Supersymmetry and
mechanisms of breaking it
Behnoosh Khavari Institute for research in Fundamental Sciences
1-Why Supersymmetry?
2-How to contain Supersymmetry?
3- Why and how to break Supersymmetry?
4-A Simple model for SUSY breaking
0-A review on QFT and SM of Particle physics
PLAN OF TALK
SYMMETRY IN PHYSICS
β’ SYMMETRY INVARIANCE
β’ CLASSICAL MECHANICS
β’ CLASSICAL ELECTRODYNAMICS
CLASSICAL PHYSICS
β’ πΏ =1
2ππ 2
β’ππΏ
ππ₯β
π
ππ‘
ππΏ
ππ₯ = 0
β’ ππ₯ = ππππ π‘.
β’ π₯ βΆ ππ¦ππππ π£πππππππ
β’ SYMMETRY CONSERVED QUANTITY
ELECTRODYNAMICS
β’ πΏ =β1
4πΉπΌπ½πΉπΌπ½ + π½πΌπ΄πΌ
β’ π΄πΌ β π΄πΌ+ππΌπΉ(π₯)
β’ ππΌπ½πΌ = 0
β’ π = ππππ π‘.
β’ SYMMETRY CONSERVED QUANTITY
QFT AS THE FUNDAMENTAL THEORY
β’ All fundamental interactions are described
through invariances.
β’ Lorentz invariance is necessary.
β’ Lagrangian of QFT has all these symmetries.
The standard model describes three fundamental
Interactions for elementary particles :
EM U(1) symmetry
electroWEAK
SU(2)*U(1) symmetry
STRONG SU(3) symmetry
OBSERVED ELEMENTARY PARTICLES UNTIL 2012
β’
β’
http://abyss.uoregon.edu/~js/ast123/lectures/lec07.html
SALAM-WEINBERG MODEL
β’ PREDICTION : HIGGS BOSON
β’ ANY PROBLEM?
β’ YES, The gauge hierarchy problem.
Hierarchy problem of the Higgs mass
β’ Quantum corrections in the frame of SM take the mass of Higgs to Planck scale.
Yukawa coupling π»4
Ξ2 divergence
A GOOD MOTIVATION FOR SUSY
What does mean supersymmetry(SUSY)?
β’ (N=1) type : for each particle in nature there is another one with a difference in spin by one half :
β’ π πππππππ = πππ ππ ,
β’ π πππ ππ = πππππππ .
How does SUSY solve Hierarchy problem?
β’ SUSY introduces new interactions due to new particles.
How to incorporate SUSY into QFT ?
A review on non-supersymmetric QFT
β’ Poincare symmetry (space-time and Lorentz transformation)
β’ Internal symmetry
β’ CPT symmetry
Poincare algebra
β’ ππ, ππ = 0
β’ πππ , ππ = π(πππππ β πππππ)
β’ πππ , πππ
= π ππππππ + ππππππ β ππππππ β ππππππ
Coleman-mandula theorem
β’ Limiting algebra to contain only commutation relations
Only QFT symmetries
β’ Loosing the constraint βonly commutation relationsβ
expanding algebra by
anti commutation relations
supercharge
β’ ππΌ ππππππ‘ππ (πππππππ‘ππ ππ ππππ)
ππππππ πππ‘ππ‘πππ ππ πΏπππππ‘π§ ππππ’π
β’ ππΌ, ππ½ = 0 , ππΌ, π π½ = 2(ππ)πΌπ½ ππ
β’ ππΌ, ππ = 0 ππΌ, πππ = (πππ)πΌ π½
ππ½
β’ ππΌ, π = 0
Massless case (N=1)
β’ πΞ± π, π ππππ‘πππ
= 0 π, π ππππ‘πππ
= Ξ©
β’ π, π , Q 2 π, π = π, π β1
2
How to incorporate N=1 SUSY into QFT ?
β’ Generalization:
β’ Space β superspace : π₯πβ π₯π , ππΌ , π πΌ
β’ Field(π₯π) β superfield(π₯π , ππΌ , π πΌ )
β’ ππΌ , π πΌ : fermionic coordinates
Chiral superfield
β’ π π₯π, π, π = π π₯ + 2ππ π₯ + πππΉ π₯
+ ππππ π₯ ππππ βπ
2πππππ π₯ πππ
β1
4πππππ(π₯)πππ π
β’ It describes quarks and squarks or leptons and sleptons.
How to make a supersymmetric lagrangian ?
πΏπ = 2ππ
πΏπ = 2ππΉ β 2ππππππ
πΏπΉ = π 2ππππππ
β’ β = (πβ π) π·
+ π π· πΉ + π». πΆ
β’ +1
4(ππΌππΌ) πΉ + π». πΆ.
A chiral model : Wess-Zumino
π =1
2ππ·2 +
1
3ππ·3
β’ β =1
2πππ΄πππ΄ β
1
2π2π΄2 +
1
2πππ΅πππ΅ β
1
2π2π΅2
β’ +1
2πΉ πΓ° β π πΉ β
ππ
2π΄ π΄2 + π΅2
β’ βπ2
4π΄4 + π΅4 + 2π΄2π΅2 β
π
2πΉ π΄ β ππ΅πΆ5 πΉ
= βππ2
44 . 3
π4π
2π 4
π
π2 β π2
= 3π2 π4π
2π 4
1
π2 β π2
= βππ2
2 2
π4π
2π 4 π
π2 β π2
= π2 π4π
2π 4 1
π2 β π2
= β2π2( π4π
2π 4 1
π2 β π2 + π4π
2π 4 1
π β π 2 β π2
+ π4π
2π 4 4π2 β π2
π2 β π2) ( π β π 2 β π2 )
ππππ¬ divergence ππππ¬ divergence
There remains no π¬2 divergence.
Take a look at nature !
Selectron, Squark have not been observed.
SUSY must be broken : SUSY
SUPERSYMMETRY BREAKING
HOW TO BREAK SUSY?
Spontaneous SUSY breaking :
β is supersymmetric, vacuum is not.
ππΌ 0 β 0
F-term SUSY breaking
Chiral superfield transformation:
πΏπ = 2ππ
πΏπ = 2ππΉ β 2ππππππ
πΏπΉ = π 2ππππππ
Only πΏπ β 0 respects Lorentz invariance.
πΏπ β πΉ β 0
Scalar potential
π = ππΉ π =ππ
ππ
2
πΉπβ = β
ππ π
πππ
Energy criterion
ππΌ, π π½ = 2(ππ)πΌπ½ ππ
π» = π0 =1
4π1π 1 + π 1 π1 + π2π 2 + π 2 π2
π― =π
ππ πΈππΈ π + πΈ π πΈπ + πΈππΈ π + πΈ π πΈπ π
SUSY π¬πππ = π
SUSY π¬πππ > π
A prototype model for susy π π·1, π·2, π·3 = ππ·2π·3 + π1π·1 π·3
2 β π2
βπΉ1β =
ππ
ππ1= π1 π3
2 β π2
βπΉ2β =
ππ
ππ2= ππ3
βπΉ3β =
ππ
ππ3= ππ2 + 2π1π1π3
π = πΉπ2
3
π=1
= π12 π3
2 β π2 2 + π2 π32
+ ππ2 + 2π1π1π32
π02 =
π πππππ π πππππ
πππππ π ππππ
ππ
π12
2 =π πππππ 0
0 ππππ ππ
π02 =
0 0 0 0 0 00 π 2 0 0 0 0
0 0 π 2 0 0 β2π2 π12
0 0 0 0 0 00 0 0 0 π 2 0
0 0 β2π2 π12 0 0 π 2
πΌ1 = πΌ2 = 0
πΌ3 = πΌ4 = π 2
πΌ5 = π 2 β 2π2 π12
πΌ6 = π 2 + 2π2 π12
π12
2 =
0 0 0 0 0 00 π 2 0 0 0 0
0 0 π 2 0 0 00 0 0 0 0 00 0 0 0 π 2 0
0 0 0 0 0 π 2
πΌ1 = πΌ2 = 0
πΌ3 = πΌ4 = πΌ5 = πΌ6 = π 2
π1 is the Goldstino
π3 =π3 + ππ3
2 ππ3
2 = π2 β 2π12π2 ππ3
2 = π2 + 2π12π2
SUSY breaking is apparent.
Cambridge Lectures on Supersymmetry and Extra Dimensions, Lectures by: Fernando Quevedo, Notes by: Sven Krippendorf, Oliver Schlotterer
Some details of SUSY breaking
A generic criterion for susy
π = W(Ξ¦π)
ππW Ξ¦π = 0, π = 1, β¦ π , π = 1, β¦ , π
Generically there is a solution.
ππΌ , π = ππΌ
β ππΌπ πππππππ β π ππΌ πππππππ = ππΌ πππππππ β π πΉ πππ ππ β π π΅ πππ ππ = πππ ππ β π πΉ = π π΅ + 1 π π = β1
β π π = 2 β W = Ξ¦1
2r1f π‘π = Ξ¦πΞ¦1
βrπr1
π + 1 πππ’ππ‘ππππ , π π’πππππ€ππ
Generically there is no solution.
Now consider some models! K.Intriligator and N. Seiberg,Lectures on supersymmetry Breaking arXiv:hep-ph/0702069v3
First example
W = fX , R(X) = 2
πΎ = XX V = πππ πππππ π πππ = ππππ πΎ π = ππ = π 2
SUSY IS BROKEN Goldstino
BUT R-SYMMETRY prohibits GAUGINO mass.
Break R-symmetry!
W = fX +1
2πX2 R(X) = 2
π = π + πX 2
X = βπ
π supersymmetric vacuum
π02 =
π 2 0
0 π 2 = π1
2
2
W = fX
K = XX βπ
π¬ 2 XX 2 + β― about X = 0
π = πΎXX β1 π 2 =
π 2
1 β4ππ¬ 2 XX 2 + β―
= 1 +4π
π¬ 2 XX 2 + β― π 2
A local minimum at π = 0
Mass spectrum
ππ2 =
4 π 2π
π¬ 2
Massless spinor field is the Goldstino.
Breaking R-symmetry
W = fX +1
2πX2, X = β
π
π : supersymmetric vacuum
β’ π = πΎXX β1 π + πX 2 = 1 +
4π
π¬ 2 XX 2 + β― π + πX 2
= π 2 + ππ₯π + π π₯ π +4π π 2
π¬ 2XX + β―
β X πππ‘π = βπ π¬ 2
4ππ
π βͺ 1 β π πππππ πππ π‘ππππ πππ‘π€πππ π‘π€π ππππππ. The possibility of the Long lived metastable state.
Mass terms
β’ ππ2 =
4 π 2π
π¬ 2
Massless spinor field is the Goldstino.
Problems of SM SUSY
Superpartners not observed SUSY breaking
F-term SUSY breaking R-symmetry
Explicit R-symmetry breaking Metastable state
SUMMARY
References β’ [1] P. Argyres, An Introduction to Global Supersymmetry, http://www.physics.uc.edu/
argyres/661/susy2001.pdf
β’ [2] M.E.Peskin, D.V.Schroeder, An introduction to quantum field theory, Addison-Wesley)1995(
β’ [3] F.Quevedo, Cambridge Lectures on Supersymmetry and Extra Dimensions , arXiv:1011.1491v1 [hep-th] 5 Nov 2010
β’ [4] P.B.Pal, Dirac, Majorana and Weyl fermions, arXiv: 1006.1718v2 [hep-ph]
β’ [5] M. E. Peskin, Duality in Supersymmetric Yang-Mills Theory, arXiv:hep-th/9702094v1
β’ [6] A.Bilal, Introduction to supersymmetry, arXiv:hep-th/0101055 v1
β’ [7] J. Lykken, Introduction to supersymmetry, arXiv:hep-th/9612114 v1
β’ [8] D.Griffiths, Introduction to elementary particles.
β’ [9] K.Intriligator and N. Seiberg,Lectures on supersymmetry Breaking arXiv:hep-ph/0702069v3
β’ [10] J. Terning, Modern supersymmetry: Dynamics and duality )Oxford University Press, 2006(.
β’ [11] A.Stergiou, New ways of supersymmetry breaking, (2007)
β’ [12] D.Bailin, A.Love, Supersymmetric gauge field theory and string theory , lOP Publishing Ltd )1994(
β’ [13] K. Intriligator, N. Seiberg and D. Shih, Supersymmetry Breaking, R-Symmetry Breaking and Metastable Vacua, arXiv:hep-th/0703281v1
β’ [14] T. Morii, C. S. Lim, S. N. Mukherjee, The Physics of the Standard Model and Beyond, World Scientific Publishing Co. Pte. Ltd. )2004(