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CPT and SUSY 1 Dezso˝ Horváth,† The Deepest Symmetries of Nature: CPT and SUSY 1 Dezso˝ Horváth,† KFKI Research Institute for Particle and Nuclear Physics, H–1525 Budapest, Hungary Institute of Nuclear Research (ATOMKI), Debrecen, Hungary Abstract. The structure of matter is related to symmetries on every level of study. CPT symmetry is one of the most important laws of field theory: it states the invariance of physical properties when one simultaneously changes the signs of the charge and of the spatial and time coordinates of particles. Although in general opinion CPT symmetry is not violated in Nature, there are theoretical attempts to develop CPT-violating models. The Antiproton Decelerator at CERN has been built to test CPT invariance. Several observations imply that there might be another deep symmetry, supersymmetry (SUSY), between basic fermions and bosons. SUSY assumes that every fermion and boson observed so far has supersymmetric partners of the opposite nature. In addition to some theoretical problems of the Standard Model of elementary particles, supersymmetry may provide solution to the constituents of the mysterious dark matter of the Universe. However, as opposed to CPT, SUSY is necessarily violated at low energies as so far none of the predicted supersymmetric partners of existing particles was observed experimentally. The LHC experiments at CERN aim to search for these particles. Keywords: CPT invariance, supersymmetry, Higgs mechanism, symmetry breaking SYMMETRIES IN PARTICLE PHYSICS Symmetries in particle physics are even more important than in chemistry or solid state physics. Just like in any theory of matter, the inner structure of the composite particles are described by symmetries, but in particle physics everything is deduced from the symmetries (or invariance properties) of the physical phenomena or from their violation: the conservation laws, the interactions and even the masses of the particles. The conservation laws are related to symmetries: the Noether theorem states that a global symmetry leads to a conserving quantity. The conservation of momentum and energy are deduced from the translational invariance of space-time: the physical laws do not depend upon where we place the zero point of our coordinate system or time measurement; and the fact that we are free to rotate the coordinate axes at any angle is the origin of angular momentum conservation. The symmetry properties of particles with half–integer spin (fermions) differ from those with integer spin (bosons). The wave function describing a system of fermions changes sign when two fermions switch quantum states whereas in the case of bosons there is no change; all other differences can be deduced from this property. 1 Invited paper presented at Workshop on Physics with Ultra Slow Antiproton Beams, RIKEN, Wako, Japan, 14-16 March 2005 TABLE 1. The elementary fermions of the Standard Model. L stands for left: it symbolizes that in the weak isospin doublets left-polarized par- ticles and right-polarized antiparticles appear, their counterparts consti- tute iso-singlet states. The apostrophes of down-type quarks denote their mixed states for the weak interaction. fermion doublets (S = 1=2) charge Q isospin I ν ν ντ 0 +1=2 leptons e µ e µ τ 1 1 2 µ ¶L µ ¶L µ ¶L ¡ ¡ = u c t +2=3 +1=2 quarks d0 s0 b0 1 3 1 2 µ ¶L µ ¶L µ ¶L ¡ = ¡ = Being an angular momentum the spin is associated with the symmetry of the rotation group and it can be described by the SU(2) group of the Special (their determinant is 1) Unitary 2 £ 2 matrices. When we increase the degrees of freedom, we get higher symmetry groups of similar properties. The next one, SU(3), which we shall use later, is the symmetry group of Special Unitary 3£3 matrices. It can be visualized the following way. A half–spin particle has two possible fundamental states (two eigenstates), spin up and spin down. In the case of the SU(3) symmetry group there are three eigenstates with an SU(2) symmetry between any two of them. We can also decrease the degrees of freedom and we get the U(1) group of unitary 1 £ 1 matrices which are simply complex numbers of unit absolute value. This is the symmetry group of the gauge transformations of electromagnetism. This gauge symmetry means, e.g., in the case of electricity a free choice of potential zero: as shown by the sparrows sitting on electric wires the potential difference is the meaningful physical quantity, not the potential itself. The U(1) symmetry of Maxwell’s equations leads to the conservation of the electric charge, and, in the more general case, the U(1) symmetry of the Dirac equation, the general equation describing the movement of a fermion, causes the conservation of the number of fermions [1]. The important role of symmetries in particle physics is well expressed by the title of the popular scientific journal of SLAC and FERMILAB: symmetry — dimensions of particle physics. SYMMETRIES IN THE STANDARD MODEL According to the Standard Model of elementary particles the visible matter of our world consists of a few point-like elementary particles: fermions, quarks and leptons, and bosons (see Table 1). The hadrons, the mesons and the baryons, are composed of quarks, the mesons are quark-antiquark, the baryons three-quark states. The three basic interactions are deduced from local gauge symmetries. By requir- ing that the Dirac equation of a free fermion were invariant under local (i.e. space-time FIGURE 1. The glory road of SM at LEP: the relative deviation of the measured quantities from the predictions of the Standard Model [3] (status of Winter 2005). At present the most deviating quantity is the forward-backward asymmetry at the decay of Z bosons to b hadrons. For the definitions see [4]. dependent) U(1) ­ SU(2) transformations one gets the Lagrange function of the elec- troweak interaction with massless mediating gauge bosons (photon γ and weak bosons Z and W§). Adding local SU(3) results in the strong interaction (quantum chromodynam- ics) with 8 massless gluons as mediating particles. And, finally, adding a two-component complex Higgs-field with its 4 degrees of freedom, which breaks the SU(2) symmetry will put everything in place: produces masses for the weak bosons (and for the fermions as well) and creates the Higgs boson [5], the scalar particle badly needed to make the theory renormalizable (to remove divergences). The Standard Model is an incredible success: its predictions are not contradicted by experiment, any deviation encountered in the last 30 years disappeared with the increasing precision of theory and experiment. For a complete comparison one should consult the tables and reviews of the Particle Data Group [4], Fig 1 presents a brief view. The only missing piece is the Higgs boson; however, it is a strong indirect evidence for its existence that the goodness-of-fitting of the electroweak parameters shows a 6 Theory uncertainty ∆α(5) had = 5 0.02761±0.00036 0.02749±0.00012 4 incl. low Q2 data 2 3 ∆χ 2 1 Excluded 0 30 100 500 [ ] mH GeV FIGURE 2. The goodness-of-fitting of the electroweak parameters as a function of the mass of the Standard Model Higgs boson [3] (status of Winter 2005). The best fit indicates a light Higgs boson and the LEP searches excluded Higgs masses up to 114.4 GeV [6]. definite minimum at light Higgs masses (Fig. 2). The direct searches at LEP excluded the Standard Model Higgs boson up to masses of 114.4 GeV (with a confidence limit of 95%), whereas the fitting seems to limit it from above as well. Thus within the framework of the Standard Model the mass of the Higgs boson should be in the interval 114:4 < MH < 260 GeV (with a 95 % confidence). ANTIPARTICLES AND CPT INVARIANCE All fermions have antiparticles, anti-fermions which have identical properties but with opposite charges. The different abundance of particles and antiparticles in our Universe is one of the mysteries of astrophysics: apparently there is no antimatter in the Universe in significant quantities, see, e.g., [2]. If there were antimatter galaxies they would radiate antiparticles and we would see zones of strong radiation at their borders with matter galaxies, but the astronomers do not see such a phenomenon anywhere. An extremely interesting property of antiparticles is that they can be treated mathe- matically as if they were particles of the same mass and of oppositely signed charge of the same absolute value going backward in space and time. This is the consequence of one of the most important symmetries of Nature: CPT invariance [1, 4]. It states that the following operations: • charge conjugation (i.e. changing particles into antiparticles), Cψ(r;t) = ψ(r;t); • parity change (i.e. mirror reflection), Pψ(r;t) = ψ(¡r;t), and • time reversal, Tψ(r;t) = ψ(r;¡t) when done together do not change the physical properties (i.e., the wave function or in the language of field theory the field function ψ(r;t)) of the system: CPTψ(r;t) = ψ(¡r;¡t) » ψ(r;t): (1) This means that, e.g., the annihilation of a positron with an electron can be described as if an electron came to the point of collision, irradiated two or three photons and then went out backwards in space-time. If we build a clock looking at its design in a mirror, it should work properly except that its hands will rotate the opposite way and the lettering will be inverted. The laws governing the work of the clock are invariant under space inversion, i.e. conserve parity. As we know, the weak interaction violates parity conservation, unlike the other interac- tions.
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