Dine m supersymmetry and string theory (2)

www.pdfgrip.com This page intentionally left blank www.pdfgrip.com SUPERSYMMETRY AND STRING THEORY Beyond the Standard Model The past decade has witnessed some dramatic developments in the field of theoretical physics, including advancements in supersymmetry and string theory There have also been spectacular discoveries in astrophysics and cosmology The next few years will be an exciting time in particle physics with the start of the Large Hadron Collider at CERN This book is a comprehensive introduction to these recent developments, and provides the tools necessary to develop models of phenomena important in both accelerators and cosmology It contains a review of the Standard Model, covering non-perturbative topics, and a discussion of grand unified theories and magnetic monopoles The book focuses on three principal areas: supersymmetry, string theory, and astrophysics and cosmology The chapters on supersymmetry introduce the basics of supersymmetry and its phenomenology, and cover dynamics, dynamical supersymmetry breaking, and electric–magnetic duality The book then introduces general relativity and the big bang theory, and the basic issues in inflationary cosmologies The section on string theory discusses the spectra of known string theories, and the features of their interactions The compactification of string theories is treated extensively The book also includes brief introductions to technicolor, large extra dimensions, and the Randall–Sundrum theory of warped spaces Supersymmetry and String Theory will enable readers to develop models for new physics, and to consider their implications for accelerator experiments This will be of great interest to graduates and researchers in the fields of particle theory, string theory, astrophysics, and cosmology The book contains several problems and password-protected solutions will be available to lecturers at www.cambridge.org/9780521858410 Michael Dine is Professor of Physics at the University of California, Santa Cruz He is an A P Sloan Foundation Fellow, a Fellow of the American Physical Society, and a Guggenheim Fellow Prior to this Professor Dine was a research associate at the Stanford Linear Accelerator Center, a long-term member of the institute for Advanced Study, and Henry Semat Professor at the City College of the City University of New York www.pdfgrip.com “An excellent and timely introduction to a wide range of topics concerning physics beyond the standard model, by one of the most dynamic researchers in the field Dine has a gift for explaining difficult concepts in a transparent way The book has wonderful insights to offer beginning graduate students and experienced researchers alike.” Nima Arkani-Hamed, Harvard University “How many times did you need to find the answer to a basic question about the formalism and especially the phenomenology of general relativity, the Standard Model, its supersymmetric and grand unified extensions, and other serious models of new physics, as well as the most important experimental constraints and the realization of the key models within string theory? Dine’s book will solve most of these problems for you and give you much more, namely the state-of-the-art picture of reality as seen by a leading superstring phenomenologist.” Lubos Motl, Harvard University “This book gives a broad overview of most of the current issues in theoretical high energy physics It introduces and discusses a wide range of topics from a pragmatic point of view Although some of these topics are addressed in other books, this one gives a uniform and self-contained exposition of all of them The book can be used as an excellent text in various advanced graduate courses It is also an extremely useful reference book for researchers in the field, both for graduate students and established senior faculty Dine’s deep insights and broad perspective make this book an essential text I am sure it will become a classic Many physicists expect that with the advent of the LHC a revival of model building will take place This book is the best tool kit a modern model builder will need.” Nathan Seiberg, Institute for Advanced Study, Princeton www.pdfgrip.com SUPERSYMMETRY AND STRING THEORY Beyond the Standard Model MICHAEL DINE University of California, Santa Cruz www.pdfgrip.com cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521858410 © M Dine 2007 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2006 isbn-13 isbn-10 978-0-511-26009-4 eBook (EBL) 0-511-26009-1 eBook (EBL) isbn-13 isbn-10 978-0-521-85841-0 hardback 0-521-85841-0 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate www.pdfgrip.com This book is dedicated to Mark and Esther Dine www.pdfgrip.com www.pdfgrip.com Contents Preface A note on choice of metric Text website Part Effective field theory: the Standard Model, supersymmetry, unification Before the Standard Model Suggested reading The Standard Model 2.1 Yang–Mills theory 2.2 Realizations of symmetry in quantum field theory 2.3 The quantization of Yang–Mills theories 2.4 The particles and fields of the Standard Model 2.5 The gauge boson masses 2.6 Quark and lepton masses Suggested reading Exercises Phenomenology of the Standard Model 3.1 The weak interactions 3.2 The quark and lepton mass matrices 3.3 The strong interactions 3.4 The renormalization group 3.5 Calculating the beta function 3.6 The strong interactions and dimensional transmutation 3.7 Confinement and lattice gauge theory 3.8 Strong interaction processes at high momentum transfer Suggested reading Exercises vii www.pdfgrip.com page xv xviii xx 9 12 18 22 25 27 28 28 29 29 32 34 35 39 43 44 51 59 61 viii Contents The Standard Model as an effective field theory 4.1 Lepton and baryon number violation 4.2 Challenges for the Standard Model 4.3 The hierarchy problem 4.4 Dark matter and dark energy 4.5 Summary: successes and limitations of the Standard Model Suggested reading Anomalies, instantons and the strong CP problem 5.1 The chiral anomaly 5.2 A two-dimensional detour 5.3 Real QCD 5.4 The strong CP problem 5.5 Possible solutions of the strong CP problem Suggested reading Exercises Grand unification 6.1 Cancellation of anomalies 6.2 Renormalization of couplings 6.3 Breaking to SU (3) × SU (2) × U (1) 6.4 SU (2) × U (1) breaking 6.5 Charge quantization and magnetic monopoles 6.6 Proton decay 6.7 Other groups Suggested reading Exercises Magnetic monopoles and solitons 7.1 Solitons in + dimensions 7.2 Solitons in + dimensions: strings or vortices 7.3 Magnetic monopoles 7.4 The BPS limit 7.5 Collective coordinates for the monopole solution 7.6 The Witten effect: the electric charge in the presence of θ 7.7 Electric–magnetic duality Suggested reading Exercises Technicolor: a first attempt to explain hierarchies 8.1 QCD in a world without Higgs fields 8.2 Fermion masses: extended technicolor www.pdfgrip.com 63 66 70 71 72 73 73 75 76 81 89 100 102 105 106 107 110 110 111 112 113 114 114 117 117 119 120 122 122 124 125 127 128 129 129 131 132 133 Appendix D The beta function in supersymmetric Yang–Mills theory We have seen that holomorphy is a powerful tool to understand the dynamics of supersymmetric field theories But one can easily run into puzzles and paradoxes One source of confusion is the holomorphy of the gauge coupling At tree level, the gauge coupling arises from a term in the action of the form: d θ SWα2 (D1) where S = −(1/4g ) + ia This action, in perturbation theory, has a symmetry S → S + iα (D2) This is just an axion shift symmetry Combined with holomorphy, this greatly restricts the form of the effective action The only allowed terms are: Leff = d θ (S + constant)Wα2 (D3) The constant term corresponds to a one-loop correction But higher-loop corrections are forbidden On the other hand, it is well known that there are two-loop corrections to the beta function in supersymmetric Yang–Mills theories (higher-loop corrections have also been computed) Does this represent an inconsistency? This puzzle can be stated – and has been stated – in other ways For example, the axial anomaly is in a supermultiplet with the conformal anomaly – the anomaly in the trace of the stress tensor One usually says that the axial anomaly is not renormalized, but the trace anomaly is proportional to the beta function The resolution to this puzzle was provided by Shifman and Vainshtein It is most easily described in a U (1) gauge theory, with some charged superfields, say φ ± Without masses for these fields, the one-particle irreducible effective action has infrared singularities In addition, we need to regulate ultraviolet divergences We can regulate the second type of divergence by introducing Pauli–Villars regulator fields, while the infrared divergence can be regulated by including a mass for φ ± The φ ± and regulator mass terms are holomorphic: d 2θ M 501 + pv − pv www.pdfgrip.com (D4) 502 Appendix D The gauge coupling term in the effective action must be a holomorphic function, now, of S and M But the effective action also includes wave function renormalizations for the various regulator fields: d θ Z −1 ( +† + + −† − ) (D5) The wave function factors, Z , are not holomorphic functions of the parameters.1 The physical cutoff is then Z pv M, and the physical infrared scale is Z φ m So to determine the coupling constant renormalization in terms of this scale, we need to compute the Z s as well One needs to be a bit careful in this computation If one works in a non-supersymmetric gauge, such as Wess–Zumino gauge, one needs to actually compute the mass renormalization; the wave function renormalizations will be different for the different component fields Starting, then, with our holomorphic expression 8π 8π = + b0 ln(m/M), g (m) g2 ( ) (D6) we have, in terms of the physical masses: 8π 8π = + b0 (ln(m/M) − ln(Z (m)/Z (M))) g (m) g2 ( ) (D7) To form the beta function, we need to take β(g) = −g ∂ b0 g b0 g −2 g (m) = − + γ 16π ∂ ln(m) 16π 16π (D8) where γ = 4g d ln(Z ) = − d ln(M) 16π (D9) So there are two- and higher-loop corrections to the beta function Plugging in, one obtains to two-loop order, for the U (1) theory: −β(g) = 4g g3 + 16π (16π )2 (D10) This is, in fact, the correct result This analysis makes clear why the holomorphic analysis is correct but subtle For non-Abelian theories, it is not quite so straightforward to introduce a holomorphic regulator One can arrive at the required modification by a variety of arguments In many ways, the most convincing and straightforward comes from examination of instanton amplitudes One can also simply make an educated guess by examining the results of two-loop computations The required relation is: 8π 8π = + b0 (ln(m/M) − ln(Z (m)/Z (M)) + C A ln(g ) g (m) g ( ) (D11) There is much dispute in the literature about whether to write the action with Z or Z −1 I have chosen Z −1 , following the convention of most field theory texts, in which propagators have a factor of Z in the numerator The reader is free to follow his or her taste www.pdfgrip.com The beta function in supersymmetric Yang–Mills theory 503 Differentiating, as before, one obtains the expression for the beta function: β(g) = g 3C A − TFi (1 − γ i ) 8π − (C A g /8π)2 (D12) Exercise (1) By examining ’t Hooft’s computation of the instanton determinant, argue that the appropriate generalization of the Shifman–Vainshtein formula is that of Eq (D11) Derive the exact expression for the beta function of Eq (D12) Verify, by comparison with published results, that this correctly reproduces the two-loop beta function of a supersymmetric gauge theory www.pdfgrip.com www.pdfgrip.com References Affleck, I (1980) Phys Lett., B92, 149 Affleck, I., Dine, M and Seiberg, N (1984) Dynamical supersymmetry breaking in supersymmetric QCD Nucl Phys B, 241, 493 Aharony, O., Gubser, S S., Maldacena, J M., Ooguri, H and Oz, Y (2000) Large N field theories, string theory and gravity Phys Rept., 323, 183 [arXiv:hep-th/9905111] Albrecht, A and Steinhardt, 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D B and Lee, C K (1978) Propagation functions in pseudoparticle fields Phys Rev D, 17, 1583 Buchmuller, W., Di Bari, P and Plumacher, M (2005) Leptogenesis for pedestrians Annals Phys., 315, 305 [arXiv:hep-ph/0401240] Carena, M., Quiros, M., Seco, M and Wagner, C E M (2003) Improved results in supersymmetric electroweak baryogenesis Nucl Phys B, 650, 24 [arXiv:hep-ph/0208043] Carroll, S (2004) Spacetime and Geometry: An Introduction to General Relativity San Francisco: Addison-Wesley Casher, A., Kogut, J B and Susskind, L (1974) Vacuum polarization and the absence of free quarks Phys Rev D, 10, 732 Cheng, T and Li, L (1984) Gauge Theory of Elementary Particle Physics Oxford: Clarendon Press Cheng, T.-P and Li, L.-F (1984) Gauge Theory of Elementary Particle Physics Oxford: Clarendon Press Chivukula, R S Technicolor and compositeness, arXiv:hep-ph/0011264 Cohen, A G., Kaplan, D B and Nelson, A E (1993) Progress in electroweak baryogenesis Ann Rev Nucl Part Sci., 43, 27 [arXiv: 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cosmological constant Nucl Phys B, 602, 307 [arXiv:hep-th/0005276] Fradkin, E and Shenker, S (1979) Phase diagrams of lattice gauge theories with Higgs fields Phys Rev D, 19, 3602 Gates, S J., Grisaru, M R and Siegel, W (1983) Superspace: or One Thousand and One Lessons in Supersymmetry San Francisco: Benjamin/Cummings Gepner, D (1987) Exactly solvable string compactifications on manifolds of SU(N) holonomy Phys Lett B, 199, 380 Giudice, G F and Rattazzi, R (1999) Theories with gauge-mediated supersymmetry breaking Phys Rept., 322, 419 [arXiv:hep-ph/9801271] Glashow, S L., Iliopoulos, J and Maiani, L (1970) Weak interactions with lepton–hadron symmetry Phys Rev D, 2, 1285 Green, M B., Schwarz, J H and Witten, E (1987) Superstring Theory, Cambridge: Cambridge University Press Greene, B R., Kirklin, K H., Miron, P J and Ross, G G (1987) A three generation superstring model Symmetry breaking and the low-energy theory, Nucl Phys B, 292, 606 Gross, D J., Harvey, J A., Martinec, E J and Rohm, R (1985) Heterotic string theory The free heterotic string Nucl Phys B, 256, 253 Gross, D J., Harvey, J A., Martinec, E J and Rohm, R (1986) Heterotic string theory The interacting heterotic string Nucl Phys B, 267, 75 Gross, D J and Wilczek, F (1973) Ultraviolet behavior of non-abelian gauge theories Phys Rev Lett., 30, 1343 Guth, A H (1981) The inflationary universe: a possible solution to the horizon and flatness problems Phys Rev D, 23, 347 Hartle, J B (2003) Gravity, an Introduction to Einstein’s General Relativity San Francisco: Addison-Wesley Harvey, J (1996) Magnetic monopoles, duality, and supersymmetry, arXiv:hep-th/9603086 Intriligator, K A and Seiberg, N (1996) Lectures on supersymmetric gauge theories and electric-magnetic duality Nucl Phys Proc Suppl., 45BC, [arXiv:hep-th/9509066] www.pdfgrip.com 508 References Intriligator, K and Thomas, S (1996) Dynamical supersymmetry breaking on quantum moduli spaces Nucl Phys., B473, 121, arXiv:hep-th/9603158 Jackson, J D (1999) Classical Electrodynamics Hoboken: Wiley Johnson, C (2003) D-Branes Cambridge: Cambridge University Press Kachru, S., Kallosh, R., Linde, A and Trivedi, S (2003) De Sitter vacua in string theory Phys Rev D, 68, 046005 [arXiv:hep-th/0301240] Kapusta, J I (1989) Finite-Temperature Field Theory Cambridge: Cambridge University Press Kolb, E W and Turner, M S (1990) The Early Universe Redwood City: Addison-Wesley Linde, A D (1982) A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems Phys Lett., 108B Linde, A (1990) Particle Physics and Inflationary Cosmology Reading: Harwood Academic Linde, A D (1994) Hybrid inflation Phys Rev D, 49, 748 [arXiv:astro-ph/9307002] Lykken, J D (1996) Introduction to supersymmetry, arXiv:hep-th/9612114 Maldacena, J M (1997) The large N limit of superconformal field theories and supergravity Adv Theor Math Phys., 2, 231 [arXiv:hep-th/9711200] Manohar, A V and Wise, M B (2000) Heavy Quark Physics Cambridge: Cambridge University Press Martin, S P and Vaughn, M T (1994) Two loop renormalization group equations for soft supersymmetry breaking couplings Phys Rev D, 50, 2282 [arXiv:hep-ph/9311340] Masiero, A and Silvestrini, L (1997) Two lectures on FCNC and CP violation in supersymmetry, arXiv:hep-ph/9711401 Mohapatra, R N (2003) Unification and Supersymmetry: The Frontiers of Quark – Lepton Physics Berlin: Springer-Verlag Nilles, H P (1984) Supersymmetry and supergravity Phys Rept, 110, Olive, D I and Witten, E (1978) Supersymmetry algebras that include topological charges Phys Lett B, 78, 97 Pais, A (1986) Inward Bound Oxford: Clarendon Press Peet, A (2000) TASI lectures on black holes in string theory [arXiv:hep-th/0008241] Peskin, M E Introduction to string and superstring theory 2, SLAC-PUB-4251 Peskin, M E (1985) SLAC-PUB-3821 Lectures Given at Theoretical Advanced Study Institute on Elementary Particle Physics, New Haven, CT, June 9–July 5, 1985 Peskin, M E (1987) In From the Planck Scale to the Weak Scale: Towards a Theory of the Universe, ed H E Haber Singapore: World Scientific Peskin, M E (1990) Theory of Precision Electroweak Measurements Lectures given at 17th SLAC Summer Inst.: Physics at the 100-GeV Mass Scale, Stanford, CA Peskin, M E (1997) In Fields, Strings and Duality: TASI 96, ed C Efthimiouo and B Greene Singapore: World Scientific Peskin, M E and Schroeder, D V (1995) An Introduction to Quantum Field Theory Menlo Park: Addison Wesley Peskin, M E and Takeuchi, T (1990) A new constraint on a strongly interacting Higgs sector Phys Rev Lett., 65, 964 Pokorski, S (2000) Gauge Field Theories Cambridge: Cambridge University Press Polchinski, J (1998) String Theory Cambridge: Cambridge University Press Politzer, H D (1973) Reliable perturbative results for strong interactions? Phys Rev Lett., 30, 1346 Ramond, P (1999) Journeys Beyond the Standard Model New York: Perseus Books www.pdfgrip.com References 509 Randall, L., Soljacic, M and Guth, A H (1996) Supernatural inflation: inflation from supersymmetry with no (very) small parameters Nucl Phys B, 472, 377 [arXiv:hep-ph/9512439] Randall, L and Sundrum, R (1999) A large mass hierarchy from a small extra dimension Phys Rev Lett., 83, 3370 [arXiv:hep-ph/9905221] Rohm, R (1984) Spontaneous supersymmetry breaking in supersymmetric string theories Nucl Phys B, 237, 553 Ross, G G (1984) Grand Unified Theories Boulder: Westview Press Salam, A and Ward, J C (1964) Electromagnetic and weak interactions Phys Lett., 13, 168 Sannan, S (1986) Gravity as the limit of the type II superstring theory Phys Rev D, 34, 1749 Seiberg, N (1993) Naturalness versus supersymmetric nonrenormalization theorems Phys Lett B, 318, 469 [arXiv:hep-ph/9309335] Seiberg, N (1994a) The power of holomorphy: exact results in 4-D SUSY field theories In International Symposium on Particles, Strings and Cosmology (PASCOS94), ed K C Wali Singapore: World Scientific [arXiv:hep-th/9408013] Seiberg, N (1994b) Exact results on the space of vacua of four-dimensional SUSY gauge theories Phys Rev D, 49, 6857 [arXiv:hep-th/9402044] Seiberg, N (1995a) The power of duality: exact results in 4D SUSY field theory Int J Mod Phys A, 16, 4365 [arXiv:hep-th/9506077] Seiberg, N (1995b) Electric–magnetic duality in supersymmetric nonAbelian gauge theories Nucl Phys B, 435, 129 [arXiv:hep-th/9411149] Seiberg, N (1997) Why is the matrix model correct? Phys Rev Lett., 79, 3577 [arXiv:hep-th/9710009] Seiberg, N and Witten, E (1994) Electric–magnetic duality, monopole condensation, and confinement in N = supersymmetric Yang–Mills theory Nucl Phys B, 426, 19 [Erratum-ibid B, 430, 485 (1994)] [arXiv:hep-th/9407087] Seiden, A (2005) Particle Physics: A Comprehensive Introduction San Francisco: Addison-Wesley Shadmi, Y and Shirman, Y (2000) Dynamical supersymmetry breaking Rev Mod Phys., 72, 25 [arXiv:hep-th/9907225] Silverstein, E and Witten, E (1995) Criteria for conformal invariance of (0, 2) models Nucl Phys B, 444, 161 [arXiv:hep-th/9503212] Susskind, L (1977) Coarse grained quantum chromodynamics In Weak and Electromagnetic Interactions at High Energy, ed R Balian and C H Llewelleyn Smith Amsterdam: North-Holland Terning, J (2003) Non-perturbative supersymmetry, arXiv:hep-th/0306119 ’t Hooft, G (1971) Renormalization of massless Yang–Mills fields Nucl Phys B, 33, 173 ’t Hooft, G (1976) Phys.Rev D14, 3432; erratum-ibid (1978) D18, 2199 ’t Hooft, G (1980) In Recent Developments in Gauge Theories, ed G ’t Hooft et al New York: Plenum Press Turner, M S (1990) Windows on the axion Phys Rept., 197, 67 Vilenkin, A (1995) Predictions from quantum cosmology Phys Rev Lett., 74, 846 [arXiv:gr-qc/9406010] Wald, R M (1984) General Relativity Chicago: University of Chicago Press Weinberg, S (1967) A model of leptons Phys Rev Lett., 19, 1264 Weinberg, S (1972) Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity New York: John Wiley and Sons www.pdfgrip.com 510 References Weinberg, S (1989) The cosmological constant problem Rev Mod Phys., 61, Weinberg, S (1995) The Quantum Theory of Fields Cambridge: Cambridge University Press Weinberg, S (2000) The cosmological constant problems [arXiv:astro-ph/0005265] Wess, J and Bagger, J (1992) Supersymmetry and Supergravity, Princeton: Princeton University Press Wilson, K G (1974) Confinement of quarks Phys Rev D, 10, 2445 Witten, E (1981) Dynamical breaking of supersymmetry Nucl Phys B, 188, 513 Witten, E (1986) New issues in manifolds of SU(3) holonomy Nucl Phys B, 268, 79 Witten, E (1995) String theory dynamics in various dimensions Nucl Phys B, 443, 85 [e-Print Archive: hep-th/9503124] Witten, E (1998) Anti-de Sitter space and holography Adv Theor Math Phys., 2, 253 [arXiv:hep-th/9802150] Yang, C N and Mills, R L (1954) Conservation of isotopic spin and isotopic gauge invariance Phys Rev., 96, 191 www.pdfgrip.com Index α expansion, 429 Abelian Higgs model, 122 ADM energy, 254 AdS space, 464 AdS-CFT correspondence, 463, 464 Affleck, I., 198 Affleck–Dine baryogenesis, 287, 292, 293, 294, 295, 296, 299 Alvarez-Gaume, L., 308 anomalies, 75 anomalies, two dimensions, 81 anomaly, 77, 78, 79 anomaly by point splitting, 79 anomaly evaluation, Fujikawa, 83 anomaly matching conditions, 238 anomaly mediation, 213 anomaly, applications, 80 anomaly, Fujikawa evaluation, 80 Appelquist, T., area law, 49 asymptotic freedom, 6, 35 Atiyah–Singer index theorem, 95 ATLAS, 475 axion, dark matter, 280 axion, finite temperature potential, 282 axion, stars, 281 axions, 103, 104, 105 background field gauge, 41 background field method, 40 Banks, T., 234, 277, 462, 479 Banks–Zaks fixed point, 234 baryogenesis, 270, 287 baryogenesis, gut, 287 baryogenesis, Sakharov conditions, 287 baryon asymmetry, 243 baryon number conservation, 63 Bekenstein, J., 255 beta function, 36, 39 beta function, non-linear sigma-model, 384, 385 beta function, sigma-model, 383 beta function, supersymmetric theories, 501 Betti numbers, 404 Bianchi identities, 261 Bianchi identity, 250 Bjorken, J., 4, 57 black holes, 255 Boltzmann equation, 284 bosonic closed string, spectrum, 319 bosonic open string, spectrum, 318 bosonic string, 313 bosonization, 387 BPS relations, 225 Calabi–Yau manifolds, spectra, 409 Calabi–Yau manifolds, three-form, 415 Calabi–Yau spaces, 401 Calabi–Yau spaces: Standard Model gauge group, 424 Casimir energy, 375 charge quantization, 113 Chern class, 405, 409 Chern–Simons number, 498 chiral basis, 483, 484 chiral superfields, 142 Christoffel connection, 247 CKM matrix, 25, 27, 33 CMS, 475 Coleman, S., Coleman–Mandula theorem, 139 collective coordinate, 122 collective coordinates, 89, 94, 121, 125 confinement, 18, 44, 45 conformal field theories, 233 conformal field theory, 323, 325 conformal field theory, commutators, 326 conformal field theory, stress tensor, 321 conformal gauge, 320 continuous symmetries in string theory, 416 contravariant vector, 246 cosmic microwave background radiation, CMBR, 266 cosmological constant, 72 cosmological principle, 259 coupling constant unification, 186 covariant derivative, gauge theory, 10 covariant vector, 245 covariantly constant spinor, 407 511 www.pdfgrip.com 512 Index CP N model, 84, 85, 86, 88, 89 curvature tensor, 249 D-brane, actions, 447 D-brane charges, 446 D-branes, 443, 444, 445 D-terms, criteria for vanishing, 195 dark energy, 72, 243, 262, 269, 300 dark matter, 72, 243, 262, 267, 269 de Sitter space, 262 Dedekind η-functions, 337 deep inelastic scattering, 57 determinant, derivative of, 247 differential forms, 401, 403 dilaton, as string coupling constant, 369 dimension-five operators and proton decay, 188 dimension-six operators, 69 dimensional regularization, 40 dimensional transmutation, 44, 85 Dine, M., 198 Dirac, P A M., 119, 122 Dirac quantization condition, 120 Dirac strings, 719 Dirichlet boundary conditions, 316, 444 discrete symmetries, 168 discrete symmetries in string theory, 416, 422 Dolbeault operators, 405 Douglas, M., 462 dsb, criteria, 209 duality, IIA–11-dimensional supergravity, 452, 453, 454 duality, perturbative, 442 duality, strong, weak coupling, 443, 451 dynamical supersymmetry breaking, 191, 192, 209 dynamical supersymmetry breaking, (3,2) model, 210 dyons, 126 Dyson, F., e+ e− annihilation, 51, 53 effective action, string theory, 334 electric dipole moment, 68, 101 electric–magnetic duality, 128, 224 electroweak baryon number violation, 497 enhanced symmetries, 381 equilibrium conditions, 266 equivalence principle, 244 extended technicolor, 134 Faddeev–Popov ghosts, 5, 20 Fayet–Iliopoulos D term, 158 Fermi theory of weak interactions, 63 Fermi, E., 3, 63 fermions, compactified, 378 Feynman rules, Yang–Mills theories, 20 Feynman, R., 4, 57 finite temperature effective potential, 497 finite temperature field theory, 492, 493, 496 fixed points, 388 flat directions, 195 Friedmann equation, 261 FRW metric, 259, 260 FRW universe, T (t), 265 FRW universe, scalar field equation, 265 Fujikawa, K., 80, 83 fundamental representation, 11 Gaillard, M., gauge fixing, theories with spontaneous symmetry breaking, 20 gauge mediation, 214 gaugino condensation, 193 Gauss law constraint, 98 Gell-Mann, M., 102 general coordinate transformations, 245 geodesic equations, 248 Georgi, H., 107, 122 Glashow, S., 6, 107, 122 global symmetries, 168 gluino condensation, 208 goldstino, 148, 160 Goldstone’s theorem, 487, 488 grand unified theories, 107 grand unified theories, mass relations, 112 grand unified theories, symmetry breaking, 111 grand unified theories, Weinberg angle, 109 Grassmann coordinates, 141 gravitational action, 250 gravitino problem, 279 Green’s function, 2-d, 324 Green, M., 308 Green–Schwarz fermions, 353, 354 Greenberg, W., Gross, D., GSO projection, 348 GSO projection, modular invariance, 349 Guth, A., 278 Hamiltonian lattice, 50 Hawking radiation, 255 Hawking temperature, 255 Hawking, black hole puzzle, 306 Hawking, S., 256, 306 heavy quark potential, 39 heterotic string, E × E , 361 heterotic string, O(16) × O(16), 363 heterotic string, bosonic formulation, 387 heterotic string, interactions, 362 heterotic string, modular invariance, 363 heterotic string, O(32), 360 heterotic theory on Calabi–Yau manifolds, 418 hidden sector, 164 hierarchy problem, 71, 131 Higgs field, 24 Higgs mechanism, 16 Higgs particle, 5, 32 Higgs, P., Hodge numbers, 405, 410 holography, 461 holomorphy, 152, 153, 208 holonomy, SU (3), 407 Horava, P., 458 www.pdfgrip.com Index Horava–Witten theory, 458 hybrid inflation, 278 hypermultiplet, 220 IIA supergravity, 367 IIB self-duality, 456 IIB supergravity, 367 ILC, 475 Iliopoulos, J., inflation, 272 inflation, flatness puzzle, 270 inflation, fluctuations, 275, 276 inflation, homogeneity puzzle, 270 inflation, reheating, 274 instanton, singular gauge, 203 instantons, 75, 86, 88, 89, 201 instantons, fermion zero modes, 93 instantons, QCD, 86 instantons, scale size integral, 94 instantons, spontaneously broken gauge theories, 96 instantons, symmetry violation, 95 Intriligator–Thomas models, 237 jets, 53 K – K¯ mixing, 182 Kahler manifolds, 408 Kaluza, F., 373, 374 Kaluza–Klein theory, 373 kink, 121 Klein, O., 373, 374 landscape, 310 large N expansion, QCD, 99 large N limit, 85 lattice gauge theory, 44, 46 Lee, B., leptogenesis, 290, 291, 292 lepton masses, 34 lepton number conservation, 63 LHC, 475 light cone gauge, 314, 315, 316, 317, 318 light element abundances, 266 lightest supersymmetric particle (LSP), 168 LSP, dark matter, 283 magnetic dipole moment, 68 magnetic monopole, 119 magnetic monopoles, 113 Maiani, L., Maldacena, J., 463 matter dominated era, 262 metric tensor, 244 Mills, R., Măobius group, 332 modular invariance, one-loop amplitudes, 337, 338 modular transformations, 336 moduli, 151, 195, 199, 375 moduli problem, 285 moduli spaces, 199 moduli spaces, approximate, 209 513 moduli, and inflation, 278 moduli, lifetime, 286 monopole problem, 270 MSSM, 167, 169 MSSM, breaking of SU (2) × U (1), 173 MSSM, Higgs mass limits, 176 MSSM, parameters, 171 mu term, 178 N = theories, 219 N = 2, vector multiplet, 220 N = Yang–Mills theory, 221 Nambu, Y., 307 Nambu–Goldstone bosons, 14 Ne’eman, Y., 102 Neumann boundary conditions, 316, 444 neutrino mass, 66 neutrino oscillations, 67 neutron electric dipole moment, 69 non-linear sigma model, 133 non-perturbative superpotential, 198 non-renormalization theorems, 151, 152, 153 non-renormalization theorems, string theory, 430, 431, 432, 433 normal ordering, 346, 347 NS 5-brane, 456 NS sector, 342 nucleosynthesis, 266 O’Raifeartaigh models, 157 O(10), 114 one-loop potential, 161 OPE, 324 open strings, 315 open superstrings, 341 orbifolds, 388, 389, 390, 391 orbifolds, discrete symmetries, 393 orbifolds, discrete symmetries and flat directions, 394 orbifolds, effective actions, 396 orbifolds, modular invariance, 394 orbifolds, N = supersymmetry, 390 orientifolds, 447 parity, 24 partons, Pauli–Villars, 501 Pauli–Villars fields, 213 Pauli–Villars regulator, 78 Peccei, R., 103 Peccei–Quinn symmetry, 103 Penzias, A., 266 perimeter law, 49 pion masses, 488 pions as Goldstone bosons, 91 Politzer, D., Prasad, M K., 124 Prasad–Sommerfield monopole, 124 proton decay, 70, 114 proton decay, supersymmetric GUTs, 188 QCD at high temperature, infrared divergences, 496 www.pdfgrip.com 514 Index QCD instantons, 92 QCD theta, 100, 101 QCD theta and massless u quark, 102 QCD theta and spontaneous CP violation, 103 QCD theta parameter, 71, 81, 89, 92 quadratic divergences, cancellations in supersymmetry, 171 quantum moduli space, 211 quark condensate, 90, 132 quark distribution functions, 58 quark mass matrix, 27 quark masses, 34 Quinn, H., 103 R-symmetries, 195 R-symmetry, 149, 170 R-parity, 168, 189 radiation dominated era, 262, 263 Ramond sector, 342, 343 Randall, R., 278 Randall–Sundrum, 475 Ricci scalar, 250 Ricci tensor, 249 right-handed neutrino, 116 S-matrix, factorization, 333 S-matrix, string theory, 331 Sakharov, A., 287 Salam, A., Scherk, J., 308 Schwarzschild metric, 252, 253, 254 Schwarzschild radius, 254 Schwarz, J., 308 Schwinger, J., Seiberg duality, 240 Seiberg, N., 152, 198, 207, 227, 230, 238, 240 Seiberg–Witten theory, 225, 226, 227, 228, 229, 230 Shenker, S., 462 Shifman, M., 501 SLAC, slow roll approximation, 272 small gauge transformations, 98, 126 soft breaking, constraints, 181, 182 soft breaking, experimental constraints, 180 soft breakings, experimental constraints, 179 soft masses, renormalization group equations, 175 soft supersymmetry breaking, 162, 169 solitons, 120 Sommerfield, C., 124 sphaleron, 499 spin connection, 257 spinor representations of orthogonal groups, 115 spinors, general relativity, 256 spontaneous supersymmetry breaking, 157 stress tensor, 250 string coupling, 368 string theory, Fayet–Iliopoulos terms, 434, 435, 436, 437 string theory, finiteness, 306 string theory, gaugino condensation, 437, 438 string theory, unification of couplings, 416, 424 strings, background fields, 382, 383 strong CP problem, 76 strongly coupled heterotic string theory, 458, 459 SU (5), 107, 108 superconformal zero modes, 204 superfields, 142 supergravity, eleven dimensions, 365, 366 superpotential, 145 superspace, 141 superspace covariant derivatives, 142 superstring action, 341 superstring mode expansion, 342 superstrings, space-time fermions, 343 superstrings, vertex operators, 355 supersymmetric guts, 185 supersymmetric qcd, 195 supersymmetry, 475 supersymmetry algebra, 140 supersymmetry breaking, early universe, 296 supersymmetry breaking, supergravity, 164 supersymmetry breaking, vanishing of the ground state energy, 148 supersymmetry currents, 147 supersymmetry generators, 142 supersymmetry representations, 140 supersymmetry zero modes, 204 supersymmetry, component Lagrangian, 146 supersymmetry, world sheet, 345 Susskind, L., 307, 462 Susy QCD, Nf < N − 1, 200 Susy QCD, Nf = N − 1, 201 ’t Hooft, G., 5, 40, 94, 96, 201, 238 ’t Hooft–Polyakov monopole, 124 T -duality, 381 T -duality, open strings and D-branes, 449 technicolor, 131, 475 theta functions, 350 Tomanaga, S., top quark, 39 top quark, symmetry breaking in MSSM, 175 toroidal compactification, 386 toroidal compactification, momentum lattice, 379 toroidal compactification, non-supersymmetric, 398 Type I-O(32) duality, 457 Type II strings, spectra, 347 U (1) problem, 99 unification of couplings, 110 unitarity triangle, 33 Vainshtein, A., 501 vector superfields, 143 Veltman, M., 5, 40 Veneziano amplitude, 333 Veneziano, G., 307 vertex operators, 329, 330 vielbein, 256, 257 Virasoro algebra, 327 Virasoro–Shapiro amplitude, 333 www.pdfgrip.com Index visible sector, 164 vortices, 122 W boson, 135 W bosons, 26, 29, 133 Weinberg, S., 5, 103, 479 Wess–Zumino model, 149 Weyl basis, 483, 484 Weyl rescaling, 375 Wilczek, W., Wilson line, 46 Wilson lines on Calabi–Yau manifolds, 420 Wilson lines, compactified, 379 Wilson loop, 47 Wilson, K., 36, 46, 71 Wilson, Ken, 47 Wilson, R., 266 winding modes, 377 Witten effect, 127 Witten index, 192 Witten, E., 127, 192, 227, 230, 308, 458 Wolfenstein parameterization, 33 Yang, C N., Yang–Mills action, 10 Yang–Mills theory, Yoneya, T., 308 Yukawa, H., Z boson, 26, 135 Z bosons, 29, 133 Zaks, A., 234 www.pdfgrip.com 515 ... symmetry SU (2)L × SU (2)R Take M to be a Hermitian, matrix field, M = σ + iπ · σ (2.39) Under the symmetry, which we first take to be global, M transforms as M → gL MgR (2.40) with gL and gR SU (2). .. Cancellation of anomalies 6.2 Renormalization of couplings 6.3 Breaking to SU (3) × SU (2) × U (1) 6.4 SU (2) × U (1) breaking 6.5 Charge quantization and magnetic monopoles 6.6 Proton decay 6.7... doublets of SU (2), with hypercharge 1/3: Q = (3, 2)1/3 The appropriate covariant derivative is: Dµ Q = ∂µ − igs Aaµ T a − igWµi T i − i g Bµ Q 23 (2.64) Here T i are the generators of SU (2), T i