American Journal of Physics and Applications
Volume 3, Issue 6-1, December 2015, Pages: 1-5

Calculating Effective Wilson Coefficients for Kaon Decays in Renormalization Scale μ=1 GeV

Khadijeh Ghasemi1, Somayeh Mehrjoo2

1Physics Department, Islamic Azad University, Central Tehran Branch, Tehran, Iran

2Physics Department, Lorestan University, Lorestan, Iran

Email address:

(Kh. Ghasemi)
(S. Mehrjoo)

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. 6-1, 2015, pp. 1-5. 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 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  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 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.


 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.


 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]:




 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].


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]:


Where  is the Fermi constant that in terms of the  weak coupling constant and  boson mass is defined as follows:


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 point-like 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 (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  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,  and  for .  is the phase which is in the range of . 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 as follows [2]


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.


In which,  is parameter that in dimensional regulation, describes dependency  model. For example, in  modelwe have Naïve Dimensional Regularization  and Hooft-Veltman:


Function  in (2-6) equations is defined as follows:


In which  is the square of the momentum carried by virtual gluons. Matrix in (2-9) 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 (2-9) equation. We will assume that  as a reliable estimate obtained of destruction effects the effective Wilson coefficients. As a result, we have [2, 4]:




3. Conclusion

Table 1.  in renormalization scale .


Table 2.  in renormalization scale .


Table 3. The effective Wilson coefficients in renormalization scale μ=1GeV.


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.


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