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 Introduction 1 Tree level processes 2 Formalism Master Formula Spin orbit decomposition Applications 3 Cross sections Constant field 4 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. 1 Phys. Rev. 80 , (1950) , 440 2 Nucl. 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, A µ ( x ) . The one-loop effective action can be written as � ∞ � T � x 2 � Γ[ A ] = − 1 dT � � ˙ 4 + 1 2 ψ · ˙ x · A ( x ( τ )) − ieψ µ F µν ( x ( τ )) ψ ν T e − m 2 T − 0 dτ ψ + ie ˙ D x D ψ e , 2 0 over periodic trajectories x µ (0) = x µ ( T ) and anti-periodic Grassmann fields ψ µ (0) + ψ µ ( 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 Formalism Tree level processes Master Formula Applications Spin orbit decomposition Constant field 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 − D 2 + m 2 + ie ′ + m S x ′ x = � − 1 � x ′ � 4 [ γ µ , γ ν ] F µν � i / �� �� � D � x where D µ = ∂ µ + ieA µ and we use the second order formalism with Feynman rules: James P. Edwards Spinor propagator in the worldline approach

Introduction Formalism Tree level processes Master Formula Applications Spin orbit decomposition Constant field Open worldlines The matrix element K x ′ x = � − D 2 + m 2 + ie x ′ � 4 [ γ µ , γ ν ] F µν � � � � x , known as the “kernel,” has path integral representation �� ∞ � � � K x ′ x [ A ] = 2 − D dT e − m 2 T 2 symb − 1 D ψ e − S [ x,ψ | A ] D x 0 with new boundary conditions x µ (0) = x , x µ ( T ) = x ′ and ψ µ (0) + ψ ν ( T ) = 2 η µ for anti-commuting variables η µ . The worldline action is unchanged � ˙ � T x 2 4 + 1 � 2 ψ · ˙ x · A ( x ( τ )) − ieψ µ F µν ( x ( τ )) ψ ν S [ x, ψ | A ] = dτ ψ + ie ˙ 0 and the symbol map is defined by √ √ √ � γ [ αβ...ρ ] � 2 η α )( − i 2 η β ) . . . ( − i 2 η ρ ) . symb = ( − i This provides our result in the Clifford basis ( D = 4 ): � 1 , γ µ , i � 4[ γ µ , γ ν ] , γ µ γ 5 , γ 5 1 . James P. Edwards Spinor propagator in the worldline approach

Introduction Formalism Tree level processes Master Formula Applications Spin orbit decomposition Constant field Open worldlines The matrix element K x ′ x = � − D 2 + m 2 + ie x ′ � 4 [ γ µ , γ ν ] F µν � � � � x , known as the “kernel,” has path integral representation �� ∞ � � � K x ′ x [ A ] = 2 − D dT e − m 2 T 2 symb − 1 D ψ e − S [ x,ψ | A ] Our symbol map provides a path integral represen- D x tation of the Feynman spin factor for open lines. 0 with new boundary conditions x µ (0) = x , x µ ( T ) = x ′ and ψ µ (0) + ψ ν ( T ) = 2 η µ for This is a matrix with Dirac indices: anti-commuting variables η µ . The worldline action is unchanged � T � e − ie 0 dτ [ γ µ ,γ ν ] F µν ( x ( τ )) � P 4 � ˙ � T αβ x 2 4 + 1 � 2 ψ · ˙ x · A ( x ( τ )) − ieψ µ F µν ( x ( τ )) ψ ν S [ x, ψ | A ] = � � dτ ψ + ie ˙ � T � = 2 − D 0 [ 1 2 ψ · ˙ ψ − ieψ µ F µν ( x ( τ )) ψ ν ] − 1 2 symb − 1 D ψ e − 2 ψ (0) · ψ ( T ) 0 ψ (0)+ ψ ( T )=2 η αβ and the symbol map is defined by which affords considerable simplification in our √ √ √ � γ [ αβ...ρ ] � 2 η α )( − i 2 η β ) . . . ( − i 2 η ρ ) . calculations due to the worldline supersymmetry symb = ( − i gained by the worldline action. This provides our result in the Clifford basis ( D = 4 ): � 1 , γ µ , i � 4[ γ µ , γ ν ] , γ µ γ 5 , γ 5 1 . James P. Edwards Spinor propagator in the worldline approach

Introduction Formalism Tree level processes Master Formula Applications Spin orbit decomposition Constant field N -point amplitudes Scattering amplitudes are extracted from the propagator by dressing it with photons : N � ε iµ e ik i · x . A µ ( x ) = i =1 Expanding to order e N 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 ) � ∞ � q ( T )=0 dT e − m 2 T e − ( x − x ′ )2 q 2 � T ˙ K x ′ x = ( − ie ) N 2 − D D q e − 0 dτ 2 4 T 4 N 0 q (0)=0 N � � � T � 0 dτ 1 2 ψ · ˙ V xx ′ symb − 1 D ψ e − ψ � [ k i , ε i ] η i =1 with insertions of vertex operators � T � x − x ′ � � � e ik · ( x +( x ′ − x ) τ T + q ( τ ) ) V xx ′ [ k, ε ] = dτ ε · + ˙ q ( τ ) + 2 iε · ( ψ + η ) k · ( ψ + η ) η T 0 The path integral is computed from the Green functions ∆( τ i , τ j ) = | τ i − τ j | − τ i + τ j + τ i τ j T ; G F ( τ i , τ j ) = σ ( τ i − τ j ) 2 2 James P. Edwards Spinor propagator in the worldline approach

To get a closed formula we make use of the N = 1 worldline supersymmetry between x µ and ψ µ . This allows us to linearise the interaction with respect to the superfield √ Q µ ( τ, θ ) = q µ ( τ ) + 2 θψ µ ( τ ) . Then using the functional determinants d 2 − 1 � 1 d = (4 πT ) − D D 2 ; � � � Det DBC Det ABC = 2 2 4 dτ 2 2 dτ we earn a generating function of tree-level pre- amplitudes: � ∞ � T e − m 2 T e − ( x − x ′ )2 dT � K x ′ x = ( − ie ) N symb − 1 dτ 1 · · · dθ N 4 T N D (4 πT ) 0 2 0 √ �� � ik i · x + x ′− x � � � N + � N ∆ ij k i · k j +2 iD i ˆ ˆ ∆ ij ε i · k j + D i D j ˆ ( θ i ε i + iτ i k i ) − 2 η · ( ε i + iθ i k i ) ∆ ij ε i · ε j � e i =1 i,j =1 T � � lin ε

To get a closed formula we make use of the N = 1 worldline supersymmetry between x µ and ψ µ . This allows us to linearise the interaction with respect to the superfield √ Q µ ( τ, θ ) = q µ ( τ ) + 2 θψ µ ( τ ) . Then using the functional determinants d 2 − 1 � 1 d = (4 πT ) − D D 2 ; � � � Det DBC Det ABC = 2 2 4 dτ 2 2 dτ we earn a generating function of tree-level pre- amplitudes: � ∞ � T e − m 2 T e − ( x − x ′ )2 dT � K x ′ x = ( − ie ) N symb − 1 dτ 1 · · · dθ N 4 T N D (4 πT ) 0 2 0 √ �� � ik i · x + x ′− x � � � N + � N ∆ ij k i · k j +2 iD i ˆ ˆ ∆ ij ε i · k j + D i D j ˆ ( θ i ε i + iτ i k i ) − 2 η · ( ε i + iθ i k i ) ∆ ij ε i · ε j � e i =1 i,j =1 T � � ǫ 1 ǫ 2 ǫ 1 ǫ 2 lin ε ǫ 1 k 1 k 1 k 2 k 2 k 1 p ′ p ′ p ′ p p p p ǫ σ (1) ǫ σ (2) ǫ σ ( N ) ... � k σ ( N ) k σ (1) k σ (2) σ ∈ S N p ′ p