Tree level processes in the worldline formalism James P. Edwards - PowerPoint PPT Presentation

Tree level processes in the worldline formalism James P. Edwards Tlaxcala Sept 2017 En colaboraci´ on con Naser Ahmadiniaz (IBS, Corea del Sur), Fiorenzo Bastianelli (INFN, Italia), Olindo Corradini (INFN, Italia) Victor Banda (IF, UASLP) and Christian Schubert (IFM, UMICH). James P. Edwards Tree level processes in the worldline formalism

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 ] . Much work has been done on one-loop effective actions , multi-loop scattering amplitudes , loop processes in a constant background field and coupling to non-Abelian gauge fields at one-loop order. 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 � ˙ 4 + 1 2 ψ · ˙ Γ[ A ] = − 1 dT � � x · A ( x ( τ )) − ieψ µ F µν ( x ( τ )) ψ ν − 0 dτ ψ + ie ˙ T e − m 2 T 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: 1 Phys. Rev. 80 , (1950) , 440 2 Nucl. Phys. B385 , (1992), 145 James P. Edwards Tree level processes in the worldline formalism

At tree level? Although Feynman gave a first quantised path integral for the scalar propagator (and Casalbuoni , Gitman , Barducci and colleagues gave 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 � − 1 � � � � � − D 2 + m 2 + ie � ′ + m S x ′ x = � 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: 1 p 2 + m 2 p e ( p − p ′ ) µ p p ′ µ ν − 2 e 2 g µν p p ′ α β − e ( σ µν ) αβ ( p + p ′ ) ν p p ′ James P. Edwards Tree level processes in the worldline formalism

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 �� ∞ � � � dT e − m 2 T K x ′ x [ A ] = 2 − D 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 Tree level processes in the worldline formalism

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

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 The path integral is computed from the Green functions and functional determinants ∆( 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 d 2 = e − ( x − x ′ )2 − 1 � 1 d (4 πT ) − D D 2 ; � � � Det P Det A = 2 4 T 2 dτ 2 4 2 dτ and we earn a generating functional of tree-level amplitudes: ǫ 1 ǫ 1 ǫ 2 ǫ 1 ǫ 2 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 (1) k σ (2) k σ ( N ) σ ∈ S N p ′ p James P. Edwards Tree level processes in the worldline formalism

Applications Our amplitudes split into leading and subleading terms. In momentum space p ′ + m S p ′ p K p ′ p ǫ i K p ′ + k i ,p � � � ( N ) = ( N ) − e / ( N − 1) . / i Remarkably, in the cross section we find that only the leading term contributes poles to LSZ when the external fermions go on-shell: 1 � 2 ∼ 1 � � K p ′ p ( N ) K † p ′ p � � T ( N ) ss ′ � � 4Tr , ( N ) 4 ss ′ written entirely in terms of the transverse N -photon kernel. 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. 3 For scalar QED see Ahmadiniaz, Bashir, Schubert Phys. Rev. D93 , (2016) James P. Edwards Tree level processes in the worldline formalism

Dressed fermion propagator in a constant background Adding a constant electromagnetic background field in Fock-Schwinger gauge N ε iµ e ik i · x ( τ ) − 1 � F µν ( x ( τ ) − x ) ν . ¯ A µ ( x ( τ )) = 2 i =1 The Green functions are modified to their constant field counterparts ∆ ⌣ ( τ i , τ j ) and G F ( τ i , τ j ) . The functional determinants become ( Z = e ¯ FT ) d 2 = e − ( x − x ′ )2 − 1 dτ 2 + ie F d 2 � sin Z (4 πT ) − D 2 det − 1 ¯ � � � Det P 4 T 4 2 dτ Z � 1 d D 2 det − 1 dτ − ie ¯ � 2 � � Det A F = 2 cos Z 2 We pick up an extra prefactor that generates the spin-coupling to the background field symb − 1 � e iη · tan Z· η � James P. Edwards Tree level processes in the worldline formalism

Dressed fermion propagator in a constant background Adding a constant electromagnetic background field in Fock-Schwinger gauge N ε iµ e ik i · x ( τ ) − 1 � F µν ( x ( τ ) − x ) ν . ¯ A µ ( x ( τ )) = 2 i =1 The Green functions are modified to their constant field counterparts ∆ ⌣ ( τ i , τ j ) and G F ( τ i , τ j ) . The functional determinants become ( Z = e ¯ FT ) d 2 = e − ( x − x ′ )2 − 1 dτ 2 + ie F d 2 � sin Z (4 πT ) − D 2 det − 1 ¯ � � � Det P 4 T 4 2 dτ Z � 1 d D 2 det − 1 dτ − ie ¯ � 2 � � Det A F = 2 cos Z 2 We pick up an extra prefactor that generates the spin-coupling to the background field symb − 1 � e iη · tan Z· η � 1 − i 4[ γ µ , γ ν ] tan Z µν + i 8 γ 5 ǫ µναβ Z µν Z αβ , = 1 ( D = 4) James P. Edwards Tree level processes in the worldline formalism

Applications 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 self energy in special background configurations Streamline the calculation of g − 2 (now via N = 2 !) ... ... ... ... James P. Edwards Tree level processes in the worldline formalism

Conclusion Tree level processes can be described using the worldline formalism. This first quantised approach has great calculational advantages over the standard first order formalism. We require open worldlines rather than the closed loops more commonly employed. 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 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 have a new way to investigate field theory processes that is specially adapted to computation of scattering amplitudes and may offer new insight into fundamental physics: James P. Edwards Tree level processes in the worldline formalism