SE9700361 Some A spects Of Chirality Fermion masses and chiral p-forms Astri Kleppe Lund 1997 Department of Mathematical Physics Lund Institute of Technology Lund, Sweden l 6 Organization Document name LUND UNIVERSITY doctoral dissertation Department of Mathematical Physics Date of issue May 30, 1997 Lund Institute of Technology P.O. Box 118, S-221 00 Lund, Sweden CODEN: Lund-MPh-97/09 Authors) Sponsoring organization Kleppe, Astri Title and subtitle Some Aspects of Chirality: fermion masses and chiral p-forms Abstract This thesis deals with chiral fields of different types. The properties of fermion mass matrices are investigated from different points of view, both within the minimal Standard Model and in extensions of the model. It is shown how mass matrix invariants are used to define the measurables of the quark mixing matrix as invariant functions of the mass matrices. One model is presented where the family pattern is suggested to originate from a kind of mass scaling. A Lagrangian density is defined for an entire charge sector, such that the existence of a Dirac field with mass mo implies the existence of other Dirac fields where the corresponding quanta have masses Smo, fl2mo,.. which are obtained by a discrete scale transformation. This suggests a certain type of democratic fermion mass matrices. Also extensions of the minimal Standard Model are investigated, obtained by including right-handed neutrinos in the model. The Standard Model extended by two right-handed neutrinos gives rise to a mass spectrum with two massive and three massless (at tree level) neutrinos. No assumptions such as the see-saw assumption of a large mass scale are necessary, provided the neutrino mass matrix assumes a certain form. The phenomenological consequences of this model are also discussed. The neutrino mass matrix in such a scheme has what is defined as a democratic texture. The demo ­ cratic family mixing matrix, with only unit entries, is found to act as a projection matrix in the sense that multiplying any 3x3 matrix with the democratic matrix is the same as multiplying an appropriate diagonal 3x3 matrix with the democratic matrix. Neutrino mass matrices with a democratic texture are studied for the cases with two and three right-handed neutrinos, respectively. I t The chiral fields that we find in the Standard Model have certain similarities with self-dual fields. l» Among other things, both chiral and self-dual fields suffer species doubling on the lattice. Chiral p-forms 19 are self-dual fields that appear in twice odd dimensions. Chiral p-forms violate manifest covariance, in SIS the same sense as manifest covariance is violated by non-covariant gauges in electrodynamics. It is shown P* that a covariant action can nevertheless be formulated for chiral p-forms, by introducing an infinite set of gauge fields in a carefully controlled way. Key words fermion masses, massive neutrinos, democratic family mixing, self-dual fields Classificationsystem and/or index terms (if any) Supplementary bibliographical information Language English ISSN and key title ISBN Some Aspects of chirality 91-628-2577-1 Recipient's notes Number of pages Price Security classification Distribution by (name and address) Dept, of Mathematical Physics, LTH, P.O. Box 118, S-221 00 Lund, Sweden I, the undersigned, being the copyright owner of the abstract of the above-mentioned dissertation, hereby grant to all reference sources permission to publish and disseminate the abstract of the above-mentioned dissertation. Signature Date May 30, 1997 Some A spects Of Chirality Fermion masses and chiral p-forms Astri Kleppe Lund 1997 Department of Mathematical Physics Lund Institute of Technology Lund, Sweden To my parents and my daughters 1 Some A spects Of Chirality Fermion masses and chiral p-forms This thesis consists of the following five papers and an introduction to the subject. The introduction consists of a first rather non-technical account, followed by a more formal part. The sections 7-12 are those directly pertaining to the articles. 1. Cecilia Jarlskog and Astri Kleppe: Measurables of the quark mixing matrix as invariant functions of mass matri ces Nucl. Phys. B286, 245 (1987) 2. Astri Kleppe: Parametrization of the fermionic mass spectra: Phys. Rev. D48, 5323 (1993) 3. Ingemar Bengtsson and Astri Kleppe: On Chiral P-forms USITP 96-12, to be published in Int. J. Mod. Phys. A 4. Cecilia Jarlskog and Astri Kleppe: Extending the Standard Model by including right-handed neutrinos Lund-Mph-97/05 5. Astri Kleppe: Neutrino mass matrices with a democratic texture Lund-Mph-97/06. Submitted for publication 1 Contents 1 . The riddle of transformation .................................3 2. Enlightened science................................................15 3. Classical mechanics................................................27 4. Quantum mechanics...............................................32 5. Symmetry................................................................34 6. Quantum field theory............................................ 41 7. Gauge symmetry....................................................45 8. Chirality and duality ..............................................52 9. Nuclear interactions ...............................................60 10. The Standard Model.............................................. 65 11. Masses..................................................................... 74 12. Neutrino masses..................................................... 79 13. The forms of substance ......................................... 85 14. Acknowledgements................................................. 86 2 1 The riddle of transformation Our existence takes place within the frame of space and time, but while spatial spots appear as reassuringly rustic and tangible, temporal locality seems much more fragile. In linguistic representations time appears as something quite concrete, ’’see you in two days ”, or "it is incredible how time flies away”, we say to each other. Statements about ’’two days ” refer to time as measured by the earthly rotation, and thus seem rather well defined. "Time flies away” is less concrete: what is it that flies away? We have learnt that it takes time to build something, a house, an understanding, any structure; just as we also know that time preys upon the world, and everything is liable to the transforming and destructive temporal forces. That the shimmering beauty of the mayfly is so touching has to do with our awareness of how strictly it is inscribed in time; it has something of the same melancholies! experience of beauty that can be evoked by the appearance of an old and worn palace. In its eroded facings, in its withering splendour, spatial robustness and temporal fragility seem to meet. Thus everything appears to be immersed in the inpalpable time, to be carried away with it, and effaced. We do not imagine that it is the "spatiality” of an object that disappears, but rather that the object residing there ’’dissolves in time”. Are these tran­ sient, dreamlike states all there is, does the world consist of flickering images without any steadiness? There is an old legend about a monk who went out in the forest an early spring morning, enticed by the wonderful trills of a bird. When he returned to the monastery after his walk, it suddenly looked strangely alien. He saw an unknown monk digging in the monastery ’s garden, and when he asked the gardener monk about the name of the place, he understood that it really was his own monastery, and felt very confused. As they continued their conversation, and the gardener told him that three hundred years ago, a Brother John had suddenly disappeared one day in May, leaving no trace; he all of a sudden exclaimed ’’but that is me!”, whereafter he fell down into a handful of dust. For three minutes he had listened to the song of the bird, but these three minutes were three minutes of God ’s time, which is three hundred years of human time. The legend is about the chimera of time. God is in reality outside time, since the divine is not subdued to the transformations of time. The holy is real, as opposed to the transient everyday reality. 3 1.1 Physics The idea of an absolute and holy reality beyond our dreamlike and arbitrary existence, has forever haunted man. Different prophets have pleaded for different truths, different doctrines have come and gone, with different representations of the essence of reality. The questions about the origin and forms of matter - about order and non-order - appear in all these explanatory systems. The notions of form and substance seem to be inherent in human thought; although nobody has ever seen formless matter, form and substance is nevertheless the everywhere appearing pair of fundamental building blocks. Physics can be conceived as yet another project in the striving to unveil "eternal ” structures. It is based on a conviction that there is a more essential level underneath the fugitive reality that we perceive with our imperfect senses; and it is accessible to man in the sense that he can create tales that reflect this more fundamental level. Such a tale is a physical model. All physical statements, all physical conclusions, refer to models. A scientific theory differs from other tales by its claim to reflect what is out there, beyond laguage. All descriptions of course take place within language, but physical descriptions claim to have an objective core. In order to be ranked as physical, a theory must have well defined observational consequences,