1. 3. Supersymmetry

2. 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

3. 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
accidental? or suggests unification of forces!?
2.5 5
7.5 10
12.5 15
energy
scale
0.02
0.04
0.06
0.08
0.1
0.12
strength
MSSM SM

4. 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 ….

5. 3.2 D=4, N=1 SUSY
supersymmetry

6. quick view of SUSY Wess-Bagger’s text book 4D N=1 supersymmetry (SUSY)
Superfields on superspace
quark/lepton/Higgs  chiral superfield (multiplet)
gauge bosons  vector superfield (multiplet)

7. Superfields on superspace
Minkowski space
Superfields on superspace
supersymmetry tr. = translation along  coordinate

10. Chiral Superfield SUSY transformation
SUSY tr. of highest component  total derivative
Counting of degrees of freedom

11. Lagrangian for chiral superfield
D-term (kinetic term)
F-term (Yukawa int., mass etc)
superpotential:

12. example　1
 scalar mass = spinor mass=m

13. example 2
SUSY relation of couplings

14. Vector Superfield: Generalized gauge transformation: U(1)
Gauge invariant Lagrangian

15. Wess-Zumino gauge

16. gauge kinetic term

17. Minimal Supersymmetric Standard Model (MSSM)
Chiral Multiplets
We need two Higgs multiplets for
anomaly cancellation.

18. Vector Multiplets

19. Superpotential -- After EW symmetry breaking  quark/lepton masses
-- m ~ weak scale is imposed.
Why? and How?  m problem

20. Absence of Quadratic Divergence
Radiative corrections to Higgs boson mass
schematic view at one loop
more sophisticated and rigorous way:
non-renormalization theorem
Superpotential does not receive radiative corrections
Cancellation between
boson and fermion loop

21. another way to understand:
SUSY
fermion
boson
chiral
sym.
mf=0
mb=0
SUSY+ chiral symmetry
 small (vanishing) boson mass

22. 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!

23. 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

24. Spontaneous SUSY breaking
SUSY must be broken some way
Probably SUSY is a fundamental symmetry of the nature, if any.  Spontaneous SUSY breaking
origin of spurious fields
Lorentz inv.
is assumed.

25. Origin of soft SUSY breaking masses
scalar masses
gaugino masses
These come from Kaehler potential and gauge kinetic function

26. Three ingredients in general SUSY theory
All interaction needed to give soft masses can be seen in
the above Lagrangian.

27. SuperHiggs mechanism supergravity gravitino (spin 3/2) 
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

28. ３.4 Mediation Mechanisms of SUSY Breaking
Soft SUSY breaking masses should
be light enough to solve the naturalness problem associated with EW scale may not be easy to quantify the statement
be heavy enough to escape detection at collider experiments
not induce too large FCNC or CP
have neutral LSP (cosmology)

29. 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”

30. New source of flavor mixing:
squark (slepton) masses
gauge inv. mass terms
Off-diagonal terms flavor mixing
Experimental constraints

31. Solutions to SUSY Flavor Problem
degeneracy
2) alignment
squarks & quarks: simultaneous diagonalization
 family symmetry?
decoupling
masses of 1st and 2nd generations~ TeV

32. 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)
……….

33. 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 non-trivial dependence on hidden sector.

34. scalar mass
Cij should be controlled appropriately. Otherwise scalar masses are flavor dependent.
How to control non-renormalizable interaction?

35. Various approaches minimal supergravity Gauge mediation:
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

36. minimal supergravity (mSUGRA)
Assume the special Kaehler potential
mSUGRA
universality  no dangerous FCNC
simple, good bench mark for phenomenology
justification of universality??

37. 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

38. Very different phenomenolgy & cosmology

39. Anomaly Meditation Mediation by superconformal anomaly
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

40. Scalar mass:
sleptons: SU(2), U(1) asymptotic non-free
 negative slepton mass^2
attempts to solve the tachyonic slepton masses

41. 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)

42. 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.

43. mass scales: little hierarchy
soft masses
gravitino mass
moduli mass

44. 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

45. General Features of Mixed- Modulus-Anomaly Mediation (or Mirage Mediation)
Endo-MY-Yoshioka 05
Choi-Jeong-Okumura 05
Compact Sparticle Mass Spectrum
small m parameter (~M1)
 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)

46. Mass Spectrum: Case Study
Endo,MY,Yoshioka 05
n=1,l=1/3
n=3,l=0
(KKLT)