Spinor propagator in the worldline approach James P. Edwards - - PowerPoint PPT Presentation

Introduction Tree level processes Applications Constant field

Spinor propagator in the worldline approach

James P. Edwards

ifm.umich.mx/∼jedwards

INFN Bologna

Dec 2017 In collaboration with Naser Ahmadiniaz (IBS, South Korea), Fiorenzo Bastianelli, Olindo Corradini, Victor Banda (UASLP, M´ exico) and Christian Schubert (UMSNH, M´ exico).

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field

Outline

1

Introduction

2

Tree level processes Formalism Master Formula Spin orbit decomposition

3

Applications Cross sections

4

Constant field Green functions Spin-coupling New 1PR contributions

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field

The worldline formalism

The worldline formalism is a first quantised approach to quantum field theory, hinted at by Feynman[1] and developed by Strassler[2]. It is often thought of as “string-inspired” and offers valuable alternative techniques to standard field theory. Much work has been done at one-loop and multi-loop order, such as Calculating one-loop effective actions Finding parameter integral representations of (multi-loop) scattering amplitudes Examining processes in constant electromagnetic backgrounds Gravitational interactions and trace anomalies Coupling matter to non-Abelian gauge potentials Propagation of higher spin fields and differential forms The main ingredients of these calculations are path integrals over point-particle trajectories, augmented by additional fields living on the particle worldlines.

• 1Phys. Rev. 80, (1950) , 440
• 2Nucl. Phys. B385, (1992), 145

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field

One-loop effective actions

The prototypical worldline path integral is for spinor QED minimally coupled to an Abelian gauge potential, Aµ(x). The one-loop effective action can be written as Γ[A] = −1 2 ∞ dT T e−m2T

• Dx
• Dψ e

− T

0 dτ

• ˙

x2 4 + 1 2 ψ· ˙

ψ+ie ˙ x·A(x(τ))−ieψµFµν(x(τ))ψν

• ,
• ver periodic trajectories xµ(0) = xµ(T) and anti-periodic Grassmann fields

ψµ(0) + ψµ(T) = 0. Gives a generating functional for all N-photon amplitudes: Worldline techniques offer various advantages over standard tools: Contributions from many Feynman diagrams combined Superior organisation of gauge invariance / covariance Different perspective for physical intuition Concrete realisation of how string theory can inform field theory calculations

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Formalism Master Formula Spin orbit decomposition

Spinor propagator

Although Feynman gave a first quantised path integral for the scalar propagator (and Casalbuoni, Gitman, Barducci and colleagues gave point-particle representations of the Dirac propagator), it is only recently that has been adequately developed to a comparable standard to the one-loop case. The matrix element in question is Sx′x =

• i /

D

′ + m

• x′
• − D2 + m2 + ie

4 [γµ, γν]Fµν −1 x

• where Dµ = ∂µ + ieAµ and we use the second order formalism with Feynman rules:

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Formalism Master Formula Spin orbit decomposition

Open worldlines

The matrix element Kx′x =

• x′

− D2 + m2 + ie

4 [γµ, γν]Fµν

• x
• , known as the

“kernel,” has path integral representation Kx′x[A] = 2− D

2 symb−1

∞ dT e−m2T

• Dx
• Dψ e−S[x,ψ|A]
• with new boundary conditions xµ(0) = x, xµ(T) = x′ and ψµ(0) + ψν(T) = 2ηµ for

anti-commuting variables ηµ. The worldline action is unchanged S[x, ψ|A] = T dτ ˙ x2 4 + 1 2ψ · ˙ ψ + ie ˙ x · A(x(τ)) − ieψµFµν(x(τ))ψν

• and the symbol map is defined by

symb

• γ[αβ...ρ]

= (−i √ 2ηα)(−i √ 2ηβ) . . . (−i √ 2ηρ). This provides our result in the Clifford basis (D = 4):

• 1

1, γµ, i 4[γµ, γν], γµγ5, γ5

• .

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Formalism Master Formula Spin orbit decomposition

Open worldlines

The matrix element Kx′x =

• x′

− D2 + m2 + ie

4 [γµ, γν]Fµν

• x
• , known as the

“kernel,” has path integral representation Kx′x[A] = 2− D

2 symb−1

∞ dT e−m2T

• Dx
• Dψ e−S[x,ψ|A]
• with new boundary conditions xµ(0) = x, xµ(T) = x′ and ψµ(0) + ψν(T) = 2ηµ for

anti-commuting variables ηµ. The worldline action is unchanged S[x, ψ|A] = T dτ ˙ x2 4 + 1 2ψ · ˙ ψ + ie ˙ x · A(x(τ)) − ieψµFµν(x(τ))ψν

• and the symbol map is defined by

symb

• γ[αβ...ρ]

= (−i √ 2ηα)(−i √ 2ηβ) . . . (−i √ 2ηρ). This provides our result in the Clifford basis (D = 4):

• 1

1, γµ, i 4[γµ, γν], γµγ5, γ5

• .

Our symbol map provides a path integral represen- tation of the Feynman spin factor for open lines. This is a matrix with Dirac indices: P

• e− ie

4

T

0 dτ[γµ,γν]Fµν(x(τ))

αβ

= 2− D

2 symb−1

ψ(0)+ψ(T )=2η

Dψ e−

T

0 [ 1 2 ψ· ˙

ψ−ieψµFµν(x(τ))ψν]− 1

2 ψ(0)·ψ(T )

• αβ

which affords considerable simplification in our calculations due to the worldline supersymmetry gained by the worldline action.

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Formalism Master Formula Spin orbit decomposition

N-point amplitudes

Scattering amplitudes are extracted from the propagator by dressing it with photons: Aµ(x) =

N

• i=1

εiµ eiki·x. Expanding to order eN leads to a path integral (shift path integral variables to absorb boundary conditions: x(τ) → x + (x′ − x) τ

T + q(τ) and ψ(τ) → ψ(τ) + η)

Kx′x

N

= (−ie)N2− D

2

∞ dT e−m2T e− (x−x′)2

4T

q(T )=0

q(0)=0

Dq e−

T

0 dτ ˙ q2 4

symb−1 Dψ e−

T

0 dτ 1 2 ψ· ˙

ψ N

• i=1

V xx′

η

[ki, εi]

• with insertions of vertex operators

V xx′

η

[k, ε] = T dτ

• ε ·

x − x′ T + ˙ q(τ)

• + 2iε · (ψ + η) k · (ψ + η)
• eik·(x+(x′−x) τ

T +q(τ))

The path integral is computed from the Green functions ∆(τi, τj) = |τi − τj| 2 − τi + τj 2 + τiτj T ; GF (τi, τj) = σ(τi − τj)

James P. Edwards Spinor propagator in the worldline approach

To get a closed formula we make use of the N = 1 worldline supersymmetry between xµ and ψµ. This allows us to linearise the interaction with respect to the superfield Qµ(τ, θ) = qµ(τ) + √ 2θψµ(τ). Then using the functional determinants DetDBC

• − 1

4 d2 dτ 2

• = (4πT)− D

2 ;

DetABC 1 2 d dτ

• = 2

D 2

we earn a generating function of tree-level pre-amplitudes: Kx′x

N

= (−ie)Nsymb−1 ∞ dT (4πT)

D 2

e−m2T e− (x−x′)2

4T

T dτ1 · · ·

• dθN

e

N

i=1

• iki·x+ x′−x

T

(θiεi+iτiki)− √ 2η·(εi+iθiki)

• +N

i,j=1

• ˆ

∆ijki·kj+2iDi ˆ ∆ijεi·kj+DiDj ˆ ∆ijεi·εj

• lin ε

To get a closed formula we make use of the N = 1 worldline supersymmetry between xµ and ψµ. This allows us to linearise the interaction with respect to the superfield Qµ(τ, θ) = qµ(τ) + √ 2θψµ(τ). Then using the functional determinants DetDBC

• − 1

4 d2 dτ 2

• = (4πT)− D

2 ;

DetABC 1 2 d dτ

• = 2

D 2

we earn a generating function of tree-level pre-amplitudes: Kx′x

N

= (−ie)Nsymb−1 ∞ dT (4πT)

D 2

e−m2T e− (x−x′)2

4T

T dτ1 · · ·

• dθN

e

N

i=1

• iki·x+ x′−x

T

(θiεi+iτiki)− √ 2η·(εi+iθiki)

• +N

i,j=1

• ˆ

∆ijki·kj+2iDi ˆ ∆ijεi·kj+DiDj ˆ ∆ijεi·εj

• lin ε

k1 ǫ1 k1 k2 ǫ1 ǫ2 k1 k2 ǫ1 ǫ2

• σ∈SN

... kσ(1) kσ(2) kσ(N) ǫσ(1) ǫσ(2) ǫσ(N) p p p′ p p′ p p′ p p′

Introduction Tree level processes Applications Constant field Formalism Master Formula Spin orbit decomposition

Momentum space

In momentum space we can write the kernel as (b := p′ + 1

T

N

i=1(τiki − iθiεi))

Kp′p

N

= (−ie)Nsymb−1 ∞ dTe−m2T T dτ1 · · ·

• dθN

e−T b2−N

i=1

√ 2η·(εi+iθiki)+N

i,j=1

• ˆ

∆ijki·kj+2iDi ˆ ∆ijεi·kj+DiDj ˆ ∆ijεi·εj

• lin ε

, Note this is the un-truncated kernel: Kp′p = 1 1 p

′2 + m2 =

⇒ Sp′p = 1 −/ p′ + m So to amputate external fermion legs we need ˆ Sp′p

N

:= (−/ p′ + m)Sp′p

N (/

p + m) (1)

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Formalism Master Formula Spin orbit decomposition

Decomposing the kernel

The second order formalism naturally separates the spin and orbital degrees of freedom, reflected in the vertex operator splitting into bosonic and Grassmann pieces. This allows for us to express the kernel as a sum over terms involving ever increasing numbers of spin interactions: Kp′p

N

=

N

• S=0

Kp′p

NS

where in Kp′p

NS a maximum of S contractions involving the spin part of the vertex

• perator,

V spin

η

[k, ε] := 2iε · (ψ + η) k · (ψ + η) = i(ψ + η) · f · (ψ + η) are taken. The remaining contribution comes from contractions in the scalar piece involving the N − S remaining photon insertions. Since the latter part is well-known we can produce a powerful iterative method for including an arbitrary number of spin interactions from simpler building blocks.

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Cross sections

Applications

Our amplitudes split into leading and subleading terms. In momentum space Sp′p

N

=

• /

p′ + m

• Kp′p

N

− e

• i

/ ǫiKp′+ki,p

N−1

. After amputation and sandwiching between spinor wavefunctions, the full N-photon amplitude for a fixed-spin process is iT p′p

(N)s′s := ¯

us′(−p′) ˆ Sp′p

N us(p)

Only the first part of Sp′p

N

has the double-pole required to contribute on-shell!

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Cross sections

Applications

Our amplitudes split into leading and subleading terms. In momentum space Sp′p

N

=

• /

p′ + m

• Kp′p

N

− e

• i

/ ǫiKp′+ki,p

N−1

. After amputation and sandwiching between spinor wavefunctions, the full N-photon amplitude for a fixed-spin process is iT p′p

(N)s′s := ¯

us′(−p′) ˆ Sp′p

N us(p)

Only the first part of Sp′p

N

has the double-pole required to contribute on-shell! 1 2

• ss′
• T(N)ss′
• 2 ∼ 1

2Tr

• Kp′p

(N)γ0K†p′p (N) γ0

, written entirely in terms of the transverse N-photon kernel. This represents a separation of manifestly gauge invariant pieces from those that vary under a gauge transformation but vanish on-shell.

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Cross sections

Compton scattering

To give an example of an explicit amplitude consider a two photon interaction: Kp′p

2

= A1 1 + Bµν 1 2[γµ, γν] − iCγ5 where A = 8pf1f2p + p · (k1 + k2)tr(f1f2) 4p · k1p · k2 B = −

• 1

2p · k1 − 1 2p · k2

• (f1f2)µν + f µν

1 pf2k1 + f µν 2 pf1k2

2p · k1p · k2 C =

• 1

2p · k1 + 1 2p · k2

• ǫµναβε1µε2νk1αk2β

Taking the trace indicated above reproduces the Compton cross section - there are two convenient gauge choices ε1 · k2 = 0 = k1 · ε2 and ε1 · p = 0 = p · ε2

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Cross sections

Summary I

In this way we can reproduce known tree-level results Feynman rules for spinor QED Linear Compton scattering cross section (N = 2) Self energy and the tensor decomposition[3]: requires subleading terms One loop contribution to g − 2 (N = 3) Note that the dressed propagator remains valid when the photons or the external fermion legs are off-shell so we can use it to sew together higher order amplitudes: ǫiµǫjν − →

• dDk

(2π)D δµν k2 Future work may incorporate wordline colour fields to consider the spinor propagator in a non-Abelian background (achieved already in the scalar case) relevant for QCD. Moreover, in ongoing work we are attempting to make this splitting of gauge invariant pieces manifest at the level of the vertex operator.

3For scalar QED see Ahmadiniaz, Bashir, Schubert Phys. Rev. D93, (2016) James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Dressed propagator in constant background

Adding a constant electromagnetic background field in Fock-Schwinger gauge Aµ(x(τ)) =

N

• i=1

εiµ eiki·x(τ) − 1 2 ¯ Fµν(x(τ) − x)ν. Green functions modified to their constant field counterparts ∆

⌣(τi, τj) and GF (τi, τj):

∆

⌣(τ, τ ′) = 1

2

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

T 2Z2

• Z

sin(Z) e−iZ ˙

GB(τ,τ′) + iZ ˙

GB(τ, τ ′) − 1

• GF (τ, τ ′) = GF (τ, τ ′)e−iZ ˙

GB(τ,τ′)

cos(Z) with (Z = e ¯ FT). The functional determinants become DetP

• − 1

4 d2 dτ 2 + ie 2 ¯ F d dτ

• = (4πT)− D

2 det− 1 2 sin(Z)

Z

• DetA

1 2 d dτ − ie ¯ F

• = 2

D 2 det− 1 2

cos(Z)

• James P. Edwards

Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Utility of symbol map

We pick up an extra prefactor that generates the spin-coupling to the background field symb−1 eiη·tan(Z)·η

4See Probing the quantum vacuum. Perturbative effective action approach in quantum electrodynamics

and its application, Dittrich and Gies, Springer Tracts 166

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Utility of symbol map

We pick up an extra prefactor that generates the spin-coupling to the background field symb−1 eiη·tan(Z)·η = 1 1 − i 4[γµ, γν] tan(Z)µν + i 8γ5ǫµναβ tan(Z)µν tan(Z)αβ , (D = 4). In fact this is a very compact representation of information known to be contained in the propagator[4] symb−1 eiη·tan(Z)·η ∼ det− 1

2 [cosh(Z)] e i 2 σ·Z

with Lorentz generators σµν := i

2[γµ, γν]. For truncation one would need the fancy

identity

• symb−1
• e

i 4 η·tan Z·η

−1 = symb−1

• e− i

4 η·tan(Z)·η

• 1 + tr(tan(Z) · tan(Z)) − det(tan(Z))

4See Probing the quantum vacuum. Perturbative effective action approach in quantum electrodynamics

and its application, Dittrich and Gies, Springer Tracts 166

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

One-particle reducible diagrams

Previously overlooked contributions to Euler-Heisenberg Lagrangians[5] and propagator self-energies[6] have recently been discovered:

kµ x0 x x0

They follow from the unusually looking identity

• dDk δD(k) kµkν

k2

= ηµν

D

so we need Γspin = δD(k)e ∞ dT T (4πT)− D

2 e−m2T det− 1 2

tan(Z) Z

• ǫ1 ·

T dτ ( ˙ GB−GF ) · k e

1 2 k·GB·k,

Sxx′

(1) = −2iǫ ·

• ∂Sxx′

∂F + ie 2 xSxx′x′

• · k + ǫ · L · k + O(k2)

5JHEP 03 (2017), 108

• 6Nucl. Phys. B923 (2017), 339

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Covariant formulae

The original addendum to the two-loop Euler-Heisenberg Lagrangian can be written in terms of the one-loop Euler-Heisenberg Lagrangian, L(1), as L1PR

(2)

= ∂L(1) ∂F µν ∂L(1) ∂Fµν . (2) Now we can give analogous versions for the one-particle reducible contribution to the fermion self energy in a constant background[7], both in configuration space, Σx′x1PR

(1)

= ∂Sx′x ∂F µν ∂L(1) ∂Fµν + ie 2 Sx′xx′µ ∂L(1) ∂F µν xν and in momentum space Σ1PR

(1) (p) = ∂Sp′p

∂F µν ∂L(1) ∂Fµν which are sufficient to extract explicit expressions for a given background.

• 7Nucl. Phys. B924 (2017), 377

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Weak field expansion

If the electromagnetic background field is considered weak we can expand perturbatively in ¯

• F. Leading or sub-leading corrections will be sufficient for

earth-based experiments. The spin part of the electromagnetic coupling is gauge invariant. We can exploit this by considering that the low energy limit of a plane wave photon as being effectively constant:

... ... ... ...

Nice approach to the calculation of g − 2 by expanding the N = 2 amplitude to linear

• rder in ¯
• F. Easier than working with a three-photon amplitude.

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Summary II

We can use the constant field propagator to Generate N-photon tree level amplitudes in a constant background Numerically investigate the Compton cross section in a constant field Derive the Gies-Karbstein linearisation formula for constant background fields Examine electron self energy in special background configurations Streamline the calculation of g − 2 The ongoing work on the structure of the vertex operator may allow us also to expand the bosonic part of the propagator as a series in the background field (already possible

• n the loop). Additional boundary terms may be required in the open line case.

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Conclusion

Tree level processes can be described using the worldline formalism. This approach retains the same calculational advantages over the standard formalism at loop order. We require open worldlines rather than the closed loops employed to date. The change in topology modifies the boundary conditions, which changes the worldline Green functions and functional determinants. Constant field backgrounds can be included using familiar techniques – we can extract physically significant information from the electron propagator. The propagator is split into a leading part that contributes to on-shell cross sections and a subleading part that is only relevant off-shell. We offer a fresh method, specially adapted to computation of scattering amplitudes, that may offer new insight into fundamental physics: Pair creation in constant background fields QCD and properties of other gauge theories Higher spin fields

James P. Edwards Spinor propagator in the worldline approach

Introduction Tree level processes Applications Constant field Green functions Spin-coupling New 1PR contributions

Conclusion

Tree level processes can be described using the worldline formalism. This approach retains the same calculational advantages over the standard formalism at loop order. We require open worldlines rather than the closed loops employed to date. The change in topology modifies the boundary conditions, which changes the worldline Green functions and functional determinants. Constant field backgrounds can be included using familiar techniques – we can extract physically significant information from the electron propagator. The propagator is split into a leading part that contributes to on-shell cross sections and a subleading part that is only relevant off-shell. We offer a fresh method, specially adapted to computation of scattering amplitudes, that may offer new insight into fundamental physics: Pair creation in constant background fields QCD and properties of other gauge theories Higher spin fields Thank you for your attention

James P. Edwards Spinor propagator in the worldline approach