Perturbative and non-perturbative infrared behavior of ...

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CONTENTS 4.3.2 The superpotential problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3.3 The solution to the superpotential problem . . . . . . . . . . . . . . . . . . 46 4.3.4 The most general gauge invariant action . . . . . . . . . . . . . . . . . . . . 47 4.3.5 One–loop renormalizability and gauge invariance . . . . . . . . . . . . . . . 52 4.4 An explicit case: the U(1) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.1 U∗(1) NAC SYM theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.2 One flavor case: Renormalization and β–functions . . . . . . . . . . . . . . 63 4.4.3 Three–flavor case: Renormalization and β-functions . . . . . . . . . . . . . 66 4.4.4 Finiteness, fixed points and IR stability . . . . . . . . . . . . . . . . . . . . 69 4.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 II Three-dimensional field theories 79 5 N = 2 Chern-Simons matter theories 81 5.1 The ABJM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6 Quantization, fixed points and RG flows 87 6.1 Quantization of N = 2 Chern–Simons matter theories . . . . . . . . . . . . . . . . 88 6.2 Two–loop renormalization and β–functions . . . . . . . . . . . . . . . . . . . . . . 93 6.2.1 One loop results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2.2 Two-loop results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3 The spectrum of fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.1 Theories without flavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.2 Theories with flavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.4 Infrared stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4.1 Theories without flavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.4.2 Theories with flavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.5 A relevant perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 III Supersymmetry breaking 113 7 Basics and motivations 115 7.1 The supertrace theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Non-renormalizable interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.3 Mediating the supersymmetry breaking effects: an overview . . . . . . . . . . . . . 119 7.4 R-parity and R-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8 Supersymmetric QCD and Seiberg duality 123 8.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.1.1 Supersymmetric Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.1.2 The vacuum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.2 Supersymmetric QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 ii
CONTENTS 8.2.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2.2 Nf = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.2.3 Nf < Nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.2.4 Nf ≥ 3Nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.2.5 3 2 Nc < Nf < 3Nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.2.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.2.7 Nc + 2 ≤ Nf ≤ 3 2 Nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2.8 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.3 SQCD with singlets: SSQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9 Non-supersymmetric vacua 139 9.1 Generalities and basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.2 The ISS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.3 Metastable vacua in the conformal window . . . . . . . . . . . . . . . . . . . . . . 150 9.3.1 A closer look to the conformal window . . . . . . . . . . . . . . . . . . . . . 150 9.3.2 Metastable vacua by adding relevant deformations . . . . . . . . . . . . . . 152 9.3.3 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.4 Supersymmetry breaking in three dimensions . . . . . . . . . . . . . . . . . . . . . 160 9.4.1 Effective potential in 3D WZ models . . . . . . . . . . . . . . . . . . . . . . 160 9.4.2 Three dimensional WZ models with marginal couplings . . . . . . . . . . . 163 9.4.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.4.4 Relevant couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Conclusions 171 Appendices 173 A Mathematical tools 175 A.1 Group theory conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 A.2 Useful integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 A.3 Notations and conventions in three-dimensional field theories . . . . . . . . . . . . 176 B Feynman rules 179 B.1 Feynman rules for the general action (4.3.124) . . . . . . . . . . . . . . . . . . . . . 179 B.2 Feynman rules for the abelian actions (4.4.146) and (4.4.147) . . . . . . . . . . . . 185 C Details on supersymmetry breaking computations 193 C.1 The bounce action for a triangular barrier in four dimensions . . . . . . . . . . . . 193 C.2 The renormalization of the bounce action . . . . . . . . . . . . . . . . . . . . . . . 196 C.3 The bounce action for a triangular barrier in three dimensions . . . . . . . . . . . . 198 C.4 Coleman-Weinberg formula in various dimensions . . . . . . . . . . . . . . . . . . . 200 iii
Page 1: Università degli Studi di Milano-B
Page 4 and 5: Riassunto della tesi dimensionale.
Page 9 and 10: List of Figures 3.1 New vertices pr
Page 11 and 12: Introduction and outline The advent
Page 13 and 14: Introduction and outline nonanticom
Page 15: Part I Nonanticommutative theories
Page 18 and 19: 1. Basics and motivations ingredien
Page 20 and 21: 1. Basics and motivations to obtain
Page 22 and 23: 1. Basics and motivations 8
Page 24 and 25: 2. Nonanticommutative superspace wh
Page 26 and 27: 2. Nonanticommutative superspace wh
Page 28 and 29: 2. Nonanticommutative superspace op
Page 30 and 31: 3. The Wess-Zumino model We stress
Page 32 and 33: 3. The Wess-Zumino model Φ 2 Φ U
Page 34 and 35: 3. The Wess-Zumino model dim U(1)R
Page 36 and 37: 3. The Wess-Zumino model • ω2 =
Page 38 and 39: 4. Gauge theories ever, a modified
Page 40 and 41: 4. Gauge theories They can be expre
Page 42 and 43: 4. Gauge theories of the U(1) field
Page 44 and 45: 4. Gauge theories and the pure gaug
Page 46 and 47: 4. Gauge theories Since for the chi
Page 48 and 49: 4. Gauge theories Figure 4.1: Gauge
Page 50 and 51: 4. Gauge theories 4.2.2 Three-point
Page 52 and 53: 4. Gauge theories 4.2.4 (Super)gaug
Page 54 and 55: 4. Gauge theories and supergauge in
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4. Gauge theories where the trace o
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4. Gauge theories Figure 4.4: One-l
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4. Gauge theories fore, one-loop re
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4. Gauge theories α ˙α Γ dim R-
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4. Gauge theories 3. Gauge sector.
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4. Gauge theories We have introduce
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4. Gauge theories the covariant pro
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4. Gauge theories We note that the
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4. Gauge theories satisfy this set
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4. Gauge theories In order to cance
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4. Gauge theories of [22] the most
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4. Gauge theories By direct calcula
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4. Gauge theories pole coefficients
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4. Gauge theories superfields, in m
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4. Gauge theories 70 φ h ∂ Γ (a
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4. Gauge theories is perfectly cons
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4. Gauge theories The system of lin
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4. Gauge theories terms are no long
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4. Gauge theories 78
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Chapter 5 N = 2 Chern-Simons matter
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5.2. Generalizations representation
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5.2. Generalizations The superpoten
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Chapter 6 Quantization, fixed point
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6.1. Quantization of N = 2 Chern-Si
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6.1. Quantization of N = 2 Chern-Si
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6.2. Two-loop renormalization and
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6.2. Two-loop renormalization and
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In order to cancel the divergences
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6.3. The spectrum of fixed points W
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6.3. The spectrum of fixed points F
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addition of flavor matter [83]. The
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6.4. Infrared stability Figure 6.5:
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6.4. Infrared stability Figure 6.6:
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6.5. A relevant perturbation Figure
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6.5. A relevant perturbation This k
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Part III Supersymmetry breaking 113
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7. Basics and motivations where the
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7. Basics and motivations It follow
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7. Basics and motivations • the h
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7. Basics and motivations 122
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8. Supersymmetric QCD and Seiberg d
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9. Non-supersymmetric vacua If we a
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9. Non-supersymmetric vacua The Pol
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9. Non-supersymmetric vacua in the
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9. Non-supersymmetric vacua by what
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9. Non-supersymmetric vacua We saw
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9. Non-supersymmetric vacua jumps f
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9. Non-supersymmetric vacua The bou
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9. Non-supersymmetric vacua where t
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9. Non-supersymmetric vacua The phy
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9. Non-supersymmetric vacua conform
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9. Non-supersymmetric vacua the lin
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9. Non-supersymmetric vacua The one
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9. Non-supersymmetric vacua and the
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9. Non-supersymmetric vacua The con
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9. Non-supersymmetric vacua 9.4.4 R
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9. Non-supersymmetric vacua We have
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conclusions 172
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174
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A. Mathematical tools A.2 Useful in
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A. Mathematical tools For SU(N) we
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B. Feynman rules as f = ∇ 2 ∗ V
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B. Feynman rules Matter sector We n
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B. Feynman rules Since terms propor
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B. Feynman rules Φ ∇ α Φ ∇
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B. Feynman rules Here we recognize
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B. Feynman rules and contain an inf
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B. Feynman rules 192 ∂ φ Γ Γ
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C. Details on supersymmetry breakin
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BIBLIOGRAPHY [14] S. Penati and A.
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BIBLIOGRAPHY [47] M. Van Raamsdonk,
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BIBLIOGRAPHY [80] D. Gaiotto and A.
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BIBLIOGRAPHY [114] T. Banks and A.
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