Part I: QEDPart II: The Calculation of a Loop Diagram

The Magnetic Moment of the ElectronJ.Hofmann

Gesellschaft f�ur Schwerionenforschung,Darmstadt2006-11-16

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramMotivationThe g -factor classicallyConsider a charge ying in circles

� = I � aI = � q = v2� r q ; jaj = � r2) � = g q2mL where g = 1 :

The g -factor in quantum mechanics:The Dirac equation(i@/� eA/�m) = 0

gives g = 2. Can we do better?J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramOutlinePart I { The QED LagrangianThe QED LagrangianFree theoryGauge symmetryFeynman diagramsPart II { Calculation of gFeynman parametersWick rotationCalculating gConventionsPeskin & Schroeder (especially ~ = c = 1 and p/= p� �)J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe QED Lagrangian

The Lagrangian DensityLQED = �i @/�m�

� 14F��F�� � e A� � (1) ; : Electron �eld (Dirac spinors)A�: Potential for the electromagentic �eldF�� = @�A� � @�A�: Electromagentic �eld-strength tensor

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe QED Lagrangian { IIFree electronsSetting A� = 0 the Lagrangian is

L = �i @/�m� :As in classical mechanics, the equation of motion can be derivedwith 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: QEDPart II: The Calculation of a Loop DiagramThe QED Lagrangian { III

Photons in vacuumSetting = 0 the Lagrangian isL = �14F��F�� :

The equations of motion can be derived similarly. This gives theMaxwell equations (in vacuum):@�F�� = 0 :

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramGauge invarianceGauge transformationsClassically, a gauge transformation is

�! �+ @�(x)@t ; A! A+r�(x)or written in a covariant form:

A� ! A� + @��(x) ; ! e�i e �(x)Reminder

LQED = �14F��F�� �m+(i @� �)� e A� �Only the combination (i @� � e A�) is gauge invariant!

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramPerturbation theoryThe interactionThe interaction term couples the equation of motion for theelectrons and the photons:

@�F�� = e � ;(i @/�m) = e A/ :How should one deal with that problem? ! Use perturbationtheory: expand in powers of e.A convenient way to visualise the contributions to thisperturbative expansion is the use of Feynman diagrams.

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramFeynman DiagramsExamples

Building Blocks

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe Photon Propagator I

Reminder: Greens Function for the wave equationThe wave equation is (in Lorentz gauge, @�A� = 0)��(x) = �(x) :

A Fourier Transformation gives�k2 ~�(k) = 1 =) ~�(k) = � 1k2 :

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe Photon Propagator II

� �How to treat the pole

�(x) = Z 1�1

dk �� 1k2� e i k�x

The photon propagator is given by�i g��k2 + i "

which guarantees causality.J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe Electron Propagator

Greens Function for the Dirac equationThe Fourier transform of the Dirac equation gives(i @� � �m)(x) = �(x) ;~(p) = 1p� � �m :

The electron propagator is given byi(p/+m)p2 �m2 + i" :

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe Interaction

The VertexThe interaction part of the Lagrangian is given byLint = �e A� � ;

correspondingly the contribution from a vertex is�i e � :

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramPart II

Part II

Calculating a Loop Diagram

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe Electron Vertex Function� ��

The full amplitudeThe 'scattering amplitude' can be read o�:Z d4k(2�)4 u(p0)(�i e �) �i g��(k � p)2 + i"� i(k/0 +m)k 02 �m2 + i" � i(k/+m)k2 �m2 + i"(�i e �)u(p)

How to calculate this monster?J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramTrick 1: Feynman Parameters I

The ideaTo combine two denominators, use the formula1AB = Z 1

0 dxdy �(x + y � 1)(x A+ y B)2 :The generalisation to 1Am11 Am22 :::Amnn is easy.

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramTrick 1: Feynman Parameters II

Simple exampleConsider1(k � p)2(k2 �m2) = Z 1

0 dxdy �(x + y � 1)[x (k � p)2 + y (k2 �m2)]2= Z 1

0 dxdy �(x + y � 1)[r2 + (x � x2)p2 � y m2]2 ;where r = k � x p.Now an integration over d4k = d4r depends only on r2.

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramIntermediate result IThe DenominatorCombining the three denominators in our original integral givesZ 1

0 dxdydz 2 �(x + y + z � 1)D3 ;where

D = r2 ��+ i" ;r = k + y q � z p ;� = �x y q2 + (1� z)2m2 :Now let's have a look at the nominator!

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe NominatorForm FactorsDue 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, ��� = i2 [ �; � ]):

u(p0) �u(p) = u(p0) �p0� + p�2m + i���q�2m� u(p)

) forget about B.J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramThe NominatorForm FactorsFinally we get

u(p0)� �F1(q2) + i���q�2m F2(q2)�u(p) :It is g = 2[F1(0) + F2(0)] :Charge conservation gives F1(0) = 1, it remains to calculate F2(0).

Our caseHere the nominator isu(p0)� �(�12 r2 + (1� x)(1� y)q2 + (1� 4z + z2)m2)

+ i���q�2m (2m2(z � z2))�u(p) :J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramTrick 2: Wick Rotation

Prescriptionr0 = i r0E ; r = rE ;dr4 = 2�2r3djr j

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramResultFinally. . .The formfactor isF2(q2) = �2�

Z 10 dxdydz �(x + y + z � 1) 2m2(z � z2)m2(1� z)2 � q2xy :

Now it is straight-forward to calculate (J. Schwinger 1948)g � 22 = F2(0) = �2� � 0:0011614 :

Experimental ValueThe currently accepted PDG g -factor isg=2 = 1:0011596521859� 38 :

J.Hofmann The Magnetic Moment of the Electron

Part I: QEDPart II: The Calculation of a Loop DiagramSummaryPart 1The QED-LagrangianFeynman DiagramsPart 2Draw the diagrams and write down the amplitudeCombine the denominators with Feynman parametersShift the momentum so that the denominator only depends onthe squared momentumPerform the integral via Wick rotation

J.Hofmann The Magnetic Moment of the Electron