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Perturbative Improvement of Fermion Operators in Strong Interaction Physics

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Perturbative Improvement of Fermion Operators in Strong Interaction Physics UNIVERSITY OF CYPRUS PHYSICS DEPARTMENT Perturbative Improvement of Fermion Operators in Strong Interaction Physics M.Sc THESIS FOTOS STYLIANOU JUNE 2010 UNIVERSITY OF CYPRUS PHYSICS DEPARTMENT Perturbative Improvement of Fermion Operators in Strong Interaction Physics M.Sc Thesis Fotos Stylianou Advisor Prof. Haralambos Panagopoulos Submitted in partial fulfillment of the requirements for the degree of Master of Science in the Physics Department at the University of Cyprus June 2010 Dedicated to my girlfriend Irene Acknowledgments The completion of this project would not have been possible without the support of family members, friends and colleagues. In this respect, I grasp this opportunity to acknowledge their contribution and to express my sincerest regards. Initially, I would like to express my deepest gratitude to my supervisor Prof. Haris Panagopoulos, for his knowledge sharing. Throughout the course of this project, his continuous guidance contributed both to my professional and personal development. I deeply appreciate his persistence in my participation in the lattice conference in China, which has been a once in a lifetime experience for me. I also take this opportunity to thank Prof. Haris Panagopoulos, for showing trust to my abilities and choosing to include me in research programs. Special thanks go to my colleague Dr. Martha Constantinou, for her uncondi- tional support, her experience and helpful ideas through difficult times. I would also like to thank my colleague Dr. Apostolos Skouroupathis, for the cooperation and his constructive calculations. For all the joyful lunch breaks as well as their meaningful advice, I would like to thank my fellow roommates Phanos, Savvas, Phillipos, Marios and Demetris. IdonothaveenoughwordstoexpressmygratitudetoIliasIlia,forallthefun time and the support through long night studying hours. A very special acknowledgement goes out to my parents, who are always proud and supportive whatever my career choices might be. Thank you father: for your resourceful mind, your endless strength andpersistence.Thankyoumother:foryour delicious cuisine, your patience and understanding through my unstable working hours. You are both an inspiration to me, in your own unique way. Last but not least, I wish to express my gratitude to my girlfriend Irene Tziakouri, for her admirable patience, support and love during the last two years. I look forward to our new future together. Fotos Stylianou June 2010 Abstract The content of this thesis lies under the broad scientific area of Quantum Field Theory (QFT). QFT provides the theoretical tool for describing the weak, electro- magnetic, and strong interactions between the elementary particles of nature; it merges successfully the principles of Quantum Mechanics and Special Relativity. In theoretical physics, Quantum Chromodynamics (QCD) is the generally accepted QFT of the strong interactions between quarks and gluons that make up hadrons. In this thesis we make use of a space-time lattice regularization in order to perform a series of perturbative analytic calculations in the framework of QCD, to second order in the coupling constant g (“1-loop”). Our aim is to minimize the (a2) systematic errors, induced by the finiteness of lattice spacing a, which arise in O non-perturbative estimates. To fulfill this task we calculate in 1-loop perturbation theory the (a2)discretizationeffectsoftherenormalization constants for various O fermionic operators (i.e., quark field, localbilinears,“extended”bilinears,andfour- fermion operators). The derived (a2 g2)correctionscanbeexplicitly subtracted O from respective non-perturbative estimates, in order to obtain reliable simulation results. The above operators find significant application in the prediction of various phys- ical properties of hadronic matter, such as: (a) masses-decay constants of hadrons (quark field and local bilinears operators),(b)structurefunctionsmeasuringspin content, momentum and charge distributions in hadrons (extended bilinears opera- tors ), (c) matrix elements of ∆S =2mesontransitions(four-fermionoperators). The subtraction procedure turns out to be significant in controlling the lattice artifacts. We demonstrate the importance ofthisprocedureviagraphicalcompari- son of the subtracted and non-subtracted lattice data. The subtracted data provide amorestableandreliablecontinuumextrapolationa 0. Another possible appli- → cation of our results is in constructing improved versions of the fermion operators under study. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, (a2). Consequently, they have addressed a number of new O issues, most notably the appearance of loop integrands with “strong” Infrared Di- vergences (convergent only beyond 6 dimensions). Such integrands are not present 1 2 Abstract in (a1) improvement calculations; there, infrared divergent terms are seen to have O the same structure as in the (a0) case, by virtue of parity under integration, and O they can thus be handled by well-known techniques. We explain how to correctly extract the full (a2)dependence;infact,ourmethodisgeneralizabletoanyorder O in a. In all our calculations we employ improved actions both for fermion and gluon fields. For fermions we use the family of (a1) improved Wilson/clover/twisted mass O actions; particular cases of these actions are currently being studied intensely by the ETMC and other collaborations. For gluons we employ a 3-parameter family of Symanzik (a2) improved gluon actions, comprising all cases which are in common O use, such as the Wilson, tree-level Symanzik, Iwasaki, DBW2, and L¨uscher-Weisz actions. In order to provide results with the widest possible applicability, we have con- sidered the following parameters of the aforementioned actions as free parameters: clover coefficient cSW,numberofcolorsNc,couplingconstantg,gaugefixingpa- rameter λ,barequarkmassmf and twisted mass parameter µ.TheSymanzik coefficients, ci, appear in a nonpolynomial way in the calculations and, thus, we tabulate the corresponding results for different choices of ci. Περίληψη Το περιεχόμενο αυτής της διατριβής Μάστερ εντάσσεται στα πλαίσια της Θεωρίας Κβαντικών Πεδίων (ΘΚΠ).Η ΘΚΠ αποτελεί το θεωρητικό εργαλείο για την περιγραφή των ασθενών,των ηλεκτρομαγνητικών,και των ισχυρών αλληλεπιδράσεων ανάμεσα στα στοιχειώδη σωματίδια που αποτελούν την ύλη.Βασίζεται στις έννοιες των πεδίων και συνενώνει με επιτυχία τις αρχές της Κβαντικής Μηχανικής και της Ειδικής Θεωρίας της Σχετικότητας.Η Κβαντική Χρωμοδυναμική (ΚΧΔ)είναι η γενικά αποδεκτή ΘΚΠ για την περιγραφή των ισχυρών αλληλεπιδράσεων ανάμεσα στα συστατικά των αδρονίων (κουάρκς και γκλουόνια). Σε αυτή τη διατριβή Μάστερ χρησιμοποιήσαμε το χωροχρονικό πλέγμα ως ομα- λοποιητή για να πραγματοποιήσουμε μια σειρά από διαταρακτικούς υπολογισμούς,στα πλαίσια της ΚΧΔ,μέχρι 2ης τάξης ως προς την σταθερά σύζευξης g.Στόχοςμας είναι να βελτιώσουμε τα αποτελέσματα των προσομοιώσεων,μειώνοντας τα συστημα- τικά σφάλματα (a2) που οφείλονται στο πεπερασμένο μέγεθος της σταθεράς πλέγ- O ματος a.Γιατηνεπίτευξηαυτούτουστόχουυπολογίσαμεσεθεωρίαδιαταραχώνενός βρόχου, (g2),τασφάλματαδιακριτοποίησης (a2) των σταθερών επανακανονικο- O O ποίησης διαφόρων φερμιονικών τελεστών,όπως:τελεστής φερμιονικού πεδίου,τοπι- κοί και εκτεταμένοι διγραμμικοί τελεστές,και τελεστές με τέσσερα φερμιονικά πεδία. Αφαιρώντας τους προαναφερόμενους διορθωτικούς όρους (g2a2) από αντίστοιχους O μη-διαταρακτικούς υπολογισμούς,καταλήγουμε σε πιο αξιόπιστα αποτελέσματα προσο- μοιώσεων. Οι εν λόγω τελεστές βρίσκουν εφαρμογή στη μελέτη της δομής και των φυσικών ιδιοτήτων της αδρονικής ύλης,όπως: (α)μάζες και σταθερές διάσπασης αδρονίων, (β) συναρτήσεις δομής που μετρούν την κατανομή φορτίου,σπιν,και ορμής στο εσωτερικό των αδρονίων, (γ)πινακοστοιχεία ∆S =2μεσονικών μεταβάσεων. Επιδεικνύουμε την σημαντικότητα της διαδικασίας αφαίρεσης των σφαλμάτων δια- κριτοποίησης,που προτείνουμε,με γραφική σύγκριση δεδομένων από προσομοιώσεις πριν και μετά την αφαίρεση.Τα δεδομένα μετά τη διαδικασία αφαίρεσης εξασφαλίζουν πιο σταθερά και αξιόπιστα αποτελέσματα στο όριο a 0.Μιαάλληδυνατήεφαρμο- → γή των αποτελεσμάτων μας είναι στην κατασκευή βελτιωμένων εκφράσεων για τους τελεστές που μελετήσαμε. Το καινοτόμο σημείο των υπολογισμών μας έγκειται στο ότι εκτείνονται μέχρι 2η 3 4 Περίληψη τάξη ως προς τη σταθερά πλέγματος, (a2).Ωςεκτούτου,βρεθήκαμεαντιμέτωποιμε O πρωτότυπα προβλήματα,με πιο αξιοσημείωτο την εμφάνιση ολοκληρωμάτων βρόχου με «ισχυρούς»υπέρυθρους απειρισμούς (συγκλίνουν μόνο πέραν των 6διαστάσεων).Τα ολοκληρώματα αυτού του είδους δεν εμφανίζονται σε υπολογισμούς βελτίωσης μέχρι (a1).Σευπολογισμούς (a1) οι όροι με υπέρυθρο απειρισμό έχουν την ίδια δομή O O με αυτούς που εμφανίζονται σε υπολογισμούς (a0),και συνεπώς αντιμετωπίζονται O με γνωστές μεθόδους.Εξηγούμε με λεπτομέρεια τη σωστή διαδικασία εξαγωγής των όρων (a2).Αξίζεινασημειωθείότιαυτήηδιαδικασίαπουαναπτύξαμεείναιγενικεύ- O σιμη σε οποιαδήποτε τάξη ως προς την σταθερά πλέγματος a. Σε όλους τους υπολογισμούς μας χρησιμοποιούμε βελτιωμένες δράσεις τόσο για φερμιονικά όσο και για γκλουονικά πεδία.Για τα φερμιόνια χρησιμοποιούμε μια οικο- γένεια από (a1) βελτιωμένες δράσεις (Wilson/clover/twistedmass).Συγκεκριμένες O περιπτώσεις αυτών των δράσεων χρησιμοποιούνται από την ETMC και άλλες ερευ- νητικές ομάδες.Για τα γκλουόνια χρησιμοποιούμε μια οικογένεια 3-παραμέτρων,από βελτιωμένες (a2) δράσεις Symanzik,ηοποίαπεριλαμβάνειτιςδράσεις:Wilson, tree O − level Symanzik, Iwasaki, DBW2, και Luescher Weisz. − Τα αποτελέσματα μας καθίστανται
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