# Volume III Supersymmetry Steven Weinberg

## Transcript Of Volume III Supersymmetry Steven Weinberg

The Quantum Theory of Fields

Volume II I Supersymmetry Steven Weinberg

PREFACE TO VOLUME III

xv i

NOTATION

xx

24 HISTORICAL INTRODUCTION

1

24.1 Unconventional Symmetries and 'No-Go' Theorems

1

S U(6) symmetry q Elementary no-go theorem for unconventional semi-simpl e compact Lie algebras q Role of relativity

24.2 The Birth of Supersymmetry

4

Bosonic string theory q Fermionic coordinates q Worldsheet supersymmetry q Wess-Zumino model q Precursor s

Appendix A S U(6) Symmetry of Non-Relativistic Quark Models

8

Appendix B The Coleman-Mandula Theorem

12

Problems

22

References

22

25 SUPERSYMMETRY ALGEBRAS

25

25.1 Graded Lie Algebras and Graded Parameters

25

Fermionic and bosonic generators q Super-Jacobi identity q Grassmann parameters q Structure constants from supergroup multiplication rules q Complex

conjugate s

25.2 Supersymmetry Algebras

29

Haag-Lopuszanski-Sohnius theorem q Lorentz transformation of fermionic generators q Central charges q Other bosonic symmetries q R-symmetry q Simple

and extended supersymmetry q Four-component notation q Superconforma l algebr a

25 .3 Space Inversion Properties of Supersymmetry Generators

40

Parity phases in simple supersymmetry q Fermions have imaginary parity q Parity matrices in extended supersymmetry q Dirac notatio n

25 .4 Massless Particle Supermultiplets

43

Known particles are massless for unbroken supersymmetry q Helicity raising an d lowering operators q Simple supersymmetry doublets q Squarks, sleptons, an d gauginos q Gravitino q Extended supersymmetry multiplets q Chirality proble m

for extended supersymmetry

25 .5 Massive Particle Supermultiplets

48

Raising and lowering operators for spin 3-component q General massive multiplets for simple supersymmetry q Collapsed supermultiplet q Mass bounds in

extended supersymmetry q BPS states and short supermultiplet s

Problems

53

References

54

26 SUPERSYMMETRIC FIELD THEORIES

55

26.1 Direct Construction of Field Supermultiplets

55

Construction of simplest N = 1 field multiplet q Auxiliary field q Infinitesimal supersymmetry transformation rules q Four-component notation q Wess -

Zumino supermultiplets regaine d

26.2 General Superfields

59

Superspace spinor coordinates q Supersymmetry generators as superspace differential operators q Supersymmetry transformations in superspace q General superfields q Multiplication rules q Supersymmetric differential operators in superspace q Supersymmetric actions for general superfields q Parity of component fields q Counting fermionic and bosonic component s

26.3 Chiral and Linear Superfields

68

Chirality conditions on a general superfield q Left- and right-chiral superfields q Coordinates q Differential constraints q Product rules q Supersymmetric Sterms q p -terms equivalent to D-terms q Superpotentials q Kahler potential s q Partial integration in superspace q Space inversion of chiral superfields q R-symmetry again q Linear superfield s

26 .4 Renormalizable Theories of Chiral Superfields

75

Counting powers q Kinematic Lagrangian q F -term of the superpotential q Complete Lagrangian q Elimination of auxiliary fields q On-shell superalgebra q Vacuum solutions q Masses and couplings q Wess-Zumino Lagrangian regained

26 .5 Spontaneous Supersymmetry Breaking in the Tree Approximation

83

O'Raifeartaigh mechanism q R-symmetry constraints q Flat directions q Gold-

stino

26.6 Superspace Integrals, Field Equations, and the Current Superfield

86

Berezin integration q D- and F -terms as superspace integrals q Potential su-

perfields q Superspace field equations q Conserved currents as components of

linear superfields q Conservation conditions in superspace

26.7 The Supercurrent

90

Supersymmetry current q Superspace transformations generated by the super symmetry current q Local supersymmetry transformations q Construction o f the supercurrent q Conservation of the supercurrent q Energy-momentum tensor and R-current q Scale invariance and R conservation q Non-uniqueness o f

supercurrent

26 .8 General Kahler Potentials*

10 2

Non-renormalizable non-derivative actions q D-term of Kahler potential q

Kahler metric q Lagrangian density q Non-linear r-models from spontaneou s internal symmetry breaking q Kahler manifolds q Complexified coset spaces

Appendix Majorana Spinors

10 7

Problems

11 1

References

11 2

27 SUPERSYMMETRIC GAUGE THEORIES

11 3

27.1 Gauge-Invariant Actions for Chiral Superfields

11 3

Gauge transformation of chiral superfields q Gauge superfield V q Extended

gauge invariance q Wess-Zumino gauge q Supersymmetric gauge-invariant kine-

matic terms for chiral superfield s

27 .2 Gauge-Invariant Action for Abelian Gauge Superfields

12 2

Field strength supermultiplet q Kinematic Lagrangian density for Abelian gauge supermultiplet q Fayet-Iliopoulos terms q Abelian field-strength spinor super field Wa q Left- and right-chiral parts of Wa q Wa as a superspace derivative o f V q Gauge invariance of Wa q `Bianchi ' identities in superspac e

27.3 Gauge-Invariant Action for General Gauge Superfields

127

Kinematic Lagrangian density for non-Abelian gauge supermultiplet q Non-

i Abelian field-strength spinor superfield WAa q Left- and right-chiral parts o f

WAa q 0-term q Complex coupling parameter

27 .4 Renormalizable Gauge Theories with Chiral Superfields

13 2

Supersymmetric Lagrangian density q Elimination of auxiliary fields q Conditions for unbroken supersymmetry q Counting independent conditions and field

variables q Unitarity gauge q Masses for spins 0, 1/2, and 1 q Supersymmetry current q Non-Abelian gauge theories with general Kahler potentials q Gaugin o mas s

27.5 Supersymmetry Breaking in the Tree Approximation Resumed

144

Supersymmetry breaking in supersymmetric quantum electrodynamics q General case : masses for spins 0, 1/2, and 1 q Mass sum rule q Goldstino component o f

gaugino and chiral fermion field s

27 .6 Perturbative Non-Renormalization Theorems

14 8

Non-renormalization of Wilsonian superpotential q One-loop renormalizatio n of terms quadratic in gauge superfields q Proof using holomorphy and new symmetries with external superfields q Non-renormalization of Fayet-Iliopoulo s constants A q For A = 0, supersymmetry breaking depends only on super potential q Non-renormalizable theorie s

27 .7 Soft Supersymmetry Breaking*

15 5

Limitation on supersymmetry-breaking radiative corrections q Quadratic diver-

gences in tadpole graphs

27 .8 Another Approach : Gauge-Invariant Supersymmetry Transformations 157

De Wit-Freedman transformation rules q Preserving Wess-Zumino gauge wit h combined supersymmetry and extended gauge transformation s

27 .9 Gauge Theories with Extended Supersymmetry*

160

N = 2 supersymmetry from N = 1 supersymmetry and R-symmetry q Lagrangian for N = 2 supersymmetric gauge theory q Eliminating auxiliary field s q Supersymmetry currents q Witten-Olive calculation of central charge q Nonrenormalization of masses q BPS monopoles q Adding hypermultiplets q N = 4 supersymmetry q Calculation of beta function q N = 4 theory is finite q

Montonen-Olive duality

Problems

17 5

References

17 6

28 SUPERSYMMETRIC VERSIONS OF THE STANDARD MODEL 17 9

28 .1 Superfields, Anomalies, and Conservation Laws

18 0

Quark, lepton, and gauge superfields q At least two scalar doublet superfield s q -term Yukawa couplings q Constraints from anomalies q Unsuppressed violation of baryon and lepton numbers q R-symmetry q R parity q p-term q Hierarchy problem q Sparticle masses q Cosmological constraints on lightes t

superparticl e

28.2 Supersymmetry and Strong-Electroweak Unification

18 8

Renormalization group equations for running gauge couplings q Effect of super-

symmetry on beta functions q Calculation of weak mixing angle and unificatio n mass q Just two scalar doublet superfields q Coupling at unification scale

28 .3 Where is Supersymmetry Broken?

19 2

Tree approximation supersymmetry breakdown ruled out q Hierarchy from nonperturbative effects of asymptotically free gauge couplings q Gauge and gravitational mediation of supersymmetry breaking q Estimates of supersymmetrybreaking scale q Gravitino mass q Cosmological constraint s

28 .4 The Minimal Supersymmetric Standard Model

19 8

Supersymmetry breaking by superrenormalizable terms q General Lagrangian q

Flavor changing processes q Calculation of K° H K q Degenerate squarks and

sleptons q CP violation q Calculation of quark chromoelectric dipole momen t q `Naive dimensional analysis' q Neutron electric dipole moment q Constraint s

on masses and/or phases

28 .5 The Sector of Zero Baryon and Lepton Number

20 9

D-term contribution to scalar potential q ft-term contribution to scalar potential q Soft supersymmetry breaking terms q Vacuum stability constraint on parameters q Finding a minimum of potential q By 0 q Masses of CP-odd neutral scalars q Masses of CP-even neutral scalars q Masses of charged scalar s q Bounds on masses q Radiative corrections q Conditions for electroweak symmetry breaking q Charginos and neutralinos q Lower bound on li I

28 .6 Gauge Mediation of Supersymmetry Breaking

22 0

Messenger superfields q Supersymmetry breaking in gauge supermultiplet propagators q Gaugino masses q Squark and slepton masses q Derivation from holomorphy q Radiative corrections q Numerical examples q Higgs scala r

masses q µ problem q A id and Cu parameters q Gravitino as lightest sparticle

q Next-to-lightest sparticle

28 .7 Baryon and Lepton Non-Conservation

23 5

Dimensionality five interactions q Gaugino exchange q Gluino exchange suppressed q Wino and bino exchange effects q Estimate of proton lifetime q

Favored modes of proton deca y

Problems

24 0

References

24 1

29 BEYOND PERTURBATION THEORY

24 8

29 .1 General Aspects of Supersymmetry Breaking

24 8

Finite volume q Vacuum energy and supersymmetry breaking q Partially broken extended supersymmetry? q Pairing of bosonic and fermionic states q Pairin g of vacuum and one-goldstino state q Witten index q Supersymmetry unbroken

in the Wess-Zumino model q Models with unbroken supersymmetry and zer o Witten index q Large field values q Weighted Witten indice s

29 .2 Supersymmetry Current Sum Rules

25 6

Sum rule for vacuum energy density q One-goldstino contribution 0 Th e supersymmetry-breaking parameter F q Soft goldstino amplitudes q Sum rul e for supersymmetry current-fermion spectral functions q One-goldstino contribution q Vacuum energy density in terms of and D vacuum values q Vacuum

energy sum rule for infinite volume

29.3 Non-Perturbative Corrections to the Superpotential

266

Non-perturbative effects break external field translation and R-conservation q Remaining symmetry q Example : generalized supersymmetric quantum chromodynamics q Structure of induced superpotential for CI > C2 q Stabilizing the vacuum with a bare superpotential q Vacuum moduli in generalized supersym-

metric quantum chromodynamics for N, > Nf q Induced superpotential is linear

in bare superpotential parameters for C l = C2 q One-loop renormalization of

[Wo,Wa] ,9 term for all C l , C2

29 .4 Supersymmetry Breaking in Gauge Theories

27 6

Witten index vanishes in supersymmetric quantum electrodynamics q C-weighted Witten index q Supersymmetry unbroken in supersymmetric quantum electro-

dynamics q Counting zero-energy gauge field states in supersymmetric quantum electrodynamics q Calculating Witten index for general supersymmetric pure gauge theories q Counting zero-energy gauge field states for general supersym-

metric pure gauge theories q Weyl invariance q Supersymmetry unbroken in general supersymmetric pure gauge theories q Witten index and R anomalies q

Adding chiral scalars q Model with spontaneously broken supersymmetry

29.5 The Seiberg-Witten Solution*

287

Underlying N = 2 supersymmetric Lagrangian q Vacuum modulus q Leading

non-renormalizable terms in the effective Lagrangian q Effective Lagrangian for component fields q Kahler potential and gauge coupling from a function h(I) q S U(2) R-symmetry q Prepotential q Duality transformation q h(1) translation q Z8 R-symmetry q SL(2, 7L)-symmetry q Central charge q Charge and magnetic monopole moments q Perturbative behavior for large lal q Monodromy a t infinity q Singularities from dyons q Monodromy at singularities q SeibergWitten solution q Uniqueness proof

Problems

30 5

References

30 5

30 SUPERGRAPHS

30 7

30 .1 Potential Superfields

308

Problem of chiral constraints q Corresponding problem in quantum electrodynamics q Path integrals over potential superfields

30.2 Superpropagators

31 0

A troublesome invariance q Change of variables q Defining property of super -

propagator q Analogy with quantum electrodynamics q Propagator for potential superfields q Propagator for chiral superfield s

30.3 Calculations with Supergraphs

31 3

Superspace quantum effective action q Locality in fermionic coordinates q D terms and . -terms in effective action q Counting superspace derivatives q N o

renormalization of S -term s

Problems

31 6

References

31 6

31 SUPER GRAVITY

31 8

31 .1 The Metric Superfield

31 9

Vierbein formalism q Transformation of gravitational field q Transformation o f gravitino field q Generalized transformation of metric superfield Hµ q Interactio n of HN, with supercurrent q Invariance of interaction q Generalized transformation of Hµ components q Auxiliary fields q Counting components q Interaction o f Hµ component fields q Normalization of actio n

31 .2 The Gravitational Action

326

Einstein superfield E µ q Component fields of E µ q Lagrangian for HN 0q Value o f

K q Total Lagrangian q Vacuum energy density q Minimum vacuum energy q De Sitter and anti-de Sitter spaces q Why vacuum energy is negative q Stability of flat space q Weyl transformation

31.3 The Gravitino

33 3

Irreducibility conditions on gravitino field q Gravitino propagator q Gravitino kinematic Lagrangian q Gravitino field equation q Gravitino mass from broke n

supersymmetry q Gravitino mass from s and p

31 .4 Anomaly-Mediated Supersymmetry Breaking

33 7

First-order interaction with scale non-invariance superfield X q General formul a for X q General first-order interaction q Gaugino masses q Gluino mass q B parameter q Wino and bino masses q A parameter s

31 .5 Local Supersymmetry Transformations

34 1

Wess-Zumino gauge for metric superfield q Local supersymmetry transformations q Invariance of action

31 .6 Supergravity to All Orders

343

Local supersymmetry transformation of vierbein, gravitino, and auxiliary field s q Extended spin connection q Local supersymmetry transformation of general scalar supermultiplet q Product rules for general superfields q Real matter superfields q Chiral matter superfields q Product rules for chiral superfields q

Cosmological constant and gravitino mass q Lagrangian for supergravity an d

chiral fields with general Kahler potential and superpotential q Elimination o f auxiliary fields q Kahler metric q Weyl transformation q Scalar field potential q Conditions for flat space and unbroken supersymmetry q Complete bosonic Lagrangian q Canonical normalization q Combining superpotential and Kahle r potential q No-scale models

31 .7 Gravity-Mediated Supersymmetry Breaking

35 5

Early theories with hidden sectors q Hidden sector gauge coupling strong at energy A q First version : Observable and hidden sectors q Separable bare superpotential q General potential q Terms of order K 4 A8 m44g q A estimated

as 10 11 GeV q µ- and Bµ-terms q Squark and slepton masses q Gaugin o masses q A-parameters q Second version : Observable, hidden, and modular sectors q Dynamically induced superpotential for modular superfields q Effective superpotential of observable sector q µ-term q Potential of observable sector scalars q Terms of order K8A12 m4g q Soft supersymmetry-breaking terms q A

estimated as .:s 10 13 GeV q Shifts in modular fields q Absence of C,, terms q

Squark and slepton masses q Gaugino masse s

Appendix The Vierbein Formalism

37 5

Problems

37 8

References

37 9

32 SUPERSYMMETRY ALGEBRAS IN HIGHER DIMENSIONS 38 2

32 .1 General Supersymmetry Algebras

38 2

Classification of fermionic generators q Definition of weight q Fermionic gen-

erators in fundamental spinor representation q Fermionic generators commute with Pµ q General form of anticommutation relations q Central charges q Anti-

commutation relations for odd dimensionality q Anticommutation relations fo r even dimensionality q R-symmetry groups

32.2 Massless Multiplets

39 3

Little group O(d - 2) q Definition of `spin' j q Exclusion of j > 2 q Missing fermionic generators q Number of fermionic generators < 32 q N = 1 supersymmetry for d = 11 q Three-form massless particle q Types IIA, IIB and heteroti c

supersymmetry for d = 1 0

32.3 p-Branes

39 7

New conserved tensors q Fermionic generators still in fundamental spinor repre-

Volume II I Supersymmetry Steven Weinberg

PREFACE TO VOLUME III

xv i

NOTATION

xx

24 HISTORICAL INTRODUCTION

1

24.1 Unconventional Symmetries and 'No-Go' Theorems

1

S U(6) symmetry q Elementary no-go theorem for unconventional semi-simpl e compact Lie algebras q Role of relativity

24.2 The Birth of Supersymmetry

4

Bosonic string theory q Fermionic coordinates q Worldsheet supersymmetry q Wess-Zumino model q Precursor s

Appendix A S U(6) Symmetry of Non-Relativistic Quark Models

8

Appendix B The Coleman-Mandula Theorem

12

Problems

22

References

22

25 SUPERSYMMETRY ALGEBRAS

25

25.1 Graded Lie Algebras and Graded Parameters

25

Fermionic and bosonic generators q Super-Jacobi identity q Grassmann parameters q Structure constants from supergroup multiplication rules q Complex

conjugate s

25.2 Supersymmetry Algebras

29

Haag-Lopuszanski-Sohnius theorem q Lorentz transformation of fermionic generators q Central charges q Other bosonic symmetries q R-symmetry q Simple

and extended supersymmetry q Four-component notation q Superconforma l algebr a

25 .3 Space Inversion Properties of Supersymmetry Generators

40

Parity phases in simple supersymmetry q Fermions have imaginary parity q Parity matrices in extended supersymmetry q Dirac notatio n

25 .4 Massless Particle Supermultiplets

43

Known particles are massless for unbroken supersymmetry q Helicity raising an d lowering operators q Simple supersymmetry doublets q Squarks, sleptons, an d gauginos q Gravitino q Extended supersymmetry multiplets q Chirality proble m

for extended supersymmetry

25 .5 Massive Particle Supermultiplets

48

Raising and lowering operators for spin 3-component q General massive multiplets for simple supersymmetry q Collapsed supermultiplet q Mass bounds in

extended supersymmetry q BPS states and short supermultiplet s

Problems

53

References

54

26 SUPERSYMMETRIC FIELD THEORIES

55

26.1 Direct Construction of Field Supermultiplets

55

Construction of simplest N = 1 field multiplet q Auxiliary field q Infinitesimal supersymmetry transformation rules q Four-component notation q Wess -

Zumino supermultiplets regaine d

26.2 General Superfields

59

Superspace spinor coordinates q Supersymmetry generators as superspace differential operators q Supersymmetry transformations in superspace q General superfields q Multiplication rules q Supersymmetric differential operators in superspace q Supersymmetric actions for general superfields q Parity of component fields q Counting fermionic and bosonic component s

26.3 Chiral and Linear Superfields

68

Chirality conditions on a general superfield q Left- and right-chiral superfields q Coordinates q Differential constraints q Product rules q Supersymmetric Sterms q p -terms equivalent to D-terms q Superpotentials q Kahler potential s q Partial integration in superspace q Space inversion of chiral superfields q R-symmetry again q Linear superfield s

26 .4 Renormalizable Theories of Chiral Superfields

75

Counting powers q Kinematic Lagrangian q F -term of the superpotential q Complete Lagrangian q Elimination of auxiliary fields q On-shell superalgebra q Vacuum solutions q Masses and couplings q Wess-Zumino Lagrangian regained

26 .5 Spontaneous Supersymmetry Breaking in the Tree Approximation

83

O'Raifeartaigh mechanism q R-symmetry constraints q Flat directions q Gold-

stino

26.6 Superspace Integrals, Field Equations, and the Current Superfield

86

Berezin integration q D- and F -terms as superspace integrals q Potential su-

perfields q Superspace field equations q Conserved currents as components of

linear superfields q Conservation conditions in superspace

26.7 The Supercurrent

90

Supersymmetry current q Superspace transformations generated by the super symmetry current q Local supersymmetry transformations q Construction o f the supercurrent q Conservation of the supercurrent q Energy-momentum tensor and R-current q Scale invariance and R conservation q Non-uniqueness o f

supercurrent

26 .8 General Kahler Potentials*

10 2

Non-renormalizable non-derivative actions q D-term of Kahler potential q

Kahler metric q Lagrangian density q Non-linear r-models from spontaneou s internal symmetry breaking q Kahler manifolds q Complexified coset spaces

Appendix Majorana Spinors

10 7

Problems

11 1

References

11 2

27 SUPERSYMMETRIC GAUGE THEORIES

11 3

27.1 Gauge-Invariant Actions for Chiral Superfields

11 3

Gauge transformation of chiral superfields q Gauge superfield V q Extended

gauge invariance q Wess-Zumino gauge q Supersymmetric gauge-invariant kine-

matic terms for chiral superfield s

27 .2 Gauge-Invariant Action for Abelian Gauge Superfields

12 2

Field strength supermultiplet q Kinematic Lagrangian density for Abelian gauge supermultiplet q Fayet-Iliopoulos terms q Abelian field-strength spinor super field Wa q Left- and right-chiral parts of Wa q Wa as a superspace derivative o f V q Gauge invariance of Wa q `Bianchi ' identities in superspac e

27.3 Gauge-Invariant Action for General Gauge Superfields

127

Kinematic Lagrangian density for non-Abelian gauge supermultiplet q Non-

i Abelian field-strength spinor superfield WAa q Left- and right-chiral parts o f

WAa q 0-term q Complex coupling parameter

27 .4 Renormalizable Gauge Theories with Chiral Superfields

13 2

Supersymmetric Lagrangian density q Elimination of auxiliary fields q Conditions for unbroken supersymmetry q Counting independent conditions and field

variables q Unitarity gauge q Masses for spins 0, 1/2, and 1 q Supersymmetry current q Non-Abelian gauge theories with general Kahler potentials q Gaugin o mas s

27.5 Supersymmetry Breaking in the Tree Approximation Resumed

144

Supersymmetry breaking in supersymmetric quantum electrodynamics q General case : masses for spins 0, 1/2, and 1 q Mass sum rule q Goldstino component o f

gaugino and chiral fermion field s

27 .6 Perturbative Non-Renormalization Theorems

14 8

Non-renormalization of Wilsonian superpotential q One-loop renormalizatio n of terms quadratic in gauge superfields q Proof using holomorphy and new symmetries with external superfields q Non-renormalization of Fayet-Iliopoulo s constants A q For A = 0, supersymmetry breaking depends only on super potential q Non-renormalizable theorie s

27 .7 Soft Supersymmetry Breaking*

15 5

Limitation on supersymmetry-breaking radiative corrections q Quadratic diver-

gences in tadpole graphs

27 .8 Another Approach : Gauge-Invariant Supersymmetry Transformations 157

De Wit-Freedman transformation rules q Preserving Wess-Zumino gauge wit h combined supersymmetry and extended gauge transformation s

27 .9 Gauge Theories with Extended Supersymmetry*

160

N = 2 supersymmetry from N = 1 supersymmetry and R-symmetry q Lagrangian for N = 2 supersymmetric gauge theory q Eliminating auxiliary field s q Supersymmetry currents q Witten-Olive calculation of central charge q Nonrenormalization of masses q BPS monopoles q Adding hypermultiplets q N = 4 supersymmetry q Calculation of beta function q N = 4 theory is finite q

Montonen-Olive duality

Problems

17 5

References

17 6

28 SUPERSYMMETRIC VERSIONS OF THE STANDARD MODEL 17 9

28 .1 Superfields, Anomalies, and Conservation Laws

18 0

Quark, lepton, and gauge superfields q At least two scalar doublet superfield s q -term Yukawa couplings q Constraints from anomalies q Unsuppressed violation of baryon and lepton numbers q R-symmetry q R parity q p-term q Hierarchy problem q Sparticle masses q Cosmological constraints on lightes t

superparticl e

28.2 Supersymmetry and Strong-Electroweak Unification

18 8

Renormalization group equations for running gauge couplings q Effect of super-

symmetry on beta functions q Calculation of weak mixing angle and unificatio n mass q Just two scalar doublet superfields q Coupling at unification scale

28 .3 Where is Supersymmetry Broken?

19 2

Tree approximation supersymmetry breakdown ruled out q Hierarchy from nonperturbative effects of asymptotically free gauge couplings q Gauge and gravitational mediation of supersymmetry breaking q Estimates of supersymmetrybreaking scale q Gravitino mass q Cosmological constraint s

28 .4 The Minimal Supersymmetric Standard Model

19 8

Supersymmetry breaking by superrenormalizable terms q General Lagrangian q

Flavor changing processes q Calculation of K° H K q Degenerate squarks and

sleptons q CP violation q Calculation of quark chromoelectric dipole momen t q `Naive dimensional analysis' q Neutron electric dipole moment q Constraint s

on masses and/or phases

28 .5 The Sector of Zero Baryon and Lepton Number

20 9

D-term contribution to scalar potential q ft-term contribution to scalar potential q Soft supersymmetry breaking terms q Vacuum stability constraint on parameters q Finding a minimum of potential q By 0 q Masses of CP-odd neutral scalars q Masses of CP-even neutral scalars q Masses of charged scalar s q Bounds on masses q Radiative corrections q Conditions for electroweak symmetry breaking q Charginos and neutralinos q Lower bound on li I

28 .6 Gauge Mediation of Supersymmetry Breaking

22 0

Messenger superfields q Supersymmetry breaking in gauge supermultiplet propagators q Gaugino masses q Squark and slepton masses q Derivation from holomorphy q Radiative corrections q Numerical examples q Higgs scala r

masses q µ problem q A id and Cu parameters q Gravitino as lightest sparticle

q Next-to-lightest sparticle

28 .7 Baryon and Lepton Non-Conservation

23 5

Dimensionality five interactions q Gaugino exchange q Gluino exchange suppressed q Wino and bino exchange effects q Estimate of proton lifetime q

Favored modes of proton deca y

Problems

24 0

References

24 1

29 BEYOND PERTURBATION THEORY

24 8

29 .1 General Aspects of Supersymmetry Breaking

24 8

Finite volume q Vacuum energy and supersymmetry breaking q Partially broken extended supersymmetry? q Pairing of bosonic and fermionic states q Pairin g of vacuum and one-goldstino state q Witten index q Supersymmetry unbroken

in the Wess-Zumino model q Models with unbroken supersymmetry and zer o Witten index q Large field values q Weighted Witten indice s

29 .2 Supersymmetry Current Sum Rules

25 6

Sum rule for vacuum energy density q One-goldstino contribution 0 Th e supersymmetry-breaking parameter F q Soft goldstino amplitudes q Sum rul e for supersymmetry current-fermion spectral functions q One-goldstino contribution q Vacuum energy density in terms of and D vacuum values q Vacuum

energy sum rule for infinite volume

29.3 Non-Perturbative Corrections to the Superpotential

266

Non-perturbative effects break external field translation and R-conservation q Remaining symmetry q Example : generalized supersymmetric quantum chromodynamics q Structure of induced superpotential for CI > C2 q Stabilizing the vacuum with a bare superpotential q Vacuum moduli in generalized supersym-

metric quantum chromodynamics for N, > Nf q Induced superpotential is linear

in bare superpotential parameters for C l = C2 q One-loop renormalization of

[Wo,Wa] ,9 term for all C l , C2

29 .4 Supersymmetry Breaking in Gauge Theories

27 6

Witten index vanishes in supersymmetric quantum electrodynamics q C-weighted Witten index q Supersymmetry unbroken in supersymmetric quantum electro-

dynamics q Counting zero-energy gauge field states in supersymmetric quantum electrodynamics q Calculating Witten index for general supersymmetric pure gauge theories q Counting zero-energy gauge field states for general supersym-

metric pure gauge theories q Weyl invariance q Supersymmetry unbroken in general supersymmetric pure gauge theories q Witten index and R anomalies q

Adding chiral scalars q Model with spontaneously broken supersymmetry

29.5 The Seiberg-Witten Solution*

287

Underlying N = 2 supersymmetric Lagrangian q Vacuum modulus q Leading

non-renormalizable terms in the effective Lagrangian q Effective Lagrangian for component fields q Kahler potential and gauge coupling from a function h(I) q S U(2) R-symmetry q Prepotential q Duality transformation q h(1) translation q Z8 R-symmetry q SL(2, 7L)-symmetry q Central charge q Charge and magnetic monopole moments q Perturbative behavior for large lal q Monodromy a t infinity q Singularities from dyons q Monodromy at singularities q SeibergWitten solution q Uniqueness proof

Problems

30 5

References

30 5

30 SUPERGRAPHS

30 7

30 .1 Potential Superfields

308

Problem of chiral constraints q Corresponding problem in quantum electrodynamics q Path integrals over potential superfields

30.2 Superpropagators

31 0

A troublesome invariance q Change of variables q Defining property of super -

propagator q Analogy with quantum electrodynamics q Propagator for potential superfields q Propagator for chiral superfield s

30.3 Calculations with Supergraphs

31 3

Superspace quantum effective action q Locality in fermionic coordinates q D terms and . -terms in effective action q Counting superspace derivatives q N o

renormalization of S -term s

Problems

31 6

References

31 6

31 SUPER GRAVITY

31 8

31 .1 The Metric Superfield

31 9

Vierbein formalism q Transformation of gravitational field q Transformation o f gravitino field q Generalized transformation of metric superfield Hµ q Interactio n of HN, with supercurrent q Invariance of interaction q Generalized transformation of Hµ components q Auxiliary fields q Counting components q Interaction o f Hµ component fields q Normalization of actio n

31 .2 The Gravitational Action

326

Einstein superfield E µ q Component fields of E µ q Lagrangian for HN 0q Value o f

K q Total Lagrangian q Vacuum energy density q Minimum vacuum energy q De Sitter and anti-de Sitter spaces q Why vacuum energy is negative q Stability of flat space q Weyl transformation

31.3 The Gravitino

33 3

Irreducibility conditions on gravitino field q Gravitino propagator q Gravitino kinematic Lagrangian q Gravitino field equation q Gravitino mass from broke n

supersymmetry q Gravitino mass from s and p

31 .4 Anomaly-Mediated Supersymmetry Breaking

33 7

First-order interaction with scale non-invariance superfield X q General formul a for X q General first-order interaction q Gaugino masses q Gluino mass q B parameter q Wino and bino masses q A parameter s

31 .5 Local Supersymmetry Transformations

34 1

Wess-Zumino gauge for metric superfield q Local supersymmetry transformations q Invariance of action

31 .6 Supergravity to All Orders

343

Local supersymmetry transformation of vierbein, gravitino, and auxiliary field s q Extended spin connection q Local supersymmetry transformation of general scalar supermultiplet q Product rules for general superfields q Real matter superfields q Chiral matter superfields q Product rules for chiral superfields q

Cosmological constant and gravitino mass q Lagrangian for supergravity an d

chiral fields with general Kahler potential and superpotential q Elimination o f auxiliary fields q Kahler metric q Weyl transformation q Scalar field potential q Conditions for flat space and unbroken supersymmetry q Complete bosonic Lagrangian q Canonical normalization q Combining superpotential and Kahle r potential q No-scale models

31 .7 Gravity-Mediated Supersymmetry Breaking

35 5

Early theories with hidden sectors q Hidden sector gauge coupling strong at energy A q First version : Observable and hidden sectors q Separable bare superpotential q General potential q Terms of order K 4 A8 m44g q A estimated

as 10 11 GeV q µ- and Bµ-terms q Squark and slepton masses q Gaugin o masses q A-parameters q Second version : Observable, hidden, and modular sectors q Dynamically induced superpotential for modular superfields q Effective superpotential of observable sector q µ-term q Potential of observable sector scalars q Terms of order K8A12 m4g q Soft supersymmetry-breaking terms q A

estimated as .:s 10 13 GeV q Shifts in modular fields q Absence of C,, terms q

Squark and slepton masses q Gaugino masse s

Appendix The Vierbein Formalism

37 5

Problems

37 8

References

37 9

32 SUPERSYMMETRY ALGEBRAS IN HIGHER DIMENSIONS 38 2

32 .1 General Supersymmetry Algebras

38 2

Classification of fermionic generators q Definition of weight q Fermionic gen-

erators in fundamental spinor representation q Fermionic generators commute with Pµ q General form of anticommutation relations q Central charges q Anti-

commutation relations for odd dimensionality q Anticommutation relations fo r even dimensionality q R-symmetry groups

32.2 Massless Multiplets

39 3

Little group O(d - 2) q Definition of `spin' j q Exclusion of j > 2 q Missing fermionic generators q Number of fermionic generators < 32 q N = 1 supersymmetry for d = 11 q Three-form massless particle q Types IIA, IIB and heteroti c

supersymmetry for d = 1 0

32.3 p-Branes

39 7

New conserved tensors q Fermionic generators still in fundamental spinor repre-