NO'J lOUUbO Ntl-NO- - Jl> I Virtual two-loop corrections to Bhabha scattering by f Knut Steinar Bjørkevoll Them pneented at the Uiiircuilj of Beigen for the dt. sdent. degree pepaxtiMnt of Thyeic* XSBxremiy otBagea Bergen, notway « ' i rs i' Virtual two-loop corrections to Bhabha scattering by Knut Steinar Bjørkevoll Bergen, March 17, 1992 Acknowledgements The work that is presented by this thesis, hu been carried out at the Department of Physics at the University of Bergen during the last three years, with financial support from the Norwegian Research Council for Science and the Humanities. I wish to thank Professor Per Osland, who has been an excellent advisor for my dr. scient. work. I also want to thank him and Docent Goran Faldt for a fruitful collaboration that has been very valuable for me. I have had many interesting and useful discussions about physics and language with Dr. Conrad Newton during his stay at the University of Bergen. I also wish to thank Amanuensis Per Steinar Iversen for being the local computer gum, always ready to help when problems arise. It it impossible to thank my family properly in a few lines. Their support has made this work much easier. To Kjersti and Skjalg Contents 1 Introduction and motivation 1 1.1 Motivation 1 1.2 An overview of our work 2 2 Gauge invariance 6 3 Extraction of infrared-divergent factors 9 3.1 General technique 9 3.2 The two-rung ladder-like diagrams 11 3.3 The three-rung ladder-like diagrams 14 3.4 Explicit expressions for the infrared parts 17 4 Multiple complex contour integrals 19 4.1 Introduction 19 4.2 The factorization formula 20 4.3 Convergence of the complex integrals 21 4.4 Feynman amplitudes 22 4.5 Closing the contours 24 4.6 Algorithms for doing the complex contour integrations 25 4.6.1 Direct integration 27 4.6.2 Algebraic reduction of the integral 29 4.7 Example: A one-loop integral 29 5 The two-loop ladder-like diagrams 32 5.1 The uncrossed diagram 32 5.2 The once-crossed diagram 35 5.3 Results 39 6 Concluding remarks 44 A Proof of two relations 45 i B Momentum integration! and manipuUtion of Dirac matrices with REDUCE 48 B.l The momentum integration! 48 B 2 Simplification of the apinor factor 49 B.3 Contraction of momentum and ipinor factor» 51 B.4 The uncrossed and once-crossed diagrams 51 B.5 Listing of procedures, operators and some other definitions 52 B.6 Output for the uncrossed diagram 56 C The MAPLE-program CINT 60 C.l Calling sequence 60 C.2 Global variables 61 C.3 Procedures 62 C.3.1 Initialization 63 C.3.2 Before the integration starts 63 C.3.3 Doing the integrations 64 C.3.4 Printing results 66 C.4 Input and output for the one-loop example 67 C.5 Listing of procedures 68 D Applications of CINT to two-loop integrals 78 D.l Listing of the main code 79 D.2 Listing of auxiliary routines 63 D.3 Output for the uncrossed diagram 89 D.4 Output for the once-crossed diagram 89 ii Chapter 1 Introduction and motivation This thesis present the first part of a study of virtual two-loop corrections to Bhabha scattering at small angles and high energies. The three papers [1, 2, 3j that we have finished so far, are enclosed at the end of the thesis. They will be referred to as paper I, II and III respectively, and equations within the three articles will be referred to by adding the prefix I, II or III to their equation number. Bhabha scattering is the reaction + e+e- - e e-(n7), (1.1) where e+ is a positron, e~ is an electron, and n is the number of photons in the final state. The present study will be limited to contributions from Feynman diagrams without real photons, and all calculations will be done with » » |t| > m2, (1.2) where a is the center of mass energy squared, t is the square of the transferred four-momentum, and m is the electron mass. In the next section we will discuss our motivation, and in section 1.2 we will give a short overview over the work that we have completed so far. 1.1 Motivation I will discuss two motivations for our work: First, higher theoretical precision is needed for the determination of luminosity at e+e~-colliders, and secondly, the study of two-loop effects is interesting from a more theoretical point of view, since it may reveal new insight into the behaviour of perturbation series of gauge theories. The kinematical region defined by (1.2), is used for luminosity measurements at present e+e~ experiments, and will also be used for future experiments. Presently, the full one-loop corrections, and some contributions from higher-order diagrams, 1 are know, see e.g. (4, 5, 8, 7, 8). The experimental and theoretical error* for lumi­ nosity measurements at LEP, are now around 0.5%, and the experimental errors are expected to decrease to around 0.1% in future experiment» [9|. Two-loop corrections will be suppressed by a factor ^j«0.5-10-' (1.3) relative to the lowest order amplitude. Extra factors of T appear at the one-loop level, e.g. in terms of the logarithm log(j/u - ie) a -in, and we may similarly get a factor of wa at the two-loop level. Thus a'/»* is replaced by a' wO.5-10"4, (1.4) and an extra factor of 20 is enough to bring the result up to the 0.1% level. Such a factor may appear as e.g.: Iog-^«24 log' i£! « 5» = 25 2irJ » 20 (1.5) at LEP energies with \t\lt « 1/100, which is relevant for luminosity measurements. Two-loop results may also give us some insight into what will happen at higher orders in the perturbation expansion. It is interesting to compare with ^-theory. Here, a study of two- and three-loop effects [10], inspired other studies of higher order effects. Presently, the leading behaviour of ladder diagrams has been summed up to all orders in four and six dimensions [11,12, 13, 14]. The one-loop results are very simple, and do not tell very much about what happens at higher orders, while two- and three-loop results give us some ideas about the structure of amplitudes at higher orders. 1.2 An overview of our work Because QED is a gauge theory, the contribution from a single diagram will depend on an unphysical gauge parameter. We are therefore interested in the contribu­ tion from a set of gauge invariant diagrams, and we have selected the set of six gauge invariant diagrams shown in fig. 1.1. We call these diagrams ladder-like, since they arise from the ladder diagram (diagram a) by crossing the photon lines in all topologically distinct ways. Gauge invariance of the six diagrams is proven in chapter 2. In chapter 3, we determine the infrared structure of the sum of the six diagrams. This result is in agreement with the results of paper II. 2 a b X s^. c d / Figure 1.1: The three-rung ladder-like diagrams. Feynman parameterization, integration orer internal momenta, and manipula­ tion of the Dirac matrices are done in paper I. We managed there to write the Feynman-parameter integrals in a very compact form. The calculations are lengthy, but straightforward, and it is therefore convenient to use an algebraic manipulation program. Appendix B shows how this may be done with the symbolic manipulation program REDUCE. The most complicated part of the calculation is the integration over the Feynman parameters, which we denote by ai ... 0:7. One of the main steps in the calculation, is to use the factorization formula which is described in chapter 4. The idea is to introduce a number of complex integrals, with contours that run parallel to the imaginary axes, in such a way that the Feynman-parameter integrals become easier to do. We have used this formula in two different ways. The first way, which is described in chapter 4, is to introduce a large number of contour integrals, such that the Feynman-parameter integrals may be done ex­ actly, yielding a product of beta-functions. It is in principle easy to find the leading behaviour of the contour integrals, but since each contour integral may get contribu­ tions from several pole singularities, we may have to add contributions from many thousands of terms. Thus, I have written a symbolic computer program that finds the leading terms of the integrals. The program needs a few hours to find the leading 3 logarithms for each diagram. By leading logarithms, we mean the terms with the largest number of logarithmic factors. For the two-loop diagrams in question, we find thai each diagram has terms with up to four logarithmic factors. This method is useful for finding terms with three or four logarithmic factors, but the coefficients of terms with only two logarithmic factors appear as sums of multiple contour in­ tegrals. Il often happens that these sums can be evaluated, but first it is necessary to convert each integral to a standard form. This is possible to do, but we have not yet made a computer program for this purpose. The symbolic program that determines leading terms, is written in MAPLE, and is described in appendix C. The second way of evaluating the Feynman-parameter integrals, is to introduce a smaller number of contour integrals.