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

4 Free-propagator: Free scalar propagator that is the Green’s function for the Klien-Gordon equation: ∂ 2 4 Dx , x ′ 1 � = � 0 | T φ ( x ) φ ( x ′ ) | 0 � = � x | − � + m 2 | x ′� , � = 0 ∂ x 2 i =1 i We exponentiate the denominator using a Schwinger proper-time parameter T . This gives � T � ∞ � ∞ � x ( T )= x d τ 1 x 2 Dxx ′ dT e − m 2 T � x | exp dT e − m 2 T � � x (0)= x ′ D x e − 4 ˙ | x ′� = − T ( − � ) 0 = 0 0 0 This is the worldline path integral representation of the relativistic propagator of a scalar particle in euclidean spacetime from x ′ to x . dT e − m 2 T e − ( x − x ′ )2 � ∞ � � T d τ 1 q 2 Dxx ′ D q ( τ ) e − 4 ˙ 4 T 0 = 0 0 DBC ⇒ Fourier transform ⇒ familiar momentum space representation � � Dpp ′ dD x ′ eip ′· x ′ Dxx ′ dD xeip · x = 0 0 1 (2 π ) D δ D ( p + p ′ ) = p 2 + m 2

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 � T � 1 � � ∞ � x ( T )= x x 2+ ie ˙ Dxx ′ dT e − m 2 T x (0)= x ′ D x ( τ ) e − d τ 4 ˙ x · A ( x ( τ )) 0 [ A ] = 0 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. � T � 1 � ∞ � � x 2+ ie ˙ dT e − m 2 T D x ( τ ) e − d τ 4 ˙ x · A ( x ( τ )) 0 Γ scal [ A ] = 0 x (0)= x ( T ) T 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) x µ x µ ( τ ) + y µ ( τ ) = 0 � � � � T 1 dD x 0 x µ d τ x µ ( τ ) D x ( τ ) = D y ( τ ) , ≡ (1) 0 0 T The remaining y ( τ ) path integral is performed using the Wick contraction rule ( τ i − τ j )2 � y µ ( τ i ) y ν ( τ j ) � = − δµν GB ( τ i , τ j ) , GB ( τ i , τ j ) ≡ GBij = | τ i − τ j | − (2) T The free Gaussian path integral gives � � T y 2 d τ 1 = (4 π T ) − D D e − 4 ˙ 0 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, N � A µ ( x ) = εµ eiki · x i i =1 After expanding the interaction term we get � ∞ dT e − m 2 T (4 π T ) − D � � V γ scal [ k 1 , ε 1] · · · V γ Γ scal [ A ] = ( − ie ) N 2 scal [ kN , ε N ] 0 T � T ⇒ V γ x eik · x scal [ k , ε ] ≡ d τε · ˙ 0

7 At this stage the zero-mode integration can be performed � N N � � dD x 0 e iki · x 0 = (2 π ) D δ D ( ki ) i =1 i =1 After some mathematical manipulations one get the following master formula which is known as Bern-Kosower master formula for external photons � ∞ � T N dT (4 π ) − D 2 e − m 2 T � � ( − ie ) N (2 π ) D δ D ( Γ scal [ k 1 , ε 1; · · · ; kN , ε N ] = ki ) d τ i 0 T 0 i =1 � ��� N � 1 1 � � GBij ( pi · pj ) − i ˙ ¨ × exp GBij ( ε i · pj ) + GBij ( ε i · ε j ) � lin ( ε 1 ··· ε N ) 2 2 i , j =1 where dGBij 2( τ i − τ j ) ˙ GBij = = sign ( τ i − τ j ) − d τ i T d 2 GBij 2 ¨ = = 2 δ ( τ i − τ j ) − (4) GBij d τ 2 T i

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 x 0 changes the path integral Lagrangian only by a term quadratic in the fields 1 iey µ F µν ˙ y ν ∆ L = (5) 2 The external field then can be absorbed by a change of the free worldline propagators, replacing GBij , ˙ GBij , ¨ GBij by T � Z e − i Z ˙ � GBij + i Z ˙ ˙ ¨ G Bij = GBij − 1 , G Bij = · · · , G Bij = · · · (6) 2 Z 2 sin Z where Z = eFT and the change in the free path integral due to the external field: � sin Z (4 π T ) − D → (4 π T ) − D 2 det − 1 � 2 2 (7) Z 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 � ∞ (4 π ) − D 2 e − m 2 T det − 1 � sin Z � � dT ( − ie ) N (2 π ) D δ D ( Γ scal [ k 1 , ε 1; · · · ; kN , ε N ] = ki ) 2 0 T Z � T � ��� N N � 1 � � 1 � ki · G Bij · kj − i ε i · ˙ ε i · ¨ × d τ i exp G Bij · kj + G Bij · ε j � lin ( ε 1 ··· ε N ) 0 2 2 i =1 i , j =1 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: � ∞ � x ( T )= x � T x 2+ ie ˙ dTe − m 2 T x (0)= x ′ D x ( τ ) e − 1 d τ [ ˙ x · A ( x ( τ ))] D [ x ′ ; x ] 4 0 = 0 After completing the square in the exponential, we obtain the following tree-level “Bern-Kosower-type formula” in configuration space � ∞ dTe − m 2 T e − 1 4 T ( x − x ′ )2 � � − D D [ x ′ ; x ; k 1 , ε 1; · · · ; kN , ε N ] = ( − ie ) N 2 4 π T 0 ε i · ( x − x ′ ) � T + iki · x ′ � � N � �� � T N � N + iki · ( x − x ′ ) τ i ∆ ij pi · pj − 2 i • ∆ ij ε i · kj −• ∆ • � ij ε i · ε j � i =1 i , j =1 × d τ i e T e � lin ( ε 1 ε 2 ··· ε N ) . 0 i =1 where we used the following derivatives of the Green function τ 2 1 1 • ∆( τ 1 , τ 2) = + sign ( τ 1 − τ 2) − T 2 2 τ 1 1 1 • ( τ 1 , τ 2) ∆ = − sign ( τ 1 − τ 2) − T 2 2 1 • ∆ • ( τ 1 , τ 2) = − δ ( τ 1 − τ 2) T where left and right dots indicate derivatives with respect to τ 1 and τ 2 respectively.

11 k 1 k 2 k 3 k N k 2 k 1 k 3 k N · · · · · · p ′ p p ′ p + + · · · + + k N k 2 k 1 k N k 1 k 2 k 3 k 3 · · · p ′ · · · p ′ p p + + · · · + + . . . . . .

12 Now, we Fourier transform the master formula � � dD x ′ eix ′· p ′ + ix · pD [ x ′ ; x ; k 1 , ε 1; · · · ; kN , ε N ] D [ p ; p ′ ; k 1 , ε 1; · · · ; kN , ε N ] = dD x After some algebra, we get the momentum-space version of our master formula (off-shell) � � ∞ D [ p ; p ′ ; k 1 , ε 1; · · · ; kN , ε N ] = ( − ie ) N (2 π ) D δ D � � dT e − T ( m 2+ p 2) p + p + ki 0 i � �� � | τ i − τ j | τ i + τ j � N � � � � T N � − 2 ki · p τ i +2 i ε i · p + − pi · pj − i sign ( τ i − τ j ) − 1 ε i · pj + δ ( τ i − τ j ) ε i · ε j � i , j =1 2 2 × d τ i e � lin ( ε 1 ε 2 ··· ε N ) 0 i =1 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)