1 Electromagnetic structure of hadrons Alexander G. Kyriakos Saint-Petersburg State Institute of Technology, St.Petersburg, Russia Present address: Athens, Greece e-mail: [email protected] Abstract In the previous papers [1,2,3], based on Dirac’s equation, we have considered the electromagnetic structure of the leptons. In the present paper, using the Yang-Mills equation, we will analyse the electromagnetic structure of the hadrons. PASC 12.10.-g Unified field theories and models. PASC 12.90.+b Miscellaneous theoretical ideas and models. Contents 1.Introduction 1.1 The Yang-Mills equation and QCD 1.2 The investigation object choice 1.3 Problem statement 2.“One quark” theory 2.1 Inner and external fields 2.2 The “one quark” equation 2.3 Electromagnetic form of “one quark” equation 2.4 The “one quark” linear equation Lagrangian 2.5 Lagrangian of non-linear “one quark” equation 3.Quantum and electromagnetic forms of “three quark” equations 3.1 Electromagnetic forms of quantum equations 3.2 Rings or knots? 3.3 “Three quarks” equation without interaction. 3.4 Interaction appearance 4. Electromagnetic “two quark” equations 5. About symmetry breaking and mass appearance Appendix 1. Hadron models A1 “Three quarks” model A2 “Two quarks” model Conclusion References 2 1. Introduction As it is known the modern Standard Model Theory of elementary particles is described by the Yang-Mills equation [4]. 1.1 The Yang-Mills equation and QCD The Standard Model Theory contains the electroweak theory, which is based on the SU(2)xU1 symmetry group, and the strong interaction theory (SU(3) symmetry group). Both theories are built on the base of the Yang-Mills equation in the same way. Let’s introduce briefly the QCD theory [4,5,6]. As it is known, the QCD theory and the QED theory forms are alike. Particularly, the Lagrangian and the equation of the QCD have the same form as the Lagrangian and the equation of the QED [4,5]. Since it is important for our theory understanding, we consider at first the theory structure of QED. In the usual notations [5] the Lagrangian of QED is represented by the sum: 1 L =ψ []()i∂ µ + eAµ γ µ − m ψ − Fµν Fνµ , (1.1) QED 4 where ∂∂∂µµ= / x is the partial derivative with respect to 4-co- ordinate xµ , FAAµν=−∂∂ µ ν ν µ is the operator of electromagnetic field strength, − e, m are the electrical charge and the mass of the electron, respectively; γ µ are the Dirac's matrices, ψ is the hermitian wave function, Aµ is the potential of the electromagneticl field (the summation is always taken with respect to same indeces). The first and the third summands of the Lagrangian describe the free motion of electron, the last summand describes the same of photons, and the term ψAµψ describes its interaction. Using the covariant derivative DieAµµ=−∂ µ, (1.2) we can write the Lagrangian in the form: 1 L =ψ []Dµγ µ − m ψ − Fµν Fνµ , (1.3) QED 4 The physical means of the present theory will be understood better, if we use the our notations [1]. In this notations we have for QED Lagrangian: = + + LQED LD Lint LM , (1.4) where =ψ + (α ε + α!ˆ !ˆ + βˆ 2 )ψ LD ˆ 0 ˆ c p mec , (1.5) is the free electron Lagrangian. The Lagrangian: = −ψ + (α ε + α!ˆ ! ) ψ Lint ˆ 0 ex c pex , (1.6) is the interaction Lagrangian, and 3 1 ! ! L = ()E 2 − H 2 , (1.7) M 8π is the Maxwell theory Lagrangian. Thus, the full Lagrangian of QED theory is: + 1 ! ! L =ψ []αˆ ()εˆ − ε + cα!ˆ ()p!ˆ − p! + βˆ m c2 ψ + ()E 2 − H 2 , (1.8) QED 0 ex ex e 8π ∂ ! where αˆ ,α!ˆ, βˆ - are Dirac's matrices, εˆ = i , p!ˆ = −i ∇ - the 0 " ∂ t " operators of energy and momentum, ψ + is the Hermitian wave e ! function, ε = eϕ, p! = A are the external electron energy and ex ex c ! ! momentum , c is the light velocity, me is the electron mass, E, H ! are the electric and magnetic field, respectively, and (ϕ, A) is 4- α α!ˆ βˆ potential of external field. The matrices ˆ 0 , , are the following Dirac’s matrices: 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 αˆ = , αˆ = , 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 ! 0 1 0 0 α ≡ βˆ = ,(1.9) − 4 − 0 0 0 i 0 0 1 0 0 0 1 0 0 0 i 0 0 0 0 −1 0 0 0 −1 αˆ = , αˆ = 2 − 3 0 i 0 0 1 0 0 0 i 0 0 0 0 −1 0 0 In Quantum Chromodynamics (QCD) we have quarks instead of electrons and gluons instead of photons, between which there are the strong interactions instead of electromagnetic interactions. The strong interactions among quarks and gluons are described by the non-abelian gauge theory, based on the gauge group SU(3)C , instead of the abelian gauge theory of the EM theory, based on the gauge group U (1) . Each quark sort (flavour) corresponds to a colour (strong interaction charge) triplet in the fundamental representation of SU(3) and the gauge fields needed to maintain the gauge symmetry, the gluons, are in the adjoint representation of dimension 8. Gauge invariance ensures that gluons are massless whose spin is equal to 1, as the photon. Since the gluons masses are equal to zero, the quark interaction radius is equal to infinity, but the sizes of the quark system (particles) are very small (in order 10−13 cm). The explanation of this fact is not known (the «confinement» problem). The QCD Lagrangian may by written in ordinary notations [4,5,6] as 1 L = ψ []Dµγ µ − m ψ − Fµν Fνµ , (1.10) QCD i i 4 4 λ a a where Fµν = Fµν , (1.11) 2 ψ stands for the gluon field tensor, i are the quark fields and the covariant derivative is defined by Dµ = ∂ µ − igAµ , (1.12) The strong coupling is represented by g and indices are summed λ over a = 1,...,8 and over i = 1,2,3 . Finally, a and f are the SU(3) 2 abc group generators and structure constants, respectively, which are related to the commutator: []λ λ = λc a , b 2if abc , (1.13) For the gluons the field strength has the view: Fµν = ∂ µ Aν −∂ ν Aµ − ig[Aµ Aν − Aν Aµ ], (1.14) or in the vector form: !!!!! FAAgAAµν=−+×∂∂ µ ν ν µ µ ν , (1.15) The photon potentials Aµ are the numbers and in the QED case the above commutator is equal to zero. In the case of the non- abelian gauge fields (gluons) Aµ are the matrices, and the above comutator is not equal to zero. Thanks to this commutator, the non-linear interaction of the gluons appears. λ The matrices a are named Gell-Mann matrices and have the view: 0 1 0 0 − i 0 1 0 0 λ = 1 0 0 , λ = i 0 0 , λ = 0 −1 0 , 1 2 3 0 0 0 0 0 0 0 0 0 − 001 00 i 000 λ = 000 , λ = 00 0 , λ = 001 , (1.16) 4 5 6 100 i 00 010 00 0 10 0 λ =−00 i , λ = 01 0 . 7 8 00i 00− 2 If we consider the quark as real fermion, the contradiction with Pauli principle appears. To eliminate this contradiction some inner degree of freedom is introduced, named colour (or colour charge). According to modern theory in strong interactions the colour charges play the same role as the electrical charges in the electromagnetic interactions. The interaction between electrons takes place through the photon exchange. In this case gluons play the role of photons, which are also electrically neutral vector particles. The basic difference between gluons and photons is that the photon is one, but gluons are 8 and have colour charges. 5 Thanks to these colour charges the gluons interact strongly with one another and can radiate one another. 1.2 The investigation object choice Apparently the simplest of the hadrons must be the scope of our investigation. There is a reason to believe that the simplest particles are particles with the lower mass of the mass spectrum of each particle family. In the case of the baryons the simplest particles are nucleons - proton (antiproton) and neutron (antineutron); among the mesons these are the pi mesons. We will name these particles the basic particles. As it is known, the basic particles consist only of two quarks u and d: 1 πππ+−===−==ud,, du0 () uu dd ,, p uud n ddu 2 which have the following charges: 2 1 q = e, q = − e u 3 d 3 The quark masses don't depend on their colour, but depend on the flavour. The u and d quarks have aproximetly the following masses: ≈ ≈ mu 4 MeV, md 7 MeV 1.3 Problem statement The problem of the present paper is to show the electromagnetic structure of the hadrons. As it is followed from the Standard Model Theory (SMT), the quark family is analogue to the lepton family. Based on this analogy we suppose that: 1) The electromagnetic structure of the hadrons is similar to the lepton structure; 2) the above hadron Lagrangian (and equations) is composed from three Lagrangians (equations) of the lepton type, i.e. by three Dirac's equation Lagrangians (three Dirac's equations).