# Quark decay top to two bodies/Decaimiento del quark top a dos cuerpos.

1. Introduction

The most accepted model in the scientific community for the phenomenological description of elementary particles is the so-called standard model. At present, any model that is designed should at least reproduce the phenomenological predictions of the standard model. With the discovery of the top quark in Fermilab in 1995 by CDF and D0 A new field of research was opened for the study of the physics of the top quark. The standard model predicts the decay of the top quark in a quark bottom and a [W.sup.+] (t[right arrow]b[W.sup.+]) which allows to carry out a research to this kind of process theoretically calculating the decay width and the decay fraction of the particle. For the theoretical calculation, the spectator model is used to compare it with the experimental data obtained from the Particle Data Group table and thus affirm the prediction of the standard model.

Hadrons are particles composed of quarks, which are divided into two classes, the baryons and the mesons. Baryons are particles whose structure is given by the combination of a trio of quarks. The mesons they are particles composed by the combination of a quark and an antiquark. The quarks have an intrinsic property called flavor (Scientific American, 1986). This property allows to account for the decays of the mesons with weak interaction. The different flavor combinations of quarks explain the existence of different kinds of hadrons.

For the physical description of the process of decay of the quark top (t) to two bodies, the quark model or free quark model will be used, in which the active quark of the meson decays independently of the other constituents of the particle as shown in the Figure 1.

2. Decay width

The width of decay of the process is defined by two terms that are the amplitude of Feynman ( M) and the phase space [1-2]. Feynman's breadth has the dynamic information of the process  and the phase space contains the kinematic information, which depends on the mass, energy and momentum of the particles participating in the interaction [4-5]. The decay width of the top quark to two bodies is by .

d[GAMMA] = [1/32[[pi].sup.2]][|M|.sup.2]|[P.sub.]|d[OMEGA], (1)

The Feynman amplitude of the process is defined as:

M = [[-ig]/[square root of 2]][V.sub.tb][bar.U](b)[[gamma].sub.[alpha]][(1 - [[gamma].sub.5])/2]U(t)[[member of].sup.[mu]]([W.sup.+]), (2)

which is calculated using the Feynman diagrams and rules for the corresponding interaction. Taking the square of the absolute value of the Feynman amplitude and replacing the spinorial products by the projection operators ,

[summation over (s)]U(t)[bar.U](t) = [[gamma].sup.[mu]] [P.sub.[mu]] + [m.sub.[tau]] = [P.sub.t] + [m.sub.[tau]]

[summation over (s)]U(b)[bar.U](b) = [[gamma].sup.[mu]] [P.sub.b] + [m.sub.b] = [P.sub.b] + [m.sub.b](3)

Is obtained,

[|M|.sup.2] = [[g.sup.2]/2][|[V.sub.tb]|.sup.2]Tr[([P.sub.t] + [m.sub.t])[P.sub.R][[gamma].sub.[mu]]([P.sub.b] + [m.sub.b])[[gamma].sub.[alpha]][P.sub.L]][-[g.sub.[mu][alpha]] + [W.sub.[mu]][W.sub.[alpha]]/[m.sup.2.sub.w]], (4)

being,[P.sub.R] = [(1 - [[gamma].sub.5])/2]y[P.sub.L] = [(1 - [[gamma].sub.5])/2]

Evaluating the traces of the previous equation, you get the dynamics of the process:

[|M|.sup.2] = [[g.sup.2]/2][|[V.sub.tb]|.sup.2][[P.sub.t]*[P.sub.b] + 2][([P.sub.t]*W][P.sub.b]*W)]/[m.sup.2.sub.w]](5)

The kinematic contribution of the process, [P.sub.t]*[P.sub.b] [P.sub.t]* W; [P.sub.b] * W it is carried out for the conservation of energy and the moment [7-8]:

[P.sub.t]*[P.sub.b] = [1/2]([m.sub.t.sup.2] + [m.sub.b.sup.2] - [m.sub.w.sup.2]); [P.sub.t]*W = [1/2]([m.sub.t.sup.2] + [m.sub.W.sup.2] - [m.sub.t.sup.2] - [m.sup.b.sub.2]);

[P.sub.t]*W = [1/2]([m.sub.t.sub.2] + [m.sup.W.sub.2] - [m.sub.b.sub.2]). (6)

Replacing the previous equations in (5), the square of Feynman's amplitude finally remains:

[|M|.sup.2] = [[g.sup.2]/2][|[V.sub.tb]|.sup.2][[m.sup.t.sub.2] + [m.sup.b.sub.2] - [m.sup.w.sub.2] + [m.sup.t.sub.4] - 2[m.sup.t.sub.2][m.sup.b.sub.2] - [m.sup.4.sub.W] + [m.sub.b.sup.4])/[m.sub.w.sup.2]]. (7)

The decay width of the top quark is obtained by replacing (7) in (1),

[GAMMA]([tau] [right arrow] b[W.sup.+]) = [G.sub.f][|[V.sub.tb]|.sup.2]/4[square root of 2][pi][m.sub.t.sup.2][[m.sub.W.sup.2][([m.sub.t.sup.2] + [m.sub.b.sup.2] - 2[m.sub.w.sup.2]) + [([m.sub.t.sup.2] - [m.sub.b.sup.2]).sup.2].sup.2]]*|[P.sub.b]|, (8)

where,

The constant |[V.sub.tb]| = 0.999146 is a parametrization element of the Cabibbo-Kobayashi-Maskawa matrix and [G.sub.f] Fermi's constant . Finally, equation (8) allows calculating the decay width of the top quark to two bodies.

3. Results

Table 1, shows the data that are used to calculate the decay width and the decay fraction of the top quark.

The theoretical decay width of the top quark is:

[GAMMA](t[right arrow]b[W.sup.+]) = 1.504453913 GeV, (9)

and the experimental is given by,

[GAMMA](t[right arrow]b[W.sup.+]) = 1.99 GeV,(10)

In terms of seconds (-1),

[GAMMA](t[right arrow]b[W.sup.+]) = 2.286769947x[10.sup.24][s.sup.-1]. (11)

The total decay width of the quark is:

[[GAMMA].sub.total] = [1/[tau]] = [1/0.5x[10.sup.-24] s = 2x[10.sup.24] [s.sup.-1]. (12)

and the decay fraction is:

Decay fraction = [[[GAMMA](t[right arrow]b[W.sup.+])]/[[GAMMA].sub.total]] = 1.143%. (13)

Table 2, presents a comparative table between the theoretical calculation of the decay of the top quark to two bodies using the spectator model and the experimental data obtained from the Particle Data Group table .

The model prediction of the decay process (t [right arrow][W.sup.+]) It can be considered good. However, the viewer model does not consider the linked states in which the quark is found, leading to a limitation of the model. Models that take into account bound states , show the equation of the system with a potential of ligature with which results are obtained more precise with respect to the experimental data.

It is important to note that from the elements of the Cabibbo-Kobayashi-Maskawa matrix 

[mathematical expression not reproducible]

the transitions between quarks q [right arrow] q' from the same family, as shown in Table 3, are more likely to occur than those carried out in different families. Experimentally, the magnitudes of the matrix elements are given by :

[mathematical expression not reproducible]

From the experimental data given by the previous matrix, the decay process (t [right arrow] b[W.sup.+]), considering the constant |[V.sub.tb]| [approximately equal to] 1, It is more likely to occur than the processes carried out between quarks of different families.

In general, the processes of decay that take place between quarks of the same family are more likely to occur than those that occur between quarks of different families.

4. Acknowledgements

This paper has been made with the support of the Universidad Pedagogica Nacional through the project DFI-307-12 of research Center CIUP.

References

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 N. Isgur, D. Scora, B. Grinstein and M. Wise, "Semileptonic B and D decays in the quark model", Physical Review D, vol. 39, no. 3, 1989, pp. 799-818. https://doi.org/10.1103/PhysRevD.39.799

Mauricio Rozo-Clavijo (1), Carlos German Cortes-Hernandez (2), Javier Alberto Olarte-Torres (3)

(1) BSc. In Physical, Universidad Pedagogica Nacional, Colombia. Specialization in Physical Sciences, Universidad Nacional de Colombia, Colombia. MSc. In Physical Sciences, Universidad Nacional de Colombia, Colombia. Current position: Professor at Universidad Pedagogica Nacional, Colombia. E-mail: mclavijo@pedagogica.edu.co. ORCID: https://orcid.org/0000-0001-6427-8608.

(2) BSc. In Physical, Universidad Pedagogica Nacional, Colombia. Current position: Professor at Colegio Parroquial Confraternidad de la Doctrina Cristiana, Colombia. E-mail: matfisast@hotmail.com. ORCID: https://orcid.orq/0000-0002-9720-1305.

(3) BSc. In Physical, Universidad estatal de Tbilisi. MSc. In Physical Sciences, Ph.D. In Physical Sciences, Universidad Nacional de Colombia, Colombia. Current position: Professor at Universidad Distrital Francisco Jose de Caldas, Colombia. E-mail: jaolartet@udistrital.edu.co. ORCID: https://orcid.org/0000-0002-9173-3685.

Cite this article as: M. Rozo-Clavijo, C. G. Cortes-Hernandez and J. A. Olarte-Torres, "Quark decay top to two bodies", Vision electronica, algo mas que un estado solido, vol. 13, no. 2, july-december 2019, pp. Xx, DOI: xx

Fecha de envio: 1 mayo de 2019

Fecha de recepcion: 15 de mayo de 2019

Fecha de aceptacion: 3 de junio de 2019
```Table 1. Experimental data obtained from the table Particle Data Group,
.

Name                   Symbol      Value

Fermi constant         [G.sub.f]     1.16639 x[10.sup.-5]Ge[V.sup.-2]
quark mass t           [m.sub.t]   173.5 GeV
quark mass b           [m.sub.b]     4.65 GeV
quark mass [W.sup.+]   [m.sub.w]    80.385 GeV
Average life of t      t             0.5x[10.sup.-24] S

Table 2. The table indicates the fraction of decay predicted by the
spectator model and the experimental data obtained from the Particle
Data Group table, .

Spectator Model (%)   Experimental Data (%)   M. Espectator--D.
Experimental
Error

1.14              0.91 [+ or -] 0.04         5[sigma]

Table 3. The table indicates the classification of the different
families of quarks according to the standard particle model, .

First family   Second family   Third family

u              c               t
d              s               b
```