: Dx,x′ = 0|Tφ(x)φ(x′)|0 = x| 1 − + m2 |x′ , = 4

• i=1

∂2 ∂x2 i We exponentiate the denominator using a Schwinger proper-time parameter T. This gives Dxx′ = ∞ dT e−m2T x| exp

• − T(−)
• |x′ =

∞ dT e−m2T x(T)=x x(0)=x′ Dx e− T dτ 1 4 ˙ x2 This is the worldline path integral representation of the relativistic propagator of a scalar particle in euclidean spacetime from x′ to x. Dxx′ = ∞ dT e−m2T e− (x−x′)2 4T

• DBC

Dq(τ) e− T dτ 1 4 ˙ q2 ⇒ Fourier transform ⇒ familiar momentum space representation Dpp′ =

• dD xeip·x
• dD x′ eip′·x′

Dxx′ = (2π)D δD (p + p′) 1 p2 + m2

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′ to x Dxx′ [A] = ∞ dT e−m2T x(T)=x x(0)=x′ Dx(τ) e− T dτ 1 4 ˙ x2+ie ˙ x·A(x(τ))

• 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] = ∞ dT T e−m2T

• x(0)=x(T)

Dx(τ) e− T dτ 1 4 ˙ x2+ie ˙ x·A(x(τ))

• 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) xµ(τ) = xµ + yµ(τ)

• Dx(τ)

=

• dD x0
• Dy(τ)

, xµ ≡ 1 T T dτxµ(τ) (1) The remaining y(τ) path integral is performed using the Wick contraction rule yµ(τi )yν (τj ) = −δµν GB (τi , τj ) , GB (τi , τj ) ≡ GBij = |τi − τj | − (τi − τj )2 T (2) The free Gaussian path integral gives

• D e−

T dτ 1 4 ˙ y2 = (4πT)− D 2 , D = spacetime dimension (3) Now if we specialize the background A(x), which so far was an arbitrary Maxwell field, to a sum of N plane waves, Aµ(x) = N

• i=1

εµ i eiki ·x After expanding the interaction term we get Γscal[A] = (−ie)N ∞ dT T e−m2T (4πT)− D 2

• V γ

scal[k1, ε1] · · · V γ scal[kN , εN ]

• ⇒ V γ

scal[k, ε] ≡ T dτε · ˙ x eik·x

7

At this stage the zero-mode integration can be performed

• dD x0

N

• i=1

eiki ·x0 = (2π)D δD ( N

• i=1

ki ) 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 ] = (−ie)N (2π)D δD (

• ki )

∞ dT T (4π)− D 2 e−m2T N

• i=1

T dτi × exp

• N
• i,j=1

1 2 GBij (pi · pj ) − i ˙ GBij (εi · pj ) + 1 2 ¨ GBij (εi · εj )

• lin(ε1···εN)

where ˙ GBij = dGBij dτi = sign(τi − τj ) − 2(τi − τj ) T ¨ GBij = d2GBij dτ2 i = 2δ(τi − τj ) − 2 T (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 ∆L = 1 2 ieyµFµν ˙ yν (5) The external field then can be absorbed by a change of the free worldline propagators, replacing GBij , ˙ GBij , ¨ GBij by GBij = T 2Z2

• Z

sin Z e−iZ ˙ GBij + iZ ˙ GBij − 1

• ,

˙ GBij = · · · , ¨ GBij = · · · (6) where Z = eFT and the change in the free path integral due to the external field: (4πT)− D 2 → (4πT)− D 2 det− 1 2 sin Z Z

• (7)

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

9

Γscal[k1, ε1; · · · ; kN , εN ] = (−ie)N (2π)D δD (

• ki )

∞ dT T (4π)− D 2 e−m2T det− 1 2 sin Z Z

• ×

N

• i=1

T dτi exp

• N
• i,j=1

1 2 ki · GBij · kj − iεi · ˙ GBij · kj + 1 2 εi · ¨ GBij · εj

• lin(ε1···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: D[x′; x] = ∞ dTe−m2T x(T)=x x(0)=x′ Dx(τ) e− 1 4 T dτ[ ˙ x2+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 D[x′; x; k1, ε1; · · · ; kN , εN ] = (−ie)N ∞ dTe−m2T e− 1 4T (x−x′)2 4πT − D 2 × T N

• i=1

dτi e N i=1

• εi · (x−x′)

T +iki ·(x−x′) τi T +iki ·x′ e N i,j=1

• ∆ij pi ·pj −2i•∆ij εi ·kj −•∆
• ij εi ·εj
• lin(ε1ε2···εN ) .

where we used the following derivatives of the Green function

• ∆(τ1, τ2)

= τ2 T + 1 2 sign(τ1 − τ2) − 1 2 ∆

• (τ1, τ2)

= τ1 T − 1 2 sign(τ1 − τ2) − 1 2

• ∆
• (τ1, τ2)

= 1 T − δ(τ1 − τ2) where left and right dots indicate derivatives with respect to τ1 and τ2 respectively.

11

k3 + p p′ + k2 k1 k3 kN · · · · · · + p p′ + k2 k1 k3 kN · · · · · · p p′ k1 k2 k3 kN · · · p p′ k1 k2 kN · · · . . . . . . + + + +

12

Now, we Fourier transform the master formula D[p; p′; k1, ε1; · · · ; kN , εN ] =

• dD x
• dD x′ eix′·p′+ix·pD[x′; x; k1, ε1; · · · ; kN , εN ]

After some algebra, we get the momentum-space version of our master formula (off-shell) D[p; p′; k1, ε1; · · · ; kN , εN ] = (−ie)N (2π)D δD p + p +

• i

ki ∞ dT e−T(m2+p2) × T N

• i=1

dτi e N i,j=1

• −2ki ·pτi +2iεi ·p+

|τi −τj | 2 − τi +τj 2

• pi ·pj −i
• sign(τi −τj )−1
• εi ·pj +δ(τi −τj )εi ·εj
• lin(ε1ε2···εN )

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: Dxx′ [A] = ∞ dT e−m2T x(T)=x x(0)=x′ Dx e− T dτ 1 4 ˙ x2+ie ˙ x·A(x)

• .

(8) where A = Aext + Aphot (9) Choosing Fock-Schwinger gauge, the gauge potential for a constant field can be written as Aµ(y) = − 1 2 Fµν (y − x′)ν , (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: ∆ ⌣(τ, τ′) ≡ τ | d2 dτ2 − 2ieF d dτ −1 | τ′DBC = 1 2

• GB (τ, τ′) − GB (τ, 0) − GB (0, τ′) + GB (0, 0)
• .

(11) The dressed propagator in momentum space: Dpp′ (F) = (2π)D δ(p + p′) ∞ dT e−m2T e−Tp( tanZ Z )p det 1 2 [cosZ] . (12) (E.S. Fradkin, D.M. Gitman, S.M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum, Springer 199)

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: Aµ phot(x) = N

• i=1

εµ i eiki ·x . (14) Each photon then effectively gets represented by a vertex operator V A[k, ε] = T dτ ε· ˙ x(τ) eik·x(τ) , (15) integrated along the scalar line. This leads to the following path integral representation of the constant-field propagator dressed with N photons: Dxx′ (F|k1, ε1; · · · ; kN , εN ) = (−ie)N ∞ dT e−m2T

• P

Dx e− T dτ[ 1 4 ˙ x2+ie ˙ x·Aext(x)] ×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: Dxx′ (F | k1, ε1; · · · ; kN , εN ) = (−ie)N ∞ dT e−m2T (4πT)− D 2 det 1 2

• Z

sin Z

• e− 1

4T x−Z cot Zx− × T dτ1 · · · T dτN e N i=1

• εi ·

x− T +iki · x−τi T +iki ·x′ ×exp

• N
• i,j=1
• ki ∆

⌣ij kj − 2iεi

• ∆

⌣ij kj − εi

• ∆

⌣

• ij εj
• +

2e T x− N

• i=1
• F ◦∆

⌣i ki − iF ◦∆ ⌣

• i εi
• ε1ε2···εN

. (17) left (right) ‘open circle’ on ∆ ⌣(τ, τ′) denotes an integral T dτ ( T dτ′). 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:

k3 + p p′ + k2 k1 k3 kN · · · · · · + p p′ + k2 k1 k3 kN · · · · · · p p′ k1 k2 k3 kN · · · p p′ k1 k2 kN · · · . . . . . . + + + +

17

and in p-space Dpp′ (F|k1, ε1; · · · ; kN , εN ) = (−ie)N (2π)D δD p + p′ + N

• i=1

ki ∞ dT e−m2T 1 det 1 2

• cosZ
• ×

T dτ1 · · · T dτN e N i,j=1

• ki ∆

⌣ij kj −2iεi •∆ ⌣ij kj −εi •∆ ⌣

• ij εj
• e−Tb( tanZ

Z )b

• ε1ε2···εN

. (18) Here we have defined Zµν = eTFµν b ≡ p + 1 T N

• i=1
• τi − 2ieF ◦∆

⌣i

• ki − i
• 1 − 2ieF ◦∆

⌣

• i
• εi
• .

(19)

• A. Ahmad, N. A, O. Corradini, S. P. Kim and C. Schubert, NPB, 919 (2017) 9.

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) spin(a, b) = − 1 8π2 ∞ dT T e−m2T e2ab tanh(ebT)tan(ebT) (20) where a and b are related to the two invariants of the Maxwell field by a2 − b2 = B2 − E2 , (ab)2 = (E · B)2 (21) 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).

19

The first radiative corrections to these Lagrangians (scalar and spinor), describibg the effect of an additional photon exchange in the loop, were

• btained in the seventies by Ritus (Sov. Phys. JETP 42 (1975) 774). He obtained L(2)

scal,spin in terms of certain two-parameter integrals which are 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 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(2)1PR EH = ∂L(1) EH ∂Fµν ∂L(1) EH ∂Fµν = F ∂L(1) EH ∂F 2 − ∂L(1) EH ∂G 2 + 2G ∂L(1) EH ∂F ∂L(1) EH ∂G (22) with the following renormalized one-loop EH Lagrangian for spinor QED L(1) EH = − 1 8π2 ∞ dT T3 e−m2T (eǫT)(eηT) tan(eǫT)tanh(eηT) − 2 3 (eT)2F − 1

• (23)

and two invariants in constant field ǫ = (

• F2 + G2 − F)1/2

, η = (

• F2 + G2 + F)1/2

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 x′ x kµ 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

• dD k δD (k)

kµkν k2 = ηµν D (24)

• ne obtains:

L(1)1PR scal = e2 ∞ dT(4πT)− D 2 e−m2T det− 1 2 sin Z Z

• ×

∞ dT′(4πT′)− D 2 e−m2T′ det− 1 2 sin Z′ Z′ 1 D tr( ˙ GB · ˙ G′B ) (25) where Zµν = eTFµν and GB = T 2Z2

• Z · cot Z − 1
• ,

˙ GB = i 2Z T GB

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 · ∂ ∂F · k

• dD x0 eik·x0

∞ dT e−m2T q(T)=0 q(0)=0 Dq e − T dτ 1 4 q·

• − d2

dτ2 +2ieF d dτ

• ·q

×

• Dψ e−

T dτ 1 2 ψ· d dτ −2ieF

• ψ.

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 Γ1 = −2ie(2π)D δD (k)

• ε ·

∂ ∂F · k L(1) + O(k3)

• .

In Feynman gauge, the sewing of two diagrams by an intermediate photon is achieved by replacing products of polarisations by a metric tensor: Γ1PR 2 =

• dD k

(2π)D Γ1[k, ε1]Γ1[−k, ε2]

• ε1µε2ν → δµν

k2 . Only the piece linear in k on both sides survives and provides

• dD k δD (k)

∂L(1) ∂Fρµ kµkν k2 ∂L(1) ∂Fν ρ − → ∂L(1) ∂Fµν ∂L(1) ∂Fµν (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′ [k, ε] = i T dτ k ·

• x + (x′ − x)

τ T + q(τ)

• ε ·

x′ − x T + ˙ q(τ)

• .

that can also be produced by differentiation of the zero-photon propagator: Dx′x 1 k = −2iε ·    ∂Dx′x ∂Fµν + ie 2 x′Dx′x x    · k + ε · L · k where Lµν is an unimportant symmetric tensor. Now we can sew this to the loop in the same way as before: Σx′x 1PR =

• dD k

(2π)D Γ1[k, ε1]Dx′x 1 [−k, ε2]

• ε1µε2ν → δµν

k2 . The result in configuration space is Σx′x 1PR = ∂Dx′x ∂Fµν ∂L(1) ∂Fµν + ie 2 Dx′x x′µ ∂L(1) ∂Fµν xν and in momentum space only the first term survives: Σ(p)1PR = ∂D(p) ∂Fµν ∂L(1) ∂Fµν .

24

The explicit expression for the momentum space version of the reducible contribution takes the form S(1)1PR (p) = e2 ∞ dTT e−T(m2+p· tan Z Z ·p) ∞ dT′(4πT′)− D 2 e−m2T′ det− 1 2 tan Zp Zp

• ×
• m − γ · (1

l + itan Z) · p

• − Tp ·

Z − sin Z · cos Z Z2 · cos2 Z · Ξ′ · p + Ξ′ µν ∂ ∂Zµν

• −iγ · sec2 Z · Ξ′ · p
• symb−1
• e

i 4 η·tan Z·η , (26) where Ξ ≡ 1 Z − 1 sin Z · cos Z = d dZ tr ln tan Z Z

• ,

(27) and Ξ′ µν ∂ ∂Zµν symb−1 e i 4 η·tan Z·η = symb−1 i 4 η · sec2 Z · Ξ′ · η e i 4 η·tan Z·η = − i 4 sec2 Z · Ξ′ µν [γµ, γν ] − ǫµναβ sec2 Z · Ξ′ µν (tan Z)αβ γ5

• .

(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,

• dDk δD(k) kµkν

k2

= ηµν

D

kµ 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, ∼ δD(k)kµ kµ that is sewn with a photon propagator to another part of a Feynman diagram,

• dDk δD(k) kµkν

k2

= ηµν

D

kµ 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

• ne-loop Euler-Heisenberg[1] (Weisskopf[2]) effective Lagrangians for spinor (scalar) QED in a

constant background: ... ... k1 kN kN k1 ∞

N=0

∞

N=0

• 1Z. Phys. 98, (1936), 714
• 2Kong. Dans. Vid. Selsk. Math-fys. 6, (1936)

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

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 ℑ L(1)

spin =

m4 8π3 eE m2 2 ∞

• n=1

(+1)n n2 e− nπm2

eE

ℑ L(1)

scal = − m4

16π3 eE m2 2 ∞

• n=1

(−1)n n2 e− nπm2

eE

.. .

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(2)

red = 2ℜ ∂L(1)

∂E ℑ ∂L(1) ∂E . with well known series representations (spinor QED). ℑL(1) = m4 8π3 eE m2 2 ∞

• k=1

e− πm2k

eE

k2 ; ℜL(1) = − 2m4 π2 eE m2 4 ∞

• n=0

cn eE m2 2n , The leading order terms are ∂E ℑL(1) = em2 8π2 e− πm2

eE

+ O(Ee− πm2

eE );

∂E ℜL(1) = − 8e4c0E 3 π2m4 + O(E 4e− πm2

eE ).

giving a leading order reducible contribution ℑL(2)

red ≈

e5E 3 360m2π4 e− πm2

eE ,

Compare this to the irreducible contribution e4E2

32π3 e− πm2

eE . The new diagram is subleading for

weak fields...but still present!

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

32

Example field configurations

Returning to the one-loop reducible contribution to the propagator in a constant background we focus on scalar QED to illustrate some of the implications of the new diagram: kµ p′ p kµ Recall that this contribution can be written covariantly (in momentum space) as Σ(1)

red = ∂Sp′p

∂Fµν ∂L(1) ∂F µν . Here we have chosen Fock-Schwinger gauge for the background field, Aµ(y) = − 1

2 Fµν(y − x)ν + ∂F . . .

To go to an arbitrary gauge, the configuration space propagator transforms as Sx′x = e−i

x′

x [A(x)−

A(x)]dx

Sx′x where variables wearing sombreros are evaluated with the potential in a fixed reference gauge.

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

33

Crossed electric and magnetic fields

Consider the situation of perpendicular electric and magnetic fields of equal magnitude so that F =    f −f −f f    and that in this frame the electromagnetic invariants vanish: − 1 4 tr (F 2) = 0 = − 1 4 tr (F · F). Moreover, for this field strength tensor it is easy to verify that F 3 = 0 is the zero matrix, so L(1) has a finite expansion in powers of F. For clarity, it is useful to distinguish the coupling to the background field into interactions of the line and the loop – we use f and f ′ respectively to denote the different contributions.

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

34

Propagator tadpole for crossed fields

Evaluating the covariant formula using the proper time representation of the Euler-Heisenberg Lagrangian L(1) leads to the result in Minkowski space (Z := eFT) D1PR

scal(p) = e2

2 ∞ dT ′(4iπT ′)− D

2 e−im2T ′ ∞

dTT e−im2T e−iTp·(1

1− Z2

3 )·p 2

9 Tp · Z · Z′ · p

• = − e4∆p2

⊥

9 ∞ dT ′ T ′(4iπT ′)− D

2 e−im2T ′ ∞

dT T 3 e−im2T e−iT[p2+ (efT)2

3

∆p2

⊥] f f ′

Note that this is linear in the loop’s coupling to the background field, f ′. This means that it is just part of the usual (necessary) renormalisation of the photon propagator: kµ p′ p kµ f ′ Repeating the analysis for spinor QED leads to the same conclusion – the tadpole contribution for constant crossed fields is not physical (only for equal |E| and |B|).

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

35

Constant magnetic field

We consider a constant B field in the z-direction (say), defining the field strength tensor by F = lim

ε→0

   −ε −B B ε    The full parameter integral form of the tadpole contribution is D1PR

scal(p) = e2

2 ∞ dT ′(4πiT ′)− D

2 e−im2T ′

eB′T ′ sin(eB′T ′)

• cot(eB′T ′) −

1 eB′T ′

• ×

∞ dT T e−im2T cos(eBT) e−iT(p2

+ tan eBT eBT

p2

⊥)

• −

T eBT tan(eBT) eBT − sec2(eBT)

• p2

⊥ + itan(eBT)

• .

This time the tadpole produces terms at higher order in the background field coupling to the

• loop. Expanding for small magnetic field and doing the parameter integrals yields

e4B′B 3(4π)

D 2

(m2)

D 2 −2

Γ

• 2 − D

2

• − 1

30 eB′ m2 2 Γ

• 4 − D

2

• + . . .
• 2p2

⊥

(m2 + p2)4 − 1 (m2 + p2)3

• kµ

p′ p kµ kµ p′ p

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

36

Photon scattering amplitudes

There is an explicit way to extract photon scattering amplitudes in the low-energy limit from constant field effective actions. We fix here K helicity + photons and L helicity − external photons attached to virtual matter loops. Then

1 Write the effective Lagrangian in terms of the invariants a, b defined by

a2 = 1 4

• FµνF µν2 +
• Fµν ˜

F µν2+ 1 4 FµνF µν b2 = 1 4

• FµνF µν2 +
• Fµν ˜

F µν2− 1 4 FµνF µν

2 Change variables from a, b to χ± via a = √χ+ + √χ− and b = −i(√χ+ − √χ−) 3 Expand the effective action L(iF) in powers of χ+, χ− and take the coefficient of the

O(χ

K 2

+ χ

L 2

−) term, C( K 2 , L 2 ). 4 Replace

χ

K 2

+ → ( K

2 )!

2

K 2

• [12]2[34]2 · · · [(K − 1)K]2 + all permutations
• := χ+

K and

χ

L 2

− → ( L

2 )!

2

L 2

• (K + 1)(K + 2)2(K + 3)(K + 42 · · · (N − 1)N2 + all perm.
• := χ−

L .

Of course the important dynamical information is in the coefficients C( K

2 , L 2 ): the scattering

amplitude A

• +K , −L

is given by (spinor QED normalisation) A

• +K , −L

= −(απ)L m4 8π2 2e m2 K+L C K 2 , L 2

• χ+

K χ− L

(29)

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

37

Two-loop reducible coefficients

A useful approach is to relate the two-loop reducible coefficients to their one-loop counterparts by applying the covariant formula to L(1)

spin(iF) = − m4

8π2

∞

• N=4

2e m2 N

N

• K=0

c(1)

spin

K 2 , N − K 2

• χ

K 2

+ χ

N−K 2

−

. Using ∂ ∂Fµν = 1 4 (F µν + i F µν) ∂ ∂χ+ + 1 4 (F µν − i F µν) ∂ ∂χ− provides a folded sum representation of the two-loop reducible coefficients[3] 2π2˜ c(2,red)spin K 2 , L 2

• =

K+L−2

• N1=4

min(N1,K+2)

• k1=max(N1−L, 0)

c(1)

spin

k1 2 , N1 − k1 2

• c(1)

spin

K + 2 − k1 2 , L − N1 + k1 2

• k1(K + 2 − k1)

+ (K ← → L), The same story carries through for scalar QED.

3JPE, Huet, A. and Schubert, C., Nucl. Phys. B. James P. Edwards Reducible Tadpole Contributions in Constant Field QED

38

• K

2 , N−K 2

• c(2,irr)

spin

c(2,red)

spin

c(2)

spin

(5, 0)

317 40320π2 467 4233600π2 4219 529200π2

(4, 1)

−8707 1814400π2 −12241 38102400π2 −12193 2381400π2

(3, 2)

−3190547 8164800π2 14837 2721600π2 −786509 2041200π2

(4, 0)

2221 403200π2 1 10080π2 323 57600π2

(3, 1)

−151379 6350400π2 1 22680π2 −151099 6350400π2

(2, 2)

−37763 282240π2 703 226800π2 −1659967 12700800π2

(3, 0)

7 960π2 1 7200π2 107 14400π2

(2, 1)

−5821 129600π2 11 12960π2 −5711 129600π2

(2, 0)

5 192π2 5 192π2

(1, 1)

−391 2592π2 −391 2592π2

For the all plus case (L = 0) we get, with K = 2n, c(2, red)

spin

(n, 0) = (−1)n+1 2π2

n−1

• m=2

B2mB2(n−m+1) (2m − 2)(2(n − m + 1) − 2) = −2 c(2, red)

spin

(n, 0) 2 .

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

39

Three loop and beyond

How do these reducible contributions generalise at higher order? Here are two examples at three loop order:

∂L(1) ∂Fµν ∂2L(1) ∂F µν∂F αβ ∂L(1) ∂Fαβ ∂L(2) ∂F µν ∂L(1) ∂Fµν

Open lines can also be included in this calculus most easily in momentum space:

∂Sp′p ∂Fαβ ∂2L(2) ∂F αβ∂F µν ∂L(1) ∂Fµν

This would be valid in Fock-Schwinger gauge for the background field – non-trivial to transform gauge in momentum space...

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

40

Ongoing work

There are a number of interesting directions for future calculation: Consider other interesting constant field configurations (parallel fields etc) Go to higher order – calculate low energy scattering coefficients and integral representations Generalise to non-constant fields (plane waves etc) Collaboration with Ahmadiniaz and Ilderton Derivative corrections to Euler-Heisenberg Understand instanton interpretation and verify AAM conjecture taking into account all contributions. Investigate corrections to the Ritus mass shift in strong fields More important would be to understand when (or if) the reducible contributions are comparable or even dominant to the familiar irreducible terms.

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

41

Summary

One particle reducible contributions to quantum field theory processes in constant background fields have long been (incorrectly!) ignored. These contributions involve tadpoles. Their insertion into a diagram can be generated by a covariant derivative traced over Lorentz indices – easy to show in the worldline formalism. At two-loop level the contribution can be written as a (double) parameter integral written explicitly in terms of invariants of the field. In the weak field constant electric field limit the contribution is subleading; this may not be the case for strong magnetic fields. We have updated two-loop low energy scattering amplitude coefficients with the reducible contribution. For crossed fields a tadpole attached to a propagator just corresponds to (photon propagator) renormalisation; already for a constant magnetic field the tadpole produces physical effects. Many past calculations should be revisited in light of this new information!

James P. Edwards Reducible Tadpole Contributions in Constant Field QED

41

Summary

One particle reducible contributions to quantum field theory processes in constant background fields have long been (incorrectly!) ignored. These contributions involve tadpoles. Their insertion into a diagram can be generated by a covariant derivative traced over Lorentz indices – easy to show in the worldline formalism. At two-loop level the contribution can be written as a (double) parameter integral written explicitly in terms of invariants of the field. In the weak field constant electric field limit the contribution is subleading; this may not be the case for strong magnetic fields. We have updated two-loop low energy scattering amplitude coefficients with the reducible contribution. For crossed fields a tadpole attached to a propagator just corresponds to (photon propagator) renormalisation; already for a constant magnetic field the tadpole produces physical effects. Many past calculations should be revisited in light of this new information! Thank you for your attention

James P. Edwards Reducible Tadpole Contributions in Constant Field QED