Loop calculations in the MSSM

Loop calculations in the MSSM

Corfu Summer School, workshop, 4th September 2009

Helmut Eberl

Thomas Bergauer

Aenderungen des Autors (wo im Moment mein Name steht) bitte unter View -> Header/Footer(dt: Ansicht -> Kopf / Fusszeile) aendern

H. EberlCorfu Summer School, workshop4th September 09

This talk concentrates on works done and in progress in the SUSY groupof 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

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Tree-level relations between Lagrangian parameters and physical observables are no longer valid - Lagrangian parameters depend on certain definitions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:


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

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 infinitybut 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 09 H. EberlCorfu Summer School, workshop

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:


DREG retains Lorentz invariance, the tensor integrals aresymmetric in the Lorentz indices – decomposition possible, e.g.

For the calculation of a complete Feynman amplitude in DREG, an extensionto 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.)


Dimensional reduction (DRED)

Usual integration momenta are D-dimensionalAll 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:

Does not break SUSY (at least at one-loop level)


An example:

Renormalisation


After regularisation we renormalise our process by introducing renormalisation counter terms (CTs) of the Lagrangian parameters and the fields. These CTs cancelthe UV divergences – the MSSM is a renormalisable theory – therefore we will getUV 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 divergencesMixed renormalisation

We will start with derivation of the wave function CTs and the mass CTsof sfermions, fermions and vector bosons in the on-shell scheme. The resultsfor the DRbar scheme are then simple derived by taking only the UV div. partsof 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.

Renormalisation of sfermions


The ren. selfenergy

takes the real part of the integrals,couplings are unaffected

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:


The on-shell ren. conditions are

The solutions for the CTs are

SM vector bosons


We have a decomposition into transverse and longitudinal part:

(subscript T is suppressed)

Mixing matrices in the MSSM


As an example,sfermions:


On-shell fixing of mixingmatrix, analogously donefor U, V, and N matrices(Hollik et al., H. E. et al.,Guasch, Sola et al.)

Electric charge renormalisation


Thomson limit - electron-positron-photon vertex at vanishing momentumRen. 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 = mZ

Contributions from light leptons and quarks are already absorbed [2,3]

-scheme

mZ-scheme


[1]

[1] F. Jegerlehner, NP Proc. Suppl. 131 (2004) 213

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


SPS1a point

H. E., W. Majerotto, Y. Yamada, PLB 597 (2004) 273


[1] Y. Yamada, PRD 64 (2001) 036008; [2] J. R. Espinosa and Y. Yamada, PRD 67 (2003) 036003


see Y. Yamada, PLB 530 (2002) 174; A. Freitas, D. Stoeckinger, PRD 66 (2002) 095014

SUSY Parameter Analysis project*


* 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 machinewill 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 repositorytheor. and exp. analyses, LHC+ILC tools, Les Houches Accord

Reference point SPS1a’

http//spa.desy.de/spa

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, α, mZ, as(mZ) and fermion masses

Decay widths/branching ratios and production cross section are calculated for the set of parameters specified above

Reference point SPS1a’


DRbar parameter at Q = 1 TeV

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

Stop production at ILC


Total one-loop corrrected cross sections at SPS1a’ for left- and right polarizedelectron (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

Works just finished and still in progress


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


• of several percent are obtained, mainly due to gluino contribution in selfenergy loop

• Measurement of this asymmetry at LHC possible



Phys. Rev. D79 (2009) 096005

A Program Package that calculates MSSM Higgs decays at

Full one-loop level• Motivation:

• Total one-loop amplitudes are necessary for for 12 and 23 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 Rξ-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


MSUGRA


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 Rξz, Rξw 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 09 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

Announcement


3rd HEPTOOLS Annual Meeting 2009

30th November – 1st December 2009

in Vienna

Registration already possible onlocal webpage http://www.hephy.at/heptools/

Loop calculations in the MSSM

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