3. Supersymmetry
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3.1 Motivations for Supersymmetry
Solution to the naturalness problem
Supersymmetry (SUSY)
– symmetry between bosons and fermions
No Quadratic Divergence in Higgs mass:
– cancellation between bosons and fermions
2
Gauge Coupling Unification
Gauge coupling constants change as energy scale changes
Minimal Supersymmetric Standard Model
Three couplings (SU(3), SU(2), U(1)) meet at one point
~1016 GeV
strength
0.12
0.06
0.1
MSSM
SM
0.05
0.04
0.08
0.03
0.06
0.02
0.04
0.01
0.02
2.5
2.5
5
7.5
10
12.5
15
5
7.5
10
12.5
15
energyscale
accidental? or suggests unification of forces!?
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Quantum Gravity
SUSY softens UV divergence of quantum gravity 
superstring theory?
Dark Matter
Lightest superparticle (LSP) is a candidate for dark
matter of the universe.
LSP: neutralino, gravitino ….
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3.2 D=4, N=1 SUSY
supersymmetry
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quick view of SUSY
• 4D N=1 supersymmetry (SUSY)
Wess-Bagger’s
text book
• Superfields on superspace
quark/lepton/Higgs  chiral superfield (multiplet)
gauge bosons
 vector superfield (multiplet)
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Superfields
• Minkowski space
• Superfields on superspace
– supersymmetry tr. = translation along 
coordinate
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8
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Chiral Superfield
• SUSY transformation
SUSY tr. of highest component  total derivative
• Counting of degrees of freedom
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Lagrangian for chiral superfield
• D-term (kinetic term)
• F-term (Yukawa int., mass etc)
– superpotential:
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• example 1
 scalar mass = spinor mass=m
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• example 2
SUSY relation of couplings
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• Vector Superfield:
Generalized gauge transformation: U(1)
Gauge invariant Lagrangian
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Wess-Zumino gauge
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• gauge kinetic term
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Minimal Supersymmetric Standard Model
(MSSM)
Chiral Multiplets
We need two Higgs multiplets for
anomaly cancellation.
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Vector Multiplets
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Superpotential
-- After EW symmetry breaking
 quark/lepton masses
-- m ~ weak scale is imposed.
Why? and How?  m problem
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Absence of Quadratic Divergence
Radiative corrections to Higgs boson mass
• schematic view at one loop
Cancellation between
boson and fermion loop
• more sophisticated and rigorous way:
non-renormalization theorem
Superpotential does not receive radiative corrections
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• another way to understand:
SUSY
fermion
boson
mf=0
mb=0
chiral
sym.
SUSY+ chiral symmetry
 small (vanishing) boson mass
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3.3. Supersymmetry Breaking
• Exact SUSY would predict
“a scalar electron which has the same mass and
charge as electron”
– Such a scalar electron is immediately ruled out.
• SUSY must be broken in some way.
– shift of coupling: quadratic div.
– shift of mass:
No effect to UV. No quadratic div.
 Take this choice!
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• Soft SUSY breaking terms:
– mass terms which do not generate quadratic
divergence
• Classification: use of spurious fields
• Superparticles (squark/slepton, gaugino) can become
heavy to escape detection.
• Origin of the spurious fields: spontaneous SUSY breaking
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Spontaneous SUSY breaking
• SUSY must be broken some way
• Probably SUSY is a fundamental symmetry of the nature,
if any.  Spontaneous SUSY breaking
Lorentz inv.
is assumed.
• origin of spurious fields
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Origin of soft SUSY breaking masses
• scalar masses
• gaugino masses
These come from Kaehler potential and gauge
kinetic function
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Three ingredients
in general SUSY theory
All interaction needed to give soft masses can be seen in
the above Lagrangian.
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SuperHiggs mechanism
• supergravity
• gravitino (spin 3/2) m
– superpartner of graviton
– gauge field associated with local supersymmetry
– gravitino is massless
• Spontaneous SUSY breaking
– Goldstino  is absorbed into the longitudinal mode of
gravitino
massive gravitino
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３.4 Mediation Mechanisms
of SUSY Breaking
Soft SUSY breaking masses should
1)
be light enough to solve the naturalness problem
associated with EW scale
--- may not be easy to quantify the statement
2) be heavy enough to escape detection at
collider experiments
3) not induce too large FCNC or CP
4) have neutral LSP (cosmology)
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SUSY flavor problem
Remember the statement:
Flavor Problem in Beyond SM
– Standard Model is too good to hide all flavor mixing
phenomena (GIM mechanism)
– Introduction of new particles/interaction may give too
large FCNCs.
This is particularly the case for SUSY:
“ SUSY flavor problem”
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New source of flavor mixing:
squark (slepton) masses
– gauge inv. mass terms
– Off-diagonal terms flavor mixing
• Experimental constraints
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Solutions to SUSY Flavor Problem
1)
degeneracy
2) alignment
squarks & quarks: simultaneous diagonalization
 family symmetry?
3)
decoupling
masses of 1st and 2nd generations~ 10-100 TeV
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Mechanisms of Mediation
The SUSY flavor problem has inspired various
mechanisms of SUSY breaking & its mediation
• gravity Mediation
– minimal supergravity
– Dilaton/moduli mediation
– gaugino mediation
• gauge mediation
• anomaly mediation
• mirage mediation (mixed moduli-anomaly medition)
•
……….
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Gravity Mediation
… a bit misleading name
• Use of non-renormalizable interaction in Kaehler
potential/gauge kinetic function
• Such interaction should always exist in supergravity
• Hidden sector (SUSY breaking sector) interacts with
visible sector (MSSM sector) via the non-renormalizable
interaction
• Scalar mass: Kaehler potential
– ~gravitino mass
– afraid of too large FCNC
• gaugino mass: Gauge kinetic function
– can be ~gravitino mass if the gauge kinetic function has nontrivial dependence on hidden sector.
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• scalar mass
• Cij should be controlled appropriately.
Otherwise scalar masses are flavor
dependent.
• How to control non-renormalizable
interaction?
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Various approaches
• minimal supergravity
– Assume
justification?
– Probably we need more fundamental theory dilaton/moduli
mediation
• Gauge mediation:
– small gravitino mass. Gravity mediation is suppressed.
– Dominant contribution from gauge interaction
• Anomaly mediation
– with sequestered sector SUSY breaking (Cij=0). maybe realized
as brane separation
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minimal supergravity (mSUGRA)
• Assume the special Kaehler potential
• mSUGRA
– universality  no dangerous FCNC
– simple, good bench mark for phenomenology
– justification of universality??
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Gauge Mediation
Messenger of SUSY breaking =SM gauge interactions
 generation universality of scalar masses
Scenario:
– messenger sector: messenger quarks/leptons
– messenger sector feels SUSY breaking
– SUSY breaking is mediated to MSSM sector through
gauge interaction
e.g. gaugino mass
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Very different phenomenolgy & cosmology
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Anomaly Meditation
Randall-Sundrum
Giudice-Luty-Murayama-Rattazzi
Mediation by superconformal anomaly
• conformal compensator:
• gauge kinetic function
• gaugino mass
one loop suppression
• Wino is lightest among gauginos
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• Scalar mass:
• sleptons: SU(2), U(1) asymptotic non-free
 negative slepton mass^2
• attempts to solve the tachyonic slepton masses
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Mirage Mediation
(mixed anomaly-moduli mediation)
Choi-Falkowski-Nilles-Olechowski ’05
Endo-MY-Yoshioka
Choi-Jeong-Okumura, …..
• Moduli mediation contribution solves the
tachyonic slepton mass problem.
• Based on KKLT-type set up (moduli stabilization
with flux and gaugino condensate)
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Set-up (in Planck unit)
superpotential:
Kaeher potential:
 supersymmetric AdS vacuum
Needs up-lifting potential to get Minkowski space
 Moduli has suppressed SUSY breaking
Moduli-mediation is comparable to anomaly-mediation.
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mass scales: little hierarchy
• soft masses
• gravitino mass
• moduli mass
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mirage mediation
Choi, Jeong, Okumura 05
RG properties: Gaugino
masses (as well as scalar
masses) are unified at a
mirage scale.
from
Lebedev, Nilles,
Ratz 05
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General Features of Mixed- Modulus-Anomaly
Mediation (or Mirage Mediation)
• Compact Sparticle Mass Spectrum
• small m parameter (~M1)
Endo-MY-Yoshioka 05
Choi-Jeong-Okumura 05
 small gluino mass/ RGE
• LSP: neutralino
– admixture of gauginos and higginos
• stau: tends to be light
• Mass Spectrum is very different from mSUGRA (CMSSM).
gauge mediation & anomaly mediation
• Testable at future collider experiments (LHC/ILC)
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Mass Spectrum: Case Study
Endo,MY,Yoshioka 05
n=1,l=1/3
n=3,l=0
(KKLT)
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