Proc. Symposia in Pure Math., 54 (1993), edited by R. E. Greene and S.-T. Yau, Part 2, 157{171 GEOMETRY OF ELEMENTARY PARTICLES Andrzej Derdzinski Contents 1. Particles and interactions in nature. 2. Bundles over the spacetime. 3. Models of free matter particles. 4. The Yang-Mills description of interactions. 5. The standard model. 6. Grand unifications. Introduction. This is a presentation of the standard model of elementary par- ticles, i.e., their generally accepted theory, discussed here on the classical level (without field quantization). Our basic reference is the book [5], for which the present paper forms a self-contained summary. The author also intends to pro- vide in [6] and [7] a more detailed treatment of particle geometry, along with its quantized version. Although this material is covered in numerous physics textbooks (such as [2] { [4], [9], [14], [15], [18], [20] { [23]), the approach adopted here may be particularly suitable for a mathematician reader due to its consistently geometric language. Pre- sentations of more general topics are also available in the mathematical literature; see, for instance, [1], [8], [10], [12], [13], [16], [17], [19]. The theory outlined below does not, by itself, provide a usable model of the real world, even though it accounts well for many qualitative effects. Its inadequacy as a source of quantitative predictions is mainly due to its classical (macroscopic) char- acter, as opposed to the predominantly quantum (small-scale) properties displayed by elementary particles in nature. However, instead of discarding the classical model for this reason, one uses field quantization to transform it into a quantum theory, which supplies the required \microscopic" corrections. Cross-references in the text are indicated by arrows ( ). ! 1. Particles and interactions in nature 1.0. Experimental data show that there are over 200 species (kinds) of sub- atomic particles, such as the electron e, proton p, neutron n, photon γ, electronic 1991 Mathematics Subject Classification. Primary 53C80; Secondary 81T13, 53B50, 81V05, 81V10, 81V15, 81V22. Key words and phrases. Standard model, electroweak model, quark model, gauge theory. Supported in part by NSF Grant DMS-8601282. Typeset by -TEX AMS 1 2 GEOMETRY OF ELEMENTARY PARTICLES 0 neutrino νe, neutral pion π , etc. ( 1.5, 1.6). Particles interact in four basic ways, known as the strong, electromagnetic,! weak and gravitational interactions (forces), ordered here by decreasing strength of the interaction, i.e., probability of its occurrence in the given circumstances. We will not discuss gravitational forces, negligibly weak on the microscopic level; see, however, 2.0. 1.1. Particle invariants assign to particle species elements of various Abelian semigroups. Here belong, for instance, the mass m R+ = [0; ), electric charge 2 11 1 Q Z ( 1.2), average lifetime τ (0; ], as well as spin s Z+ = 0; ; 1;::: 2 ! 2 1 2 2 f 2 g and parity " Z2 = 1; 1 . The spin measures the capacity of the particle to carry internal2 angularf momentum− g (as if it were rotating about its axis), while the parity of the particle describes symmetry properties of its configurations with respect to space reflections ( 3.3, 3.5). Particles with s Z (resp., s Z) are known as bosons (resp., fermions! ). 2 62 1.2. Quantization of the electric charge. According to experimental ev- idence (Millikan, 1909) and theoretical arguments ( 5.1.iv), the electric charge always comes in integral multiples of the electron charge! q (by convention, q < 0). Charges of individual particle species may be indicated by superscripts such as 0 + ++ + 0 0 0 −; ; ; ; so that e = e−; p = p ; n = n ; γ = γ ; νe = ν , etc. ( 1.0). e ! 1.3. Interaction carriers are the few particle species that serve as agents mediating interactions. These are: the photon γ ( 5.1.ii) for electromagnetism, + 0 ! the weak bosons W ; W−; Z for the weak interaction ( 5.3.ii.b,c), eight kinds of gluons ( 5.2.ii) for the strong force and, probably, gravitons! for gravity. Except for gravitons! with s = 2, their spins ( 1.1) all equal 1. ! 1.4. Matter particles, i.e., those particle species which are not interaction carriers, may in turn be classified into hadrons and leptons, depending on whether they can or cannot participate in the strong interaction. 1 1.5. Leptons consist of 12 known species, all of spin 2 ( 1.1): the electron e, muon µ and tauon τ (with positive masses and electric charge! 1), the electri- − cally neutral, massless neutrinos νe; νµ; ντ , as well as their antiparticles ( 3.2.i): + + + ! e ; µ ; τ ; νe; νµ; ντ . 1.6. Hadrons which are fermions (resp., bosons, 1.1) are called baryons (resp., mesons), with about 100 known species of either.! For instance, π0 is a meson. Baryons can further be divided into the disjoint classes of baryons proper (such as p; n), and their antiparticles ( 3.2.i), the antibaryons. ! 1.7. Classification of elementary particles, as outlined above: interaction carriers 8 > > leptons > all particles > 8 <> > ··· > matter particles <> mesons > ··· 8 > hadrons < baryons proper > > ··· baryons > > ··· antibaryons : :> : PROC. SYMPOSIA IN PURE MATH., 54 (1993), 157-171 3 2. Bundles over the spacetime 2.0. Spacetime. We work with a fixed spacetime ( ; g), which is a 4-manifold endowed with a pseudo-Riemannian metric g of signatureM + + + and with aMtime orientation, i.e., a continuous choice x C+ of one (\future")− component 7! x of the timelike cone Cx = v Tx : g(v; v) < 0 at each x . Although one often assumes that fis an2 affineM space and ggis translation2 M invariant (the Minkowski spacetime), nonflatM spacetimes are also used, as models of gravity (in general relativity). 2.1. Vector bundles over the spacetime ( ; g), used below, include the M product line bundle 1 = C , the tangent bundle T = T , its dual T ∗, the 4 M × M bundle Λ T ∗ of volume forms (pseudoscalars) on and, for each k Z+, the k M 2 bundle S T ∗ whose fibre over x consists of all real k-linear symmetric forms 0 2 M on Tx with g-contraction zero (i.e., of all pseudo-spherical harmonics in Tx ). M M 2.2. Weyl spinors. Whenever necessary, the spacetime ( ; g) will be assumed to be an orientable spin manifold. Denoting L; R the orientationsM of , we may M then choose fixed Weyl spinor bundles σL ; σR over ( ; g), with σR = σL ( 3.2). They are complex vector bundles of fibre dimensionM 2, obtained from a common! spin structure over ( ; g) via two mutually conjugate nontrivial representations of M Spin(3; 1) = SL(2; C) in C2. The (Levi-Civita) spinor connection ◦ in σ and σ r L R then gives rise to the Dirac operator sending sections of σL to those of σR and vice versa. D 2.3. Dirac spinors. If ( ; g) is orientable and spin, we choose a fixed Dirac M spinor bundle over ( ; g) to be the direct sum σ = σL + σR of the Weyl spinor bundles selected as inM 2.2, with unordered summands, so that no orientation of M is distinguished. The operators ◦ and now are also defined in σ. r D 3. Models of free matter particles 3.0. Particle models. Each particle species is represented by ("lives in") a specific fibre bundle η, endowed with some additional geometric structure, over the spacetime manifold ( 2.0). The sections of η are to be thought of as \semiclassical states" ofM the particle,! evolving with time in a manner described by suitable field equations (which form a part of the geometry of η). 3.1. Models of matter particles ( 1.4) are vector bundles and their field equations are linear. For those matter particles! which are free, i.e., not subject to interactions, the choices of bundles and equations are quite specific ( 3.3 { 3.5). On the other hand, interaction carriers are represented by a special type! of affine bundles ( 4.0) which, for many "practical" purposes, may also be regarded as vector bundles! ( 4.4, 5.3.ii,iii). ! 3.2. Physical meaning of vector bundle operations. Equalities between vector or affine bundles, such as (η∗)∗ = η, stand for natural (functorial) isomor- phisms, the category in question being usually clear from the context. For a complex vector bundle η over , let η be its conjugate bundle, with each fibre η ; x , M x 2 M 4 GEOMETRY OF ELEMENTARY PARTICLES consisting of all antilinear maps ηx∗ C . Thus, we have η = η∗ whenever the ge- ometry of η involves a fixed Hermitian! fibre metric (which may even be indefinite), as is the case for all models of free matter particles except neutrinos ( 3.4). ! i. For a particle species represented by a complex vector bundle η ( 3.0, 3.1), the conjugate η, along with the corresponding \conjugate geometry",! may be expected to host another, related particle species, called the an- tiparticle of the original one. The resulting antiparticle formation (denoted ), leaves most of the relevant particle invariants ( 1.1) either completely unchanged (e.g., " for bosons, m; τ; s), or just changes! their signs (as in the case of " for fermions and Q ; 3.5, 5.1.iii). Antiparticles also make sense for interaction carriers ( 4.4,! 5.3.ii.b,c), and, in fact, turn out to exist for all particles known in! nature. However, some species (referred to as strictly neutral) coincide with their antiparticles and then it is natu- ral to represent them by real rather than complex bundles. For instance, 0 0 0 0 + + π = π ; γ = γ; Z = Z , while (W ) = W−; e = e = e; νe = νe; p = p; n = n (notation of 1.0, 1.2, 1.3).