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The Supersymmetric Standard Model� Exercise� Compute Q hepthyymmnnn The Sup ersymmetric Standard Mo del a b c Jan Louis Ilka Brunner and Stephan J Hub er a MartinLutherUniversitat Hal leWittenberg FB Physik D Hal le Germany emailjlouisphysikunihallede b Department of Physics and Astronomy Rutgers University Piscataway NJ USA emailibrunnerphysicsrutgersedu c Universitat Heidelberg Institut f ur Theoretische Physik Philosophenweg D Heidelberg Germany emailshuberthphysuniheidelbergde ABSTRACT This set of lectures introduces at an elementary level the sup ersymmetric Stan dard Mo del and discusses some of its phenomenological prop erties Lectures given by J Louis at the summer school Grund lagen und neue Methoden der theoretischen Physik Saalburg November We intend to regularly up date this review Therefore we would b e very grateful for your corrections comments and suggestions Introduction The Standard Mo del of Particle Physics is an extremely successful theory which has b een tested exp erimentally to a high level of accuracy After the discov ery of the top quark the Higgs b oson which is predicted to exist by the Standard Mo del is the only missing ingredient that has not b een directly observed yet However a number of theoretical prejudices suggest that the Standard Mo del is not the nal answer of nature but rather an eective description valid up to the weak scale of order O GeV The arbitrariness of the sp ectrum and gauge group the large number of free parameters the smallness of the weak scale compared to the Planck scale and the inability to turn on gravity suggest that at higher energies shorter distances a more fundamental theory will b e necessary to describ e nature Over the past years various extensions of the Standard Mo del such as Technicolor Grand Unied Theories Sup ersymmetry or String Theory have b een prop osed In recent years sup ersymmetric extensions of the Standard Mo del b ecame very p opular also among exp erimen talists not necessarily b ecause of their convincing solution of the ab ove problems but rather b ecause most other contenders have b een more or less ruled out by now Another reason for the p opularity of sup ersymmetric theories among the orists is the fact that the low energy limit of sup erstring theory a promising candidate for a unication of all interactions including gravity is by and large sup ersymmetric This set of lectures give an elementary introduction to the sup ersymmetric Standard Mo del Section contains some of the necessary background on generic sup ersymmetric eld theories while section develops sup ersymmetric extensions of the Standard Mo del and discusses sp ontaneous breaking of sup ersymmetry In section extensions of the Standard Mo del with softly broken sup ersymmetry are presented and some of the phenomenological prop erties are discussed Section contains a summary and our conventions which follow rather closely ref are recorded in an app endix These lectures are not meant to review the latest developments of the su p ersymmetric Standard Mo del but rather attempts to give an elementary in tro duction from a mo dern p oint of view Many excellent review articles on sup ersymmetry and the sup ersymmetric Standard Mo del do exist and have b een heavily used in these lectures In addition a collection of some of the classic pap ers concerning the sub ject can b e found in ref Introduction to Sup ersymmetry Sup ersymmetry is a symmetry b etween b osons and fermions or more precisely it is a symmetry b etween states of dierent spin For example a spin particle particle under a sup ersymmetry transformation Thus is mapp ed to a spin the generators Q Q of the sup ersymmetry transformation must transform in representations of the Lorentz group These new fermionic generators the spin form together with the fourmomentum P and the generators of the Lorentz m mn transformations M a graded Lie algebra which features in addition to commu tators also anticommutators in their dening relations The simplest N sup ersymmetry algebra reads m fQ Q g P m fQ Q g f Q Q g Q P Q P m m mn mn Q Q M mn mn Q Q M m where we used the notation and convention of ref are the Pauli matrices mn y and the are dened in the app endix The particle states in a sup ersymmetric eld theory form representations su p ermultiplets of the sup ersymmetry algebra We do not recall the entire representation theory here see for example refs but only highlight a few generic features a There is an equal number of b osonic degrees of freedom n and fermionic B degrees of freedom n in a sup ermultiplet F n n B F I y I In general it is p ossible to have N sets of sup ersymmetry generators Q Q I N _ in which case one refers to N extended sup ersymmetry Such extended sup eralgebras have b een classied by Haag Lopuszanski and Sohnius generalizing earlier work of Coleman and Mandula who showed that the p ossible b osonic symmetries of the Smatrix of a four dimensional lo cal relativistic quantum eld theory consist of the generators of the Poincare group and a nite number of generators of a compact Lie group which are Lorentz scalars In ref this theorem was generalized to also include symmetry transformations generated by fermionic op erators and all p ossible sup eralgebras were found In extensions of the Standard Mo del N extended sup ersymmetries have played no role so far since they cannot accommo date the chiral structure of the Standard Mo del b The masses of all states in a sup ermultiplet are degenerate In particular z the masses of b osons and fermions are equal m m B F c Q has mass dimension and thus the mass dimensions of the elds in a sup ermultiplet dier by The two irreducible multiplets which are imp ortant for constructing the su p ersymmetric Standard Mo del are the chiral multiplet and the vector multiplet which we discuss in turn now The chiral sup ermultiplet The chiral sup ermultiplet contains a complex scalar eld Ax of spin and and mass dimension and an mass dimension a Weyl fermion x of spin auxiliary complex scalar eld F x of spin and mass dimension Ax x F x has oshell four real b osonic degrees of freedom n and four real B fermionic degrees of freedom n in accord with The sup ersymme F try transformations act on the elds in the multiplet as follows p A p p m A F i m p m F i m where we used the conventions of ref and the app endix The parameters of the transformation are constant complex anticommuting Grassmann parameters ob eying The transformations can b e thought of as generated by the op erator Q Q with Q and Q ob eying This can b e explicitly checked by evaluating the commutators on the elds A and F z 2 This follows immediately from the fact that P is a Casimir op erator of the sup ersymmetry 2 2 mn algebra P Q P M m m Exercise Show i by using and m Exercise Evaluate the commutator using for al l three elds A and F and show that this is consistent with the results of the previous exercise The eld F has the highest mass dimension of the members of the chiral multiplet and therefore is called the highest comp onent As a consequence it cannot transform into any other eld of the multiplet but only into their deriva tives This is not only true for the chiral multiplet as can b e seen explicitly in but holds for any sup ermultiplet This fact can b e used to construct La grangian densities which transform into a total derivative under sup ersymmetry transformations leaving the corresp onding actions invariant We do not review here the metho d for systematically constructing sup ersymmetric actions which is done most eciently using a sup erspace formalism Since these lectures fo cus on the phenomenological prop erties of sup ersymmetry we refer the reader to the literature for further details and only quote the results For the chiral multiplet a sup ersymmetric and renormalizable Lagrangian is given by m m A A F F LA F i m m mAF AF Y A F A F A A where m and Y are real parameters This action has the p eculiar prop erty that no kinetic term for F app ears As a consequence the equations of motion for F are purely algebraic L L F mA Y A F mA YA F F Thus F is a nondynamical auxiliary eld which can b e eliminated from the action algebraically by using its equation of motion This yields m m LA F mA Y A i A A m m m Y A A V A A where V A A is the scalar p otential given by V A A j mA YA j m AA mY AA AA Y A A F F j L L F F As can b e seen from and after elimination of F a standard renormaliz able Lagrangian for a complex scalar A and a Weyl fermion emerges However is not the most general renormalizable Lagrangian for such elds Instead it satises the following prop erties L only dep ends on two indep endent parameters the mass parameter m and the dimensionless Yukawa coupling Y In particular the AA coupling is not controlled by an indep endent parameter as it would b e in non sup ersymmetric theories but determined by the Yukawa coupling Y x The masses for A and coincide in accord with V is p ositive semidenite V The vector sup ermultiplet The vector sup ermultiplet V contains a gauge b oson v of spin and mass m dimension a Weyl fermion called the gaugino of spin and mass dimension and a real scalar eld D of spin and mass dimension V v x x D x m Similar to the chiral multiplet
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