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Oscillations of Neutral K Mesons in the Theory of Dynamical Expansion Of XJ9900029 E2-98-125 Kh.M.Beshtoev OSCILLATIONS OF NEUTRAL K MESONS IN THE THEORY OF DYNAMICAL EXPANSION OF THE WEAK INTERACTION THEORY OR IN THE THEORY OF DYNAMICAL ANALOGY OF THE CABIBBO-KOBAYASHI-MASKAWA MATRICES 30-07 1998 1 INTRODUCTION At the present time, the theory of electroweak interactions, has a status of theory which is confirmed with a high degree of precision. However, some experimental results (the existence of the quarks and the leptons families, etc.) did not get any explanation in the framework of the theory. One part of the electroweak theory is the existence of quark mixings introduced by the Cabibbo-Ivobavashi-Maskawa matrices (i.e., these matrices are used for parametrization of the quark mixing). In previous works [1] a dynamical mechanism of quark mixing by the use of four doublets of massive vector carriers of weak interaction B±,C±,D±,E±, i.e., expansion of the standard theory of weak inter­ action (the theory of dynamical analogy of the Cabibbo-Cobayashi- Maskawa matrices) working on the tree level, was proposed. This work is devoted to the study of A'0, A'0 oscillations in the the­ ory of dynamical analogy of the Cabibbo-Cobayashi-Maskawa matrices. At first, we will give the general elements of the theory of dynamical analogy of the Cabibbo-Cobayashi-Maskawa matrices, then the A'0, A'0 oscillations will be considered in this theory. 2 The Theory of Dynamical Analogy of the Cabibbo- Kobayashi-Maskawa matrices In the case of three families of quarks the current -J1' has the following form: ( d\ J11 = (uct) L YV (1) \b)L f Vud Vus V = Vcd Vcs Vch K Vtd Vts Vtt j where V is Kobayashi-Maskawa matrix [2], 1 Mixings of the d, s, b quarks are not connected with the weak in­ teraction (i.e., with Mz±, Z° bosons exchanges). From equation (1) it is well seen that mixings of the d, s, b quarks and exchange of W±,Z° bosons take place in an independent manner (i.e.. if matrix V were diagonal, mixings of the d, s, b quarks would not have taken place). If the mechanism of this mixings is realized independently of the weak interaction (W±,Z° boson exchange) with a probability deter­ mined by the mixing angles #, /3, 7,6 (see below), then this violation could be found in the strong and electromagnetic interactions of the quarks as a clear violations of the isospin, strangeness and beauty. But, the available experimental results show that there is no clear violations of the number conservations in strong and electromagnetic interac­ tions of the quarks. Then we must connect the non-conservation of the isospins, strangeness and beauty (or mixings of the d. s. b quarks) with some type of interaction mixings of the quarks. We can do it intro­ ducing (together with the 117±. Z° bosons) the heavier vector bosons B±, C±, D±,E± which interact with the d. s, b quarks with violation of isospin, strangeness and beauty. We shall choose parametrization of matrix V in the form offered by Maiani [3] ' cl3 ( <-9 (1 0 0 N 0 s.jexp(—id) ^ So 0 0 Cy Sy 0 1 0 —$o Co 0 { 0 VO ~Sy Gy y 1 -Sp exp(tTS) 0 cp j 0 !/ eg = cos#, s#) = sin#, exp(z"6) = cos6 + isin6. (2) To the nondiagonal terms in (2), which are responsible for mixing of the d, s, b- quarks and CP-violation in the three matrices, we shall make correspond four doublets of vector bosons B±,C'±,D±, E± whose contributions are parametrized by four angles 0. /3, . It is supposed that the real part of Re(sp exp (id)) = sp cos 6 corresponds to the vec­ tor boson C± , and the imaginary part of Im($p cxp(ib)) = sp sin6 corresponds to the vector boson E± (the couple constant of E is an imaginary value !). Then, when q2 « m'2v , we get: 2 tan 9 = inwOB ™%gw' tan f3 = mw9c mbalv ' >n2wgb tan 7 = m2nglw ‘ (3) ™e9w If qb ± = gc± = 9d ± = gE± — Qw± - then ™w tan 5 = O t mB tan (3 = mW m9lc i tan 7 = ™w m2D' m'w tang = (4) rnE Concerning the neutral vector bosons B°, C°, D°. E°, the neutral scalar bosons B'°, C °, D'°, E'°and the GIM mechanism [4] we , can repeat the same arguments which were given in the previous work [1]. The proposed Lagrangian for expansion of the weak interaction theory (without CP-violation) has the following form: = + (5) where J*-° = ipi,L7°‘T(Pi,L, 0 1 T = -10/' 1 i = 2 i — 3 V’i.L = 3 i = 1 i z= 2 / = 3 d\ d 'r’i.L — h b L L The weak interaction carriers A‘a, which are responsible for the weak transitions between different quark families are connected with the B. C, D bosons in the following manner: (6) Using the data from [5] and equation (4) we have obtained the following masses for B±, C*, D±, E± bosons: mB± ^ 169.5 4-171.8 GeV . mc± = 345.2 4- 448.4 GeV . (7) mD± “ 958.8 4-1794 GeV , Tri£± = 4170 4- 4230 GeV. 3 Oscillations of Neutral K°, K° Mesons in the Theory of Dynamical Analogy of Kabibbo-Koba- yashi-Maskawa Matrices The oscillations of A'0, A'0 mesons are characterized by two parametrs: angle mixing-0 and length- R or time oscillations-r. At first, we will consider mixings of A'° mesons and then obtain the Lagrangian for mass differences of A'®, I\-l arising for existence of the weak interaction through B boson violating strangeness, and further mass differences of A'®, A'® mesons is estimated. In the end, the general scheme of A'0, A'0 meson oscillations is given. In the further considerations, for transition from the standard model to our model, the following values are used for sin6,cosQ and Gf: 4 >»'\r fiiiiB = »'W\Y >»h cos? ft = 1 — sii?f). (8 ) ■i _ 9\v G F ~ 2 1 32 w IV A) Mixings of K°,K° mesons Let us c onsider mixings of A’°. A'0 mesons. K° and A"0 consist of j.s.d.s. (quarks and have the same masses (it is a consequence of the C'PT invariance) hut their strangenesses are different s^,, = — l..s/;-i> = 1. Since the weak interaction, through D boson exchanges, changes the strangeness, then the A'11. A’° are mixed. The mixings of A'11 mesons can be considered using the following nondiagonal mass matrix of A’" mesons: f T'" '"W. (0) V n,K«K>' mK« J Since m2I<0 — the angle O' of A"'1. A’0 mixing, or the angle- rotation for diagonalization of this matrix ( 1,1 i o \ l 0 mi j ' given by the expression: m •2 tcj20' = A'"A" 1,11<° equal (O' =)j and 7ni,2 — 2 (('"A" ~ m A" ) ^ \j(n> A" — 777 h 1’)" + )J)- (lb) m. nr, = mA"A' (ID 5 Then the following new states A'". A’.® arise: A"0 + A* 0 A"® -K" A'? = At = (12 ) s/2 s/2 Below the C'.P-invariance is supposed to he strong conscrvated (the consequences which arise of the CP violation were discussed in work [6] ), and then the following decays are possible: (CP parity of A"® is P(A'°) == +1, and of K$ is P(A'®)' = -1) A',0 —> 2tt. A'.® —> 3/r. (13) B). A Lagrangian for Mass differences of A'®. A'" Mesons and Esti­ mation of the Value of this Difference If the masses of A'®, A'® mesons are the average value of Lagrangian L (the evident form of L will be obtainned below) then: m\ =< A'® | L | A'? >. m\ =< A'® | L | A.® >. (14) and, if to take into account equation (12) then: m\ - m\ =< A'® | L | A'® > + < A'® | L | A ® > . (15) The calculation of the value Am. = m , — m? will be executed in the framework of the weak quark interactions using the following diagram: s u, c d 6 where B is a boson changing the strangeness; it. c are quarks (for sim­ plification the t quark is not taken into account). We do not give here the details of the calculation on this diagram since they were widely discussed in literature [7,8]. Using the standard Feynman rules for the weak interaction (after integrating on the inside lines, twice using the Firtz rules and without the outside momenta), we obtain the following equation for the amplitude of A"0 —» A'0 transition: G?jpm^sin28cos'20 M(A'° A'0) = (lQasdQ as (16) 8tt2 = GdQ asd.Q a s. G? pjn~sin20cos 2$ where G = 8tt2 Qa = 7q (1 - 7o) • Then, the Lagrangian for this process with changing the strangeness on As = —2 is: Las=-2 — —GdQ asdQ a s. (17) and the Lagrangian with As = +2 is Lagrangian conjugated by Ermit to Lagrangian (17). Using equations (14), (15) and quark Lagrangian (17), Am is com­ puted. For this purpose the following phenomenological matrix element is used: < 0 | dQ tt.s | A >= ffcPc (18) where pa is four-momentum, <P k is the wave function of A'0 meson and fj< = 1.27/* . If to take into account that the quarks have colour, then there appears the factor (1 + |). 7 So, the following equation for matrix element of L is obtained: < #° | 2 | >= _ (19) For the full matrix element of L, when also the inverse process K° —> K° is taken into account , the following equation is obtained: < A'0 | L | A'0 > + < K° | L | A'0 >= yGml fl.
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