1. Loop calculations in the MSSM Corfu Summer School, workshop, 4 th September 2009 Helmut Eberl

2. H. EberlCorfu Summer School, workshop4th September 09 This talk concentrates on works done and in progress in the SUSY group of the HEPHY Vienna. Regularisation and Renormalisation Linear R  gauge SPA project Running projects doing loop calculations List of relevant publication of our group Conclusions An announcement

3. Inclusion of higher orders – two complications: Tree-level relations between Lagrangian parameters and physical observables are no longer valid - Lagrangian parameters depend on certain definitions Loop diagrams can be divergent for large momenta (= small distances) - UV divergence + An example: H. EberlCorfu Summer School, workshop4th September 09 Renormalization in the MSSM* Treatment of such divergent integral: 1. Regularisation 2. Renormalisation * see W. Hollik et al., hep-ph/0204350, NPB 639 (2002) 3 (on-shell scheme) + IR divergence will not be treated in this talk

4. Regularisation Several regularisation schemes are known Cut-off scheme: Physically best motivated, Introduction of an Energy cut-off  Integral now  dependent and divergent for  to infinity but breaks Lorentz invariance! Dimensional regularisation (DREG) Analytical continuation of four-vectors (momenta and vector fields) from 4 to D dim.  = 4 – D The one-loop Feynman diagrams can be defined in terms of Passarino-Veltman Integrals: 4th September 09H. EberlCorfu Summer School, workshop

5. The UV divergence parameter is This convention is used: The scalar integrals up to four propagators in the convention of A. Denner are Two simple analytic results are: 4th September 09H. EberlCorfu Summer School, workshop

6. DREG retains Lorentz invariance, the tensor integrals are symmetric in the Lorentz indices – decomposition possible, e.g. For the calculation of a complete Feynman amplitude in DREG, an extension to D dims. of the Lorentz covariants is necessary. For arbitrary D the metric tensor obeys BUT DREG violates Supersymmetry, not applicable to MSSM calculations! (Vector fields cannot be combined with fermionic partner fields to superfield in D dims.) 4th September 09H. EberlCorfu Summer School, workshop

7. Dimensional reduction (DRED) Usual integration momenta are D-dimensional All objects which are related to vector fields are kept 4-dimensional. Therefore, two metric tensors are necessary To retain gauge invariance and field equations it must hold: 4th September 09H. EberlCorfu Summer School, workshop At one-loop level we can do a nice trick: Does not break SUSY (at least at one-loop level)

8. H. EberlCorfu Summer School, workshop4th September 09 An example:

9. Renormalisation H. EberlCorfu Summer School, workshop4th September 09 After regularisation we renormalise our process by introducing renormalisation counter terms (CTs) of the Lagrangian parameters and the fields. These CTs cancel the UV divergences – the MSSM is a renormalisable theory – therefore we will get UV finite amplitudes. We use the technique of multiplicative renormalisation – the bare parameters are split into the renormalised ones and their CTs, On-shell renormalisation – CTs have also finite parts DRbar renormalisation – CTs only UV divergences Mixed renormalisation We will start with derivation of the wave function CTs and the mass CTs of sfermions, fermions and vector bosons in the on-shell scheme. The results for the DRbar scheme are then simple derived by taking only the UV div. parts of the CTs. But for the external particles the wave function CTs remain on-shell in that scheme. Wave function CTs for internally propagating particles always drop out.

10. Renormalisation of sfermions H. EberlCorfu Summer School, workshop4th September 09 The ren. selfenergy takes the real part of the integrals, couplings are unaffected

11. Renormalisation of fermions H. EberlCorfu Summer School, workshop4th September 09 The ren. selfenergy now has a richer structure and consists of four scalar coefficients: The UV convergent coefficients again have one-loop irreducible parts and CTs:

12. H. EberlCorfu Summer School, workshop4th September 09 The on-shell ren. conditions are The solutions for the CTs are

13. SM vector bosons H. EberlCorfu Summer School, workshop4th September 09 We have a decomposition into transverse and longitudinal part: (subscript T is suppressed)

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15. Mixing matrices in the MSSM H. EberlCorfu Summer School, workshop4th September 09 As an example, sfermions:

16. H. EberlCorfu Summer School, workshop4th September 09 On-shell fixing of mixing matrix, analogously done for U, V, and N matrices (Hollik et al., H. E. et al., Guasch, Sola et al.)

17. Electric charge renormalisation H. EberlCorfu Summer School, workshop4th September 09 Thomson limit - electron-positron-photon vertex at vanishing  momentum Ren. condition: The counter term for electric charge is given by Two problems: Scale of MSSM processes > 100 GeV – far away from Thomson limit Contributions of light hadrons in - large uncertainties [1,2] [1] H. Burkhardt et al., Z. Phys.C 43 (1989) 497; [2] H. Eberl et al., NPB 625 (2002) 372; [3] Oeller et al., PRD 71 (2005) 115002 Possible solution: Input is an effective MSbar running coupling at Q = m Z Contributions from light leptons and quarks are already absorbed [2,3]  -scheme  m Z  -scheme

18. H. EberlCorfu Summer School, workshop4th September 09 [1] [1] F. Jegerlehner, NP Proc. Suppl. 131 (2004) 213

19. The G Fermi scheme H. EberlCorfu Summer School, workshop4th September 09 The Fermi constant G F = 1.16637(1) 10 -5 GeV -2 is defined by the muon life time. It is related to the fine-structure constant by

20. H. EberlCorfu Summer School, workshop4th September 09 SPS1a point H. E., W. Majerotto, Y. Yamada, PLB 597 (2004) 273

21. H. EberlCorfu Summer School, workshop4th September 09

22. H. EberlCorfu Summer School, workshop4th September 09 [1] Y. Yamada, PRD 64 (2001) 036008; [2] J. R. Espinosa and Y. Yamada, PRD 67 (2003) 036003

23. H. EberlCorfu Summer School, workshop4th September 09

24. H. EberlCorfu Summer School, workshop4th September 09 see Y. Yamada, PLB 530 (2002) 174; A. Freitas, D. Stoeckinger, PRD 66 (2002) 095014

25. SUSY Parameter Analysis project* H. EberlCorfu Summer School, workshop4th September 09 * J. A. Aguillar-Saavedra et al., EPJ C46 (2006) 43; see also J. Kalinowski, Acta Phys. Polon. B37 (2006), 1215 In order to get information on fundamental SUSY parameters and SUSY-breaking mechanism in the MSSM: observables shall be measured with high accuracy. LHC – explorer machine will see SUSY with masses at ~ 1 TeV scale, squark and gluino decays ILC - high precision machine – requires equally theor. calc. including higher orders Need of a well-defined theoretical framework: SPA convention provides a clear base for calculating masses, couplings, mixing, decay widths and production cross sections. Program repository theor. and exp. analyses, LHC+ILC tools, Les Houches Accord Reference point SPS1a’ http//spa.desy.de/spa

26. SPA convention H. EberlCorfu Summer School, workshop4th September 09  Masses of SUSY particles and Higgs bosons defined as pole masses All SUSY Lagrangian parameters are in the DRbar scheme at Q = 1TeV ✠ All elements in mass matrices, rotation matrices and corresponding mixing angles are def. DRbar at Q, except (h 0 –H 0 ) mixing angle is defined on-shell with p = m h0 SM input parameters: G Fermi, α, m Z, a s (m Z ) and fermion masses Decay widths/branching ratios and production cross section are calculated for the set of parameters specified above

27. Reference point SPS1a’ H. EberlCorfu Summer School, workshop4th September 09 DRbar parameter at Q = 1 TeV

28. Chargino- Neutralino production at ILC H. EberlCorfu Summer School, workshop4th September 09 Total one-loop corrrected cross sections [1,2] at SPS1a’. The Born cross sections (dashed lines) are shown only for two channels. [1] T.Fritzsche, W. Hollik, NP Proc. Suppl. 135 (2004) 102 [2] W. Oeller, H. E., W. Majerotto, PRD 71 (2005) 115002; PLB 590 (2004) 273

29. Stop production at ILC H. EberlCorfu Summer School, workshop4th September 09 Total one-loop corrrected cross sections at SPS1a’ for left- and right polarized electron (P(e-) = 0.8) and positron (P(e+) = 0.6) beams [1,2]. The Born cross section (dashed line) is shown for comparison. [1] K. Kovarik, H. E., W. Majerotto, C. Weber, PRD 72 (2005) 053010; PLB 591 (2004) 242 [2]A. Arhrib, W. Hollik, JHEP 0404 (2004) 073

30. Works just finished and still in progress H. EberlCorfu Summer School, workshop4th September 09

31. CP violating asymmetry in stop decay into bottom and chargino In MSSM with complex parameters, loop corrections to decay can lead to CP violating decay rate asymmetry Studied this asymmetry at full one-loop level, analyzing dependence on parameters and phases Yukawa couplings of top and bottom quark running Consider constraints (EDM, DM, ) on the parameters Diploma Thesis by S. Frank, to be published together with H. E. and W. Majerotto 4th September 09H. EberlCorfu Summer School, workshop

32. of several percent are obtained, mainly due to gluino contribution in selfenergy loop Measurement of this asymmetry at LHC possible 4th September 09H. EberlCorfu Summer School, workshop

33. H. EberlCorfu Summer School, workshop4th September 09

34. H. EberlCorfu Summer School, workshop4th September 09 Phys. Rev. D79 (2009) 096005

35. A Program Package that calculates MSSM Higgs decays at Full one-loop level Motivation : Total one-loop amplitudes are necessary for for 1  2 and 2  3 processes with resonant propagators Light SUSY particles in loops can change branching ratios The program package: All amplitudes are generated using FeynArts and FormCalc SUSY spectrum is calculated using SPHENO Implementation of R ξ -gauge The renormalization will be done in the DRbar-scheme following the SPA convention The output will be in the Les Houches Format PhD Thesis by W. Frisch, in progress 4th September 09H. EberlCorfu Summer School, workshop

36. MSUGRA 4th September 09H. EberlCorfu Summer School, workshop

37. MSUGRA 4th September 09H. EberlCorfu Summer School, workshop

38. MSUGRA 4th September 09H. EberlCorfu Summer School, workshop

39. MSUGRA 4th September 09H. EberlCorfu Summer School, workshop

40. Package: sfermion decays at full one-loop level within the MSSM goal: we will use the package to study the corrections (including EW) decay width – needed in resonant propagators package: we use FeynArts, FormCalc, LoopTools, SPheno packages renormalization in DRbar scheme implementation of linear R ξz, R ξw gauge automatic split to gluon, gluino, photon, Susy-QCD, Susy-EW, SM-EW corrections link to Mathematica for easy manipulation and plotting Corfu Summer School, workshop PhD Thesis by H. Hlucha, in progress 4th September 09H. Eberl

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42. H. EberlCorfu Summer School, workshop4th September 09

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46. Final comments and conclusions H. EberlCorfu Summer School, workshop4th September 09 common renormalisation procedure established – SPA but scheme translator still missing SUSY-QCD corr. known for all important processes many SUSY processes at full one loop level done 1 to 2, 2 to 2 processes, 2 to 3 processes started 2 to 3 processes with possibly resonant propagators, (Drees, Hollik …) – C- and D- functions with general set of complex arguments necessary leading two-loop corrections done for two-point functions no public code for SUSY processes at full one-loop level up to now

47. Announcement H. EberlCorfu Summer School, workshop4th September 09 3 rd HEPTOOLS Annual Meeting 2009 30 th November – 1 st December 2009 in Vienna Registration already possible on local webpage http://www.hephy.at/heptools/