Journal of Modern Physics, 2012, 3, 15621571 http://dx.doi.org/10.4236/jmp.2012.310193 Published Online October 2012 (http://www.SciRP.org/journal/jmp) CP Violation in Kaon Decay in the Scalar Strong Interaction Hadron Theory F. C. Hoh Dragarbrunnsg. 55C, Uppsala, Sweden Email: hoh@telia.com Received August 14, 2012; revised September 15, 2012; accepted September 22, 2012 ABSTRACT CP conservation and violation in neutral kaon decay are considered from a first principles’ theory, recently published as “Scalar Strong Interaction Hadron Theory”. The arbitrary phase angle relating K0 and 0 in current phenomenology is identified to be related to the product of the relative energy to the relative time between the s and d quarks in these kaons. The argument of the CP violating parameter is predicted to be 45˚ without employing measured data. The 0 S decay rate is twice the 00 S K 0 masss difference, in near agreement with data, and both are proportional to the square of the relative energy 29.44 eV. Any pion from decay will also have a mass shift of 1.28 105 eV. The present first principles’ theory is consistent with CP conservation. To achieve CP violation, the relative time cannot extend to both and but is bounded in at least one direction. The values of these bounds lie outside the present theory and it is unknown how they can be brought forth. 00 BB mixing is also considered and the relative energy is 663.66 eV. Keywords: CP Violation; Kaon Decay; Relative Energy; Relative Time; Scalar Strong Interaction 1. Introduction CP vioaltion in neutral kaon decay has been treated phe nomenologically [13] but its origin remains a mystery ever since its discovery nearly 50 years ago. This prob lem is now treated employing the recently developed first principles’ theory [4] in which this mysterious origin is shown to be connected to the relative energy and relative time between the s and d quarks in the kaons. Section 2 reproduces some phenomenological results. In Section 3, the arbitrary phase angle relating 0 to K0 in Section 2 is identified to be connected to the relative energy and time among the quarks in the kaon. When applied to S, the phase of the CP violating pa rameter in Section 2 is predicted by means of the de generacies of SU(3) gauge fields to known SU(2) ones. The decay rate depends upon the relative energy. In Sec tion 4, the CP violating L is found to be for bidden unless the relative time is bounded between cer tain finite values. The mass shift of 02πK 0 K2π 0 depends upon the relative energy 29.44 eV and is half the S decay rate. Any pion from 02πK 0 decay will also have a mass shift of ≈1.28 105 eV. In Section 5, the semilep tonic decays of kaons are treated and CP violation also requires that the relative time be bounded in one direc tion. Section 6 summarizes the roles of relative energy and time and considers the possible origins of CP viola tion. B0B0 mixing is similarly treated in Section 7. 2. Phenomenology [2,3] The decays S 0 , L, 3 have been considered using the timedependent Schrödinger equation without specifying the Hamiltonian in connection with CP non conservation [2]. The starting point is the ansatz [2 (15.28)] 02πK 00 exp i SCPTK (2.1) where S denotes strangeness, CPT the conventional dis crete operator and an arbitrary phase angle. Under CPT invariance, [2] gives 0 0 1 1, 1 2 1 1 1 2 S L K K (2.2) where the upper row refers to K0 and the low row to 0 and is a small, complex quantity [2]. Using the meas ured [1] C opyright © 2012 SciRes. JMP
F. C. HOH 1563 6 06 3.483 10eV 2π7.352 10eV2 KL KS mm SK LKS Km m (2.3) the argument of ≈ 45˚ or 225˚ has been deduced. Also, unitarity provides an upper bound for  . Thus, 3 1i 2, 10 2i 4.3 (2.4) Further, (2.1, 2) with (7.2.18) leads to (2.5) so that the argument of ≈ 45˚ or 225˚. [2 (15.105, 109, 113)] define the ratio of the semilep tonic decay ampitudes 0 0 π π isL isL L L SK xSK (2.6a) where L stands for the lepton species, as in (7.1.1113), and s the lepton helicity. [2 (15.116, 117)] give the ratios of the decay rates to first order in , 2 11 1* 1 e π π Ls L LsL 0 0 0 0 π π 14R SsL SsL KL x KL KL KL (2.6b) where x turns out to be small and has been dropped. From the measured semileptonic decay ratios [1], (2.6b, 4) yields 33 2.28 10 Re1.612 10, (2.7) Furthermore, [1] gives the nonleptonic CP violating amplitude ratios 000 00 000 0 0 ππ ππ ππ ππ L s L s K K K K The numbers in (2.7) and (2.8) show that 3 3 2.221 10 2.232 10 (2.8) 00 (2.9) This numerical agreement suggests that CP noncon servation for the semileptonic and nonleptonic decays can be characterized by one parameter . However, the above results say nothing about the nature of this CP violation. 3. Relative Energy and Time and K0 2, 3 The references in form of (x, y, z) or §x, y, z below refer to those in [4], mostly Chapter 7. This chapter has earlier appeared in a more general form as the first article in another volume [5]. In the ansatz (2.1), the nature of the phase angle is not specified. Here, it is naturally identified to be related to the product of the relative energy 0 and the relative time x0 in (A7), as will be specified in (6.1) ff below. In the absence of guage field, (7.1.9) reverts to (A4) which led to (A8). With the association of with 0x0 above, however, 0 0 and (A9) has to be relaxed for the pre sent application. In the absence of weak interaction, g 0 in (7.1.5) and K0 and 0 are complex conjugate of each other in (7.2.19) and are stable, physical states with the same mass given by (A10). The phase angle in (2.1), hence also the relative energy 0, drops out. Turning on the weak interaction igWU or i U Wg, K0 and 0 are no longer physical states and and 0 0 become small quatitites of the same first order as igWU. The physical states are now S 0 and 0 shown in (2.2) or (7.2.18) incorporating the phase factor 0 0 exp i according to (A7). Therefore, 0 in (A8) cannot be dropped but is of the same order as the igWU term. Consider one of the two first order terms in (7.1.9a) for 00 π U W of (7.3.27b). Let the first operator in (7.1.9a) be of first order g included in (7.1.45) and the second operator be of zeroth order given by the last of (3.1.4). By (A5), (qr) = (32) in (7.1.9a) refers to 0 02 S K which by (7.2.18) becomes . Using the wave function (A7), this first order term takes the form 000 00 00 00 11 i2expi i 42 2 , be be abab be UXX r WXEX x rxxr (3.1) With the degeneracy (A15), (3.1) can be written as 000 0000 11i1 iiiexpi i 4cos2 22 ab be ab be W ZX r EEXx (3.2) Copyright © 2012 SciRes. JMP
F. C. HOH 1564 Since 0 is generated by the weak interaction g term, it must have the same phase in (3.2); 000 1 i, 00 real constant 000 1i (3.3) Turning to the first order term in the second operator in (7.1.9a) and repeat the above procedure. It is found that 02πSK 022 (3.4) These two relations are related to the two phases in (2.4) but without using the empirical data (2.3); they are the consequences of the necessary degeneracy (A15). There is no conflict between (3.3) and (3.4) because they cancel out upon summing the two first order terms in (7.1.9a), as is reflected in (A8) which contains no linear 0 term. Follow now Section 7.3.24 and obtain the first order decay amplitude fi S. In the source term (7.3.5a), the first term in the braces is replaced by (3.2) multiplied by a final state wave function : 00 00 0 4 11 i ii 4cos 2 1iexpii 22 ab fe lllII rp easr bf ps ab ab ea W be be W ZX g r EE Xx 0 00 pi1 00 0 022 0 ig (3.5) The final state 2 comes from the decay of the Z boson via (7.7.2, 7) and the final state 022 represents a vac uum meson state 0 given by (7.3.20) with which does not cntain any phase factor. Consider the integral over the relative time x0 in the source (7.3.5a) with the replacement (3.5), 22 11 000 0 210 dexpi dex xx x 0 exp i (3.6) where (3.3) has been consulted. In the surface term (7.3.4), the exponential 0 in (3.5) also enters but, in addition, also its complex conjugate 0 0 exp i 00*0 00 00 00 i i 2 contained in 023 in (7.3.4). The integral over the rela tive time x0 in the surface integral (7.3.4) reads 2 1 2 1 dexp dexp xx xx (3.7) Equations (7.3.4) and (7.3.5) are sandwiched between the initial stae 0 iKS and final state 0, 2πfZ 02πSK as was mentioned beneath (7.3.13). The 0 that follows < f = refers to the abovementioned vacuum meson state 0. By (7.3.3  5, 13), fi S is proportional to the ratio of (3.6) to (3.7) which vanishes for 0 in (3.6); 2 1 2 1 000 00 00 00 00 00200 1 00200 1 dexp dexp2 expi 1expi 1 1i 0 exp2 exp2 xixx xx 02πK (3.8) This also holds if (3.3) is replaced by (3.4). Therefore, S is forbidden to first order in igWU. Even if this first order 02π S SK fi does not vanish, for in stance by letting 1, 2 be finite such that (3.8) becomes of the magnitude of unity, it is expected to lead to a de cay rate 02πKS of the same magnitude as 2pK 8 1. 110e V from phase space consid erations. This rate is 668 times smaller than the observed rate in (2.3). This is “one of the remaining unsolved problems of the weak interaction” [2]. However, there is now a second order term ( 0)2 = i2( 00)2 in (A8). Let E00 be the K0 mass in (A10) and E0 = E00 + E0S in (A8), the exponent in (3.5) will contain i E0SX0 corresponding to the decay rate 02 00000 2πi4 SS EE (3.9) where the lower sign in (A10) is chosen. Note that the relative energy 00 is a hidden variable but its square in (3.9) is visible. Although (3.9) comes from a second or der term, its amplitude is of first order in 0, hence of order g. In (3.2), the ratio of 0 to the g term that leads to the last mentioned 02π fi S SK is not fixed or known. 02π S K 2πK to be ; it This allows may be regarded that the nonvanishing 02πSK fi S in the last paragraph is greatly amplified by the relative energy 00. 000 2ππ 2π SS KK The factor [1] comes from the factor 12 in front of 0 in (A5) that enters (7.7.2) via (7.7.7). Instead of 00 π U W of (7.3.27b) considered above, 00 πKW 0 02πK 03πK U of (7.3.29) can be treated in the same way. The only differences are that (1 + i) (1 i) in (3.3  4) and the upper sign in (A10) is used when deriving the corresponding (3.9); the above results remain unchanged. If the vacuum meson state 0 assigned to 22 in (3.5) is replaced by a real final state 0, S there goes over to S. Because and 0 form a triplet in the limit of SU(2) symmetry, they must have the same phase. Since cannot contain a phase factor with com plex argument, 0 00 expi 1 02πK , which would cause them to decay like S in (3.9), this final state 0 can at most contain a phase factor of the form 0 expi 00 0 , similar to that for following (4.6) below. The 0 00 exp 03πK factor in (3.6) remains un changed and the ratio (3.8) remains 0; S is for bidden. However, if 1 S is finite, (3.8) becomes Copyright © 2012 SciRes. JMP
F. C. HOH 1565 2exp T 03πK 012 2.57 10 00 S and some S will be seen. This form is the same as that for the semileptonic decay in (5.4) below. The difference is that TK in (5.4) given by (5.8) is small. Here, S is much larger and is estimated to be ~3 1015 sec if the small 3π S K 00 LS 0 L K eV [1] is to be accounted for. 4. 2, 3 and K Mass Difference The couplings 0U W and 0 U WK of (7.3.27b) and the second of (7.3.29) are the same with respect to S 0 but not to 0 according to (7.2.18). This leads to results mentioned in the next to last paragraph of Section 3 for 0 S . For 0 , the WU and U W contributions to needs be summed, as has been done in re vised (7.3.27, 29) [unpublished]. The first order (3.1) for 02π L 0 S K can be taken over here for 0 if its sign is changed, as is seen in (7.2.18). Take the hermitian ad joint of (7.1.9a) and consider the first order term corre sponding to (3.1) for 0 , 0 000 00 11 i2 42 exp ii 2 be be ab U gWX rEX x 0 ab ab XX 000 (4.1) Add this expression to the negative of (3.1) and work out the resulting expression using (A15A16). The result shows that the imaginary part is of the form ig(real op erator). Since 0 i 0 must have the same phase, analogous to that mentioned above (3.3), 0 must be real here. The summation removes the imaginary parts in the above (real operator). The expressions (3.3, 4) are re placed by 000 (4.2a) 02πK 02πK (4.2b) respectively, for L. The above (real operator) contains both W6 and W7 which are the same by (A15 A16). The treatment of for S in (3.35), using (4.2) instead, can be taken over and the ratio (3.8) is here re placed by 00 00 00 00 0 dexpi sin dexp ii 2 xx xx T 00 00 0 T T 02πK 02πK 22 (4.3) where (7.3.16) has been noted. Thus, L, just like L, is also forbidden to first order in igWU. This result agrees with the requirement of CP invariance [2]. Being a first principles’ theory, consequences of conservation laws, including CP conservation, are in cluded in the equations of motion (7.1.89). Turning to the second order term 00 0 in (A8) via (4.2). Let E00 be the K0 mass in (A10) and E0 = E00 + E0L in (A8), the equvalent of (3.9) becomes the mass shift 2 00000 2 L EE 2 (4.4) when the upper sign in. in (A10) is chosen. The square of the relative energy among the s and d quarks in the hid den space x is visible in the laboratory space X in form of the mass shift (4.4), just like it is in form of the decay rate (3.9). Again, (4.4) comes from the second order 0 in (A8) but its amplitude is of first order in 0, hence of order g. Comparison of (4.4) to (3.9) yields 0 0 Γ2π2 SL E (4.5) which is 5.25% smaller than data (2.3). This discrepancy has been used to modify the argument 45˚ in (2.4) to about 43.5˚ [13]. Here, this 45˚ cannot be changed due to the constraint (3.3, 4). This discrepancy cannot be ac counted for in the present theory. Similar to the CP vio lation cases mentioned above, finite 2, 1 in (3.8) will render the first order 02πSK 00 29.44 eV fi S mentioned below it to contribute and to reduce this discrepancy. Using (2.3) for E0L, (4.4) gives the relative energy between the s and d quarks in neutral kaons, (4.6) The upper sign is used. The lower sign can also be used if it is accompanied by x0 x0, as is evident from the phase factor 0 exp i0 0 in (3.1), noting (3.3, 4). This leads to that the ’s in (3.6), at the end of Sec. 3 and in (5.8) and (6.1) below also change sign. If the vacuum meson state 0 in (3.5) adapted for is changed to a real 0, L, forbidden by (4.3) above, turns into . This real final state 0 must have a phase factor 02πK 03π S K 0 exp i00 0 which is to be inserted into the upper integral in (4.3) to make this ratio to be come unity, as is implicit in [4]. This implies that any 0 from decay will also have a mass shift analogous to (4.4). According to the last paragraph of Section 3, and 0 have the same phase in the limit of SU(2) symme try so that any from 0 decay will also have a mass shift. With (4.6) and the masses [1], (4.4) yields the shifts 5 π0 5 π 1.28410eV 1.277 10eV E E (4.7) Although these shifts are far less than the error margin 0.35 kev for the masses and are not observable, it sig nifies that there exists two different species of triplets Copyright © 2012 SciRes. JMP
F. C. HOH 1566 with slightly different masses. 03πK 02π,3π L K 2π 02πK 2πK 02πK S has been treated in [4] employing the null relative energy condition (A9). A relaxation of this con dition here however does not affect the results obtained there. To achieve CP violation, let L in (4.3), like those mentioned in Section 3 for . For L 0, (4.3) becomes unity and like S mentioned beneath (3.8), from phase space considerations. This rate is much greater than data. To account for the mearsured L, L is chosen such that (4.3) becomes of the magnitude 0 L K 00 R 00 Re , e ,668 0.0407, where are given by (2.79) and 668 has been mentioned beneath (3.8). Thus, CP is correctly violated if 00 00 sin 0.0407 0.1025, 0.226, LL L T T 1 0.712 eV (4.8) where (4.6) has been used. The angles 00 L are close to multiples of . These relative times are very short, of the order of 1016 sec. These values, like the finite ’s men tioned in Section 3, cannot be predicted from the present theory and have to come outside of it. 5. Semileptonic Decay of and 0 L K K 0 The amplitude for 0π W is given by (7.3.27a) and for 0π W π by the first of (7.3.29). Subse quently, WL and π WL , where L , e. But the amplitudes for 0π W and 0π W do not follow from the internal index com binations in (7.3.1819) and hence do not appear in (7.3.2629). These decays are therefore forbidden, in agreement with the selection rule S = Q [2]. This rule is a consequence of the present theory and has been veri fied by that x of (2.6a) is consistent with 0 [1]. The amplitudes (7.3.27a, 29) assume the null relative energy condition (A9). The rates of these decays have not been evaluated because the pions are relativistic and their wave functions unknown; the coarse nonrelativistic (3.5.23), (7.7.13) used for L is insufficient here and in other K 2 decays. Nevertheless, the forms of (7.3.27a) and the first of (7.3.29) are the same so that the 03πK 0 decay rates π KL and 0π KL 0 are the same. By (7.2.18), S and 0 contain equal amount of 0 and K0, they contribute about equally to these semileptonic decays. This is approximately verfied by data [1] and corresponds to the CP conserving part of these decays. As was mentioned in the beginning of Section 3, (A9) has been relaxed here. Since the amplitudes for 0π W and 00 πU W have the same form according to (7.3.27), the developments (3.14) can with some modifications be taken over. The expression corre sponding to (3.1) reads 012 0 000 00 1 i2 i 4 1 2 exp ii 2 abab ab be be be XX gWXWX rEX x (5.1) where the degeneration (A14) for V W together with (A13) have been used. Analgous to W6(X) = W7(X) in (A15A16), W1(X) = W2(X) here inasmuch as the both WX I have the same X dependence, so that the lower sign form of (5.1) turns to a form nearly the same as (3.2). Thus, (3.3) holds for 0πW 0π . For W , the upper sign of (5.1) lead to 000 1i 03πK (5.2) The phase factor considerations for S in the last paragraph of Sec. 3 can be taken over here. The phase factor 0 expi 00 for the final state 0 there can also be used for the final state in 0π W 03πK here. Such a factor is requried for in L men tioned beneath (4.6). Inserting this factor into (3.6) turns (3.8) into 2 1 2 1 00 00 00 00 00200 1 00200 1 210 dexp dexp2 exp exp 20 exp2 exp2 xx xx TT TT TTT 02πK (5.3) Analogous to that S is forbidden by (3.8), the semileptonic decay 0ππ WL is also forbidden when the relative energy 00 0, contrary to observation. If 00 0 or the null relative energy condi tion (A9) holds, (5.3) turns into 1 and these semileptonic decays can take place, as was implied in the second paragraph of this section. These decays however conserve CP. CP violation can be obtained in a way similar to that achieved in the last paragraph of Section 4, in which the limits of the relative time x0 were not allowed to extend to . Let 2 = 0 = as in (5.3) and (3.6) but 1 = K finite, (5.3) turns into 00 00 00 00 00 dexp 2exp dexp2 K K xx T xx 0ππ (5.4) For WL , (5.2) replaces (3.3). Let 1 = 0 = as in (5.3) and (3.6) but 2 = K, the Copyright © 2012 SciRes. JMP
F. C. HOH 1567 0 0 exp i equivalent of (5.4) reads 00 2e xp K T 0 00 00 00 00 dexp dexp2 K K xx xx (5.5) As was mentioned in the second paragraph of this sec tion, S and 0 contain equal amount of K0 and 0 and 0 ππ 0 L KL KL for the domi nating CP conserving part. This together with (5.45) give the expression corressponding to (2.6b) 00 00 2 00 00 00 exp 14 exp SsL ππ ππ LsL L Ls K KL KL Ss K K KL KL TT T 00 Re K T (5.6) Comparison with (2.6b) gives the identification 11 1.210 s 0 (5.7) This together with (2.7) and the lower of (4.6) yields 51 5.510 eV K T (5.8) which is more than 10 times shorter than the S decay time. Thus, by suitable choices of the limits of the rela tive times, the present theory reproduces the rather suc cessful phenomenological (2.6b). 6. On the Origin of CP Violation in Neutral Kaon Decay From the semileptonic decays in Section 5, the phe nomenological phase angle in (2.1) by (2.45) and (5.7) becomes 00 i 21i 2 00 i2 i2 K TT 02πK 0 (6.1) Here, is identified with the product of a relative en ergy and a relative time, both finite. For the CP violating L in Section 4, the phase angle 00 L in (4.8) via (4.3) associated with is real. This corresponds to that the angle in (2.1) is also real for this case and is related to 00 L, also a product of a finite relative energy and a finite relative time. Equations (5.45) show that it is the imaginary part of the relative energy 0 in (3.3) and (5.2) that causes (5.6) to deviate from unity and thereby causes CP violation in the semileptonic decays, which takes place when the relative time x0 between the s and d quarks does not ex tend to both and but is finite at one end. This does not conflict with that in (7.4.6b) via (7.3.16) to generate the MW mass. There, the actions in (7.4.34) involve stationary K0 and 0 x 0 and not their decay so that (A9) can and has been applied in that stage. The phase factor 0 x here reduces to 1 there which allows for , similar to that 00 0 in (5.3) ff renders it to be . 0 02πK 0 For the CP violating L, the relative time x0 also does not extend to both and but is finite at both ends as is shown in (4.3) with L given in (4.8). Summarizingly, CP conservation is related to the phase angle 0x0. The relative energy 0 gives rise to the mass shifts for in (4.4) and pions in (4.7) and to that the large S decay rate in (3.9) is twice the 02πK0 mass shift in (4.5). These results are derived within the frame work of the present theory. The relative energy 0 is necessary but not sufficient to account for the CP violating S in Section 4 and the semileptonic decays in Section 5. To achieve these CP violations, bounds need be put on the values of the relative time x0 so that it cannot run from 02πK to . The values of these bounds, given in (4.8) and (5.8), re spectively, lie outside the present theory. It not known how such relative time bounds can be brought forth. One observation is that the relative time I 000 II xx must be shorter than twice the laboratory time I 000 II 2 xx in (3.1.3a). Now this X0 is limited by the finite decay time so that the relative time is also limited. Another one is that the relative time 0 y has been set to the finite 2/dm in the last line of §7.7.2. Fur ther, how the kaons are produced may also enter here. In any case, the CP violations here are not related to the CP violating phase in neutrino oscillation phenomenology [1 Neutrino Mixing]. In this connection, it may be pointed out that the boundary condition of the wave functions and in (A1), hence also in the action (7.1.8), at 0 x, like 0 and in (3.67), (4.3) and (5.3), has been assumed to be the same as the corresponding ones (fixed) in the laboratory time 0 , as was pointed out in Section 6.1.1. The validity of this assumption has not been fully inves tigated and it is not clear whether this may impact upon the above relative time considerations. 7. 00 BB Mixing The 00 BB mixing differs fundamentally from the 00 K mixing considered above. In Table 1, the u, d and s quarks have about the same mass. This leads to that mpr in (2.4.1) differ by 12%, as was mentioned above (2.4.5). Therefore, the kaons and pions are mem bers of an approximate SU(3) octet. In the limit of SU(3) symmetry, the u, d and s quark masses coalesce. 2 M 0 and K0 belong to the same octet providing basis vectors of the regular representation of SU(3). They can be transformed into each other by suitable choice of the Copyright © 2012 SciRes. JMP
F. C. HOH 1568 It is proportional to the phenomenological probability for remaining in the original bottom state [3] Table 1. Quark masses and d obtained from (5.1.14) using the masses , K , K 0, D0, 0 m + s and B 0, and quark contents of [P1]. m1 (GeV) m2 m1 m3 m4 m5 0 m d(GeV2) 0.6592 0.00215 0.7431 1.6215 4.7786 0.24455 transformation U3qs(X) of (7.1.7). Their linear combina tions S 0 and 0 in (7.2.18) are also vectors in this octet space on par with the pion vectors and represent physical mesons suitable for describing some weak de cays. The c and b qaurks are much heavier and the D and B mesons cannot meaningfully be accomodated in SU(4, 5) multiplets together with the kaons and pions, as was in dicated above (2.4.5). In the limit of SU(3) symmetry, B0 belongs to the triplet (B+, B0, S) providing basis vec tors of the first funda mental representation of SU(3). But 0 B 0 B belongs to the antitriplet (B , 0 B, 0 BS) providing basis vectors of the second fundamental representation of SU(3) transforming differently [6]. Thus, B0 and 0 B cannot be transformed into each other and linear combi nations of them of the type (7.2.18) do not have definite transformation properties and hence are not members of any SU(n) multiplet. They remain as physical mesons in two different triplets. The D meson triplets behave analogously. As in Section 3, the null relative energy condition (A9) is also relaxed for B0 and 0 B. The complex 0 in (3.3) is due to the degeneration (A15) involving the complex neutral gauge boson WU. Data [1] however show that the semileptonic decays of B0 and 0 B proceed dominantly via charged gauge bosons W charged leptons; the branching ratio of 0 2, 2πDK 0 00 BDZ is very small. Therefore, 0 = 00 = real as in (4.2) for KL. Therefore, a mass shift E equivalent to E0L in (4.4) also holds for B0. For 0 B, the lower sign in (A10) is chosen. The laboratory time X0 dependent part of the B0 and 0 B wave functions is 0 0 0 0 00 0 00 exp i 2 exp i 2 B B B B 0 00 0 0 i i B B EEX EEX (7.1) where E00 is the mass and 0 the decay rate of B0 and 0 B. The X0 dependent part of the probability of mixed B0 and 0 B is 00 0 2 00 2 00 00 2exp1co 2 BB B XX X EE 00 00 s2 BB E X (7.2) 00 00 1exp Γ1cos 2d PB B PBB tmt (7.3) Here, = 0 , t = X0 and the oscillation frequency md = 0 2 E = 3.337 10−4 eV [1]. With the B0 mass E00 = 5.2796 GeV, the second of (7.2) gives the relative energy 00 = 663.66 eV between the b and d quarks. This value is 22.5 times greater than 29.44 eV in (4.6) for that between the s and d quarks in K0. Note that the ratio be tween the B0 and K0 masses is 10.6, nearly half the above ratio. The mass difference 3.337 10−4 eV is somewhat less than the B0 decay rate 0 = 4.43 10−4 eV [1], similar to that the 00 S K 0 mass difference is about half of the S decay rate in (2.3). These mass differences, apart from their values, are outcomes of the present theory, irrespective CP violation. Analogously, CP violation in the 00 BB system is also attributed to relative energy and time between the b and d quarks. Such a treatment however requires the knowledge of the amplitudes of 5 specific decays [3] and is beyond the scope of this paper. In passing, it may be noted that while mixings of d, s and b quarks take place, the c and u quarks with charge 2e/3 do not seem to mix, as 00 DD 0 S mixing is absent in [1]. 8. Conclusions In the standard model, quarks in meson decay are treated [3] largely as leptons are in QED. The effects of quark confinement in the relative space and of the relative time between the quarks are practically lost. In the paper, the relative energy between the quarks gives rise to the mass shift between 0 and , equalling half the S 0 decay rate, and 00 BB mixing. Hadron spectra stem from the relative space [4] and the relative time generates the W and Z boson masses [4] without Higgs. Here, CP violation is achieved by limiting the relative time to certain regions. REFERENCES [1] J. Beringer, et al., “Particle Data Group,” Physical Re view D, Vol. 86, No. 1, 2012, Article ID: 010001. doi:10.1103/PhysRevD.86.010001 [2] T. D. Lee, “Particle Physics and an Introduction to Field Theory,” Harwood Academic Publisher, Newark, 1981. [3] K. Kleinknecht, “Uncovering CP Violation: Experimental Clarification in the Neutral K Meson and B Meson Sys tems,” Springer, Berlin, 2003. [4] F. C. Hoh, “Scalar Strong Interaction Hadron Theory,” Nova Science Publishers, New York, 2011. Copyright © 2012 SciRes. JMP
F. C. HOH Copyright © 2012 SciRes. JMP 1569 https://www.novapublishers.com/catalog/product_info.ph p?products_id=27069 [5] F. C. Hoh, “Scalar Strong Interaction Hadron Theory (SSI)—Kaon and Pion Decay,” In: C. J. Hong, Ed., The Large Hadron Collider and Higgs Boson Research, Nova Science Publishers, New York, 2011, pp. 177. [6] D. B. Lichtenberg, “Unitary Symmetry and Elementary Particles,” Academic Press, Waltham, 1978.
F. C. HOH 1570 Appendix Relevant Equations from the Book [4] Parts of the book [4] often referred to are reproduced here. The references in form of (x, y, z) or §x, y, z below refer to those in [4]. The starting point is the equations of motion for meson including internal coordinates given by (2.3.19, 21, 23, 27), IIIIII III 2 III III ,, ,, ab fp ef r b mm e xx zz Mxxx III ,0 ap r xzz III ,0 dp r xzz (A1a) III IIIII I 2 III III ,, ,, de cp er bc mm b xx zz Mxxx (A1b) I II , sa b III III 4 II , 1Re , 2 m ba xx xx xx (A2) 2 mpr Mmm III III 2 III III , ,,0 ab feprbf ae m mpr pr xx Mxxxx (A3) Here, xI and xII are the quark and antiquark coordinates, respectively, I = /xI, II = /xII, and are the meson wave functions with the spinor indices a, b, ··· run ning from 1 to 2, m the scalar interquark potential, p the quark flavor, q the antiquark flavor, and zI and zII respect tively the internal coordinates for the quark and the anti quark in an abstract complex three dimensional space. p, r = 1, 2 and 3 refer to the u, d and s quarks, respectively. Because the quark masses mp and mr are different for different mesons, (A1) can, after cancelling out the functions be written in the form (2.4.3) (A4a) III II I 2 III III , ,,0 ce ed pr bc m mpr pr bd xx Mxxxx (A4b) where (pr) indicates the dependence upon the quark fla vors. The matrix form of is shown in (2.4.18), π0π 11 1213 21222311 22 33ππ00 31 3233 0 11 26 111 , 326 2 6 K pr K KK ab (A5) Follow (3.1.3a) and (3.5.7) and introduce the relative and laboratory coordinates, II I II , 1, mm xxxX ax ax I12 m a (A6) where the relative space time x = (x, x0) are hidden va riables [4]. The wave functions of a pseudosclar meson at rest reads (3.1.1, 5, 7, 9) III0 III 00 000 ,, exp ii, be be be xx xx xEXx (A7) where E0 is the mass of the meson and 0 the relative energy among the quarks. Inserting (A7) into (A4a) for a given (pr) and taking the trace, one obtains from the first line of (3.1.8), 22 2 00 00 2 2 4mm ExMx x 00 (A8) The null relative energy condition (3.5.6) (A9) With this ansatz, (A8) together with m obtained from (3.2.8, 20) and (3.4.1), (4.3.2), (4.4.1) leads to the steady state meson mass (5.1.1) 22 00 0 4 rmm Emmdd (A10) where dm = 0.864 GeV is given by (5.2.3) and dm0 and the quarks masses mp are given in Table 1. In the presence of SU(3) gauge fields, the operators in (A4) are generalized according to (7.1.4, 5) and (7.2.14), II I 1111 ii 2224 abab abababababab XpsX psllpspsl ps ps DgWX ab l gWX IIII II 11 i 24 fefe fefefefe Xpspsll ps DgWX (A11) Copyright © 2012 SciRes. JMP
F. C. HOH 1571 47 47 47 47 ii 22 2sincos 2cos22 i2 cos 2 222sin2sin l llabllab ps ps WWW I IW U VW g gWX WX AZW gg gWZ g ggWggWAZ 2 2 cos V U WW W gW (A12) 12 67 2i, 2i, I U WWW WWW 45 67 2 i 2 i V U WWW W WW (A13) where A and Z are gauge fields (7.2.1), W the Weinberg angle (7.2,12) and g and g 47 weak coupling constants (7.2.7). There are four known gauge fields I, Z and A in (7.2.2) but eight I W, W 3, W8, V, WU, and W WU W in (7.1.5). The last four gauge fields are new, have not been observed and are converted into the four known ones via the following degeneracy scheme Section 7.2.3. Thus, (7.2.20) reads I 2 ab WX 47 V 2 ab gWX g (A14) Equations (7.2.21, 22) have been written down heuris tically and are now replaced by the formal analog of (A14) as is evident from (A12), 47 47 67 1i 2i cos 2 UW gg Z WWW gg (A15) 47 47 67 1i 2i cos 2 U W gg Z WWW gg (A16) where the magnitudes on both sides are equal. Copyright © 2012 SciRes. JMP
