Jan Kalinowski

1. SUSY 1 Jan Kalinowski Supersymmetry, part 1

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

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

4. Literature • 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 • Signer, arXiv:0905.4630 • and many others Warning: be aware of many different conventions in the literature Disclaimer: cannot guarantee that all signs are correct Supersymmetry, part 1

5. The Standard Model Why do we believe it? Why do we not believe it? Supersymmetry, part 1

6. Why do we believe the SM? • 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 Why then we do not believe the Standard Model? Supersymmetry, part 1

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

8. The hierarchy problem 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 Supersymmetry, part 1

9. The hierarchy problem 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 We will consider supersymmetry Supersymmetry, part 1

10. Symmetries Supersymmetry, part 1

11. 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 Supersymmetry, part 1

12. Gauge symmetries 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 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 Hironari Miyazawa (’68) first who considered mesons and baryons in the same multiplets Supersymmetry, part 1

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

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

15. The SUSY algebra Supersymmetry, part 1

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

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

18. Product of two spinoirs is defined as in particular Technicalities: For Dirac spinors Lorentz covariants Supersymmetry, part 1

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

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

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

22. Supermultiplets 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 Supersymmetry, part 1

23. Superspace and superfields 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 Supersymmetry, part 1

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

25. 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) (dimensions!) using Baker-Campbell-Hausdorff i.e. under SUSY transformation non-trivial transformation of the superspace Supersymmetry, part 1

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

27. Most general superfield in terms of components (in general complex) Scalar fields Vector field Weyl spinors note different dimensions of fields • 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 Supersymmetry, part 1

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

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

30. Vector superfields 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 Supersymmetry, part 1

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

32. Constructing SUSY Lagrangians Supersymmetry, part 1

33. SUSY Lagrangians 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 Supersymmetry, part 1

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

35. 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 Supersymmetry, part 1

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

37. Vector superfields 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 Supersymmetry, part 1

38. Gauge theory: Abelian case 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 is also a LHxSF Then the gauge interaction is invariant since (for Abelian) Supersymmetry, part 1

39. 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 Supersymmetry, part 1

40. 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 Supersymmetry, part 1

41. 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 Supersymmetry, part 1

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

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

44. Non-renormalisation theorem 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 Supersymmetry, part 1

45. Summary on constructing SUSY Lagrangians 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 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 Supersymmetry, part 1