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An Effective Theory on the Light Shell An effective theory on the light shell The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Sajjad, Aqil. 2014. An effective theory on the light shell. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:13064982 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA An effective theory on the light shell A dissertation presented by Aqil Sajjad to the Department of Physics Center for the Fundamental Laws of Nature in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts September 2014 c 2014 Aqil Sajjad All rights reserved. Dissertation Advisor: Professor Howard Georgi Aqil Sajjad An effective theory on the light shell Abstract We describe work on the construction of an effective field theory on a spherical light shell. The motivation arises from classical electromagnetism: If a collision produces charged particles with zero net charge emerging simultaneously from a point and instantaneously accelerating to the speed of light, then the electromagnetic fields due to these charges lie entirely on a spherical shell expanding at the speed of light. We show that this also applies to classical color radiation from high-energy collisions that produce colored particles. Specifically, the color fields produced in such a process are associated with a non-linear σ-model on the 2D light shell with specific symmetry- breaking terms. The quantum version of such a picture exhibits asymptotic freedom and should therefore be a useful starting point for a light-shell effective theory for QCD. We start in the simplified context of zero-flavor scalar quantum electrodynamics. Our effective theory has 3 major ingredients: breaking down the fields into soft and hard sectors with the large energy of the hard fields in the radial direction scaled out, a special gauge called light-shell gauge in which the picture simplifies, and a gauge-invariant source defined on a spherical light shell having infinitesimal radius. We match the fields between the effective theory and the full theory, meaning zero-flavor scalar QED. This allows us to compute the amplitude for the production of any number of scalars from the gauge-invariant source. We then find the tree-level iii amplitude for the emission of a photon using our effective theory and show that our result agrees with the full theory. To calculate loop effects in our effective theory, we need the photon propagator in light-shell gauge. We derive this propagator and use it to calculate the 1-loop correction to the amplitude for the production of a scalar and anti-scalar pair arising from virtual photon effects. This reduces to a pair of purely angular integrals in the effective theory and reproduces the familiar double logs of the full theory subject to an appropriate interpretation of an angular cutoff. iv Contents 1 Introduction 1 2 Classical motivation 6 2.1 The retarded potential calculation . .6 2.2 The non-abelian case . 10 3 An amusing calculation 22 3.1 The amplitude squared in the quantum theory as an intensity . 22 3.2 The classical fields as probability amplitudes . 24 3.2.1 Energy in a plane wave . 24 3.2.2 Fourier transform for a spherical wave . 25 3.3 Applying to our classical theory . 27 3.4 The QED calculation for comparison . 28 4 Constructing Our Effective Theory 30 4.1 Constructing the light-shell effective theory Lagrangian . 30 4.2 The LSET source . 35 5 Tree-level matching 38 5.1 Matching LRE scalars . 38 5.2 Photon emission . 41 6 The Light-shell gauge propagator 45 6.1 Notation for vector derivatives . 46 6.2 The Kinetic Energy Matrix in LSG . 47 6.3 Inversion on a subspace . 52 6.4 Returning to the LS gauge propagator . 52 7 Radiative corrections 56 7.1 The relevant LSG Green's function on the sphere . 57 7.2 Computing the integral . 59 7.2.1 Computing the off-diagonal (j 6= k) integral . 59 7.2.2 The j = k integral . 61 7.3 Combining the results and comparing with the full theory . 62 8 Some open questions 64 8.1 Finding an exact 2-dimensional theory . 64 8.2 Understanding the σ-model on the sphere . 65 8.3 Multiple photon emissions with large energies . 65 8.4 The meaning of angular cutoffs and jets . 66 8.5 Understanding LSET in terms of holography . 67 9 Conclusion 68 v A The strange looking fourier transform in spherical coordinates 70 B Conventions 72 C 0-Flavor Scalar Quantum Electrodynamics 73 D LRE Canonical Quantization 74 E Operator Notation 75 F Functional forms for inverse operators 76 F.1 Trying canonical quantization . 76 F.2 Using a different approach . 78 G Derivation of C 80 H Does L−2 make sense? 84 I Useful results involving C 85 J Last step for the LSG propagator on the sphere 87 K The last integral in the loop calculation 89 vi Citations to Previously Published Work H. Georgi, G. Kestin, and A. Sajjad, \Color Fields on the Light-Shell," http://arxiv.org/abs/1004.1404arXiv:1004.1404 [hep-ph]. H. Georgi, G. Kestin, and A. Sajjad, \The Photon Propagator in Light-Shell Gauge," http://arxiv.org/abs/1312.1741arXiv:1312.1741 [hep-ph]. H. Georgi, G. Kestin, and A. Sajjad, \Towards an Effective Field Theory on the Light- Shell Effective," http://arxiv.org/abs/1401.7667arXiv:1401.7667 [hep-ph]. vii Acknowledgements I owe a major debt of gratitude to a number of people who have played a major role in making it possible for me to get to the point of writing this thesis. These include family, friends, numerous teachers I have had throughout my life, many helpful people in the administration of the educational institutes I have attended, especially after the loss of my eye-sight, a number of very friendly and supportive staff members working in these institutes, and last but not the least, those who have contributed in one form or another to making science and education accessible to people with disabilities. I write this section with the apprehension that I will not be able to do justice to everyone, and it is possible that I might miss a few names. First, I am really grateful to my Ph.D adviser Howard Georgi for all his guidance and support. I have always been impressed by how much physics he knows and how he keeps on coming up with exciting new ideas. But more importantly, in addition to physics, he loves to mentor students. Thank you for everything. I would also like to thank my thesis committee members Matthew Schwartz and Subir Sachdev for all their support and encouragement, especially in moments of self-doubt. You may not know, but it really meant a lot. I am really grateful to Greg Kestin who worked with Georgi and myself on this project. He has become a valued friend and we have had many long hours of conversations working through the intricacies of our theory. In addition to being really good at physics, Greg is also very enthusiastic in his outreach to the wider community about science and research. Wish you all the best wherever you go. I should also thank Matthew Reece, Andrey Katz, John Mason, Tongyan Lin and Thomas Ryttov for their research collaborations on other projects during the course of my Ph.D. While these collaborations may have been on projects other than light-shell ef- fective theory, they were a major source of help and confidence as I worked through my thesis research. viii The Harvard physics department and the university's disability support services deserve a special mention for all the support they provided which have enabled me to carry out my studies at Harvard with my visual impairment. Several really great individuals have been involved through this process. In the physics department, David Norcross, Anne Trubia, David Morin, Jacob Barandes, Prof John Huth and Prof Melissa Franklin at the physics department have especially played a major role in making sure that I got all the accessibility related support that I needed. Graduate students David Thompson, Itay Yavin, WANG Yihua, Jonathan Ruel, Eunju Lee and Nicholas Manzi played a crucial role in the transcription of reading material in accessible reading format for me. At the adaptive technology lab, Rob Eveleigh in the initial years, and later Tanya Washburn, were closely involved in the scanning of reading material that needed to be converted for me, with Curtis Wilcox always readily available at the administrative end. Katherine Callaghan, Louise Russell and Sheila Petruccelli at the Harvard Accessible Education Office were always there to make sure that the entire system was running properly. This list would be totally incomplete if I did not mention Joseph Kolb, the mobility instructor who taught me the routes on campus with my white cane. The staff at the physics department has been exceptionally nice and makes it really fun to walk into the physics building. Dayle Maynard has throughout been a motherly figure looking after my needs, helping me do any paperwork, reading my mail (since I cannot read it due to my visual impairment), and above all, being always there as a great friend. Grad- uate student administrators Sheila Ferguson initially and then Lisa Cacciabaudo towards the end of my Ph.D have both been very supportive and helpful.
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