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ON QUANTIZATION OF THE DECAY PRODUCTS MOMENTA IN THE ELEMENTARY PARTICLE RESONANCES Fangil A. Gareev Joint Institut for Nuclear Research, Dubna, Russia Jaime Keller∗ o ımica and Divisi´n de Estudios de Posgrado, Facultad de Qu´ a Facultad de Estudios Superiores–Cuautitl´n o e Universidad Nacional Aut´noma de M´xico e e A. Postal 70-528, 04510 M´xico, D. F., M´xico e-mail: keller@servidor.unam.mx and Robert M. Yamaleev a Facultad de Estudios Superiores–Cuautitl´n o Universidad Nacional Aut´noma de M´xicoe On leave: Joint Institut for Nuclear Research, Dubna, Russia e-mail: iamaleev@servidor.unam.mx (Received: September 28, 1998, Accepted: October 30, 1998) Abstract. Based on the comparison of our analysis of the relativistic spherical sym- metric top and the experimental evidence we suggest a formula to calculate families of mass distributions of elementary particle resonances from the masses and expected predecay momenta of the binary decay products. Good description of experimetal data is achieved once we accept the hypothesis that predecay decay products mo- menta is quantized. Using this principle the resonance decay channels may be col- lected into families. Inside these families decay product momenta are quantized using a characteristic length for each family, that in the model is the radius of the sphere of the topan should correspond to a predissociation radius, equivalent to the range of the potential. Elementary particle resonances are systematically analyzed within ∗ Author to whom all correspondence should be addressed Advances in Applied Cliﬀord Algebras 8 No. 2, 255-270 (1998) 256 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev all available experimental data. From the good agreement with those observations we can suggest a strategy for experimental searches of new resonances besides the systematization of the already known ones. 1. Introduction The possibility of the construction of a fundamental theory is often created by the discovery of an empirical formula describing the phenomena with good accuracy. In this context let us remind one of the best examples: the Balmer formula, which preceeded the Bohr model for the hydrogen atom. In the context of strong interactions Balmer-like formulae have been suggested several times using group theory, the groups are considered and named as Dynamical Groups. In those approaches, hadrons are basically considered as composite extended relativistic objects, where the motion of the center of mass of the extended objects as a whole adds to the internal motions, the last ones consist of some collective motions and the relative motion of the assumed constituents. The relativistic elementary particles are classiﬁed according to the conti- nouos unitary irreducible projective representations of the Poincar´ group P. e The eigenvalues m2 and j(j +1) of invariant operators of P uniquely character- ize these representation spaces of P. Thus elementary particles are character- ized with two parameters m2 and j(j + 1) with interpretation as the mass and e the spin, corresponding to the the generators Pµ and Jµν of the Poincar´ group considered as the momentum and angular momentum operators. But it may happen that the distinction between the notions of momentum and angular momentum is only conventional, one well known example is the case in a space with constant curvature where both notions are undistinguishable. Indeed, a translation on the surface of the sphere is also a rotation. The equivalence be- tween the two types of motion is established via a constant with dimension of length: the radius of the sphere. Well known results in high-energy physics indicate that there is a profound connection between spins and masses for the strongly interacting elementary particles, hadrons. The spin J of some baryons and mesons appears to be nearly proportional to the square of their mass m: m2 ∝ J. The correlation between spin and mass of experimentally known low mass hadrons is well known to be represented by a straightline Regge trajectory. Some masses of hadrons follow a simple empirical formula [1],[2]: m2 = m2 + J/R2 , ( = 1, c = 1), where (m, j) 0 is (mass, spin), and R is a constant with dimension of length— an elementary length, equivalent to the range of the potential. The large number of hadrons discovered and the relations observed among them, such as the above mass-spin formula, do not permit to consider all of them as truly elementary particles. Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 257 Instead, one is led to the possibility that they are better considered as states of one single structured relativistic quantum system. Thus, the hadron mass spectrum can be connected to a hadron structure. The actual calculation of the dynamics of a resonance is an impracticable task, but in this paper we consider the particular moment when the two decay products are ﬁrst out of range for a mutual interaction and no potential energy is involved in the calculation. This happens at a distance R for a given family of resonances, distance at which the resonance ceases to exist and two decay products are the only objects to consider. In that moment there is still no translational energy of the decay products and only the orbital kinetic energy is relevant for each of the two decay products. This spin rotational energy is described with the analysis of the relativistic spherically symmetric top, where that kinetic energy is quantized. Otherwise said: we do not attempt to describe the resonance but the total energy at the threshold of the decay, this is dominated by the symmetry as the state is described as a collection of two tops in the same rotational state. From this we extend the consideration made in [3] we have considered the possibility of commensurability of the decay momenta of the elementary particle resonanses. In this paper we give a formal foundation of that assumption. In particular, we suggest for binary decay of the resonances formula for the mass operator M of those elementary particle resonances in terms of the two particle decay products, the eigenvalues of which are M= 2 m2 + Pn + a m2 + Pn , b 2 (1.1) where ma and mb are the masses of decay products of the resonance, and Pn is the decay products asympthotic momenta. We show, that a good description of the experimetal data is achieved when we use as a result of the model the hy- pothesis of quantization of decay product momenta. Within this principle the resonance decay channels may be classiﬁed into families, which have the prop- erties of the model quantum objects consisting of two relativistic spherically symmetric tops. Inside each family the decay products momenta are quantized using only one parameter R as a characteristic length of the family. The theoretical analysis for the quantization of momentum for the spinning particles on the surface of a sphere was done in [5] and further developed in [6]. In Sec. 2 we develop the top model as applied to the decay products. We obtain the concept of quantization of decay product momenta, according to which the dynamics of the system is derived from SU (2) group generators. Twistor and Cliﬀord algebra is used. 258 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev In Sec. 3 we present the tables where the invariant masses of resonances decaying through binary channels and the results of sets of calculations are compared with experimental data. 2. Action of SU (2) Group on Yhree-dimensional Sphere and Quan- tization of Momentum In the procedure of quantization of some classical systems it is found that the SU (2) group plays a fundamental role in the process of quantization of energy o and momentum. Let us recall for example, the Schr¨dinger-Coulomb problem where the SO(4) symmetry is used to obtain the spectrum of hydrogen atom. o The SO(4) symmetry of the Schr¨dinger-Coulomb problem, generated by the angular momentum of the central system M = [r × p] and the normalized Runge-Lenz vector r A = (−2mH)−1/2 ( + (2mα)−1 (M × p − p × M )) r results in 1) That the eigenvalues follow direcly from the invariant operator (Casimir operator) of SO(4) by rewriting the Hamiltonian as 1 α2 H= → −mc2 , n = 0, 1, 2, ...; (2.1) 2(((M ± A), σ) + 2 )2 2(n + 1)2 these eigenvalues (including degeneracy) are given by well known standard group theoretical arguments [7]. 2) The eigenstates for a ﬁxed value of n can be constructed by employing shift operators of SO(4). In an appropriate basis classiﬁed by the quantum numbers (l, m) these shift operator have the form [10].: K = A(D + 1) − 1 2 2 1 ( 1 2 [L , A] with D = (L + 2 ) 1/2) − 2 . The component K3 of K acts on the three-dimensional eigenstates |nlm > of the Coulomb problem with |nlm >= ξnl (r)Ylm (θ, φ) as ∂ l 1 K3 ξnl (r)Ylm (θ, φ) = Yl−1,m (θ, φ)( + − )ξnl (r) = Yl−1,m (θ, φ)ξnl−1 (r). ∂r r r The operators K± = K1 ± iK2 lead to K± |nlm >= |nl − 1m ± 1 > . In this form A and M suﬃce to derive the bound states of the nonrelativistic Coulomb problem. Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 259 Now we should remember the known isomorphism between the SO(4) and the SU (2) groups, which, in the vector- parametrization of the SO(4) group, is given by ˆ (1+ˆ T [SO(4)] = √ a+ )(1+b−2) = T+ (a)T− (b), 2 (1+a )(1+b ) it is important that we can also obtain the geometrical interpretation of this process. For that purpose let us deﬁne the set of generators of the SO(4) group M = [r × p], N = r4 p − rp4 . The linear combinations of these orthonormal operators M± = (M ±N ), contribute two set of generators of the SU (2) group. Thus the SU (2) group generate in fact the action on a three-dimensional sphere S 3 . This action consists of the translation with whirling arround the direction of translation. The operator N on S 3 can be written as N = Rp + r(rp)/R, where R is the radius of the four dimensional sphere. Comparing this vector with the normalized Runge-Lenz vector we ﬁnd for Z = 1 case of the hydrogen atom 2 R0 = , (2.2) me2 or the Bohr radius. The mass spectrum formula, which we are using here, is based on applying to each of the two decay products a top-model for the notion of the resulting particles [5], [6]. According to this concept the quantum top equations have to be formulated on S 3 . In this formulation the Hamiltonian of the spinning particle is expressed purely in terms of the generators of the SU (2) group. We get 1 H= {2 + (M± , σ)}{2 + (M± , σ)}, (2.3) 2mR2 where M± = (M ± N ). The spectrum of H may easily be found 2 HΨn = (n + 1)2 Ψn , n = 0, 1, 2, ... . (2.4) 2mR2 As in fact this Hamiltonian corresponds at the classical level to the kinetic energy of a spherical symmetric classical top with Hamiltonian on S 3 : J2 H= 2I , where I is a moment of inertia. 260 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev The discreteness of the energy spectrum is a consequence of the compactness of the group SU (2), the space of which is the space of the solutions. When R → ∞, the Hamiltonian tends to the Hamiltonian of the Pauli equation. In this case M± /R = (M ± N )/R → ±p, and 1 1 H= {2 + (M± , σ)}{2 + (M± , σ)} → (p, σ)2 . 2mR2 2m Let us now generalize this concept to the relativistic case in the same form as the generalization of the Pauli equation into the Dirac equation. This procedure proceeds as the following scheme: (σ M + 2 ) (σ M + 2 ) 2mH = → R R HD HD (σ M + 2 ) (σ M + 2 ) →( − mc)( + mc) = → c c R R HD c − mc (σ M± + 2 )/R → Det HD =0→ (σ ± + M)/R 2 c + mc HD αM± 2 0 I → Ψ± = + βmc + γ5 Ψ± , γ5 = . (2.5) c R R I 0 From (2.4) the spectrum of this Hamiltonian (2.5) is found: 2 (n+ 1)2 E = c m2 c2 + , n = 0, 1, 2, ... . (2.6) R2 From a physical point of view, the formula (2.6) corresponds to the quantum properties of a classical top. Indeed, at the classical limit we obtain c E= m2 c2 R2 + J 2 . (2.7) R However, let us remind an essential diﬀerence between (2.6) and (2.7), because at the quantum level the spectrum of (2.7) would be given by c E= m2 c2 R2 + n(n + 1). R We show below, that the use for each resonance’s decay product of the for- mula (2.6) gives to experimental accuracy, the spectrum of the mass of the Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 261 original resonances. The resonances are postulated to decay into two relativistic spherically symmetric tops in the same quantum angular momentum states “n” each, in principle they could have opposite azimuthal quantum number (which does not contribute to the energy). 3. Formula for Mass Distribution of Elementary Particle Resonances Let us consider in accordance with our model the decay chanels of resonances consisting of two decay products. For example, the mesons ρ(770), ρ(1450) and ρ(2150) decay through two pions π ± π with momenta 358, 719 and 1075.99 MeV/c. Let us as a ﬁrst approximation calculate the ratio of these values, to acceptable accuracy to ﬁnd that these values are commensurable, i.e. 719 1075.99 358 ≈ ≈ . 2 3 Further, the simple numerical analysis of the values of the momenta of the de- cay process of the resonances f2 (1430), f2 (1565), f2 (1640), f2 (1810), f2 (2150) through two pions π ± π also leads to conclusion on commensurability of these values within the accuracy of experimental data. Some resonances have a dom- inant decay channel, and we suggest that the momentum of this channel should manifest itself in properties of decays through other channels. For example, it is known that the pion π ± decays into a muon and a neutrino with a probability near unity and asymptotic momentum P1 = 29.7918M eV /c. A simple calcu- lation shows that the momenta of the decay products of the resonances above are proportional to P1 . Thus, within the accuracy of experimental data we conclude that we can start from the working hypothesis that Pn = nP1 , where n is integer. We suppose, that our observation reﬂect the common property of the decay, not of the actual very complicated constitution dynamics, of all elementary particle resonances. Taking into account this proposition, and our analysis above, the mass spectrum operator for resonances can be formulated in the following way 1 1 M (R) = m2 + a ((σ M) + 2 )2 + m2 + b ((σ M) + 2 )2 , (3.1) R2 R2 where ma and mb are the masses of the decay products of the resonance, R is the characteristic length of the group of resonances. The spectrum of this operator is obviously given by Mn = 2 m2 + Pn + a m2 + Pn , b 2 (3.2) 262 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev where here Pn = nP0 , P1 = R . We have then considered that there is a universal form of decay of the resonances. A theoretical model not of the resonance state itself but of its decay process would be that the potential has a well deﬁned range and that once the decay products are farther than the distance they correspond now to a non interacting couple of relativistic spherically symmetric tops. In Table 1 we join experimental data (taken from [8]) into one group, decay product momenta of which are quantized accordingly value P1 = 29.7918M eV /c. In this table the usefulness and accuracy of formula (3.2) is clearly demon- strated. Let us remark that the model results in the decay products momenta commensurability which does not depend on the type of interaction between resonance decay products, quantum numbers of resonances or type of particles. It is a universal property, as a symmetry of a process requires, and not a result of the details of the particles or their interactions. Table 1. Invariant masses of resonances decaying through binary channels with momenta Pn = n ∗ 29.7918 M eV /c, ∆M =| Mexp − Mth |. Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 263 The ideas leading to table 1 are further corroborated by the study of other groups of resonances. In the following group we join the X and B resonances decaying through pp and pp. Results are in Table 2. Table 2. Invariant masses of resonances decaying through binary channels with momenta Pn = n ∗ 29.7918 MeV/c. ∆M =| Mexp − Mth |. We see from tables 1, and 2 that our simple postulate, leading to a phe- nomenological method correctly describes the experimental data within the accuracy of measurements. This approach is unusual and unexpected for this branch of physics. The results of the χ2 calculations are given in [4] (see, Fig. 1) as a function of decay momentum P including about four hundred experi- mental data. Well pronounced deep minimum χ2 is found at P1 = 29.79 MeV/c which corresponds to the above-mentioned decay momentum. 264 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev If we change P1 = 29.79 MeV/c to P1 = 29.79 ± 0.3 MeV/c, the magnitude of χ2 is increased three times. Therefore the mass distribution of resonances is a sensitive function of the basic decay momentum and the observed good de- scription of the experimental data is not accidental. It means that our postulate manifests a simple physics of the decay of resonances hitherto unnoticed. Now let us consider another family of resonances. Namely, let us exam- ∗ ∗ ∗ ine the resonances D2 (2460)0 , BJ (5732), BsJ (5850) and Υ(1S) with decay ± channels through µ e with momenta 1227.18, 2848.02, 2925.55 and 4729.59, correspondingly. Thus we can ﬁnd with good accuracy, that 1227.18 2848.02 2925.55 4729.59 ≈ ≈ ≈ ≈ 26.12. 47 109 112 181 This ratio is now very close to the momentum of the low probability decay channel π 0 → µ± e is equal 26.1299 MeV/c. However the fraction of this channel is equal ≈ 10−8 %. Of course, in that case a question arises on the role of resonance decay channels with a very small probability. We can reason here, in analogy with nuclear physics (see, for example, [9]), that basic vectors with small (large) weights in the ground states of nuclei become the dominant (small) ones in highly excited states. The same phenomenon is expected in the physics of resonances. Thus we choose as a basic channel the decay channel π 0 → µ± e . The change in P1 results in a change in R = /P1 , indicating that most probably the potential between charged leptons has a slightly larger eﬀective range than the potential between charged and uncharged leptons, but our method can not directly prove anything about the interaction itself. The results of calculations are presented in Table 3. Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 265 Table 3. Invariant masses of higher excited resonances decaying through binary channels with momenta Pn = n ∗ 26.1299 M eV /c, ∆M =| Mexp − Mth |. 266 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev Continuation of Table 3. Tables 1, 2, 3 contain rich information, and it is possible to make several fundamental conclusions based on it. We think that the results contained in the tables convincingly demonstrate that resonance decay product momenta are quantized. It is clear from the Tables that commensurability of momenta does not depend on the type of interaction between resonance decay products, quantum numbers of resonances and type of particles. Moreover, commensurability of decay products momenta is justiﬁed for all considered resonances. It seems to be a universal property of resonances. An excellent possibility for prediction of new resonances and veriﬁcation of masses of the existing ones arises in any case. The method presented here may also be used to calculate mass distribu- tions of “resonances” with hypothetical decay channels as chanels of vanish- ing probability. Let us consider one important example. The proton is prac- tically stable (mean lifetime τ > 1.6 ∗ 1025 years — independent of the de- cay mode [8]). According to some proposals like, for example, the minimal SU (5) Grand Uniﬁed Theory it can decay via diﬀerent channels. For example, p → e+ π 0 , p → µ+ γ, p → νK ∗ (892)+ , .... But the proton does not decay despite there is no restriction arising from energy-momentum conservation. We decided to investigate some of these channels, for example, p → νK ∗ (892)+ . The masses of p, ν, K ∗ (892)+ are known. So we are able to calculate the de- cay momentum P1 and then to evaluate the masses of resonances that could be related in a family through a formula (1.1) where now m1 , m2 are the masses of hypothetical decay particles of this family (ν and K ∗ (892)+ ) and P1 is the momentum of their relative motion. The results of our calculations and the correspondence with experimental data [8] are illustrated in Figure 1 and Figure 2. The X-axis characterizes the families of resonances (baryonic Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 267 or mesonic) and the Y -axis presents their masses (in MeV). The ﬁgures show that momentum P1 proposed as generator of the families of resonances. Figure 1. The mass distribution of baryonic resonances with momenta mul- tiples of 45.52 MeV/c. The basic momentum is taken from the hypothetical channel p → νK ∗ (892)+ The method presented above is able to describe large amounts of existing experimental data. A comparison of our calculations with experimental data of the momenta and the masses of the elementary particle resonances suggests immediately practical strategies for experimental searches of still nonobserved resonances. 268 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev Figure 2. The mass distribution of mesonic resonances with momenta mul- tiples of 45.52 MeV/c. The basic momentum is taken from the hypothetical channel p → νK ∗ (892)+ 4. Conclusions We showed that the systematical analysis of available experimental data of elementary particle resonances gives the empirical fact that resonance decay product momenta and mass of resonances have discrete spectrum. As it has been demonstrated above the mass-spectrum formula (3.2) de- Advances in Applied Cliﬀord Algebras 8, No. 2 (1998) 269 scribes the existing experimental data with high accuracy. In this context this formula indeed can pretend on a role of the Balmer-like formula for masses of elementary particle resonances in accordance with the systematic analysis of experimental data. An excellent possibility for the prediction of new resonances and veriﬁcation of masses of existing ones arises in any case. In fact, we gave the new scheme of systematization of the resonances. The resonance decay channels had been joinned into groups with a charasteristic length. Inside the group the resonanse channels are classiﬁed by the quantum numbers n = Pn /P1 and characteristic length R = /P1 , in the model this should correspond to a predissociation radius, equivalent to the range of the potential. In that sense n is the internal quantum number of the resonance channel and the latter is considered as some range for dissociation. It is justiﬁed for all considered resonances becoming their universal prop- erty. Therefore there arises an excellent possibility for prediction of new reso- nances and veriﬁcation of masses of the existing ones. The physics of our model, decay into two relativistic spherically symmetric tops of opposite, cancelling, angular momenta is extremely simple and the use of the direct analysis with Cliﬀord algebra is shown to be a very powerful tool to analyse physical problems. 5. Acknowledgments We are indebted to M. Yu. Barabanov, G. S. Kazacha (Dubna, JINR) for help in calculations. J. Keller is a member of the National Research System of Mexico. We are grateful to the constructive suggestions of the referee and the CONACYT (Mexico) for ﬁnancial support. This paper, in its present form was presented in the symposium “Algebraic Methods in Mathematical Physics”, September 8-12, 1998, IIMAS-UNAM, Mexico, a complete, longer version will be submitted elsewhere. 270 On Quantization of the Decay . . . F. A. Gareev, J. Keller, R. M. Yamaleev References [1] Regge T., Nuovo Cimento, 14 (1959)951. [2] Chew G. F., S. C. Frautschi, Phys. Rev. Lett. 7, 394 (1961). [3] Gareev F. A., G. S. Kazacha, Yu. L. Ratis, Particles and Nuclei, 27, 97 (1996); Gareev F. A., Yu. L. Ratis, Isv. Ran, ser. ﬁz., 60, 121 (1996); Gareev F. A., JINR Communications P2-96-456, Dubna, 1996; Gareev F. A., JINR Commu- nications E4-97-25, Dubna, 1997. [4] Gareev F. A., M. Yu. Barabanov, G. S. Kazacha, P. P. Korovin, JINR Com- munications E2-98-120, Dubna, 1998. [5] Yamaleev R. M., JINR Communications P2-84-727, Dubna, 1984; P2-85-722, Dubna, 1985; E2-84-197, Dubna, 1984. [6] ıguez and R. M. Yamaleev, Advances in Applied Cliﬀord Algebras Keller J. Rodr´ ıguez and R. M. 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