Part I: QED Part II: The Calculation of a Loop Diagram The Magnetic Moment of the Electron J.Hofmann Gesellschaft f ur Schwerionenforschung, Darmstadt 2006-11-16 J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Motivation The g -factor classically Consider a charge ying in circles =I a v I =q = 2q r q ; jaj = r ) = g 2 m L where g = 1 : The g -factor in quantum mechanics: The Dirac equation (i @/ eA/ m) = 0 gives g = 2. Can we do better? 2 J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Outline Part I { The QED Lagrangian The QED Lagrangian Free theory Gauge symmetry Feynman diagrams Part II { Calculation of g Feynman parameters Wick rotation Calculating g Conventions Peskin & Schroeder (especially ~ = c = 1 and p/ = p) J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The QED Lagrangian The Lagrangian Density LQED = i @/ m 1 F F e A 4 ; : Electron eld (Dirac spinors) A : Potential for the electromagentic eld F = @ A @ A : Electromagentic eld-strength tensor J.Hofmann The Magnetic Moment of the Electron (1) Part I: QED Part II: The Calculation of a Loop Diagram The QED Lagrangian { II Free electrons Setting A = 0 the Lagrangian is L = i@ / m : As in classical mechanics, the equation of motion can be derived with a variational principle. This gives @L @ = @ @ (@@L) : Calculating this one gets the Dirac equation for and ! J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The QED Lagrangian { III Photons in vacuum Setting = 0 the Lagrangian is = 14 F F : The equations of motion can be derived similarly. This gives the Maxwell equations (in vacuum): @ F = 0 : L J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Gauge invariance Gauge transformations Classically, a gauge transformation is @(x ) ; A ! A + r(x ) !+ @t or written in a covariant form: A ! A + @ (x ) ; ! e i e (x ) Reminder 1 F F m + (i @ ) e A 4 Only the combination (i @ e A) is gauge invariant! LQED = J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Perturbation theory The interaction The interaction term couples the equation of motion for the electrons and the photons: @ F = e ; (i @/ m) = e A/ : How should one deal with that problem? ! Use perturbation theory: expand in powers of e . A convenient way to visualise the contributions to this perturbative expansion is the use of Feynman diagrams. J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Feynman Diagrams Examples Building Blocks J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Photon Propagator I Reminder: Greens Function for the wave equation The wave equation is (in Lorentz gauge, @A = 0) (x ) = (x ) : A Fourier Transformation gives 1: k ~(k ) = 1 =) ~(k ) = 2 k2 J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Photon Propagator II How to treat the pole Z 1 1 (x ) = e i k x dk k 1 The photon propagator is given by 2 which guarantees causality. i g k2 + i " J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Electron Propagator Greens Function for the Dirac equation The Fourier transform of the Dirac equation gives (i @ m)(x ) = (x ) ; ~ (p) = p 1 m : The electron propagator is given by i ( p/ + m) : p m + i" 2 J.Hofmann 2 The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Interaction The Vertex The interaction part of the Lagrangian is given by Lint = e A ; correspondingly the contribution from a vertex is i e : J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Part II Part II Calculating a Loop Diagram J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Electron Vertex Function The full amplitude The 'scattering amplitude' can be read o: Z i g k u (p0 )( i e ) 4 (2) (k p )2 + i " 0 k 02i ( k/ m+2m+) i " k 2i ( k/m+2m+) i " ( i e )u(p) 4 d How to calculate this monster? J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Trick 1: Feynman Parameters I The idea To combine two denominators, use the formula 1 = Z dx dy (x + y 1) : AB (x A + y B ) The generalisation to Am Am :::Amn n is easy. 1 2 0 1 1 1 2 2 J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Trick 1: Feynman Parameters II Simple example Consider Z 1 (x + y 1) = dx dy (k p ) ( k m ) [x (k p) + y (k m )] Z (x + y 1) = dx dy ; [ r + ( x x )p y m ] where r = k x p. Now an integration over d k = d r depends only on r . 1 2 2 2 2 0 2 2 1 2 0 4 J.Hofmann 4 2 2 2 2 2 The Magnetic Moment of the Electron 2 Part I: QED Part II: The Calculation of a Loop Diagram Intermediate result I The Denominator Combining the three denominators in our original integral gives Z 2 (x + y + z 1) ; dx dy dz 1 where D3 0 = r + i" ; = k +y q zp; = x y q + (1 z ) m Now let's have a look at the nominator! D r 2 2 J.Hofmann 2 2 : The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Nominator Form Factors Due to Lorentz-invariance, the result must be of the form u (p0 ) A + (p0 + p ) B + q C + q D u (p) : This must vanish when contracted with q (no longitudinal photons) ) D = 0 : Gordon identity (from Dirac equation, = i [ ; ]): 0 p + p i q 0 0 u (p ) u (p ) = u ( p ) 2 m + 2 m u (p ) ) forget about B . 2 J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram The Nominator Form Factors Finally we get i q F2 (q 2 ) u (p) : 2m It is g = 2[F1 (0) + F2 (0)] : Charge conservation gives F1(0) = 1, it remains to calculate F2(0). u (p0 ) F1 (q 2 ) + Our case Here the nominator is 1 u (p0 ) ( r + (1 x )(1 y )q + (1 4z + z )m ) 2 + i 2 mq (2 m (z z ))u(p) : 2 2 2 2 J.Hofmann 2 2 The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Trick 2: Wick Rotation Prescription r = rE ; r 0 = i rE0 ; 4 2 3 dr = 2 r djr j J.Hofmann The Magnetic Moment of the Electron Part I: QED Part II: The Calculation of a Loop Diagram Result Finally. .. The formfactor is 2 m (z z ) : 1) m (1 z ) q xy Now it is straight-forward to calculate (J. Schwinger 1948) g 2 2 = F (0) = 2 0:0011614 : Experimental Value The currently accepted PDG g -factor is g =2 = 1:0011596521859 38 : F 2 (q 2 )= 2 Z 1 x y z (x + y + z d d d 0 2 2 2 2 2 J.Hofmann The Magnetic Moment of the Electron 2 Part I: QED Part II: The Calculation of a Loop Diagram Summary Part 1 The QED-Lagrangian Feynman Diagrams Part 2 Draw the diagrams and write down the amplitude Combine the denominators with Feynman parameters Shift the momentum so that the denominator only depends on the squared momentum Perform the integral via Wick rotation J.Hofmann The Magnetic Moment of the Electron

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