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Particle Physics and the Standard Model, Part 2

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Particle Physics and the Standard Model, Part 2 LectureNotes jiom simple jield theories to the standard model by RichardC. Slansky he standard model of electrowcak and strong interactions determines the form of the interaction bclween panicles. or fields, consists of Iwo relativistic quantum field theories. one to iha[ carry the charge assoclalcd with the symmclry (no~ ncccssarll! describe [he strong interactions and one to describe [he the cleclnc charge). The interaction is mcdia{ed by a spin-l partlclc. T clec!romagnetic and weak interactions, This model. which the vector boson. or gauge parliclc. The second conccpl IS spon - incorporates all Ihe known phcnomenology of these fundamental [ancous symmclr> breaking. where the \acuum ([he slate wilh IIL) ]nterac[ions. descnbcs splnless. spin-lb, and spin-1 fields interacting particles) has a nonzero charge dls[rlbution. In [hc standard model twth onc ano[hcr in a manner delcrmlned by its Lagrangian. The [he nonzcro weak-interaction charge dis~ributton of [he vacuum IS {hcory is rclalivlstically Invariant. so the mathematical form of the the source of most masses of the parliclcs in ~hc theory. These Iwo Lagranglan is unchanged by Lorcntz transformations, basic ideas. local symmetry and spontaneous symmct~ breaking. arc ,411hough rather complicated in detail. the standard model La- exhibited by simple field lhcorws. Wc hcgin [hcsc Icc[urc noms t~llh a grangian IS based on JUSI IWO basic Ideas beyond those necessary for a Lagrangian for scalar tlclds and then. through the cs~cnslcrns and quantum field [hcory. One is the concep! of local symmetry, which is generalizations indicalcd by [hc arrow in the diagram below. build encountered in its simples! form in electrodynamics. Local symmetry up [he formalism ntedcd to construc[ [hc standard model, Fields, I.agrangians, and Equations a of Motion Continuous p S} mmetrit+ ? ) . Spontaneous I.agrarrgians Breaking ofa with I,arger @ Global Symmetr} ~ @ Global S! mmetries K7 w v I.ocal Phase Invariance and @ Electrodynamics 4 Spontaneous }’ang-%liils Breaking of I.ocal “1’heories @ Phase Invariance - @ 7 J ‘I’be S[’(2) X(”(l) Elect rweak - @ llodel # Quarks Future “1’btwitw ‘! .@ 9 Set Articleon *Q“ o● 1‘nified ‘1’heoriw 54 Summer/Fall 1984 LOS ALAMOS SCIENCE —— (?s4’ d@pfa’4x, (2) + d~) 1 We begin this introduction to field theory with one of the simplest theories, a complex scalar field theory with independent fields rp(x) where the variation is defined with the restrictions &p(x,tl) = &p(x,t2) and rpt(x). (pt(x) is the complex conjugate of q(x) if rp(x) is a classical = &pt(x,tl) = iSrpt(x,t2)= O,and &p(x) and lkp+(x)are independent. The field, ancl, if q(x) is generalized to a column vector or to a quantum last two terms are integrated by parts, and the surface term is dropped field, qf(x) is the Hermitian conjugate of q(x).) Since q(x) is a since the integrand vanishes on the boundary. This procedure yields complex function in classical field theory, it assigns a complex the Euler-Lagrange equations for qt(x), number to each four-dimensional point x = (et, x) of time and space. The symbol x denotes all four components. In quantum field theory (3) q(x) is an operator that acts on a state vector in quantum-mechanical Hilbert space by adding or removing elementary particles localized around the space-time point x. and for p(x). (The Euler-Lagrange equation for q(x) is like Eq. 3 In this note we present the case in which p(x) and Vt(x) correspond except that rpt replaces q. There are two equations because SW(x) and respectively to a spinless charged particle and its antiparticle of equal tipt(x) are independent.) Substituting the Lagrangian density, Eq. 1a, mass but opposite charge. The charge in this field theory is like into the Euler-Lagrange equations, we obtain the equations of mo- electric charge, except it is not yet coupled to the electromagnetic tion, field. (The word “charge” has a broader definition than just electric charge.) In Note 3 we show how this complex scalar field theory can Way(p + Fn*(p+ 2k((ptq)rp= 0, (4) describe a quite different particle spectrum: instead of a particle and its antiparticle of equal mass, it can describe a particle of zero mass plus another equation of exactly the same form with p(x) and and one ofnonzero mass, each of which is its own antiparticle. Then Vt(x) exchanged. the scalar theory exhibits the phenomenon called spontaneous sym- This method for finding the equations of motion can be easily metry breaking, which is important for the standard model. generalized to more fields and to fields with spin. For example, a field A complex scalar theory can be defined by the Lagrangian density, theory that is incorporated into the standard model is elec- trodynamics. Its list of fields includes particles that carry spin. The electromagnetic vector potential A!(x) describes a “vector” particle with a spin of 1 (in units of the quantum of action h = 1.0546 X 10–27 erg second), and its four spin components are enumerated by the where dP~ = @/d.#. (Upper and lower indices are related by the space-time vector index K ( = O, 1, 2, 3, where O is the index for the metric tensor, a technical point not central to this discussion.) The time component and 1, 2, and 3 are the indices for the three space Lagrangiam itself is components). In electrodynamics only two of the four components of A!(x) are independent. The electron has a spin of 1/2, as does its antiparticle, the positron. Electrons and positrons of both spin pro- jections, *1/2, are described by a field ~(x), which is a column vector with four entries. Many calculations in electrodynamics are com- The first term in Eq. la is the kinetic energy of the fields q(x) and plicated by the spins of the fields. qt(x), anti the last two terms are the negative of the potential energy. There is a much more difficult generalization of the Lagrangian Terms quadratic in the fields, such as the –m2qtq term in Eq. la. formalism: if there are constraints among the fields, the procedure are callecl mass terms. If m2 >0, then q(x) describes a spirdess yielding the Euler-Lagrange equations must be modified, since the particle and rpt(x) its antiparticle of identical mass. If m2 <0, the field variations are not all independent. This technical problem theory ha:] spontaneous symmetry breaking. complicates the formulation of electrodynamics and the standard The equations of motion are derived from Eq. 1 by a variational model, especially when computing quantum corrections. Our ex- method. Thus, let us change the fields and their derivatives by a small amination of the theory is not so detailed as to require a solution of amount 89(x) and &3Pq(x) = dV&p{x).Then, the constraint problem. (6b) Continuous ‘Symmetries with 5X = x’ — x. The Ta are the “generators” of the symmetry transformations of q(x). (We note that &p(x) in Eq. 6a is a small symmetry transformation, not to be confused with the field varia- tions Zipin Eq. 2.) II. is often possible to find sets of fields in the Lagrangian that can The space-time point x’ is, in general, a function of x. In the case be rearranged or transformed in ways described below without where x’ =x, Eq. 5 is called an internal transformation. Although our changing the Lagrangian. The transformations that leave the La- primary focus will be on internal transformations, space-time sym- grangian unchanged (or invariant) are called symmetries. First, we metries have many applications. For example, all theories we de- will look at the form of such transformations, and then we will scribe here have Poincar6 symmetry, which means that these theones discuss implications of a symmetrical Lagrangian. In some cases are invariant under transformations in which x’ = Ax + b, where A is symmetries imply the existence of conserved currents (such as the a 4-by-4 matrix representing a Lorentz transformation that acts on a electromagnetic current) and conserved charges (such as the electric four-component column vector x consisting of time and the three cha:rge), which remain constant during elementary-particle collisions. space components, and b is the four-component column vector of the The conservation of energy, momentum, angular momentum, and parameters of a space-time translation. A spirdess field transforms electric charge are all derived from the existence of symmetries. under Poincar6 transformations as rp’(x’) = q (x) or E@= –bKdPqI(X). Let us consider a continuous linear transformation on three real Upon solving Eq. 6b, we find the infinitesimal translation is repre- spinless fields qi(~) (where i = 1, 2, 3) with qi(.x) = q](x). These three sented by idy. The components of fields with spin are rearranged by fields might correspond to the three pion states. As a matter of Poincar6 transformations according to a matrix that depends on both notation, q(x) is a column vector, where the top entry is q](x), the the e’s and the spin of the field. second entry is p2(x), and the bottom entry is 93(x). We write the We now restrict attention to internal transformations where the linear transformation of the three fields in terms of a 3-by-3 matrix space-time point is unchanged; that is, axy = O. If &a is an in- U(s), where finitesimal, arbitrary function ofx, s.(x), then Eqs. 5 and 6a are called q’(1~) = u(&)@) , (5a) local transformations.
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