Calculating Effective Wilson Coefficients for Kaon Decays in Renormalization Scale =

The decay rates of mesons, consisting of a quark-anti 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 = 1 are calculated.


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 cosmic-ray 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.
s 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. k and k are not eigenstate of CP. However, when CP acts on them, they are conjugate of each other.
But theorists can make a pair kaon with determined CP from combination of wave function k and k .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]: (1-4) 1 k just can decays to CP 1 = + state, while 2 k should go to CP 1 = − 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 C 1 = + . As a result, 1 k decays to two pions and 2 k decays to three pions [11].
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 10 10 s − and 7 10 s − , respectively [3,12].
K 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]: Where F G is the Fermi constant that in terms of the w g weak coupling constant and W boson mass is defined as follows: And i Q are the local operators that decays discussed in

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 (1-6) equation which has already been mentioned in the introduction. Effective Hamiltonian of the → transition is defined as follows [2].
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: In this equation is defined as follows: 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] ! matrix is the 3 × 3 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.
In which, ij ij c cos = θ and ij ij s sin = θ for i, j 1, 2, 3 = . δ is the phase which is in the range of 2 ≤ δ ≤ π . Matrix elements are calculated by using the following data [14]: 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]: Here $ %&& ( ) are defined )*++*, [2]: In these equations, 1 is matrix transpose operator. Matrix with variable dimensions 2 3 (4) and constant matrix V r are obtained from correction of 5 − 54 operators' vertex. Also, the values of % , 7 and are extracted from 8 Penguin diagram, 5,9 operators and QCD Penguin diagram, : − ; operators and electroweak penguin diagrams < − 54 , respectively.
In which, = is parameter that in dimensional regulation, describes dependency 2 > model. For example, in 2 > model we have Naïve Dimensional Regularization (?8@) and Hooft-Veltman (A ): Function (B, , ) in (2-6) equations is defined as follows: In which 9 is the square of the momentum carried by virtual gluons. Ĉ Matrix in (2-9) equations gives constant terms; they are independent of the momentum that is based on the 2 > behavior in the dimensional regularization. For the kaon decays, there was no heavy quark mass scale between B E and B F . Therefore, logarithmic term arising from corrections four quarks operations vertex to ln & ⁄ form will be like (2-9) equation. We will assume that & = 1 as a reliable estimate obtained of destruction effects the effective Wilson coefficients. As a result, we have [2,4]:      Table 3 shows calculated values for the effective Wilson coefficients for the decay of quark and antiquark in renormalization scale µ=1GeV.