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Calculation of the Photon Vacuum Polarization Without Divergent Loop Diagrams

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Calculation of the Photon Vacuum Polarization Without Divergent Loop Diagrams Calculation of the photon vacuum polarization without divergent loop diagrams Bachelor-Arbeit zur Erlangung des Hochschulgrades Bachelor of Science im Bachelor-Studiengang Physik vorgelegt von MALTE STIENECKER geboren am 03.06.1993 in LENGERICH Institut für Kern- und Teilchenphysik Fachrichtung Physik Fakultät Mathematik und Naturwissenschaften Technische Universität Dresden 2016 Eingereicht am 03. Juni 2016 1. Gutachter: Prof. Dr. Dominik Stöckinger 2. Gutachter: Prof. Dr. Walter Strunz 3 Summary Abstract As powerful as the Feynman calculus for quantum electrodynamics might be, loop diagrams lead to divergent integrals. The strategy called renormalization, which deals with this problem, seems at rst like another rule of thumb estimation. In order to give this strategy a physical and mathematical basis we take a step back and start with the fundamental physical concepts of scattering processes. This will lead us to the optical theorem and in the end to an expression for the contribution of photon vacuum polarization in QED. Contents 1 Introduction 1 2 Basic physical assumptions 3 2.1 Unitarity of the S-matrix through conservation of ptobability . .3 2.2 The optical theorem . .3 2.3 Analytic properties . .5 3 Calculation of the imaginary part from the vacuum polarization 7 3.1 The Feynman diagrams . .7 3.2 Calculation of the left side of the optical theorem, 2 · Im(Tii) ..........9 3.2.1 The amplitude Tii ..............................9 3.2.2 Casimir's trick . 10 3.2.3 The result of 2 · Im(Tii) .......................... 11 3.3 Calculation of the right side of the optical theorem, P 2 .......... 12 f jTif j 3.3.1 The right side written explicitly . 12 3.3.2 Calculation of the Trace . 13 3.3.3 The delta distributions . 14 3.3.4 Angular Integration . 15 3.3.5 Ward identity . 17 3.3.6 The result of P 2 ........................... 18 f jTfij 3.4 The imaginary part from the vacuum polarization . 19 4 Calculation of the vacuum polarization 21 5 Summary and Outlook 25 6 Bibliography 27 1 Introduction The vacuum polarization is an eect of quantum electrodynamics (QED). The name presents a literal description for the phenomenon. In vacuum polarization the mediator of the elec- tromagnetic eld creates virtual electron positron pairs. Since the photon is that mediator, it is responsible for the electromagnetic interaction between charged particles and the virtual electron positron pairs create a so called screening eect. Just like a dielectric medium within a plate capacitor leads to a lower eld between the plates due to polarization of the atoms, the virtual pairs also weakens eects from primarily charged particles. This means that the existence of virtual pairs of charged particles does in a sense polarize the vacuum. The result of this polarization is that all measured charges are just eective ones which are dependent on the distance to particles. In this paper we are therefore going to study the inuence of vacuum polarization. In order to predict scattering processes Richard P. Feynman introduced a powerful calculus. The process is then described by Feynman diagrams which lead to a recipe for calculating the scattering amplitude. This calculus is based on perturbation theory. The order of perturbation is given by the coupling constant which is connected to the involved charges and the vertices in Feynman diagrams. Therefore the order of perturbation can easily be read o any possible diagram. Obviously the amount of dierent diagrams for the same process increases rapidly with higher orders. But already for lower orders we can construct loop diagrams which symbolize vacuum polarization. Applying the Feynman calculus to such diagrams leads to a problem: The integrals which need to be computed are divergent. In order to solve this problem a strategy called renormalization was established. One could say that this goal-oriented method solves the problems with divergent integrals. But this is not just mathematically unsatisfying. Even if the experiments are consistent with the results, it is desirable to conrm the method. In order to do so, we go back to the basic principles of scattering as well as quantum eld theory and we will see that these general exact properties already lead to the expressions we are interested in. 2 Basic physical assumptions 2.1 Unitarity of the S-matrix through conservation of ptobability The rst of these basic physical assumptions we will use is the conservation of probability and norm. In order to describe scattering processes we introduce the S-matrix. This matrix describes all the changes from the initial state h ij at time ti = −∞ to the nal state h f j at time tf = +1 ([1]: page 14). j i(ti)i = S j f (tf )i (2.1) Conservation of probability means that if we start in the xed state h ij the sum of the probabilities for all possible processes leading to the nal states h f j must be equal to one. X 2 X y 1 = j h ijSj f i j = h ijSj f i h f jS j ii (2.2) f f No we use the completeness relation 1 P . = f j f i h f j X y y 1 = h ij S j f i h f j S j ii = h ijSS j ii (2.3) f Since i is normalized, h ij ii = 1, the upper equation enforces the S-matrix to be unitary. SSy = 1 (2.4) 2.2 The optical theorem One of the consequences of the unitarity of the S-matrix is the optical theorem. If we construct the new matrix iT from S−1 it describes all the scattering except the trivial one where nothing happens. So now the S-matrix has the form S = 1 + iT (2.5) 4 2 Basic physical assumptions and we write Sif for h ijSj f i or in a shorter form hijSjfi. The unitarity then becomes, as already seen in equation 2.2., X 1 = hijii = hijSSyjii = hijSjfi hfjSyjii (2.6) f If we now insert the expression for S = 1 + iT , we get X X (︁ )︁ (︁ )︁ hij1 + iT jfi hfj1 − iT yjii = hijfi + i hijT jfi · hfjii − i hfjT yjii (2.7) f f (︁ )︁ X y y * X * (2.8) 훿if 훿if + 푖훿fiTif − 푖훿if Tfi + Tif Tfi = 1 + iTii − iTii + Tif Tif f f For the last step we used that y * . Since the symmetric is the discrete Kronecker Aij = Aji 훿if Delta, its values are 1 for i = f and 0 for i 6= f and the product of two Deltas is therefore well dened. Since i · (z* − z) = 2 · Im(z) we obtain by cancelling the one on both sides of the equation X * X 2 (2.9) 2 · Im(Tii) = Tif Tif = jTif j f f This is one version of the so called optical theorem. To express explicitly that we can just consider processes which full energy and momentum conservation we introduce the four di- mensional delta distribution 4 4 P P . The factor 4 is a conventional factor (2휋) ·훿 ( i pi − f pf ) (2휋) which is also used in the Feynman calculus and comes from the Fourier transformation. The energy and momentum conservation for the forward scattering amplitude Tii is self-evident. X 4 4 X X 2 2 · Im(Tii) = (2휋) · 훿 ( pi − pf )jTif j (2.10) f i f Here it es important to note that the two summations P are in fact dierent ones. Inside f (:::) the delta distribution the summation runs over the xed momenta in the selected nal state. For the scattering process of two particles , the summation would become P4 1+2 ! 3+4 f=3 pf and the value of p3 and p4 is given by the nal state we are looking at. This nal state is selected by the second summation which runs over all possible nal states. In our example it would become a sum over the spins and an integral of the momenta from p3 and p4. The most important fact to note is that the matrix elements Tii and Tif , which we just dened by subtracting the unit matrix from the S-matrix, are the same complex numbers as the Feynman amplitudes. We will not go deeper into this, but more information on this topic are given in Bogoliubov ([2] chapter 18). 5 2.3 Analytic properties Analyticity is an important property of the photon vacuum polarization. In comparison to the perturbation theory of the Feynman calculus these are exact results of quantum eld theory and not limited to a specic order. Analyticity is based on causality which is the consequence of information not travelling faster than with the speed of light. More specic it says that the commutator of two events which are space like separated is zero. [A(x);A(y)] = 0 (2.11) This can also be written in terms of the time ordered product [9]. T (A(x)A(y)) = Θ(x0 − y0)A(x)A(y) ± Θ(y0 − x0)A(y)A(x) (2.12) where plus (minus) refers to Bose (Fermi) statistics. Based on causality one can show that the vacuum polarization (in [5] Π(v)) has certain analytic properties. In Itzykson and Zuber ([5] from page 245) it is shown that Π(v) is analytic in the upper half v plane except for cuts along the real axis. With the help of the Källen-Lehmann representation ([5] page 203) one can also show that Π(v) is analytic for Im(v) = 0 and Re(v) < 0. The representation says that Π is proportional to the integral over d(m2) of a density function 휌(m2) multiplied by a factor 1=(q2 − m2 + i").
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