Why Quarks are Different from Leptons – An Explanation by a Fermionic Substructure of Leptons and Quarks H. Stumpf Institute of Theoretical Physics, University Tuebingen, Germany Reprint requests to Prof. H. S.; E-mail: [email protected] Z. Naturforsch. 59a, 750 – 764 (2004); received August 14, 2004 To explain the difference between leptons and quarks, it is assumed that electroweak gauge bosons, leptons and quarks are composites of elementary fermionic constituents denoted by partons (not to be identified with quarks) or subfermions, respectively. The dynamical law of these constituents is assumed to be given by a relativistically invariant nonlinear spinor field theory with local interaction, canonical quantization, selfregularization and probability interpretation. According to the general requirements of field operator algebraic theory, this model is formulated in algebraic Schroedinger representation referred to generating functionals in functional state spaces. The derivation of the corresponding effective dynamics for the composite particles is studied by the construction of a map between the spinor field state functionals and the state functionals of the effective theory for gauge bosons, leptons and quarks. A closer examination of this map shows that it is then and then only selfconsistent if certain boundary conditions are satisfied. The latter enforce in the case of electroweak symmetry breaking the difference between lepton and quark states. This difference can be analytically expressed as conditions to be imposed on the wave functions of these composite particles and leads ultimately to the introduction and interpretation of color for quarks, i.e., the characteristic of their strong interaction. Key words: Substructure of Quarks and Leptons; Effective Dynamics; Difference of Leptons and Quarks. 1. Introduction At present the general tendency prevails to embed the standard model into theories with grand unifica- In the standard model of elementary particle physics tion, supersymmetry, superstrings, etc., see [1]. But to leptons and quarks are assumed to constitute the solve the above problem we prefer the compositeness fermionic part of its basic particle set, i.e. in this model hypothesis as a possible approach of model building. leptons and quarks are considered to be the ultimate While the former model building leads to an en- fermionic constituents of matter without substructure. largement of symmetries and (or) dimensions, etc., the In spite of the enormous success of this model with latter approach should lead to a simplification of the respect to the experimental verification, it raises a lot of basic particle set and of the interactions. Thus from an theoretically and experimentally unsolved questions, economic point of view the compositeness hypothesis see [1], Sect. 3.10, [2], Sect. 5.7, [3], Sect. 12, and is very attractive. Nevertheless, before starting a the- these have led to further alternative model building oretical discussion one should have a look on experi- since several decades, see [1]. ments. Does there exist any experimental hint in favour One of these unsolved problems is the existence of of this hypothesis? lepton and quark families, or more simply: Why quarks In high energy physics deep inelastic scattering pro- are different from leptons and why do they exist in gen- cesses of photons by photons and similarly of quarks erations? by photons seem to provide signals of composite- If one tries to answer these special questions, one ness of the photons themselves, [4 – 6], as well as is confronted with the basic problem by which kind of the quarks, [7 – 9]. In addition an unexpected con- of model the standard model should be replaced, be- firmation of compositeness comes from the study of cause within the standard model itself one cannot find superfluids with an enormous variety of experimen- any reason for the difference of leptons and quarks and tal facts and analogies to high energy physics and their replication. even cosmology, [10]. We consider these facts as a 0932–0784 / 04 / 1100–0750 $ 06.00 c 2004 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com H. Stumpf · Why Quarks are Different from Leptons 751 justification of theoretical research in compositeness models cited in [15], because the mathematical for- models. mulation and evaluation is different from previous at- In the past decades such a model was developed tempts in this field. which we use as a suitable candidate to treat quanti- In previous attempts, see [15], the difference be- tatively the substructure of elementary particles. This tween leptons and quarks was generally explained by model is based on a relativistically invariant nonlin- attributing to quarks special algebraic and group the- ear spinor field theory with local interaction, canoni- oretical degrees of freedom, whereas in the case un- cal quantization, selfregularization and probability in- der consideration we start with a globally invariant terpretation. It can be considered as the quantum field SU(2) ⊗ U(1) spinor theory, and the difference be- theoretic generalization of de Broglie’s theory of fu- tween leptons and quarks is generated by the dynamics sion, [11], and as a mathematical realization and phys- of the model itself. Being composite particles, the ef- ical modification of Heisenberg’s approach, [12], and fective dynamics of leptons, quarks and gauge bosons is expounded in [13, 14]. will be studied. In this case the consistence of the map- Its basic particle set consists of partons (not to be ping from the basic parton theory to the corresponding identified with quarks) or subfermions, respectively, effective theory imposes subsidiary conditions which which are assumed to be the elements of the substruc- enforce the difference between lepton and quark states ture of the elementary particles of the standard model. and hence offers an explanation of the difference of In this model the formation of additional new partonic leptons and quarks themselves. bound states is not excluded, where the latter can act as Thus, before discussing the latter problem we first additional “elementary particles” in the corresponding develop the formal proofs for the derivation of the ef- effective theory replacing the standard model. fective dynamics of these composite particles and, sub- In analyzing the problem why quarks are different sequently, based on the results of these investigations, from leptons, we refer to this model. In particular, we can treat the actual problem in the last section. in its simplest version the electroweak gauge bosons are assumed to be two-parton bound states, while the 2. Algebraic Representation of the Spinor Field fermion families are assumed to arise from three par- ton bound states. The latter assumptions are made in The algebraic representation is the basic formula- several compositeness models, [15]. Among other au- tion of the spinor field model and the starting point for thors, the three fermion substructure of leptons and its evaluation. In order to avoid lengthy deductions we quarks was proposed by Harari [16] and Shupe [17]. give only some basic formulas of this formalism and But apart from this assumption our model has no- refer for details to [13, 14, 18]. The corresponding La- thing in common with the Harari-Shupe model or other grangian density reads, see [14], Eq. (2.52) 3 −1 ¯ µ L(x):= λi ψAαi(x)(iγ ∂µ − mi)αβδABψBβi(x) i=1 (1) 1 2 3 − h h ¯ ( ) ( ) ¯ ( ) ( ) 2g δABδCDvαβvγδ ψAαi x ψBβj x ψCγk x ψDδl x h=1 i,j,k,l=1 with v1 := 1 and v2 := iγ5. The field operators are To obtain a uniform transformation property with re- assumed to be Dirac spinors with index α =1, 2, 3, 4 spect to Lorentz transformations, the adjoint spinors and additional isospin with index A =1, 2 as well as are replaced by formally charge conjugated spinors, auxiliary fields with index i =1, 2, 3 for nonperturba- which are defined by tive regularization. The algebra of the field operators is c ¯ ψ (x)=CαβψAβi(x), (3) defined by the anticommutators Aαi and the index Λ is introduced by [ + (r ) (r )] = (r−r) ψAαi ,t ψBβj ,t + λiδij δABδαβδ (2) ( ); =1 ( )= ψAαi x Λ ψAΛαi x c ( ); =2 . (4) resulting from canonical quantization. All other anti- ψAαi x Λ commutators vanish. Then the set of indices is defined by Z := (A, Λ, α, i). 752 H. Stumpf · Why Quarks are Different from Leptons The operator formulation of the theory by means bra with corresponding duals ∂I := ∂Z (r) which sat- of (1) and (2) cannot be evaluated without further isfy the anticommutation relations preparation. In order to obtain definite results from this theory a state space is needed in which the dynamical [jI , ∂I ]=δZZ δ(r − r ), (8) equations can be formulated. This is achieved by the use of the algebraic Schroedinger representation. For while all other anticommutators vanish. ∂ |0 =0 a detailed discussion we refer to [13, 14, 18]. Here we With I f the basis vectors for the generat- present only the results and definitions. ing functional states can be defined. The latter are not To ensure transparency of the formalism we use the allowed to be confused with creation and annihilation symbolic notation operators of particles in physical state spaces, because generating functionals are formal tools for a compact | (ψI1 ...ψIn ):=ψZ1 (r1,t) ...ψZn (rn,t) (5) algebraic representation of the states a : To each state |a in the physical state space we associate a functional with Ik := (Zk, rk,t). Then in the algebraic Schroe- state |A(j; a) in the corresponding functional space. dinger representation a state |a is characterized by the The map is biunique, and the symmetries of the orig- set of matrix elements inal theory are conserved. Nevertheless this map does not induce any equivalence of the kind indicated above. ( ):=0|A( )| =1 ∞ τn a ψI1 ...ψIn a ,n ..