Introduction
Worldline approach
Overview

Obtaining fully polarised amplitudes in gauge invariant form
James P. Edwards

ifm.umich.mx/∼jedwards

20th Lomonosov Conference
August 2021
Based on arXiv:2107.00199 and arXiv:2004.01391.
In collaboration with Christian Schubert (UMSNH), Olindo Corradini (Modena),
Naser Ahmadiniaz (Helmholz Zentrum, Dresden), Fiorenzo Bastianelli (Bologna),
Victor Guzmán (UMSNH) and others.
With financial support from Mexico’s Sistema Nacional de Investigadores andCONACyT’s “Frontier Science” program.
James P. Edwards

Gauge invariant amplitudesIntroduction
Worldline approach
Overview

Outline1

Introduction
Scattering in QFT
Amplitude calculations

2

Worldline approach
Worldline representation
Amplitudes

3

Overview

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Scattering in QFT
Amplitude calculations

Introduction

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Scattering in QFT
Amplitude calculations

Quantum Field theory
We will work within standard QED: a Dirac field coupled to a gauge potential
A(x) = Aµ (x)dxµ described by the familiar Dirac action with “minimal coupling ”
Zh 1
 i/ −m ΨS[Ψ, A] = dD x − trF 2 + Ψ̄ iD
4where the covariant derivative is Dµ := ∂µ + ieAµ , which couples the fields togetherand produces the interaction vertex of the quantum theory−ieγ µµ/−eΨ̄AΨ
We can interpret “particles” as excitations of the quantum field about “the vacuum,”a state of minimal energy denoted by |0i:|p, σi = ba†p,σ |0iJames P. Edwards†|k, hi = αbk,h|0i

Gauge invariant amplitudes

Introduction
Worldline approach
OverviewScattering in QFT
Amplitude calculations

Interactions
Working with canonical quantisation in the interaction picture the fundamental objectsof interest are correlation functions:ˆ (x ) . . . Â (x ) |Ωi .
hΩ|T Ψ̂(x ) . . . Ψ̄1nµN

NIn standard perturbation theory we split the action into a “free” part and an“interaction” part which couples the fields together – these indicate the interaction
vertices of the theory. Some examples from the standard model include:σ, agf abc ×
η σµ (p1 − p3 )ρ + . . .
−ieγ µ−igW γ µ cos ϑcµµµ, c2// VCKM Ψ−eΨ̄AΨ
−eΨ̄W− 41 tr[Fµν]ρ, bWe use the above vertices to form Feynman diagrams that contribute order by order inthe coupling constants:

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Scattering in QFT
Amplitude calculations

Diagram complexity
Perturbative calculations of amplitudes often involve tedious manipulation of longexpressions....
But lead to surprisingly simple final results:
Weisskopf was so unhappy with the conventional calculation of the Compton
cross section that he asked me, and several other Ph.D. students, to find abetter way to get the simple final result.
(R. Stora).
There is also a factorial explosion in the number of diagrams at a given loop orderthat makes organisation of the calculation progressively more difficult.
Example: electron g − 2 (QED):
1 diagram at one-loop order: g−2+ = 12 α(Schwinger[1] )2π
2g−2
7 diagrams at two-loops: 2 + = −0.328 . . . α(Petermann / Sommerfeld[2] )π372 diagrams at three-loops: g−2+ = 1.181 . . . α(Laporta, Remiddi[3] )2π

4891 diagrams at 4-loops: g−2+ = −1.912 . . . α(Laporta[4] )2π12672 diagrams at 5-loop order (some numerical evaluations)1

Schwinger, J.S. Phys. Rev. 73, (1948), 416–417.
Petermann, Helv. Phys. Acta 30, (1957), 407–408 & Sommerfield, C.M. Ann. Phys. 5 (1958), 26–57
3
Laporta, S., Remiddi, E. Phys. Lett. B 379, (1996), 283–291
4Laporta, S. Phys. Lett. B 772 (2017), 232–238
2James P. Edwards

Gauge invariant amplitudesIntroduction
Worldline approach
Overview

Scattering in QFT
Amplitude calculations

Miraculous cancellations
Natural question: ¿Why are these final results so small?
Aside from this, for g − 2 there is also a cancellation of spurious U V and IR
divergences between diagrams[5] due to gauge symmetry.

5See G. V. Dunne et al., J. Phys. Conf. Ser.37, 59–72 (2006) for a discussion on the influence of gauge
cancellations on divergence structure in QFT which is still not very well understood.
James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Scattering in QFT
Amplitude calculations

Miraculous cancellations
Natural question: ¿Why are these final results so small?
Aside from this, for g − 2 there is also a cancellation of spurious U V and IR
divergences between diagrams[5] due to gauge symmetry.
Is this gauge invariance also responsible for the spectacular cancellations of the finite
part of the amplitude?
Motivated P. Cvitanović to examine the contributions from individual “gauge-sets” –sets of gauge invariant diagrams. The contributions of different gauge sets are veryclose to being integer multiples of ± 21 × ( α)n !
πCvitanović also made the conjecture that this suggests that the perturbative series forg − 2 may in fact converge, having asymptotic form∞Xg−2
α 2n=cn,
2πn=1|cn | ∼n,2(1)rather than the usual factorial growth.
5See G. V. Dunne et al., J. Phys. Conf. Ser.37, 59–72 (2006) for a discussion on the influence of gauge
cancellations on divergence structure in QFT which is still not very well understood.
James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Worldline approach

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

The worldline representation
The worldline formalism[6] is an alternative method for the quantising fields, based on
the first quantisation of relativistic particles.
In this worldline approach to field theory the dressed electron propagator has path
integral representation (Dµ = ∂µ + ieAµ )


0ie
/ −1 x = m + iD
/ x0 m2 − D2 + γ µ Fµν γ ν −1 x
S x x = x0 m − iD2 0/ K x x [A]≡ m + iD

6Phys. Rev. E80, 3 (1950), 440,7Nucl. Phys. B385, (1992), 145

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

The worldline representation
The worldline formalism[6] is an alternative method for the quantising fields, based on
the first quantisation of relativistic particles.
In this worldline approach to field theory the dressed electron propagator has path
integral representation (Dµ = ∂µ + ieAµ )
0ie/ −1 x = m + iD
/ x0 m2 − D2 + γ µ Fµν γ ν −1 xS x x = x0 m − iD2 0/ K x x [A]≡ m + iD0

Here we defined the kernel K x x whose path integral / proper-time representation is0

DK x x = 2− 2 symb−1

Z ∞2

dT e−m T0

Z x(T )=x0 Z
Dxx(0)=xDψ e−S[x,ψ]

(2)

AP C

with the worldline action of a minimally coupled relativistic point particle 2Z TẋiS [x, ψ] =dτ+ ψ · ψ̇ + eA(x) · ẋ + ie(ψ + η)µ Fµν (x)(ψ + η)ν .
420√
on which the symbol map acts according to symb(γ α1 ...αn ) = (−i 2)n η α1 · · · η αn .
6Phys. Rev. E80, 3 (1950), 440,

7Nucl. Phys. B385, (1992), 145

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Scattering Amplitudes
For N -photon scattering amplitudes we decompose the background into plane waves,Aµ (x) =NXiki ·xεµ.
iei=0Interactions with photons are represented by vertex operators under the path integral:Z T0Vηx x [k, ε] =dτ ε · ẋ + 2ieε · (ψ + η)k · (ψ + η) eik·x
0similarly to in string perturbation theory.
In momentum space we recover the photon-dressed propagator from
00p p0p pSN[{ki , εi }] = (p/ + m)KN [{ki , εi }] − eNX

p0 +k ,p/εj KN −1 j [{ki , εi }, k̂j , ε̂j ]j=1

which splits into parts we call leading and subleading
James P. Edwards

Gauge invariant amplitudes(3)Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Scattering Amplitudes
For N -photon scattering amplitudes we decompose the background into plane waves,Aµ (x) =NX
iki ·xεµ.
ie
i=0Interactions with photons are represented by vertex operators under the path integral:
Z T0Vηx x [k, ε] =dτ ε · ẋ + 2ieε · (ψ + η)k · (ψ + η) eik·x
0similarly to in string perturbation theory.
In momentum space we recover the photon-dressed propagator from00p p0p pSN[{ki , εi }] = (p/ + m)KN [{ki , εi }] − eNX

p0 +k ,p/εj KN −1 j [{ki , εi }, k̂j , ε̂j ]j=1

which splits into parts we call leading and subleading
James P. Edwards

Gauge invariant amplitudes(3)Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Gauge invarianceTo manifest the gauge invariance of the on-shell scattering amplitudes – which mustsatisfy the QED Ward identity – we rewrite the vertex operator asZ T0ε·r ddτ ε · ẋ + iVηx x [k, ε] =− ie(ψ + η) · f · (ψ + η) eik·x
k · r dτ0Z T
 r · f · ẋ=dτ
− ie(ψ + η) · f · (ψ + η) eik·x
r·k0
where introduce an auxiliary “reference vector” r (the boundary terms will vanish
when the LSZ formula is applied as they will be missing the required pole structure).
In the on-shell limit, only the leading term in the propagator contributes to theS-matrix (here we “amputate” the external fermion legs):

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Gauge invarianceTo manifest the gauge invariance of the on-shell scattering amplitudes – which mustsatisfy the QED Ward identity – we rewrite the vertex operator asZ T
0ε·r ddτ ε · ẋ + iVηx x [k, ε] =− ie(ψ + η) · f · (ψ + η) eik·x
k · r dτ
0Z T r · f · ẋ=dτ
− ie(ψ + η) · f · (ψ + η) eik·x
r·k0
where introduce an auxiliary “reference vector” r (the boundary terms will vanish
when the LSZ formula is applied as they will be missing the required pole structure).
In the on-shell limit, only the leading term in the propagator contributes to theS-matrix (here we “amputate” the external fermion legs):00

p p
MpN p = ū(−p0 )(−p/ + m)(p/ + m)SN (p/ + m)u(p)0= ū(−p0 )KpN pu(p) .
2mwhich will now be a function of the field strength tensors of the individual photons.
James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Simple examples
The kernel admits a natural decomposition onto the even sub-algebra of the Dirac
representation of the Clifford algebra (see arXiv:2004.01391):00p pKN≡ (−ie)NKpN p,(p02 + m2 )(p2 + m2 )0KpN p ≡ AN 11 + BN αβ σ αβ − iCN γ5 .

Example: N=1 (vertex):
A1 = −2

r·f ·p,r·kJames P. Edwards

B1αβ =

1fαβ2Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
Amplitudes

Simple examples
The kernel admits a natural decomposition onto the even sub-algebra of the Dirac
representation of the Clifford algebra (see arXiv:2004.01391):00p pKN≡ (−ie)NKpN p,(p02 + m2 )(p2 + m2 )0

KpN p ≡ AN 11 + BN αβ σ αβ − iCN γ5 .

Example: N=1 (vertex):
A1 = −2r·f ·p,r·k

B1αβ =1fαβ2Example: N=2 (Compton):r1 · f1 · f2 · r21h 11 itr f1 · f2−+ 00r 1 · k1 r 2 · k2
2 2p · k1
2p · k2
αβ
1 r1 · f1 · k2 f2αβ + r2 · f2 · k1 f1αβ
1h 11 iB2αβ = −+− 0f1 , f2
02r 1 · k1 r 2 · k2
2 2p · k1
2p · k2
1 i
1 µναβ1h 1µνee
C2 =+ 0tr f1 · f2 ,(f := fαβ ) .
2 2p0 · k1
2p · k22
A2 = −2

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Worldline representation
AmplitudesBenefits of worldline approach
Using our first quantised representation we gain certain advantages over standard
techniques:
1Simple formula fori
h unpolarised amplitudes:
αβ ?
|MN |2 = e2N |AN |2 + 2BN
BN αβ − |CN |2

2

Manifest gauge invariance of the on-shell amplitudes, expressed in terms of fieldstrength tensors at the level of the (worldline) integrand.

3

Allows a complete separation of photon and electron polarisations, conveniently
formulated in the spinor helicity framework.

4

Facilitates a “spin-orbit decomposition” that differentiates the photons’
interactions with the spin and orbital degrees of freedom of the electron.

5Simple proofs of the vanishing of massless equal-helicity amplitudes in themassless case (c.f. Mahlon Phys.Rev.D 49 (1994) 2197)...

6...and unexpected relations between these amplitudes in scalar and spinor QED forthe massive case.

7

Relations between the coefficients AN , BN αβ and CN , and recursion relations
relating these to their N − 1 counterparts.
James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Overview

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Conclusion
First quantised representations of field theory processes can be powerful alternatives tostandard techniques that can make certain structural aspects of the theory clearer.
See following talk by José Nicasio!
We have given a worldline description of the dressed electron propagator and produced
a generating function for N -photon amplitudes that is manifestly transverse.
Future work:
1 Extend the first quantised representation to propagation in the presence ofelectromagnetic or gravitational background fields
2 Generalise to the non-Abelian case using “worldline colour fields” – Master
Formula for gluon amplitudes
3 Apply to study higher-order processes, such as to calculation of the electron’s
anomalous gyromagnetic moment.

James P. Edwards

Gauge invariant amplitudes

Introduction
Worldline approach
Overview

Conclusion
First quantised representations of field theory processes can be powerful alternatives tostandard techniques that can make certain structural aspects of the theory clearer.
See following talk by José Nicasio!
We have given a worldline description of the dressed electron propagator and produced
a generating function for N -photon amplitudes that is manifestly transverse.
Future work:
1 Extend the first quantised representation to propagation in the presence ofelectromagnetic or gravitational background fields
2 Generalise to the non-Abelian case using “worldline colour fields” – Master
Formula for gluon amplitudes
3 Apply to study higher-order processes, such as to calculation of the electron’s
anomalous gyromagnetic moment.
For more information on worldline techniques see
Classic review: C. Schubert Phys. Rept. 355 (2001) 73 [arXiv:0101036 [hep-th]]More up-to-date report: JPE and C. Schubert [arXiv:1912.10004 [hep-th]]¡Thank you for your attention!
James P. Edwards

Gauge invariant amplitudes