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Basic Ideas of the Standard Model BASIC IDEAS OF THE STANDARD MODEL V.I. ZAKHAROV Randall Laboratory of Physics University of Michigan, Ann Arbor, Michigan 48109, USA Abstract This is a series of four lectures on the Standard Model. The role of conserved currents is emphasized. Both degeneracy of states and the Goldstone mode are discussed as realization of current conservation. Remarks on strongly interact- ing Higgs fields are included. No originality is intended and no references are given for this reason. These lectures can serve only as a material supplemental to standard textbooks. PRELIMINARIES Standard Model (SM) of electroweak interactions describes, as is well known, a huge amount of exper- imental data. Comparison with experiment is not, however, the aspect of the SM which is emphasized in present lectures. Rather we treat SM as a field theory and concentrate on basic ideas underlying this field theory. Th Standard Model describes interactions of fields of various spins and includes spin-1, spin-1/2 and spin-0 particles. Spin-1 particles are observed as gauge bosons of electroweak interactions. Spin- 1/2 particles are represented by quarks and leptons. And spin-0, or Higgs particles have not yet been observed although, as we shall discuss later, we can say that scalar particles are in fact longitudinal components of the vector bosons. Interaction of the vector bosons can be consistently described only if it is highly symmetrical, or universal. Moreover, from experiment we know that the corresponding couplings are weak and can be treated perturbatively. Interaction of spin-1/2 and spin-0 particles are fixed by theory to a much lesser degree and bring in many parameters of the SM. Moreover, at present it is not known whether self-interaction of the Higgses is weak or strong and we should reserve for both possibilities. Standard model is too extended a subject to be covered in a series of four lectures. The outline of the lectures was shaped to a great extent by an internal message to lecturers at the present School which in the part relevant to us reads: Standard Model: SU (2) U (1) gauge theory but no QCD (!); Higgs mechanism and spontaneous symmetry breaking; Goldstone and equivalence theorems; electroweak radiative corrections, LEP physics, triple gauge boson coupling; brief introduction to CKM matrix but no detailed discussion of flavour physics and CP violation. In the written version, the notes consist of three parts. First we consider general cosequences from a current conservation. In Part 2 we review Lagrangian models where the general relations of Part 1 are realized on concrete exmaples. There are also some model predictions. In Part 3 radiative corrections are discussed briefly. Both general relations of Part 1 and specific predictions of Part 2 are utilized here. After these preliminaries we proceed to a systematic presentation. 59 1 PART I: CONSERVED CURRENTS The outline of this part is as follows: (a) Noether’s theorem (symmetries vs. conserved currents) (b) Conserved currents as sources of massless vector bosons (c) Realizations of current conservation: (i) Degeneracy of states, or minimal coupling (ii) Goldstone mode (iii) Higgs mode 1.1 Noether’s Theorem Noether’s theorem states that to every continuous symmetry of a Lagrangian there corresponds a con- served current. (i) (i) L(' ;' ) S To prove the theorem consider a Lagrangian ; and the corresponding action : Z 4 (i) (i) = d xL ' (x);' (x) S (1) ; (i) ' where labels various fields (and will often be omitted in fact) while ; denotes a derivative: @' ' : ; (2) @x Moreover, we assume that there exists a symmetry transformation on the fields which leaves the action invariant. We distinguish between two possible kinds of transformations. Namely, discrete (or “large”) such as ! ' ' (3) and continuous transformations when the change in the fields can be made infinitesimally small. As an example of the second type, consider the Lagrangian which is just a kinetic term of a spinor particle, 1 (@ )+(@ ) L = kin (4) 2 Then there are two independent transformations which leave the Lagrangian (4) invariant: i i i i 1 1 2 5 2 5 ! e ; ! e ; ! e ; ! e ; (5) ;2 where 1 are two independent infinitesimally small parameters. Turning back to the general consideration, we note that symmetry means that the action is not changed: = 0: S (6) Let us evaluate, on the other hand, the change in the action directly: Z X X @L @L 4 (i) (i) S = d x L; L = ' + ' (7) ; (i) (i) @' @' ; i i '; ' where ; are variations of the fields and their derivatives. Moreover, since the parameter of the x transformation does not depend on (i.e. we consider what is called global symmetry transformations) (' ) = @ ('): ; (8) Applying chain rule we get then: ! ! X @L @L @L (i) (i) ' + @ : L = @ ' (9) (i (i) (i) @' ) @' @' ; ; i 60 (i) Expressions in front of ' are in fact the Lagrange equations of motion and therefore we have for the ! variation of the action: Z X @L 4 (i) S = d x @ = 0 ' (10) (i) @' ; i =0 where S because of the assumption that we deal with a symmetry transformation. Substitute (i)) X ' (i) ' = a) ( (11) (a) a ! 0 (a) x a) (a) where ( and the index labels independent transformations. Since does not depend on : Z 4 d x(@ j ) = 0: S = a) ( (12) (a) Therefore @ = 0 j (13) (a) @x where (i) X @L ' : = j (14) (a) (i) (a) @' ; i This is the Noether current. 3 x Integrating (13) over d and assuming that there are no fields at spatial infinity: Z +1 @ i 3 j x = 0; d (15) i @x 1 we get conservation of charge: d (t) Q (16) dt where Z 3 Q(t) = j (x;t)d x: 0 (17) So far we considered a classical field theory. It is important to emphasize that the construction generalizes in fact to case of QFT (quantum field theory) as well. Let me outline the steps in this generalization. QFT is introduced in close analogy to the case of quantum mechanics (QM), see also lectures on QFT at this School. In quantum mechanics we have the fundamental commutation relation: q; p^] = ih [^ (18) q; p) where ( are canonically conjugated variables and the hat reminds that we dealing now with operators. ^ Time dependence of the operators is fixed in terms of commutators with the Hamiltonian H : dq^ dp^ ^ ^ q;^ H ; = p;^ H : = (19) dt dt If there are a few degrees of freedom then q^ ; p^ = ih : k ik i (20) Moreover, QFT deals with systems with infinite number of degrees of freedom which are fields at each spatial point x: @L q (t) ! '^(x;t); p^(t) ! ^(x;t) ^ (21) @' (x;t) ;0 while the fundamental commutation relation has the same form: 0 3 0 ' (x;t);^(x ;t) = ih (x x ) ^ (22) 61 Using these canonical commutation relations it is straightforward to prove that ^ dQ ^ ^ Q; H = 0; = (23) dt which is conservation of charge. Also, (i) (i) ^ = Q '^ Q; '^ i) ( (24) ' e e ' where ' is the charge of the field so that is an ordinary number (not an operator). The crucial point j (x;t) of the derivation (24) is that the zeroth component of the current, 0 is constructed as a product of the canonical momenta and coordinates: (i) @L ' j (x;t) = 0 (25) (i) (a) @' (x;t) ;0 (i) since variations of the fields ' is reexpandable in the fields themselves in the examples we considered. 1.2 Conserved Currents as Sources of Massless Vector Bosons The importance of conserved currents is that they can be coupled to massless vector particles, as we will demonstrate now. Note first that conservation of a current, @ j = 0; (26) (q ) implies that the corresponding vertex in momentum space, is transversal: q (q ) = 0 (27) q where is the momentum carried in by the current and is the Fourier transform of the matrix elements of the current: Z 4 d x iq x (q )= h1jj (x)j2i e (28) 4 (2 ) 1; 2i where j aresomestates. Let us now couple the current to a massless vector field: L = ej A (29) e A where is the (gauge) coupling and is the vector potential. Furthermore, consider S-matrix element corresponding to an exchange of a vector particle: g 0 0 2 S (1 + 1 ! 2+2 ) = e 0 0 (30) 1 !2 1!2 2 q where the three factors describe emission, propagation and absorption of a massless vector particle, g = g = g = g =1 11 22 33 respectively, and 00 . The expression (30) should be consistent with the fact that there exist only two independent po- larization states for the vector particle (left- and right-handed polarized photons). This is not a trivial g condition since to ensure Lorentz invariance of the matrix element (30) we had to use the tensor ) where ( run from 0 to 3 and, al least superficially there are four degrees of freedom involved . Con- servation of the current is crucial at this point. In more detail, if a particle is absorbed far oof the point of ots emission then 2 0: q (31) Or, choosing the third axis along the momentum of the photon, q q ; 3 0 (32) 62 and the transversality condition (27) reads: q 0 (3) (0) = : (33) q 3 This allows us to rewrite: 2 q (0) (0) (3) (3) (0) (0) 0 = 1 0 : 0 0 0 0 0 0 at the pole (34) 1!2 1!2 1!2 1 !2 1 !2 1 !2 2 q 3 Thus, 2 1 g q !0 ? ? ! : 0 0 0 0 (35) 1!2 1 !2 1!2 1 !2 2 2 q q ? 1 2 where includes only the transversal indices, and .
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