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

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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parameters and physical observables are no
longer valid - Lagrangian parameters depend on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
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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
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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
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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.)
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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
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An example
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Renormalisation
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.
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Renormalisation of sfermions
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Renormalisation of fermions
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The on-shell ren. conditions are
The solutions for the CTs are
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SM vector bosons
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Mixing matrices in the MSSM
As an example, sfermions
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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.)
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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
a(mZ)-scheme
Possible solution Input is an effective MSbar
running coupling at Q mZ 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
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1 F. Jegerlehner, NP Proc. Suppl. 131 (2004)
213
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The GFermi scheme
The Fermi constant GF 1.16637(1) 10-5 GeV-2 is
defined by the muon life time.
It is related to the fine-structure constant by
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SPS1a point
H. E., W. Majerotto, Y. Yamada, PLB 597 (2004)
273
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1 Y. Yamada, PRD 64 (2001) 036008 2 J. R.
Espinosa and Y. Yamada, PRD 67 (2003) 036003
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see Y. Yamada, PLB 530 (2002) 174 A. Freitas, D.
Stoeckinger, PRD 66 (2002) 095014
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SUSY Parameter Analysis project
http//spa.desy.de/spa
J. A. Aguillar-Saavedra et al., EPJ C46 (2006)
43 see also J. Kalinowski, Acta Phys. Polon. B37
(2006), 1215
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SPA convention

• 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 (h0 H0) mixing angle is
  defined on-shell with p mh0
• SM input parameters GFermi, a, mZ, as(mZ) and
  fermion masses
• Decay widths/branching ratios and production
  cross section are calculated for the set of
  parameters specified above

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Reference point SPS1a
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Chargino- Neutralino production at ILC
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
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Stop production at ILC
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 2A.
Arhrib, W. Hollik, JHEP 0404 (2004) 073
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Works just finished and still in progress
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CP violating asymmetry in stop decay intobottom
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
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• of several percent are obtained, mainly
  due to gluino contribution in selfenergy loop
• Measurement of this asymmetry at LHC possible

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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?-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
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MSUGRA
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MSUGRA
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MSUGRA
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MSUGRA
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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?z, R?w gauge
• automatic split to gluon, gluino, photon,
  Susy-QCD, Susy-EW, SM-EW corrections
• link to Mathematica for easy manipulation and
  plotting

PhD Thesis by H. Hlucha, in progress
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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

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Announcement
3rd HEPTOOLS Annual Meeting 2009
30th November 1st December 2009
in Vienna
Registration already possible on local webpage
http//www.hephy.at/heptools/