Volume III Supersymmetry Steven Weinberg

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Volume III Supersymmetry Steven Weinberg

Transcript Of Volume III Supersymmetry Steven Weinberg

The Quantum Theory of Fields
Volume II I Supersymmetry Steven Weinberg


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24.1 Unconventional Symmetries and 'No-Go' Theorems


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


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


Appendix B The Coleman-Mandula Theorem








25.1 Graded Lie Algebras and Graded Parameters


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


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


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


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


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







26.1 Direct Construction of Field Supermultiplets


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


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


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


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


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


26.6 Superspace Integrals, Field Equations, and the Current Superfield


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


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


26 .8 General Kahler Potentials*

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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

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27.1 Gauge-Invariant Actions for Chiral Superfields

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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

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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


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

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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


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

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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*

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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*


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


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28 .1 Superfields, Anomalies, and Conservation Laws

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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

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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?

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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

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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

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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

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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

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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


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29 .1 General Aspects of Supersymmetry Breaking

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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

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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


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

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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*


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


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30 .1 Potential Superfields


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

30.2 Superpropagators

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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

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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


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31 .1 The Metric Superfield

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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


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

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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

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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

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

31 .6 Supergravity to All Orders


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

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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

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32 .1 General Supersymmetry Algebras

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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

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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

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