Perturbative and non-perturbative infrared behavior of ...

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4. Gauge theories We have introduced the following group tensors to take into account all possible color structures (we use the shorten notation Tr(T A ) = (A) for any group matrix) C ABC 1 = (ABC) C ABC 2 = (AB)(C) C ABC 3 = (A)(B)(C) D ABCD 1 D ABCD 3 D ABCD 5 = (ABCD) D ABCD 2 = (A)(BCD) D ABCD 4 = (AB)(CD) D ABCD 6 = (ACBD) = (C)(ABD) = (AC)(BD) D ABCD 7 = (AB)(C)(D) D ABCD 8 = (AC)(B)(D) D ABCD D 9 ABCD 10 = (A)(B)(C)(D) E ABCDE 1 E ABCDE 4 = (A)(B)(CD) = (ABCDE) E ABCDE 2 = (ABCD)(E) E ABCDE 3 = (BCDE)(A) = (ABC)(DE) E ABCDE 5 = (BCD)(AE) E ABCDE 6 E ABCDE 7 = (BCD)(A)(E) E ABCDE 8 = (AB)(CD)(E) E ABCDE 9 E ABCDE 10 = (A)(BC)(D)(E) E ABCDE 11 E ABCDE 12 G ABCDE 1 = (A)(B)(C)(D)(E) = (ABCDE) G ABCDE 2 = (AB)(C)(D)(E) = (ACBDE) G ABCDE 3 = (ABCD)(E) G ABCDE 4 = (ACBD)(E) G ABCDE 5 G ABCDE 6 = (ABC)(DE) G ABCDE 7 = (BCD)(AE) G ABCDE 8 G ABCDE 9 = (ABC)(D)(E) G ABCDE 10 = (BCD)(A)(E) G ABCDE 11 G ABCDE 12 = (AB)(CD)(E) G ABCDE 13 = (BC)(DE)(A) G ABCDE 14 G ABCDE 15 = (A)(BC)(D)(E) G ABCDE 16 G ABCDE 17 = (A)(B)(CD)(E) G ABCDE 18 H ABCD 1 H ABCD 3 H ABCD 6 = (ABCD) H ABCD 2 = (ACBD) = (AB)(C)(D)(E) = (A)(B)(C)(D)(E) = (A)(BCD) H ABCD 4 = (C)(ABD) H ABCD 5 = (AB)(CD) H ABCD 7 = (AC)(BD) = (ABC)(D)(E) = (BC)(DE)(A) = (BCDE)(A) = (AB)(CDE) = (D)(ABC) = (A)(B)(CDE) = (BC)(AD)(E) H ABCD 8 = (AB)(C)(D) H ABCD 9 = (AC)(B)(D) H ABCD 10 = (AD)(B)(C) H ABCD 11 = (A)(B)(CD) H ABCD 12 = (A)(B)(C)(D) (4.3.128) Whenever in the action the ∗–product is not explicitly indicated the products are indeed ordinary products. This happens in most terms above because of the presence of the ¯ θ 2 factor. 4.3.5 One–loop renormalizability and gauge invariance In this Section we will provide general arguments in support of the one–loop renormalizability of the action (4.3.124). 52
4.3. Renormalization with interacting matter The action (4.3.124) has been obtained by including all possible divergent structures which can appear at one–loop. Therefore, one might be tempted to conclude that it is a fortiori renormalizable. However, some of these terms need enter particular linear combinations in order to insure gauge–invariance. Such terms are identified by couplings (κ−1) and t3,t4,t5. Therefore, proving one–loop renormalizability amounts to prove that quantum corrections maintain the correct gauge–invariant combinations. In what follows we will be mainly focused on these terms and find the conditions under which gauge invariance is maintained at quantum level. In order to perform one–loop calculations we use the background–field method revised in Section 4.1 and Appendix B.1 and applied to the general action (4.3.124). In Appendix B.1 the necessary Feynman rules are collected. When drawing possible divergent diagrams we make use of the following observations: First of all, from the dimensional analysis performed in Section 4.3.4, one-loop divergences may be proportional to the non–anticommutation parameter F at most quadratically. Therefore, we do not take into account diagrams which give higher powers of F. Moreover, the structures we are mainly interested in (the ones associated to the couplings (κ − 1) and t3,t4,t5) are proportional to F αβ , so they cannot receive corrections from diagrams which contain vertices proportional to F 2 . For each supergraph we perform ∇–algebra [13, 15, 16, 17] in order to reduce it to an ordinary momentum graph and read the background structures associated to the divergent integrals. We discuss renormalizability of the different sectors, separately. Pure gauge sector In the absence of a superpotential term, the one–loop effective action for the gauge sector has been already computed in [18]. With the addition of the cubic superpotential and the related modifications of the classical action, the gauge effective action could, a priori, get corrected because of two different reasons: The modification of the chiral propagators to include different couplings for the abelian superfields which might affect the evaluation of ∆ in (4.3.86), and the presence of new mixed gauge–chiral interaction vertices from Sint in (4.3.85) as coming from S Γ and S ¯ W and the second line of (4.3.125). The former modification is harmless because of the reparametrization invariance of ∆ under the change of variables Φ A → Φ ′A ≡ (Φ a ,κ1Φ 0 ), ¯Φ A → ¯Φ ′A ≡ (¯Φ a ,κ2 ¯Φ 0 ), κ = κ1κ2 ∆ = DΦD¯Φ exp ∼ DΦ ′ D¯Φ ′ exp d 4 xd 4 θ κ − 1 Tr¯ΦΦ + N Tr¯ΦTrΦ d 4 xd 4 θ Tr¯Φ ′ Φ ′ (4.3.129) The κ–independence of ∆ can be also checked by explicit calculations, noting that in its one–loop expansion abelian superfields never enter. The other source of possible modifications for the gauge effective action is the appearance of new gauge–chiral vertices in S Γ and S ¯W , eqs. (4.3.126) and (4.3.127), and second line of (4.3.125). In any case the new vertices produce tadpole–like diagrams when contracting the matter superfields leaving gauge fields as background fields. After ∇–algebra, the tadpole provides 53
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Università degli Studi di Milano-B
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Riassunto della tesi dimensionale.
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CONTENTS 4.3.2 The superpotential p
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List of Figures 3.1 New vertices pr
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Introduction and outline The advent
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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
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Page 64 and 65: 4. Gauge theories 3. Gauge sector.
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Page 72 and 73: 4. Gauge theories satisfy this set
Page 74 and 75: 4. Gauge theories In order to cance
Page 76 and 77: 4. Gauge theories of [22] the most
Page 78 and 79: 4. Gauge theories By direct calcula
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Page 82 and 83: 4. Gauge theories superfields, in m
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Page 95 and 96: Chapter 5 N = 2 Chern-Simons matter
Page 97 and 98: 5.2. Generalizations representation
Page 99 and 100: 5.2. Generalizations The superpoten
Page 101 and 102: Chapter 6 Quantization, fixed point
Page 103 and 104: 6.1. Quantization of N = 2 Chern-Si
Page 105 and 106: 6.1. Quantization of N = 2 Chern-Si
Page 107 and 108: 6.2. Two-loop renormalization and
Page 109 and 110: 6.2. Two-loop renormalization and
Page 111 and 112: In order to cancel the divergences
Page 113 and 114: 6.3. The spectrum of fixed points W
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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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