Relations between the baryon quantum numbers of the Standard Model and of the rotating neutrino model
Constantinos G. Vayenas, Athanasios S. Fokas, Dimitrios P. Grigoriou
aa r X i v : . [ phy s i c s . g e n - ph ] J un Relations between the baryon quantum numbers of theStandard Model and of the rotating neutrino model
C.G. Vayenas , , ∗ , A.S. Fokas , , ∗ & D.P. Grigoriou School of Engineering, University of Patras, GR 26504 Patras, Greece Division of Natural Sciences, Academy of Athens, 28 Panepistimiou Ave.,GR-10679 Athens, Greece Department of Applied Mathematics and Theoretical Physics, University ofCambridge, Cambridge, CB3 0WA, UK Viterbi School of Engineering, University of Southern California, Los Angeles,California, 90089-2560, USA
Abstract
We discuss the common features between the Standard Model taxonomyof particles, based on electric charge, strangeness and isospin, and the taxon-omy emerging from the key structural elements of the rotating neutrino model,which describes baryons as bound states formed by three highly relativistic elec-trically polarized neutrinos forming a symmetric ring rotating around a centralelectrically charged or polarized lepton. It is shown that the two taxonomiesare fully compatible with each other.
PACS numbers: ± q, 12.60. ± i, 14.20.Dh, 14.65. ± q After it was established experimentally that baryons contain three quarks there wereintense efforts to identify a satisfactory taxonomy for a variety of baryons. This goalwas satisfactorily achieved in the seventies [1, 2], by employing electric charge as wellas the concepts of isospin and strangeness, see Figures 1 and 2 [1, 2].In several recent works [3, 4] we have introduced a relativistic neutrino modelwhich treats quarks as electrically polarized neutrinos, see Figure 3. This model doesnot contain adjustable parameters and allows for the computation of baryon masseswithin 1% accuracy. It is a Bohr-type model and is based on the use of the deBroglie wavelength equation, together with the Newtonian gravitational law utilizinggravitational rather than rest masses [3, 4], namely ∗ E-mail: [email protected]; [email protected] u , d or s quarks form baryons with a spin-1 / uds baryon octet . Q is the charge in units of e , S is the strangeness and I is theisospin. For example, the proton has charge 1, isospin 1/2 and strangeness zero [1, 2]. F = Gm g, m g, √ r = Gm o γ √ r , (1)where m g, and m g, are the gravitational masses of the two attracting particles which,according to the equivalence principle [5] are equal to the inertial masses, m i, and m i, , and which according to Einstein’s pioneering special relativity paper [6] equal γ m o , where γ (= (1 − v /c ) − / ) is the Lorentz factor and m o is the rest mass ofeach particle. This result has been proven by Einstein for linear motion [6, 7, 8] andwas shown recently [3, 4] that it also holds for arbitrary motion. The factor √ F = γm o v r , (2)and together with the de Broglie wavelength equation¯ λ = ¯ hγm o v , (3)and the assumption ¯ λ = r , leads to the expression F = γm o v r = ¯ h v r . (4)2igure 2: Combinations of u , d or s quarks form baryons with a spin-3 / uds baryon decuplet [1, 2].Equations (1) - (3) imply [3, 4] that v ≈ c and γ = 3 / ( m P l /m o ) / , where m P l (=(¯ hc/G ) / ) is the Planck mass. Thus the mass, 3 γm o , of the proton and of the neutroncan be approximated by m B = 3 / ( m P l m o ) / = 938 . M eV /c , (5)where we have used m o = 0 . eV /c [3, 4] for the highest neutrino mass [9, 10].More recently the approach has been extended to describe the mass of W ± [11], Z o [12] and H o bosons, modeled as rotational e ± ν e structures, for which the followingexpressions have been obtained: m W ± = 2 / ( m P l m e m o ) / = 81 . GeV /c , (6) m Z = 2 / ( m P l m e m o ) / = 91 . GeV /c , (7) m H = 2 / ( m P l m e m o ) / = 122 . GeV /c . (8)In the recent works [13, 14] it is argued that hadrons may be viewed as microscopicblack holes [15, 16], where gravitational collapse is prevented by the uncertaintyprinciple. Furthermore, in [17, 18] Bohr-type models are used in order to establish alink between black holes and quantum gravity.The relativistic Newtonian gravitational law (eq. (1)) used to derive the aboveexpressions has also been shown recently to yield exactly the same results as generalrelativity in the well known problem of the perihelion advancement of Mercury [19].3igure 3: Schematics of a proton and a neutron according to the rotating neutrinomodel, showing the electric dipoles induced on the rotating neutrinos; δ denotes theinduced partial dipole charge [11]. Reprinted with permission from Elsevier.In the present work we compare the predictions of equation (1) and of the asso-ciated rotating-neutrino-lepton model, with experimental results and with the corre-sponding description of the Standard Model. The taxonomy of hadrons according to the Standard Model is based on charge, Q ,isospin, I , and strangeness, S . This is manifested in the octet of baryons shownin Figure 1, named by Gell-mann the “eightfold way” [20, 21] and the decuplet ofbaryons [20, 22] shown in Figure 2.Multiplets of the octet and of the decuplet can be obtained by employing higherrepresentations of the symmetry group SU(3), a higher symmetry than the SU(2) ofisospin theory.One observes in these figures that I increases with increasing electrical charge,i.e. ∂I /∂Q ≥ , (9)which is consistent with the Gellmann-Nishijima formula [23, 24] Q = I + (1 / A + S ) , (10)where A is the baryon number and S is the strangeness. The maximum I value in a4igure 4: Dependence of baryon mass on strangeness S .multiplet, I ,max , is related to the number of members of a multiplet, N , via I ,max = N − . (11)Strangeness ( S ) is attributed to the number of S (strange) quarks in the hadron.As shown in Figures 1 and 2, strangeness increases with baryon mass. This isshown more clearly in Figure 4 which depicts the dependence of baryon masses onstrangeness. Indeed m increases with S but there is considerabe scattering in the dataand it is clear that S alone cannot provide a satisfactory fit to the baryon masses.It appears from Fig. 4 that at least a second parameter is needed to provide a gooddescription of the mass spectrum.In fact, Figure 4 already shows that there appear to exist two families of baryons,one starting from S = 0 with the proton and the neutron, and the other starting from S = 0 with the ∆ baryons and containing the ∆, Σ ∗ , Ξ ∗ and Ω − baryons.5able 1: Computed baryon masses via (eq. 12), i.e. m = [ n B (2 ℓ + 1)] / m p andexperimental values of baryon masses, the latter shown in parenthesis. n B ℓ B m P article S I Spin − , Ξ o (1318) 2 1/2, -1/2 1/21 4 1356 Σ ∗ (1384) 1 -1,0,1 3/22 0 1183 Σ − , Σ o , Σ + (1192) 1 -1,0,1 1/22 1 1420 - - -2 2 1547 Ξ ∗ , − , Ξ ∗ ,o (1532) 2 -1/2,1/2 3/22 3 1636 Ω(1672) 3 0 3/2 As shown in Table 1 and in Figure 5, the masses of all uncharmed baryons, consistingof u , d and s quarks, fall, with a linear correlation coefficient R better than 0.9966,on two parallel straight lines, corresponding to the equation m = m p (cid:2) n B (2 ℓ B + 1) (cid:3) / , (12)where n B is a positive integer (1,2,3,....) and ℓ B is a non-negative integer (0,1,2,....).In view of equation (5) equation (12) can also be written in the form m = (cid:2) n B (2 ℓ B + 1) (cid:3) / / m / P l m / o , (13)where m p (= 3 / m / P l m / o = 938 .
32 MeV/c ) is the proton mass.The family n B = 1 includes, in addition to the proton and the neutron, the Λ, ∆,Ξ and Σ ∗ baryons. On the other hand, the family n B = 2 contains the Σ, Ξ ∗ and Ωbaryons. 6igure 5: Comparison of the masses, m B , of the uncharmed baryons, consisting of u,d and s quarks, with equation (12), i.e. m = m p [ n B (2 ℓ B + 1)] / . Numbers next toeach baryon denote strangeness and spin. With the exception of Ξ − and Ξ o , whichhave spin 1/2, all baryons with ℓ ≥ n B bears some similaritieswith the principal quantum number, n , of the Bohr model for the H atom and todistinguish it here we use the subscript “B” (for Baryon). Also the second number ℓ B , may have some similarities with the second quantum number of the H atom Bohrmodel, thus we use the symbol ℓ with the same subscript “B”.Regarding the spin, we note that Figure 5 shows that all baryons with ℓ B ≥ − and Ξ o baryons whichhave spin 1/2. Also all baryons with ℓ < n , which in viewof Figure 5 and Table 1, we replace here by the two quantum numbers, n B and ℓ B which provide a quantitative fit to the mass spectrum (Fig. 5).7 .2 Analogy with Bohr’s H atom model In order to get some insight into the experimental results, which are described semi-quantitatively by equations (12) and (13), it is useful to recall some key aspects ofthe Bohr model for the H atom. Bohr’s original model published in 1913 [25] wasbased on the equation of motion for the electron, i.e. m e v r = e εr , (14)coupled with the assumption that the angular momentum, L , (= m e v r ) is an integermultiple fo the Planck’s constant ¯ h , i.e. L = m e v r = n ¯ h, (15)where n , now called the principle quantum number, is a positive integer. These twoequations lead to the well known results v = αc/n ; α = e /εc ¯ h, (16) E n = − m e α c n = − T = V / , (17)where E n are the energy levels of the electron and T and V denote its kinetic andpotential energy respectively.Some ten years later, after de Broglie published his famous de Broglie wavelengthequation [26], i.e. λ = hp ; ¯ λ = ¯ hp = ¯ hm e v , (18)it became clear that equation (15) can be interpreted as the “standing wave con-dition”, namely that the electron is described by a wave where a whole number ofwavelengths is required to fit along the circumference of the electron orbit. Indeedsubstituting the relation 2 πr = nλ ; r = n ¯ λ, (19)into equation (18) and accounting for the definition of L it follows that L = m v r = m e v ¯ λn = n ¯ h, (20)which is Bohr’s assumption of Eq. (15). 8 .3 Interpretation of the rotating neutrino taxonomy According to the rotating neutrino model [3, 4], a hadron consists of two buildingelements: a. A rotating neutrino ring consisting of three (in the case of baryons) or two (inthe case of mesons) relativistic neutrinos or antineutrinos held in orbit by therelativistic gravitational force. This bound rotational state is found here tobe characterized by an integer number, n B ( ℓ B + 1), consisting of two integers n B and ℓ B . These integers may be viewed as two quantum numbers whichbear some similarity with the principal quantum number, n , of the Bohr modelfor the H atom [11]. The fact that in the present case two, rather than one,principal quantum numbers are need to define the ground state may be relatedto the Pauli exclusion principle which does not allow more than two particles(e.g. neutrinos) to occupy the same state. b. A central mass consisting of one or rarely two leptons. These lepton(s) can becharged (e.g. e + in the case of a proton or e − in the case of the Σ − or Ξ − baryons), and are stabilized at the center of the rotating ring via ion-induceddipole or induced dipole-induced dipole electrostatic interactions [11, 12].Denoting by L + and L − the numbers of positively and negatively charged leptonsat the central position, it follows that the charge Q is given by Q = L + − L − ≡ L .Equation (8) implies ∂I /∂ L ≥ S = 0, then I = L − / r centered at the centerof gravity of the three neutrinos (denoted by O in Fig. 6), by the force expressed viathe relativistic gravitational law, i.e. F = γm o v r = Gm o γ √ r , (21)where γ is the Lorentz factor (1 − v /c ) − / , the factor √ γ m o is the inertial mass of each neutrino.Noting that γm o v r = L v r , (22)9nd that the angular momentum, L , equals γm o r v it follows that equation (21) implies √ L v = Gm o γ . (23)This equation yields equation (12) provided that L is given by L = n B (2 ℓ B + 1)¯ h. (24)This equation again can be interpreted as a standing wave condition where nowinstead of equation (19) we have r = n B (2 ℓ B + 1)¯ λ, (25)where n B is a positive integer ( n B = 1 , , , ... ) and ℓ B is a non-negative integer (Fig.6).Figure 6: Rotating neutrino model of baryon showing the Bohr-type description ofthe neutrino orbits in terms of the two quantum numbers n B and ℓ B .Indeed substituting equation (25) into the de Broglie wavelength equation, i.e.¯ λ = ¯ hγm o v (26)and using the definition of the angular momentum L , we find equation (24) L = γm o r v = n B (2 ℓ B + 1)¯ h. (27)10s already noted the integer ℓ B is related via 2 ℓ B + 1 = 2 n − n ( n = 1 , , .... ) which we have used in previous studies of the rotating neutrinomodel [3, 4].If γ >>
1, then v ≈ c , thus replacing in (23) v by c and L by (24) we find √ n B (2 ℓ B + 1)¯ hc = Gm o γ . (28)Thus γ = h √ n B (2 ℓ B + 1) i / (cid:18) m P l m o (cid:19) / , (29)where m P l (= ¯ hc/G ) / is the Planck mass. Hence m = 3 γm o = (cid:2) n B (2 ℓ B + 1) (cid:3) / / m / P l m / o . (30)The excellent agreement between equation (30) and the experimental baryon massspectrum is shown in Table 1 and figure 5. In summary, the three quantum numbers of the SU(3) symmetry group of the Stan-dard Model, namely charge, Q , Isospin, I , and strangeness, S , are analogous to thethree quantum numbers, ( Q, n B and ¯ λ B ), which emerge from the rotating neutrinomodel. The baryon charge is determined by the charge, L , of the positron(s) or elec-tron(s) located at the center of the rotating neutrino ring. The maximum isospinvalue is also closely related to the number of leptons, L + and L − , which can beaccomodated at the center of the rotating ring.Strangeness, on the other hand, is closely related both to the first quantum num-ber, n B , and to the second quantum number, ℓ B , which both dictate the value ofthe angular momentum and thus, via the Lorentz factor γ , the mass of the rotatingneutrino baryon state given by m = (cid:2) n B (2 ℓ B + 1) (cid:3) / / m / P l m / o . This expression provides a very good fit ( R > . − (2 / m p [ n B (2 ℓ B + 1)] / c obtained via the rotating model, may be found in thefuture to be in reasonable agreement with the experimental values. However, the11nergies associated with transitions between such states are very high, in the far γ − ray range, and thus may be difficult to obtain. References [1] Griffiths D (2008) Introduction to Elementary Particles. 2nd ed. Wiley-VCHVerlag GmbH & Co. KgaA, Weinheim.[2] Tully CG (2011) Elementary Particle Physics in a Nutshell, ISBN13: 978-0-691-13116-0, Princeton University Press.[3] Vayenas CG, Souentie S (2012), Gravity, special relativity and the strong force:A Bohr-Einstein-de-Broglie model for the formation of hadrons. Springer, ISBN978-1-4614-3935-6.[4] Vayenas CG, Souentie S, Fokas A (2014) A Bohr-type model of a compositeparticle using gravity as the attractive force. arXiv:1306.5979v4 [physics.gen-ph];Physica A 405:360-379.[5] Roll PG, Krotkov R, Dicke RG (1964) The equivalence of inertial and passivegravitational mass. Annals of Physics 26(3):442-517[6] Einstein, A.: Z¨ur Elektrodynamik bewegter K¨orper. Ann. der Physik., Bd. XVII,S. , 891-921 (1905); English translation On the Electrodynamics of MovingBodies ( http : ) by G.B. Jefferyand W. Perrett (1923).[7] French, A.P. in Special Relativity (W. W. Norton and Co., New York, 1968).[8] Freund, J. in
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