1. Jan KalinowskiSupersymmetry, part 1 SUSY 1 Jan Kalinowski

2. Supersymmetry, part 1 Three lectures: 1.Introduction to SUSY 2.MSSM: its structure, current status and LHC expectations 3.Exploring SUSY at a Linear Collider

3. Jan KalinowskiSupersymmetry, part 1 Outline What’s good/wrong with the Standard Model? Symmetries SUSY algebra Constructing SUSY Lagrangian

4. Jan KalinowskiSupersymmetry, part 1 J. Wess, J. Bagger, Princeton Univ Press, 1992 H. Haber, G. Kane, Phys.Rept.117 (1985) 75 S.P Martin, arXiv:hep-ph/9709356 H.K. Dreiner, H.E. Haber, S.P. Martin, arXiv:0812.1594 M.E. Peskin, arXiv:0801.1928 D. Bailin, A. Love, IoP Publishing, 1994 M. Drees, R. Godbole, P. Roy, World Scientific 2004 A.Signer, arXiv:0905.4630 and many others Disclaimer: cannot guarantee that all signs are correct Warning: be aware of many different conventions in the literature Literature

5. Jan KalinowskiSupersymmetry, part 1 Why do we believe it? Why do we not believe it?

6. Jan KalinowskiSupersymmetry, part 1  Renormalizable theory  predictive power  18 parameters (+ neutrinos): coupling constants quark and lepton masses quark mixing (+ neutrino) Z boson mass Higgs mass  for more than 20 years we try to disprove it  fits all experimental data very well up to electroweak scale ~ 200 GeV (10 –18 m)  the best theory we ever had

7. Jan KalinowskiSupersymmetry, part 1 inspite of all its successes cannot be the ultimate theory: Higgs mass unstable w.r.t. quantum corrections SM particles constitute a small part of the visible universe WMAP neutrino oscillations mater-antimater asymmetry does not contain gravity can be valid only up to a certain scale Hambye, Riesselmann

8. Jan KalinowskiSupersymmetry, part 1 Loop corrections to propagators 1. photon self-energy in QED U(1) gauge invariance  2. electron self-energy in QED Chiral symmetry in the massless limit  Mass hierarchy technically natural

9. Jan KalinowskiSupersymmetry, part 1 3. scalar self-energy Even if we tune, two loop correction will be quadratically divergent again Presence of additional heavy states can affect cancellations of quadratic divergencies  scalar mass sensitive to high scale In the past significant effort in finding possible solutions of the hierarchy problem

10. Jan KalinowskiSupersymmetry, part 1

11. Jan KalinowskiSupersymmetry, part 1 Noether theorem: continuous symmetry implies conserved quantity In quantum mechanics symmetry under space rotations and translations imply angular momentum and momentum conservation Generators satisfy Extending to Poincare we enlarge space to spacetime Poincare algebra Explicit form of generators depends on fields

12. Jan KalinowskiSupersymmetry, part 1 In 1960’ties many attempts to combine spacetime and gauge symmetries, e.g. SU(6) quark models that combined SU(3) of flavor with SU(2) of spin generators fulfill certain algebra Electroweak and strong interations described by gauge theories invariance under internal symmetries imply existence of spin 1 Gravity described by general relativity: invariance under space-time transformations -- graviton G, spin 2 Hironari Miyazawa (’68) first who considered mesons and baryons in the same multiplets Gauge symmetries

13. Jan KalinowskiSupersymmetry, part 1 However, Coleman-Mandula theorem ‘67: direct product of Poincare and internal symmetry groups Here all generators are of bosonic type (do not mix spins) and only commutators involved we have to include generators of fermionic type that transform |fermion> |boson> and allow for anticommutators Particle states numerated by eigenvalues of commuting set of observables Haag, Lopuszanski, Sohnius ’75: no direct symmetry transformation between states of integer spins {a,b}=ab+ba

14. Jan KalinowskiSupersymmetry, part 1 Gol’fand, Likhtman ’71, Volkov, Akulov ’72, Wess Zumino ‘73 Graded Lie algebra, superalgebra or Remarkably, standard QFT allows for supersymmetry without any additional assumptions transforms like a fermion

15. Jan KalinowskiSupersymmetry, part 1

16. Jan KalinowskiSupersymmetry, part 1 only one fermionic generator and its conjugate Reminder: two component Weyl spinors that transform under Lorentz where spinors transform according to Dirac spinor requires two Weyl spinors Simplest case: N=1 supersymmetry

17. Jan KalinowskiSupersymmetry, part 1 Raising and lowering indices using antisymmetric tensor We will also need Dirac matrices Variables with fermionic nature with Grassmann variables

18. Jan KalinowskiSupersymmetry, part 1 Technicalities: Product of two spinoirs is defined as For Dirac spinorsLorentz covariants in particular

19. Jan KalinowskiSupersymmetry, part 1 The Lagrangian for a free Dirac field in terms of Weyl The Lagrangian for a free Majorana field in terms of Weyl We will also use Frequently used identities:

20. Jan KalinowskiSupersymmetry, part 1 or in terms of Majorana Normalization, since Spectrum bounded from below If vacuum state is supersymmetric, i.e. then For spontaneous SUSY breaking andnon-vanishing vacuum energy Supersymmetry algebra

21. Jan KalinowskiSupersymmetry, part 1 SUSY multiplets – massless representations fermionic and bosonic states of equal mass Since Then only Equal number of bosonic and fermionic states in supermultiplet

22. Jan KalinowskiSupersymmetry, part 1 Most relevant ones for constructing realistic theory Chiral: spin 1/2 and 0 Weyl fermion complex scalar Vector: spin 1 and 1/2 vector (gauge) Weyl fermion (gaugino) Gravity: spin 2 and 3/2 graviton gravitino and CPT conjugate states Supermultiplets

23. Jan KalinowskiSupersymmetry, part 1 Reminder: when going from Galileo to Lorentz we extended 3-dim space to 4-dim spacetime When extending to SUSY it is convenient to extend spacetime to superspace with Grassmannian coordinates and introduce a concept of superfields Taylor expansion in superdimensions very easy, e.g. scalar Weyl auxiliary Superspace and superfields

24. Jan KalinowskiSupersymmetry, part 1 Derivatives with respect to Grassmann variable one has to be very careful: since Derivatives also anticommute with other Grassmann variables Integration defined as

25. Jan KalinowskiSupersymmetry, part 1 With Grassmann variables SUSY algebra can be written as like a Lie algebra with anticommuting parameters Reminder: for space-time shifts: Extend to SUSY transformations (global) using Baker-Campbell-Hausdorff i.e. under SUSY transformation non-trivial transformation of the superspace (dimensions!)

26. Jan KalinowskiSupersymmetry, part 1 In analogy to, we find a representation for generators Convenient to introduce covariant derivatives Check that satisfy SUSY algebra transform the same way under SUSY Properties:

27. Jan KalinowskiSupersymmetry, part 1 Most general superfield in terms of components (in general complex) Scalar fields Vector field Weyl spinors Not all fields mix under SUSY => reducible representation Too many components for fields with spin < or = 1 For the Minimal Supersymmertic extension of the SM enough to consider chiral superfield vector superfield note different dimensions of fields

28. Jan KalinowskiSupersymmetry, part 1 left-handed chiral superfield (LHxSF) right-handed chiral superfield (RHxSF) Invariant under SUSY transformation Since is LHxSF Expanding in terms of components: RHxSF: contains one complex scalar (sfermion), one Weyl fermion and an auxiliary field F (dimensions: ) Chiral superfields

29. Jan KalinowskiSupersymmetry, part 1 Transformation under infinitesimal SUSY transformation, component fields  boson  fermion fermion  boson F  total derivative The F term – a good candidate for a Lagrangian Product of LHxSF’s is also a LHxSF comparing with gives

30. Jan KalinowskiSupersymmetry, part 1 General superfield We need a real vector field (VSF)  impose and expand (dimensions: ) In gauge theory many components are unphysical Important: under SUSY a total derivative Vector superfields

31. Jan KalinowskiSupersymmetry, part 1 By a proper choice of gauge transformation we can go to the Wess-Zumino gauge it is not invariant under susy, but after susy transformation we can again go to the Wess-Zumino gauge Many unphysical fields have been „gauged away”

32. Jan KalinowskiSupersymmetry, part 1

33. Jan KalinowskiSupersymmetry, part 1 Supersymmetric Lagrangians F and D terms of LHxSF and VSF, respectively, transform as total derivatives Products of LHxSF are chiral superfields Products of VSF are vector superfields Use F and D terms to construct an invariant action SUSY Lagrangians

34. Jan KalinowskiSupersymmetry, part 1 Consider one LHxSF (using ) Introduce a superpotential We also need a dynamical part a D-term can be constructed out of Kaehler potential Example: Wess-Zumino model superfields

35. Jan KalinowskiSupersymmetry, part 1 Both scalar and spinor kinetic terms appear as needed. However there is no kinetic term for the auxiliary field F. F can be eliminaned from EOM Terms containing the auxiliary fields read Here superpotential as a function of a scalar field Finally Scalar and fermion of equal mass All couplings fixed by susy

36. Jan KalinowskiSupersymmetry, part 1 Generalising to more LHxSF Yukawa-type interactions couplings of equal strength Alternatively, Lagrangian can be written as kinetic part and contribution from superpotential D-terms only of the type Terms of the type forbidden – superpotential has to be holomorphic

37. Jan KalinowskiSupersymmetry, part 1 General superfield We need a real vector field (VSF)  impose and expand (dimensions: ) In gauge theory many components are unphysical Important: under SUSY a total derivative Vector superfields

38. Jan KalinowskiSupersymmetry, part 1 Remember that chiral superfield contains with complex Therefore define gauge transformation for the vector superfield where is a LHxSF with proper dimensionality Now define gauge transformation for matter LHxSF Then the gauge interaction is invariant since is also a LHxSF (for Abelian) Gauge theory: Abelian case

39. Jan KalinowskiSupersymmetry, part 1 General VSF contains a spin 1 component field Products of VSF are also VSF but do not produce a kinetic term Notice that the physical spinor can be singled out from VSF by where means evaluate at But is a spinor LHxSF since In terms of component fields – photino, photon and an auxiliary D Note that is gauge invariant, i.e. does not change under

40. Jan KalinowskiSupersymmetry, part 1 Drawing the lesson from the construction of chiral superfield theory No kinetic term for D – auxilliary field like F D field appears also in the interaction with LHxSF For Abelian gauge symmetry one can also have a Fayet-Iliopoulos term Now the auxiliary field D can be eliminated from EOM

41. Jan KalinowskiSupersymmetry, part 1 But, i.e. there are other terms In the Wess-Zumino gauge expanding Term with 1 contains kinetic terms for sfermion and fermion The other two contain interactions of fermions and sfermions with photon and photino An Abelian gauge invariant and susy lagrangian then reads

42. Jan KalinowskiSupersymmetry, part 1 The VSF must be in adjoint representation of the gauge group For matter xSF Explicitly Extending to non-Abelian case

43. Jan KalinowskiSupersymmetry, part 1 Feynman rules: relations among masses and couplings

44. Jan KalinowskiSupersymmetry, part 1 R-symmetry -- rotates superspace coordinate Define R charge Terms from Kaehler are invariant since are real For to be invariant component fields of the SF have different R charge Consider Wess-Zumino Assume as vev’s of heavy SF (spurions) For global symmetry Renormalised superpotential must be of But must be regular Only Kaehler potential gets renormalised Non-renormalisation theorem

45. Jan KalinowskiSupersymmetry, part 1 Construct Lagrangians for N=1 from chiral and vector superfields Multiplets containing fields of equal mass but differing in spin by ½ Fermion Yukawa and scalar quartic couplings from superpotential Gauge symmetries determine couplings of gauge fields  Many relations between couplings Summary on constructing SUSY Lagrangians Comment on N=2: more component fields in a hypermultiplet contains both + ½ and – ½ helicity fermions which need to transform in the same way under gauge symmetry N>1  non-chiral