Calculating Effective Wilson Coefficients for Kaon Decays in Renormalization Scale μ=1 GeV
Khadijeh Ghasemi^{1}, Somayeh Mehrjoo^{2}
^{1}Physics Department, Islamic Azad University, Central Tehran Branch, Tehran, Iran
^{2}Physics Department, Lorestan University, Lorestan, Iran
Email address:
To cite this article:
Khadijeh Ghasemi, Somayeh Mehrjoo. Calculating Effective Wilson Coefficients for Kaon Decays in Renormalization Scale . American Journal of Physics and Applications. Special Issue: Physics Quantum. Vol. 3, No. 61, 2015, pp. 15. doi: 10.11648/j.ajpa.s.2015030601.11
Received: September 25, 2015; Accepted: October 24, 2015; Published: March 14, 2016
Abstract: The decay rates of mesons, consisting of a quarkanti quark, as a weak decay in the presence of strong interactions have been studied by means of the Effective Hamiltonian Theory. One of the most important key factors for calculating Effective Hamiltonian is Wilson coefficients. In this paper, effective Wilson coefficients in renormalization scale are calculated.
Keywords: K Meson, Effective Hamiltonian, Wilson Coefficients, CKM Matrix
1. Introduction
One of the successful models in particle phenomenology is the quark model which is applied to calculate the decays of various particles with a few differences. The particles called kaons, or K mesons, were first observed in the late 1940s in cosmicray experiments. By today’s standards, they are common, easily produced, and well understood. Over the last four decades research into how kaons decay has played a major role in the development of the Standard Model. Yet, after all this time, kaon decays may still prove to be a valuable source of new information on some of the remaining fundamental questions in particle physics.
When first observed, kaons seemed quite mysterious. Experiments showed that they were produced in reactions involving the strong force, or strong interaction—the most powerful of the four fundamental forces in nature—but that they did not decay (that is, transform into two or more less massive particles) through the strong interaction. This is because kaons have a property, ultimately labeled "strangeness," which is conserved in the strong interaction [10].
One of the most interesting and unique observed particles in the nature is kaon. There are two neutral kaons which are, in fact, strange mesons.
(11)
is the Eigenvalue of the strange state. Since each kaon under CP effect turns into another kaon, neither of these kaons have determined CP number. and are not eigenstate of CP. However, when CP acts on them, they are conjugate of each other.
(12)
But theorists can make a pair kaon with determined CP from combination of wave function and .According to Quantum Mechanics rules, these combinations corresponding with real particles and have a mass and determined lifetime. Therefore normalized eigenstate CP are [3, 9]:
(13)
So,
(14)
just can decays to state, while should go to state. Neutral kaons usually decay to two or three pions. Arrangement of two pions has +1 parity and three pions system has 1 parity and both of them have a . As a result, decays to two pions and decays to three pions [11].
(15)
Since a kaon has hardly enough mass to produce three pions, two pion decays are fast but three pion decays are longer. Observed lifetimes are about and , respectively [3, 12].
mesons decay as a weak decay in the presence of strong interactions requires a special approach. The main tool to investigate these decays is the effective Hamiltonian theory. Beginning of any phenomenological weak decay of hadrons is the effective weak Hamiltonian that its structure is as follows [4, 6]:
(16)
Where is the Fermi constant that in terms of the weak coupling constant and boson mass is defined as follows:
(17)
And are the local operators that decays discussed in turn controlled. Cabibbo – Kobayashi – Maskawa factors and Wilson Coefficients are described the force with which an operator enters the Hamiltonian. In fact, the effective pointlike vertices are represented by local operators can correct picture of the decay of hadrons with a mass of the order of a better way to provide. The Wilson coefficients to be used as coupling constants (depending on scale) corresponding to the vertices are considered. Select the scale is optional, but it is customary that to choice the order of the mass of hadrons decay, eg forand mesons decays, the value of are respectively the order of and . For kaon decays the common choice of is the order of instead of order [1].
2. Theoretical Framework
In this paper, Wilson coefficients of quark and antiquark decays are calculated [2]. General framework of how to calculate Wilson coefficients is based on that (16) equation which has already been mentioned in the introduction. Effective Hamiltonian of the transition is defined as follows [2].
(21)
In this equation, is the Fermi constant and is the local operator which controls the decay. coefficients are showed Wilson coefficients. The overall structure of the Wilson coefficients is as follow:
(22)
In this equation is defined as follows:
(23)
In the τ equation , and are the elements of the Cabibbo – Kobayashi – Maskawa matrix. Cabibbo – Kobayashi – Maskawa matrix is a unitary matrix which contains information on the strength of flavor changing weak decays. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions [3, 13].
(24)
matrix is the matrix, since there are three generations of quarks, which Kobayashi and Maskawa in 1973 stated that the third generation of quarks to the matrix, mixed phases that, if not zero, it is symmetry breaking. If this phase is virtually zero, to explain the CP failure must seek something beyond the standard model.
Several methods have been proposed for matrix parameterization which among them to discuss the introduction of standard parameterization.
(25)
In which, and for . is the phase which is in the range of . Matrix elements are calculated by using the following data [14]:
(26)
(27)
To obtain quark decay rate, we need the effective Wilson coefficients of the tree and penguin decay. The effective Wilson coefficients can be defined as follows [3]:
(28)
Here are defined as follows [2]
(29)
In these equations, is matrix transpose operator. Matrix with variable dimensions and constant matrix are obtained from correction of operators’ vertex. Also, the values of, and are extracted from Penguin diagram, operators and QCD Penguin diagram, operators and electroweak penguin diagrams , respectively.
(210)
In which, is parameter that in dimensional regulation, describes dependency model. For example, in modelwe have Naïve Dimensional Regularization and HooftVeltman:
(211)
Function in (26) equations is defined as follows:
(212)
In which is the square of the momentum carried by virtual gluons. Matrix in (29) equations gives constant terms; they are independent of the momentum that is based on the behavior in the dimensional regularization. For the kaon decays, there was no heavy quark mass scale between and . Therefore, logarithmic term arising from corrections four quarks operations vertex to form will be like (29) equation. We will assume that as a reliable estimate obtained of destruction effects the effective Wilson coefficients. As a result, we have [2, 4]:
(213)
(214)
(215)
3. Conclusion
Table 1. in renormalization scale .






























Table 2. in renormalization scale .
















































By using the effective Lagrangian density of the weak interaction, we can calculate decay rate in tree level. Furthermore, decay rates of S quark –anti quark can be calculated in tree and penguin level by the use of the effective Hamiltonian Theory. This is possible by the means of Effective Wilson coefficients.
In this paper, Effective Wilson coefficients are calculated. In table 1 values were calculated. Moreover, the numerical values of are showed in the table 2 [5]. In conclusion, Table 3 shows calculated values for the effective Wilson coefficients for the decay of quark and antiquark in renormalization scale μ=1GeV.
References