Lecture 5
A Vector superfield obeys the constraint: Vector Superfields Note: still more degrees of freedom than needed for a vector boson. Some not physical! Remember Vector bosons appears in gauge theories! Supergauge invariance of superfields means many excess degrees of freedom! Can fix gauge to Wess-Zumino gauge:
Gauge boson Gaugino Auxilliary D Note: 1) Wess-Zumino gauge has only gauge boson, gaugino and auxilliary D degrees of freedom. 2) Wess-Zumino gauge does not fix the ordinary gauge freedom! 3) SUSY transforms will spoil Wess-Zumino gauge fixing constraints. Mainfest SUSY invariance lost in this gauge. 4) After each SUSY transform field dependent gauge transformation can restore us to Wess-Zumino gauge
Gauge invariant Kahler potential Lagrangian
But what about Kinetic terms for the gauge superfields?
Kinetic term of the vector superfield The vector superfield is the generalisation of a gauge potential, but from this we still need to construct a field strength term which is invariant under supergauge transformations, to form the kinetic term for the gauge field. Try
One can also show (home exercise) that, Then one can then show that, Finally note that from both the kinetic term for chiral and vector super fields the auxilliary fields F i and D a have appeared without derivatives. Are not dynamical degrees of freedom, eliminated by E-L eqns:
General SUSY invariant Lagrangian density Superpotential: Gauge invariant Kahler potential Supersmymetric field strengths
Spontaneous SUSY breaking Recall: OR
O Raighferataigh SUSY breaking: (F-term breaking example) VEV breaks SUSY
Masses Fermion masses: VEV breaks SUSY Dirac mass term coupling Goldstino: (required by Goldstone s theorem) SUSY mass relations split by F-VEV!
Note Supertrace: True generally for this type of breaking, If used directly to split SM particles from SUSY partners: Solution: SUSY breaking transmitted indirectly: Hidden sector and SUSY breaking mediation
Hidden sector and SUSY breaking mediation Transmit breaking from Hidden sector where SUSY is spontaneously broken to visible sector.
Gravity Mediation (for example)
Soft SUSY breaking Broken symmetries play a very important role in physics, e.g. Electroweak symmetry breaking and the Higgs Mechanism and can be well motivated. None of the important motivations for SUSY, Hierarchy problem (HP), gauge coupling unifincation, dark matter, require it to be an exact symmetry of nature. SUSY may be broken in such a way that the mass relations between superpartners are broken, but the relationship between couplings required to solve HP can be mainteined! It is models of supersymmetry with Soft SUSY breaking which are motivate new physics at the TeV scale and are being searched for at the LHC right now. Models of how supersymmetry is broken is also a very active area of theoretical research. More info from (where I stole some parts of the previous slides!) : http://iktp.tu-dresden.de/iktp/seminare/is2011/steveabel.pdf But here we evade the precise details of how SUSY is broken, and simply construct models without breaking prejudice, writiing down all possible ways SUSY can be softly broken. To construct such models we need:
Terms in the soft SUSY breaking Lagrangian [Shown to be soft to all orders, L. Girardello, M. Grisaru] All dimension 3 or less, ) all coefficients have mass dimension! ) relationships between dimensionless couplings maintained!