Loop calculations in the MSSM Helmut Eberl Corfu

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

This talk concentrates on works done and in progress in the SUSY group of the HEPHY Vienna. Regularisation and Renormalisation Linear Rx gauge SPA project Running projects doing loop calculations • List of relevant publication of our group Conclusions An announcement 4 th September 09 Corfu Summer School, workshop H. Eberl

Renormalization in the MSSM* 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of higher orders – two complications: Tree-level relations between Lagrangian parameters and physical observables are no longer valid - Lagrangian parameters depend on certain definitions Loop diagrams can be divergent for large momenta (= small distances) - UV divergence + An example: 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 th September 09 Corfu Summer School, workshop H. Eberl

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

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

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. ) 4 th September 09 Corfu Summer School, workshop H. Eberl

Dimensional reduction (DRED) Does not break SUSY (at least at one-loop level) 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: At one-loop level we can do a nice trick: 4 th September 09 Corfu Summer School, workshop H. Eberl

An example: 4 th September 09 Corfu Summer School, workshop H. Eberl

Renormalisation 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. 4 th September 09 Corfu Summer School, workshop H. Eberl

Renormalisation of sfermions The ren. selfenergy takes the real part of the integrals, couplings are unaffected 4 th September 09 Corfu Summer School, workshop H. Eberl

Renormalisation of fermions 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: 4 th September 09 Corfu Summer School, workshop H. Eberl

The on-shell ren. conditions are The solutions for the CTs are 4 th September 09 Corfu Summer School, workshop H. Eberl

SM vector bosons We have a decomposition into transverse and longitudinal part: (subscript T is suppressed) 4 th September 09 Corfu Summer School, workshop H. Eberl

4 th September 09 Corfu Summer School, workshop H. Eberl

Mixing matrices in the MSSM As an example, sfermions: 4 th September 09 Corfu Summer School, workshop H. Eberl

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. ) 4 th September 09 Corfu Summer School, workshop H. Eberl

Electric charge renormalisation a-scheme Thomson limit - electron-positron-photon vertex at vanishing g momentum Ren. condition: The counter term for electric charge is given by Two problems: • Scale of MSSM processes > 100 Ge. V – far away from Thomson limit • Contributions of light hadrons in - large uncertainties [1, 2] a(m. Z)-scheme Possible solution: Input is an effective MSbar running coupling at Q = m. Z Contributions from light leptons and quarks are already absorbed [2, 3] [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 4 th September 09 Corfu Summer School, workshop H. Eberl

[1] F. Jegerlehner, NP Proc. Suppl. 131 (2004) 213 4 th September 09 Corfu Summer School, workshop H. Eberl

The GFermi scheme The Fermi constant GF = 1. 16637(1) 10 -5 Ge. V-2 is defined by the muon life time. It is related to the fine-structure constant by 4 th September 09 Corfu Summer School, workshop H. Eberl

SPS 1 a point H. E. , W. Majerotto, Y. Yamada, PLB 597 (2004) 273 4 th September 09 Corfu Summer School, workshop H. Eberl

4 th September 09 Corfu Summer School, workshop H. Eberl

[1] Y. Yamada, PRD 64 (2001) 036008; [2] J. R. Espinosa and Y. Yamada, PRD 67 (2003) 036003 4 th September 09 Corfu Summer School, workshop H. Eberl

4 th September 09 Corfu Summer School, workshop H. Eberl

see Y. Yamada, PLB 530 (2002) 174; A. Freitas, D. Stoeckinger, PRD 66 (2002) 095014 4 th September 09 Corfu Summer School, workshop H. Eberl

SUSY Parameter Analysis project* http//spa. desy. de/spa 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 Te. V 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 SPS 1 a’ * J. A. Aguillar-Saavedra et al. , EPJ C 46 (2006) 43; see also J. Kalinowski, Acta Phys. Polon. B 37 (2006), 1215 4 th September 09 Corfu Summer School, workshop H. Eberl

SPA convention v Masses of SUSY particles and Higgs bosons defined as pole masses All SUSY Lagrangian parameters are in the DRbar scheme at Q = 1 Te. V ✠ 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 = mh 0 SM input parameters: GFermi, α, m. Z, as(m. Z) and fermion masses • Decay widths/branching ratios and production cross section are calculated for the set of parameters specified above 4 th September 09 Corfu Summer School, workshop H. Eberl

Reference point SPS 1 a’ DRbar parameter at Q = 1 Te. V 4 th September 09 Corfu Summer School, workshop H. Eberl

Chargino- Neutralino production at ILC Total one-loop corrrected cross sections [1, 2] at SPS 1 a’. 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 4 th September 09 Corfu Summer School, workshop H. Eberl

Stop production at ILC Total one-loop corrrected cross sections at SPS 1 a’ 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 4 th September 09 Corfu Summer School, workshop H. Eberl

Works just finished and still in progress 4 th September 09 Corfu Summer School, workshop H. Eberl

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, ) on the parameters Diploma Thesis by S. Frank, to be published together with H. E. and W. Majerotto 4 th September 09 Corfu Summer School, workshop H. Eberl

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

4 th September 09 Corfu Summer School, workshop H. Eberl

Phys. Rev. D 79 (2009) 096005 4 th September 09 Corfu Summer School, workshop H. Eberl

A Program Package that calculates MSSM Higgs decays at Full one-loop level • Motivation: • Total one-loop amplitudes are necessary 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 Feyn. Arts and Form. Calc SUSY spectrum is calculated using SPHENO Implementation of Rξ-gauge The renormalization will be done in the DRbar-scheme following the SPA convention • The output will be in the Les Houches Format Ph. D Thesis by W. Frisch, in progress 4 th September 09 Corfu Summer School, workshop H. Eberl

MSUGRA 4 th September 09 Corfu Summer School, workshop H. Eberl

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 Feyn. Arts, Form. Calc, Loop. Tools, SPheno packages renormalization in DRbar scheme implementation of linear Rξz, Rξw gauge automatic split to gluon, gluino, photon, Susy-QCD, Susy-EW, SM-EW corrections link to Mathematica for easy manipulation and plotting Ph. D Thesis by H. Hlucha, in progress 4 th September 09 Corfu Summer School, workshop H. Eberl

4 th September 09 Corfu Summer School, workshop H. Eberl

Final comments and conclusions 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 4 th September 09 Corfu Summer School, workshop H. Eberl

Announcement 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/ 4 th September 09 Corfu Summer School, workshop H. Eberl