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The Standard Model Lagrangian The Standard Model Lagrangian Elementary Particle Physics Strong Interaction Fenomenology Diego Bettoni Academic Year 2011-12 Dirac Formalism i m 0 j Conserved Current 0 0 1 1 0 i i 0 5 i 0 1 0 0 1 R L i m L D. Bettoni Fenomenologia Interazioni Forti 2 Gauge Invariance U i 1 1 igA A U U UA U Abelian D g D U D 1 W W D ig2 W i i ijk j k Non 2 g2 Abelian Gauge invariance requires the introduction of vector bosons, which act as quanta of new interactions. In gauge theories the symmetries prescribe the interactions. D. Bettoni Fenomenologia Interazioni Forti 3 The Symmetries of the Standard Model • U(1) invariance. All particles appear to have this kind of invariance, related to electromagnetism. It requires a vector boson, B, whose connection with the photon will be determined later. • SU(2) invariance. Non abelian gauge invariance (electroweak isospin). It requires three vector bosons, Wi , one for each generator of SU(2). The physical W particles have definite electromagnetic charges. 1 2 1 2 0 3 W W iW 2 W W iW 2 W W • SU(3) invariance. It requires eight vector bosons, Ga , the gluons, the quanta of the strong interaction, described by Quantum ChromoDynamics (QCD). D. Bettoni Fenomenologia Interazioni Forti 4 The Lagrangian • In order to obtain the Standard Model Lagrangian we start from the free particle Lagrangian and replace the ordinary derivative by the convariant derivative. It will contain two parts: Lgauge kinetic energies of the gauge fields Lferm covariant derivative fermion kinetic energies • Next we must specify the particles and their transformation properties under the three internal symmetries. • Notation D. Bettoni Fenomenologia Interazioni Forti 5 Leptons e P e P R R e L L e Left-handed and right-handed particles behave differently under electroweak SU(2) transformations: the electrons R are SU(2) singlets, whereas the electrons L are put in doublets together with the L neutrinos. e eR SU(2) singlet L SU(2) doublet e L • Rotations in electroweak SU(2) turn L electrons in L neutrinos and vv. • Ordinary spin: raising and lowering operators (vectors). • Strong isospin: pions (vectors). • Weak isospin: the W bosons connect the members of an electroweak doublet • eR is not connected to any other state by electroweak transitions. - • p,q,r=1,2 e.g.: Lp L1=eL, L2=e L. D. Bettoni Fenomenologia Interazioni Forti 6 Quarks u QL d R ,uR d L • the index describes how the quark transforms under color SU(3). • The basic representation is a triplet: ,, = 1,2,3 or r,g,b. • Color (e.g. r) and anticolor (e.g. r ). Singlet (rr+ gg+ bb) • All leptons are color singlets. • All quarks are color triplets. • The gluons generate the transitions from one color to another: they are the quanta of the strong interaction, but unlike photons they carry color charge. • There are eight “bi-colored” gluons (e.g. bg): octet representation of color SU(3). D. Bettoni Fenomenologia Interazioni Forti 7 • In the Standard Model there are no R neutrinos: – Experimentally ony L are observed. – Neutrino masses are very small (but non-zero, oscillations). – If right-handed neutrinos R exist either they are very heavy or they interact very weakly. • R and L fermions were put in different electroweak SU(2) multiplets: that implies a violation of parity, since clearly the theory is not invariant under the reversal of the component of spin in the direction of motion. This is the way in which parity violation is described in the standard model, but it is not explained in a fundamental sense. • The same theory can be applied to the two other fermion families: (,, c,s) and ( ,,t,b). – The universe consists of fermions from the first generation. – The other families are produced in high-energy cosmic ray collisions and in particle accelerators. – No reason has been found for the existence of three families of particles with identical quantum numbers and interactions. D. Bettoni Fenomenologia Interazioni Forti 8 The Quark and Lepton Lagrangian D i a Y i a D ig B ig W ig G 1 2 2 2 3 2 • B is the spin-one field needed to maintain the U(1) gauge invariance. g1 is the coupling strength (to be measured experimentally. Y is the generator of U(1), transformations, a constant, but in principle different for the different fermions. • Analogous remarks describe the SU(2) and SU(3) terms. We introduce 3 and, respectively, 8 vector bosons which are needed to maintain the local i i 1 1 2 2 3 3 gauge invariance. W W W W •D gives a zero result when it acts on a term of different matrix form. For i i example W is a 22 matrix in SU(2) and it gives zero acting on eR, uR ,dR. Lferm f D f f L,eR ,QL ,uR ,d R f D. Bettoni Fenomenologia Interazioni Forti 9 Gauging the Global Symmetries Dirac kinetic energy Lagrangian for the first generation: L eR eR eL eL L L We put eL and L in a doublet, eR in a singlet. L ei 2 L Global SU(2) symmetry eR eR L ei L Global U(1) symmetry i eR e eR . We make these symmetries locals by introducing potentials Wi and B and by replacing with the covariant derivative D . Thus we obtain the same result. Some attempts to extend the Standard Model proceed along these lines, by adding particles and symmetries and then gauging the symmetries. D. Bettoni Fenomenologia Interazioni Forti 10 The Electroweak Lagrangian • Since the term is always present we will not write it. • All the calculations in SU(2) will be done only for the leptons. • Since the color labels of the quarks do not operate in the U(1) or SU(2) space, quarks will behave the same way as leptons for U(1) and SU(2) interactions. D. Bettoni Fenomenologia Interazioni Forti 11 The U(1) Terms YL YR Lferm U 1 ,leptoni Li ig1 B L eRi ig1 B eR 2 2 L L L L eL eL g L U 1 ,leptoni 1 Y e e Y e e B ferm 2 L L L L L R R R D. Bettoni Fenomenologia Interazioni Forti 12 The SU(2) Terms i SU2 ,leptoni Li ig W i L Lferm 2 2 g W 3 W 1 iW2 2 e L L L 1 2 3 2 W iW W eL g W 0 2W 2 e L L L 0 2 2W W eL g W 0 2W e 2 e L L L L 0 2 2W L W eL g 2 W 0 2 e W 2e W e e W 0 2 L L L L L L L L D. Bettoni Fenomenologia Interazioni Forti 13 The Neutral Current Electromagnetic interaction of particles of charge Q: LEM QA eL eL eR eR g1 g2 0 There are terms involving neutrinos YL B W L L 2 2 We assume the the electromagnetic field A is the orthogonal combination: 0 0 A g2 B g1YLW Z g1YL B g2W 0 0 g2 B g1YLW g1YL B g2W A Z g 2 g 2Y 2 g 2 g 2Y 2 2 1 L 2 1 L D. Bettoni Fenomenologia Interazioni Forti 14 g1 g2 0 g1 Terms involving electrons: eL eL YL B W eR eR YR B 2 2 2 g A g Y Z 0 g1YL A g2Z B 2 1 L W 2 2 2 g 2 g 2Y 2 g g Y 2 1 L 2 1 L g1g2YL g1g2YR A eL eL eR eR g 2 g 2Y 2 2 g 2 g 2Y 2 2 1 L 2 1 L 2 2 2 2 g1 YL g2 g1 YRYL Z eL eL eR eR 2 g 2 g 2Y 2 2 g 2 g 2Y 2 2 1 L 2 1 L The term in A must be the usual electromagnetic current. The term in Z can be an additional interaction, to be checked experimentally. D. Bettoni Fenomenologia Interazioni Forti 15 g g Y g g Y e 1 2 L e 1 2 R g 2 g 2Y 2 2 g 2 g 2Y 2 2 1 L 2 1 L YR 2YL We can choose YL=-1, since any g 2 g 2Y 2 2 1 L change in YL can be absorbed by YL e a redefinition of g . g1g2 1 g g Y 1 e 1 2 L 2 2 g2 g1 The theory we have been writing can be interpreted to contain the usual electromagnetic interaction, plus an additional neutral current interaction with Z for both electrons and neutrinos. D. Bettoni Fenomenologia Interazioni Forti 16 g1 Define: sinW g 2 g 2 2 1 W weak mixing angle g (Weinberg angle) cos 2 W 2 2 g2 g1 e g1 cosW e g2 sinW 2 g1 and g2 are written in terms of the known e (e /41/137) and the electroweak mixing angle, which needs to be measured or calculated some other way. 2 sin W 0.23 D. Bettoni Fenomenologia Interazioni Forti 17 -Z Coupling 2 2 g2 g1 g2 Z L L Z L L 2 2cosW g2 quantity to be associated to each L-Z vertex. 2cosW “electroweak charge” of the left-handed neutrino. 1 2 e2 e2 g 2 g 2 2 1 2 2 cos W sin W 1 2 e2 2 2 cos W sin W e cosW sinW D.
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