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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. Yamaleev, Advances in Applied
       6(2), 275 (1996); Keller J. Rodr´
       Clifford Algebras 8(2), 235 (1998).
 [7]   Biedenharn L. C. , Remarks on relativistic Kepler problem, Phys. Rev.126 845-
       851 (1961); Biedenharn L. C. , The Sommerfeld Puzzle revisited and resolved,
       Foundations of Physics 13 13-34 (1983).
 [8]   Phys. Rev. D 54, Part. 1 (1996); Phys. Rev. D 50, Part.1 (1994).
 [9]   Bang J., F. A. Gareev, W. T. Pinkston, J. Vaagen, Phys. Rep. 125, 253 (1985).
[10]   Stahlhofen A. A., Algebraic solutions of relativistic Coulomb problems, Hel-
       vetica Physica Acta 70 (1997) 372.

								
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