Baryon Wave Functions and Free Neutron Decay in the Scalar Strong Interaction Hadron Theory (SSI)

Vol.2, No.9, 929-947 (2010) Natural Science http://dx.doi.org/10.4236/ns.2010.29115

Baryon wave functions and free neutron decay in the scalar strong interaction hadron theory (SSI)

F. C. Hoh

Dragarbrunnsg. 55C, Uppsala, Sweden; [email protected]

Received 16 June 2010; revised 19 July 2010; accepted 25 July 2010.

ABSTRACT coefficients. Nonconservation of angular moemntum and detachment of weak and electromagnetic couplings are From the equations of motion for baryons in the pointed out. scalar strong interaction hadron theory (SSI), two coupled third order radial wave equations 2. BARYON WAVE FUNCTIONS for baryon doublets have been derived and published in 1994. These equations are solved 2.1. Conversion to Six First Order numerically here, using quark masses obtained Equations from meson spectra and the masses of the neutron, 0 and 0 as input. Confined wave Although the equations of motion for mesons [1] and for functions dependent upon the quark-diquark baryons [2] in the scalar strong interaction hadron theory distance as well as the values of the four inte- (SSI) were both published in the early 1990’s, subse- gration constants entering the quark-diquark quent work took place wholly in the meson sector. This interaction potential are found approximately. due to that the meson equations can be decomposed into These approximative, zeroth order results are second order differential equations Eqs.6.7-8 of [1] or employed in a first order perturbational treat- Eqs.3.2.3-4 of [3] which have analytical solutions in the ment of the equations of motion for baryons in rest frame providing the zeroth order background upon SSI for free neutron decay. The predicted mag- which higher order perturbational problems can be built nitude of neutron’s half life agrees with data. If and treated. Much success and new insights about mes- the only free parameter is adjusted to produce ons have been achieved, as are seen in Chapters 4-8 in the known A asymmetry coefficient, the pre- [3]. Of current interest is that no Higgs boson is needed dicted B asymmetry agrees well with data and to generate the mass of the gauge boson [4]. The linear vice versa. It is pointed out that angular mo- Dalitz plot slope parameters in kaon decaying into three mentum is not conserved in free neutron decay pions [5] and electromagnetic and strong decays of some and that the weak coupling constant is detached vector mesons [6] have been treated. The epistemologi- from the much stronger fine structure constant cal foundation of SSI is given in the recent [7]. of electromagnetic coupling. On the other hand, the two coupled third order radial equations for baryon doublets Eq.A20 cannot be decou- Keywords: Scalar Strong Interaction; Baryon pled and reduced to lower order equations. They are eve- Radial Wave Functions; Free Neutron Decay ntually reduced to the six first order equations Eq.1 below which have to be solved numerically. Further, the 1. INTRODUCTION interquark potential Eq.A15 contains four unknown integration constants that enter Eq.1. Very lengthy work This paper consists of two parts. has been spent in finding these constants and solving Eq.1 In Part 2, the equations of motion of ground state dou- below by computer. This is the reason that the baryon re- blet baryons in the scalar strong interaction hadron the- sults to be presented below come so many years later. ory (SSI) derived earlier are solved numerically to yield Such numerical computations have been carried out and approximate baryon wave functions and quark-diquark the four integration constants in Eq.A15 and Eq.1 are interaction potential. obtained together with the radial wave functions. In Part 3, these results are employed in free neutron The coupled third order equations Eq.A20 cannot be decay to obtain decay time and the A and B asymmetry solved analytically and have to be treated numerically.

The standard procedure is to convert them into a first sy- where + are given by Eq.10.2.8a of [3]. The three cases stem according to Eq.10.7.5 of [3] after putting the or- with possible +  0 are excluded because they lead to bital angular momentum L = 0 there (see Eq.1), diverging w (r = 0). The remaining cases yield In arriving at Eq.1, Eq.A15 has been used. E0 is the 3   w400  w100  0,1,0 (5) neutron mass, Mb is the quark mass term defined in

Eq.A4 and the four db constants are integration constants   w401  ,0,1 101  ww 1 (6) in the solution to the homogenized Eq.A14  (r) = b   ,2  www  0, (7) 0 and are independent of flavor, i.e., baryon species, as  402  1022 was pointed out below Eq.10.1.6 of [3]. where the amplitude in Eq.5 has been normalized to These four db constants can therefore not be fixed by unity. w(1) and w(2) are amplitudes for the remaining two using four baryon masses as input. The procedure ado- solutions. The general solution near r  0 reads pted is as follows. The quarks masses in Eq.A4 are mu      0    1   2  rwrwrwrw (8) 0.6592, md  mu  0.00215 and ms  0.7431 in units of Gev taken from Table 1 of [8] or Table 5.2 of [3] obtain- Substituting this into Eq.1, which is now solved as an ned from meson spectra. Three known baryon masses initial value problem with initial conditions Eqs.5-7. 0 are E0  0.9397, 1.1926 and 1.3148 Gev for neutron,  0 2.2. Numerical Solution and Results and  , respectively [9]. These are inserted into Eq.1 and the four db’s are varied over suitable ranges such that The six unknown parameters db, db0, db1, db2, w(1), and the solutions satisfy the boundary condition at r   or w(2) are adjusted so that the six boundary conditions w 3 converge there. (r  )  0 for a given Mb and the associated baryon Near the origin r  0, the baryon radial wave func- mass E0. The calculations are done via a Fortran progr- tions can be expanded in power series in r according to am employing Runge-Kutta integration subroutine on Eq.10.7.6 of [3], computers of the Department of Information Technology

  at Uppsala University. In the paramater range of interest,   o     , o    rbrgrarf (2) it was found that integration of Eq.1 and summation of  0  0 the power series Eq.2 give the nearly the same results where  satisfy the indicial equation Eq.10.7.4 of [3] and for r  7-8 Gev1. Beyond this value, Eq.2 become unre- is related to + in Eq.4 below. a and b are constants liable due to accumulated computational errors. Simi- satisfying the recursion formula Eq.10.7.7 of [3]. At larly, Eq.1 leads often to diverging solutions at r  7-12. large r, Eq.10.2.7a of [3] gives This is because that Eq.3 without the minus sign in the exponent is also a solution which tends to overshadow rg rf  0 0 Eq.3 due to accumulated errors and takes over at large r.  3 1 5  (3) 3 3 Near the origin r  0 and at large r, the potential in  constant exp b2 rd  2  5  Eq.1 is dominated by the db/r and db2r , respectively, and the flavor dependent quark mass term M 3 there can be The presence of r1/3 is incompatible with the power b dropped. The solutions f (r) and g (r) in these r regions series Eq.2 and Eq.1 has to be solved numerically. 0 0 are independent of the baryon flavor. Thus, d and d Equivalent to Eq.2 is the expansion around the regular b b2 are flavor independent constants on par with the corresp- singular point r = 0 in Eq.1, onding meson sector’s dm and dm0 which are independent     of flavor according to Subsection 4.4 of [3]. The small r wr  w r 0 rw  r, (4) region is very small and the large r region determines the   1,2,....6 asymtotic behavior of f0(r) and g0(r).

1  0 rfrw , 4    0 rgrw  w w w w w w w w  21 ,  32 ,  54 ,  65 r r r r r r r r w  E 2  2 E 2  1   E3   E (1) 3 w  0 2 wr  0 wr w r 0 3  r 2 3  4  ww 0 1   2   3 4 db  dM bb 0  b1 b2rdrd  65 r  4   r 4  r   8   2 w   E3   E 2 E 2  1 6 w r 0 3  r 2 3 4   ww 0 w  0  wr 1 db  dM bb 0  b1 b2 Erdrd 0  32 5   6 r   8   2  r 4  r

Therefore, db2, which determines the “confinement st- Repeat this step for db0 and db1 and define rength” via Eq.3, is used to lable a set of the six unkn-    dbs   db0   db1 (10) own parameters and chosen first. Extensive computer calculations showed that the remaining five constants are which is a measure of how much the db, db0 and db1 values deviate from the averages db, db0 anddb1. uniquely fixed if the solutions g0 and f0 are to converge at r  . Among a huge numbers of combinations of Near the origin r = 0, the potential Eq.A15 is dominated by the d /r terms and the solutions f (r) and g (r) the six parameters, only a very narrow range of values b 0 0 are independent of the baryon flavor. Thus, d is a flavor for these parameters leads to converging w (r). Some b  independent constant on par with the corresponding me- examples of the results are given in Table 1. son sector’s d and d which are independent of flavor The wave functions for the d  0.4462 case are m m0 b2 according to Subsection 4.4 of [3]. Conversely, one ex- plotted in Figure 1 below. pects calculations produce a common db value which is Convergent solutions have been found for continuous confirmed in Table 1. ranges of db2 values largely in the region –0.1 to –1.4, If a set of db2, db1, db0, and db values that lead to con- although some values outside this region seem also to le- vergent solutions for all three baryons in Table 1, a solu- ad to convergence and some values inside this region, tion to the present baryon spectra problem has been fou- for instance from –0.23 to –0.27, do not. It may note that nd. Table 1 shows that this is not the case but some sets it is to some degree arbitrary to regard a set of solutions possess values that are rather close to each other for the to be convergent or not. Due to accumulation of com- three baryons and may therefore be regarded as appro- puter errors at large r, all solutions eventually diverge ximate solutions to the baryon spectra problem here. Po- for sufficiently large r. Convergence is regarded as good ssible nature of these approximations are consisdered in if f00(r) and g00(r) are nearly zero over a “sufficiently” the next section. large range of r when r is large. The set that yields db2, db1, db0, and db values which The mean spread s in Table 1 is defined as follows. are closest to each other for the three baryons in Table 1 Let is the db2 = 0.4462 case with a minimum spread db  1.4% for db. The mean spread s  6.2% is also a minim- 1 2    dd ,3/ um. This set and may therefore presently be regarded as db d baroyn species bb b (9) representing an approximate solution to the baryon spec-

db  average value of db tra problem here.

Table 1. Values of the four db constants in Eq.1 with Gev as basic unit, the spread db1, db0 and db from the mean values of the four db constants and the sum spread s according to Eq.10 below and w(1) and w(2) in Eqs.5-7 for some converging solutions in Eq.1.

db2 db1 db0 db s w(1) w(2) Neutron 0.140 1.0439 3.0 0.695 0.016 0.0288 0  0.1411 1.336 4.06 1.085 0.0415 0.0691 0  0.141 1.111 3.568 0.924 0.0019 0.0851

d 10.7% 12.2% 17.7% 40.6% Neutron 0.1641 1.334 3.719 1.013 0.0668 0.045 0  0.1641 1.279 3.706 0.990 0.0353 0.0756 0  0.1641 1.220 3.691 0.9177 0.0523 0.0721

d 3.6% 0.3% 4.2% 8.1% Neutron 0.3202 2.272 4.922 1.024 0.1827 0.0674 0  0.3202 2.167 4.783 1.032 0.0586 0.0662 0  0.3202 2.142 4.899 1.083 0.1191 0.1061

d 2.6% 1.2% 2.5% 6.3% Neutron 0.4462 2.968 5.749 1.035 0.247 0.0859 0  0.4462 2.831 5.541 1.066 0.1624 0.090 0  0.4462 2.768 5.516 1.033 0.121 0.0974

d 2.9% 1.9% 1.4% 6.2% Neutron 0.575 3.658 6.55 1.064 0.2943 0.102 0  0.575 3.471 6.224 1.073 0.203 0.997 0  0.575 3.383 6.147 1.029 0.195 0.1164

d 2.2% 2.8% 1.8% 6.8% Neutron 0.975 5.572 8.328 0.8173 0.4278 0.1576 0  0.975 5.319 7.895 0.9048 0.316 0.1298 0  0.975 5.090 7.441 0.6859 0.2902 0.1391

d 3.7% 4.6% 11.2% 19.5%

1.2

Neutron d b2 = -0.4462 1

g00(r) 0.8

0.6

-f00(r) y(1) 0.4 y(4)

0.2

r(1/Gev) 0 052.5 7.5 -0.2

1.2

Sigma0 d b2 = -0.4462

0.8 g00(r)

0.6 y(1) y(4) 0.4

-f00(r) 0.2 r(1/Gev)

0 0 2.55 7.5

-0.2 1.2

Xi0 d b2 = -0.4462 1

g00(r) 0.8

0.6 y(1)

0.4 y(4)

-f00(r) 0.2

r(1/Gev) 0

0 2.55 7.5 -0.2

Figure 1. Baryon radial wave functions f0(r) and g0(r) in Eq.1 normalized according to Eq.A23 for the db2  0.4462 case in Table 1. r is the quark-diquark distance.

a p However, Table 1 also shows that this minimum is 0 (x)r (zI,zII) in Eq.9.3.7b of [3] needs be generalized a p very shallow one and sets of db2, db1, db0, and db values to the nonseparable form 0 r (x,zI,zII). A conjecture is with db2 values in the range around 0.32 to 0.45 may now made that the mass operator m3op(zI,zII) of Eq.9.3.14 also be qualified to yield approximate solutions. This ra- of [3] is also generalized to a nonseparable, as yet un- nge is supported by the approximative agreement of the known form m3op(zI,zII,x), analogous to the generalization calculated and experimental values of the half life and A of the masses operator m2op(zI,zII) to m2op(zI,zII,x) in Sub- and B asymmetry coefficients of free neutron decay for section 5.4 of [7] or Appendix G of [3]. This generaliza- the db2  0.3202 case in Part 3 below. tion may for instance be such that when r  |x| falls out- The value of the constant db corresponding to meson side some region of values, m3op(zI,zII,x) degenerates sector’s dm in Eq.5.2.3 of [3] and dm0 in Table 5.2 of [3] back to m3op(zI,zII). a p is close todb for db2 values in the range around db2  The product wave fucntion 0 (x)r (zI,zII) in Eq.9.3. 0.32 to 0.45 in Table 1 and is 7b of [3] has 10 components, two from a  1, 2 and eight d  Gev04.1 2 (11) from p  r  1, 2, 3(less the singlet). The generalized, b a p nonseparable 0 r (x,zI,zII) has 2  8  16 wave fucntion The above results have been obtained using Eq.10. components, six more than 10 components. These six 2.3a of [3] or for j  l  1/2. If Eq.10.2.3b of [3] or j  l  extra components are associated with this presently un- 1/2 were chosen, the d values for the three baryons in b0 known dependence of m3op(zI,zII,x) upon x. They may be Table 1 would deviate so much from each other that db0 the cause of the approximative nature of the results in can no longer be considered as an approximately flavor Table 1. Actually, the generalization from separable to independent constant. nonseparable forms can already be formally introduced The “reduced order” baryon spectra Eq.10.4.7 of [3] at the quark level, as was pointed out at the end of Sub- was obtained assuming non-relativistic qaurks, i.e., 2 section 11.1.2 of [3]. E0d /4 >> 0, 1 mentioned above Eq.10.4.1 of [3]. However, it can be estimated from Figure 1 that the op- 3. FREE NEUTRON DECAY AND posite holds. Therefore, Eq.10.4.7 of [3], hence also POSSIBLE NONCONSERVATION Eq.10.5.20 of [3] for the quartet, cannot be used. The OF ANGULAR MOEMNTUM quark and diquark in the baryons are highly relativistic just like the quarks in the mesons are at the end of Sub- 3.1. Background secction 5.3 of [3]. The present theory of nuclear -decay is based upon the 2.3. On Functions Nonseparable in Relative electroweak part of the standard model [9]. The origin of Space x and Internal Space z this part is the four fermion point interaction Lagrangian Consider a two electron system. When the both electrons density are far apart, it is described by two independent Dirac F  CL   enpV   (12) wave functions  (xI) and  (xII) totaling eight wave function components. When they are closer to each other for neutron decay the product wave fucntions  (xI) (xII) must be gener- epn    (13) alized to the nonseparable  (xI,xII) having 16 components governed by the Bethe-Salpeter equation which first proposed by Fermi in 1934. Here Cv is a constant. includes the interaction between the both electrons. The Subsequently, Eq.12 has been generalized to the -interaction Hamiltonian currently in use Eq.13.9 of [10], eight extra wave function components are associated with this interaction and are small perturbations when 1 3 HW   pOXd  ni the both electrons are not too close to each other. This 2 i (14) example illustrates the conjecture below.  '   chOCOC .. The approximate solutions in Subsection 1.2 are based i e i  i e i 5  upon the construction that the total baryon wave func- Here, i refers to the scalar, vector, tensor, axial vector, tions are separable in the relative space wave function and pseudoscalar interactions between the nucleon and a p 0 (x) and internal space functions r (zI,zII) in Eq.9.3. lepton currents. The Oi’s contain  and 5 and the C´s 7b of [3]. According to the epistemological considera- are generally complex. tions in Subsection 5.4 of [7] or Appendix G of [3], these Based upon Eq.14, lepton kinematics and convention- both spaces are “hidden”, on par with each other, and at nal conservation laws, including angular momentum co- the so-called “level logic2” and can be combined to form nservation, Jackson et al. [11] derived in 1957 a number a larger manifold (x,zI,zII). In this case the product form of decay rate formulae. The most frequently used one is

33 dW  constant  e   e   EEEEKdKd np  half life of the neutron in approximate agreement with data and a relatively good prediction of the B asymmetry  KK J n  K K 1  1  a e    A e  B  (15) coefficient. EE J  E E  e  n  e  3.2. Introduction of Vector and Tensor  KK K     Gauge Fields e  e e   D  R   ... eEE  Ee   3.2.1. Action-Like Integrals for Doublet Baryons The starting point is the equation of motion for doublet in which the final spins have been summed over for a baryons Eqs.A7-A8. Multiply Eq.A7 from the left by given initial neutron polarization J . Here, the K’s de- n  * and Eq.A8 by  *b , subtract the resulting expres- note momenta, E energy and  the electron polarization. 0a 0 e sions from their complex conjugates and integrate over The constants a, A, B, D, R, and  depend upon the C´s x and x to obtain and the nucleon current consisting of a vector and an I II axial vector part in the so-called V-A theory. Experi- ' 1 44  ef he ba  1 S  dxdx     ments on neutron decay have since then largely been  2i  0 IIaIII I I 2 fhb (16) devoted to determine these constants in this nearly 50 3  a Mi abb  00  cc ..  year old Eq.15 and related formulae [12]. These have, however, yielded little physical insight into the decay ' 1 44 *b 1 eck  S  dxdx III   0 heII    mechanism.  cbIkhI 2 2i (17) The standard Hamiltonian Eq.14 is a phenomenology- 3 *b Mi  bb 0    cc ..  ical model, not derivable from any first principles theory. 0b

It treats nucleon as a point particle and hence ignores its where * is an extra sign denoting complex conjugate, I  quark structure. There are in principle 20 real C con- /xI and II  /xII and their positions have been chan- stants in Eq.14, leaving the theory with little predictive ged so that the summations over the e and h indices con- power. This hints at a superfluousness of Eq.14, which is form to matrix multiplication convention and that the bo- not invariant under SU(2) gauge transformations. Such th I’s appear next to each other to reflect that they oper- an invariance would give rise to an intermediate vector ate on the diquark part indicated by the braces. * boson W which couples to a left-handed e pair. These integrals will not be varied with respect to 0a Therefore, despite its noble origin and general accep- *b and  0 in an attempt to reproduce Eqs.A7-A8; han- tance, this model Eq.14 does not differ in principle from dling of the last c.c. terms and the necessary boundary the large number of recent phenomenological models, conditions will require efforts beyond the scope of this constructed for different and narrow application angles, chapter. Nor is such a variation necessary for the present present in a vast body of literature. purposes. If the solutions to Eqs.A7-A8 is inserted into Angular momentum conservation has not been establ- Eqs.16-17, we obtain ished experimentally in free neutron decay. Most of the ' ' ,0 ' SSSS '  0 experiments make use of nuclei and the conclusion is that   0   0 (18) angular momentum is conserved. But a nucleus poses a 3.2.2. Non-Minimal Substitution and Tensor highly complex, unsolved many body problem and exp- Gauge Field eriments with it cannot lead to any firm conclusion on angular momentum conservation in free neutron decay. The minimal substitution of Eq.12 of [4] or Eq.7.1.4 of In this part, free neutron decay will be treated using [3] led to the introduction of the vector gauge boson the equations of motion for doublet baryons Eqs.A7-A8 field W which is naturally associated with the vector part and its development in Chapters 9-11 of [3], incorporating V of the V-A theory mentioned beneath Eq.15. The axial vector gauge fields of Eqs.12-13 of [4] or Subsection 7.1.2 vector part A is an asymmetrical part of a tensor which of [3] and new tensor gauge fields. The results obtained, can be introduced by the following non-minimal substi- not reachable from Eq.15, include a prediction of the tution (see Eqs.19-20),

i  1 i  he ba   he ba    he baba he   bahe  XTXWXWg  he  ba  XWXgW  (19) I I  fbh I I  fbh  I I     fhb 4  2 4 

i  1 i  ekc  ekc  XWXWg XT XWXgW eck   cbIkhI   cbIkhI      khIcbcbIkh   cbkh   cbkh  (20) 4  2 4 

Copyright © 2010 SciRes. OPEN ACCESS F. C. Hoh / Natural Science 2 (2010) 929-947 935 where Eq.A10 has been noted. The right side of Eq.19 of [3] has been consulted. The right side of Eq.20 trans- can readily be shown to be invariant under the U(1) gau- form analogously. ge transformations, 3.2.3. SU(3) Tensor Gauge Fields and Gauge ba  ba  ba   XWXW  X s X , Invariance These expressions are now generalized to include SU(3) bahe  bahe  he ba   XTXT X  X s X (21) gauge fields analogous to Eq.7.1.4 of [3]. Limiting our- i selves to baryon doublets in Eq.9.3.7b of [3], Eqs.16-17  fbh    fbh exp2 s Xg with Eqs.19-20 are generalized, with the sign of the where s(X) is a local phase and Eq.12 of [4] or Eq.7.1.4 tensor term changed in Eq.20, to

 ef i ef    II ps   l psWg l     4      1    he ba  ba he  bahe      lsq qr Wl I  l sqWl  Iqr  l sq  l qr T ll   1   i  2   44  he ba    S  dxdx  0atpIII    I I  qrsq  g    fhbrt  2i  4     i he  ba      l sqWg l  l qr Wl     4      Mi 3 2  a  cc ..   bb pt    (22)  i     psheII   l psWg hel     4        1     lsq qrW  cbIkhl  l sqW cbl  qr  khI  l sq  l qr T   cbkhll  1   i  2   S  dxdx 44  *b    g   eck     III  tp 0    cbIkhI qrsq  rt  2i   4  i     l Wg   l W       4 sq khl qr cbl     3  Mi     cc ..2   bb 0 bpt    (23) Equation Eq.22 is invariant with respect to the SU(3) baryon indices associated with  in Eq.22, together with gauge transformations Eq.7.1.7 of [3], with the obvious a generalization of the second of Eq.21, replacement of the two meson indices by the three  2i   bahe  XT )(     bahe  XT )( he  ba  1 XUXU  (24) l sq l qr ll   fhbrt l sq l qr ll  X X 3sq  3qr   fhbrt  g  Analogously, the same invariance also holds for Eq.2.3 with Eq.2.4 replaced by  2i   XT )(  eck     XT )(  1 XUXU  eck  (25) l sq l qr   cbkhll rt l sq l qr   cbkhll  cbXkhX 3sq  3qr  rt  g  For application to neutron decay, only the SU(2) part corresponding to H has been given in Subsection 4-5 of of Eqs.7.1.4-5 of [3] is needed and l and l´ run from 1 to [13] which are found by means of the invariant aymmet- bahe  3. Apart from the flavor indices l and l´, the tensor T ll  rical operator kl of Eq.B15 of [3]. These are has 16 components, 10 symmetrical and 6 asymmetrical, bh bahe  1 bh  21 bh  12 which in its turn is grouped into a vector E(electric field TT     TT  ll ll ea 2 ll ll in electromagnetism) and an axial vector H(magnetic 21 12  ,2 11 2  iHHiTiHTT , (26) field), which is assigned to the axial vector A mentioned  llll  3 ll  21 22 above Eq.19. Identification of the tensor components ll  2  iHHiT 21

bahe  The remaining 13 components of T ll  do not enter evaluation of the type 2) terms, the final state  f| in here and are put to zero. Eq.28 will contain the gauge fields,

ba  ba  3.3. First Order Relations  1    2 XiWXW  2  The gauge boson and tensor field in Eqs.22-23 are decay   ba XW  products of the neutron whose wave functions now ac- W 0 ba    ba W  (31) quire a weak time dependence. Follow Eq.6.4.1 or Eq.7. ba  0 3.1 of [3], noting Eq.A10, and let the nucleon wave fun-  expW  W XKiXiEw  ctions in Eqs.22-23 take the form ba  ba 0 w0   expW  W XKiXiEw

  , xX fhb   bahebahebahebahe   13 31 32  iTiTTT 23  2  01 0 (32)  opop  fhb exp XKiiEXxXaa bahe  bahe  0   expW  W XKiXiEtXT   Xa 01  op  where Eq.12-13 of [4] or Eq.7.1.4-5 of [3] limited to its 1    , xX   0fhb  aop  SU(2) part has been consulted. The initial and final nu- 0 cleon states in Eq.28 will have a laboratory space time    expXKiiEXx    xX ,,   0fhb 1fhb dependence of the form given in Eq.27 with subscripts p (27) for proton and n for neutron attached to the variables there. Here, use has been made of Eq.9.3.7b and Eq.9. where E is the energy, K the momentum, aop  1 and (1)  3.18a of [3] which gives tp = 31 for proton and rt = 23 aop (X ) is a first order quantity varying slowly with time. Both can be elevated to operators in quantized case, for neutron in Eq.22. After summing over the flavor in- as are described beneath Eq.6.4.2 of [3]. Ordering of dices t, p, s, q, and r and carrying out the integration (1)  over X, the type 2) terms become these small quantities aop (X ), g, W etc is the same as that in Eqs.18-19 of [4] or Eq.7.3.2 of [3]. The subsc- g 4 2   Wp   EEEKK nWp ripts 0, 1 denote zeroth order and first order quantities, 24 respectively.     E 2    Following the rudimentary quantization procedure of 4 *  n 2      00 ap xwxd  2      0 ann   ccxE ..  Eqs.23-25 of [4] or Eqs.6.4.12-15 of [3], let the initial     4    and final states be denoted by * ef  eabh   0  IIap txi  0  fhbn  ccx ..  Kni n  ,0 (28) (33)  p W W W ,,,, KETKEWKpf W where Eq.A6 has been used and EW and KW terms have respectively. n, p, W, T denote the neutron, proton, gauge been neglected because they are small relative to |I,II boson, and tensor gauge field, respectively. Further, |/||. With Eq.26, Eq.32 and Eq.B5 of [3], the 16 t’s in Eq.33 reduce similarly to three for an axial vetor: 0 0 iffi  ,0 (29) 1 00 ffii  1 tt bahebh   21  tt bhbh  12 , ea 2 + Let aop in Eq.27 and its hermetian conjugate aop be 1 elevated to annihilation and creation operators. Insert baheea  ea ea 2112  tt  bh   tt Eq.27 into Eq.22 and sandwich the resulting expression 2 between . There are two types of first order  2211  1122 (1) 0 22 11, 11  tttttttt 22 , (34) terms: 1) those containing aop (X ) and 2) those linear in gW and gT. 2112 21 21  ttttt 12 Carrrying out integration over the time X0, the type 1) bh  bh  0 terms read  expW  W XKiXiEtXT ,

  E 2   ea ea 0 3 4  n  expW  W XKiXiEtXT fi   0a xixdXdS   0a x ,  4    (30)  0)1( 3.4. Decay Amplitude fi  opop  iXaafS The decay amplitude Sfi for the  function is found by 3 in which the c.c. term in Eq.22 contributes equally. Sfi is putting Eq.30 to the negative of Eq.33. Letting   d X, the decay amplitude and Eq.A6 has been used. For the the result is

ig 4 S fi  2   Wp  EEEKK nWp  24

    E 2     3  * xwxd  2 n   2    ..  *  eftxiccxE  eabh   ccx .. (35)    00 ap     0 ann    0 IIap 0 fhbn       4     2 3    En    0a xxd   0a x    4  

In an analogous fashion, The decay amplitude Sfi for the  function is obtained from Eq.23 and reads

ig 4 S fi  2   Wp  EEEKK nWp  24

    E 2     3  *a xwxd  2 n   2  a  ..  *a   ebh   ccxtxiccxE .. (36)    p00     nn 0   p0  deII hbad n0       4        2 3  *a  En  a    0 xxd    0 x    4  

The starred wave functions in nominators of Eqs.35- neutron, as  f| has been included in Sfi of Eq.30. 36 represent final states or proton, irrespective the nucl- The proton may have a different m or spin value in eon lables; the c.c. terms will turn out to contribute eq- Eqs.A16-A19 relative to that pertaining to the initial ually and can be dropped together with an overall factor neutron in Eqs.35-36. There are four combinations 2 multiplying the right of Eqs.35-36. The wave functi- which are denoted by ons in denoiminators of Eqs.35-36 are those of the initial

1 1 1 1 neutron m  2 2  2  2 proton m 1  1 1  1 2 2 2 2 (37) notation F  GT  GT  F

As there are two equally correct solutions Eqs.A16- tray to the measured A asymmetry coefficient [9]. In- A17 and Eqs.A18-A19, Eq.30 and Eq.33 are to be av- serting Eqs.A16-A17 and Eqs.A18-A19 into Eq.35 and eraged over these two equally probable solutions. The Eq.36 for the four combinations of Eq.37, making use of averaging will turn out to not affect Eq.30 so that it can Eq.A10, summing over the spinor indices, employing be carried out for Sfi and Sfi of Eqs.35-36 directly to Eq.34, and integrating over the angles  and  in the obtain Sfiav and Sfiav. It removes terms of containing relative space, one finds that the both averaged decay am- f0(r)g0(r) or their derivatives that will appear in Eqs.35- plitudes Sfiav and Sfiav are the same, as may be expected 36 after application of Eqs.A16-A17 or Eqs.A18-A19 from the symmetry between the  part Eq.22 and the  and Eq.37. Such terms will for instance differentiate part Eq.23; between neutron spin up and spin down decay rates con- S   fiF   S  4  fiGT   g 2 1  SS avfiavfi   i   nWp   KKEEE Wp S  22  N dr  fiGT     S fiF    (38)  1   t 12       0  I  t 22   2 2 f 0   2 0  fgndr 00     IEiIIENw gn 0       11   0 3  3  t      12   1   t 

     2 2 ,  2 2 , IrfdrrIrgdrrI  2 2 , Irgdrr  2 2 rfdrr g0 0 0 f 0 0 0 g00 0 00 f 00 0 00 (39)

ab where Ndr is given by Eq.A22 and Ig00 and If00. are defi- via Eq.31, expressions for the other unknowns t there ned Eq.A23. will be obtained in this and the next paragraph. The vector gauge boson fields Eq.40 was found by varying the 3.5. Expressions for Vector and Tensor total action Eq.7.1.1 of [3], after removing its last term, Gauge Fields or Eq.5 of [4] with respect to W in Subsection 7.4.2 of 3.5.1. Mass Generation of W Boson Via Virtual 0 [3] or Eqs.30-32 of [4]. For neutron decay, the meson  In the analogous meson case, the pion beta decay   action Sm3 in Eq.7.1.1 of [3] is to be replaced by the cor- 0   e e of [4] or Subsection 7.4.5 of [3], the gauge boson responding baryons actions Eqs.22-23, S  S, and the W decay into a pair of leptons. The mass of the W boson vector gauge boson action SGB be replaced by a gauge 2 comes from the (gW) terms in the meson action integral boson-tensor field action SGBT. Equation Eq.5 of [4] or Sm in Eq.11 of [4] via Eq.32 and Eq.35 of [4] or Sm3 in Eq.7.1.1 of [3] is here replaced by Eq.7.1.8 of [3] via Eq.7.4.4 and Eq.7.4.6b of [3] and is       SSSSSS (41) generated by the integral of the |pion wave functions|2 Tn GBT avb  LmLav over the relative space x. It can be seen from Eq.7.4.3a Here, the lepton actions SL + SLm are the same as those and Eq.7.4.4 of [3] that the energy of pion does not enter in Eqs.8-10 of [4] or Eqs.7.1.16-17 of [3]. b is a dim- and can be zero. In this case, the pion is virtual. ensionless proportional constant that, for instance, signi- In the corresponding action-like integrals S and S fies that the action-like baryon integrals S and S differ 2 of Eqs.22-23, there is no such (gW) mass term, only basically from the meson action Sm in Eq.11 of [4] or Sm3 terms of the form (gW)(gI,IIW). Therefore, the gauge in Eq.7.1.8 of [3] in that the normalization of the wave boson W from neutron decay cannot decay into a pair of function amplitudes are different. Sav(Sav) denotes that leptons via the integrals Eqs.22-23 for neutrons. S(S) has been averaged over the both equally pro- The interpretation of this situation is that gauge bos- bable solutions Eqs.A16-A17 and Eqs.A18-A19 for the onW in the neutron case here decays into a pair of lep- baryon wave functions that enter it, just like that Sfi and 0 tons via a virtual  action as has been considered in Sfi of Eqs.35-36 have been averaged to Sfiav and Sfiav Subsection 7.6.2 3 of [3] and employed earlier for muon in Eq.38. decay in Subsection 7.6.3 4 of [3]. In the pion beta decay, The  sign in Eq.41 is chosen because Eq.41 will lead the W is positively charged and decays into a lepton pair to an expression for tab in Eq.38; if this sign is replaced via Eqs.44-45 of [4] or Eq.7.4.10 of [3]. In neutron de- by , the needed Eq.42 below will vanish. Further, this   cay, the W is negatively charged and WI in Eq.7.4.5 of sign will remove terms linear in gW, as is implied by + [3] is replaced by WI so that Eq.7.4.5 and Eq.7.4.6a [3], Sfiav  Sfiav  0 from Eq.38. This will cause Sav  Sav neglecting the first three terms there and noting in Eq.41 to possess (gW)2 terms, apart from the tab terms, Eq.7.4.10 of [3], are modified to to that order and render it to have an extremum when solutions near the correct ones are inserted into S 2 2 av  M W MW  30  iWWWW 21  W  ab     Sav. 2 I 2    WWiWW   3021  Unlike SGB in Eqs.6-7 of [4] or Eq.7.1.2 of [3], SGBT in Eq.41 is unknown. If the tensor gauge field is to have an  ba  g    L  WS   La 22 bL equation of motion like that for the gauge boson Eqs.34- (40) 35 of [4] or Eq.7.4.6 of [3], SGBT is expected to contain       g L1 L1 L1  L2 terms of the form ( W)( T). Physical existence and   , X X       interpretation of the tensor gauge field are not known 22  L2 L1 L2 L2  and will be left to eventual future work. For the present 0 ab dx  2 purpose, it is sufficient to obtain an expression for t  gM 22   Gev42.80 W  from Eq.41 for use in Eq.38. where the gauge boson mass M is given by Eq.35 and Follow the steps of Eqs.30-33 of [4] or Subsection W  ba  Eq.41 of [4] or Eq.7.4.6b and Eq.7.4.9 of [3], x0 is the 7.4.2 of [3] and vary Eq.41 with respect to W . The 0 unknown SGBT is expected to give rise to en- relative time between the quarks in  and  a large nor- 2 0 ergy-momentum terms of the form ( T) corresponding malization volume of the  . X to the first terms on the left of Eq.34 of [4] or Eq.7.4.6a 3.5.2. Variation of the Total Action for Neutron of [3], which have been neglected because they are much Decay smaller than the gauge boson mass term that follows 2 Having found an expression for w0 in Eq.38 from Eq.40 them. Similarly, the obtained (X T) terms can also be

Copyright © 2010 SciRes. OPEN ACCESS F. C. Hoh / Natural Science 2 (2010) 929-947 939 dropped on the same ground so the exact but unknown Eq.7.4.4 of [3]. form of SGBT is of no concern here. Variation of the SL + Comparing the X dependence of W’s and T’s and the  ba  SLm terms in Eq.41 with respect to W is analogous ’s in Eqs.43-46 via Eq.31, Eq.34 and Eq.7.4.19 of [3], to that given by Eq. 33 of [4] or Eq.7.4.5 of [3] and leads replacing L and L there e and , we find to Eq.40.   W   e , eW  KKKEEE  (47) 3.5.3. Expressions for Tensor and Vector Gauge Removing the X dependence in Eqs.43-46 and obser- Fields ving Eq.31 and Eq.34, we obtain  ba  Variation of SavSav in Eq.41 with respect to W is 2 limited to terms of order g . When evaluating the aver- 1 g     0     uuuu 2  L1 L1 L2 L2  age Sav(Sav) using Eqs.A16-A17 and Eqs.A18- A19, it M W  24   e is practically sufficient to use one of them, for instance (48) 7  1 Eqs.A16-A17 for S(S) of Eqs.22-23, and drop the   3  wIIE   00 bfgn 3   f0(r)g0(r) terms, as was mentioned beneath Eq.37. 96  1 In the evaluation of Eqs.22-23, use is made of Eq.27, 12 24 g     Eq.A6 and Eq.A10. When summing over the flavor in-  it    uuuu 2  IIM L1 L1 L2 L2 dices t, p, s, q, and r, tp = 32 and rt = 23 for neutron and fgW 00 24   e Eqs.31-32 are employed. Carrying out the angular inte- 3  II 3  1 grations, one finds  Ei fg 00  w   n b 0   4  II fg 00  1  avb  SS av    iUUUU 2130  U     (42)  ba  ab   (49) W    UUiUU 3021 

 I f 0 32  24 g 2 1 11    b g  1     t     i  uu ,    b 2 L1 L2 U 0    gn 00  f 7 30 XWIIE     II 3 M  24  96  3   1  fg 00  W  e (43)   II fg 00 3 g     22 24 g        t     i  uu L1 L1 L2 L2 b  I 32  M 2  L2 L1 24  f 0  W 24   e  g 2  1   1 U  b    3WIIE    (50) 3 96 gn 00 3 f 00  1     The w´s in Eq.31 are similarly found from Eq.40 and  g 2 1 read  i b   12 XTII   (44) 24 fg 000 1   1 g     w     uuuu , 0 2 L1 L1 L2 L2 g     MW 22   e   L1    L2 24 L1 L2 (51) 1 g     w     uuuu 3 2 L1 L1 L2 L2 2 M 22   g  I f 0 32  W  e  iiUU b   11 XT   21 24 0   II 3  fg 00  (45) 3.5.4. Decay Amplitude as Function of Lepton g   Wave Functions    L2 24 L1 Eqs.48-51 can now be inserted into Eq.38. Here, it is noted that the neutron spin is up or m  1/2 for the upper 2  g   II fg 00 3 two amplitudes corresponding to the upper case in  iiUU b  22 XT   21 0   Eqs.48-50 and down or m  1/2 for the lower two am- 24 I f 0 32   (46) plitudes corresponding to the lower case in Eqs. 48-50. g    ,   dx0 Thus, the w0 in the third component of a triplet Eq.44 L2 L1 0  24 and Eq.49 become a singlet to be combined to the w0 in Eq.51. Analogously, the singlet Eq.48, when inserted where the upper and lower rows in the U´s refer to neu- into Eq.38 becomes the third component of a triplet to be tron spin up m  1/2 and spin down m  1/2, respec- combined with Eq.49. Effectively, the 7 W3 in Eq.43 and tively, in Eqs.A16-A19. Further, Eq.42 has been equated 3 W0 in Eq.44 change place after insertion into Eq.38. to the negative of Eq.40 as is prescribed in Eq.41. Note The two terms are not the dominating ones but will lead that U0 in Eq.43 does not contain W0, which however to the two c’s (< 0.4 or so) in Eq.56 below. The resulting 0 enters Eq.40 and stems from the virtual  action in decay amplitude, noting Eq.47 and Eq.39, reads

 S  16E   II 3  fiF   bbb  n fg 00 (55)  mGTpGT GT 2   b  IIa fgg 00 3 S 4  fiGT   g 2 21  S   i fi   2 1 2 S 8 M W  F 0   00  IIaEc fggbn 00 ,3  fiGT    e 16   (56)  S  7  fiF    2  IIaEc 3 F 3 48 00 fggn 00   1  p  e  n  ep  KKKEEEE  Because  in Eq.41 can be incorporated into the wa- N b dr ve functions  and , it can be absorbed into the normal-     ization condition Eq.10.3.13 of [3] by chosing a different b 00 b  L1   uuuu L2   F 0 F 3  L1 L2  normalization constant or equivalently a different nor-         malized amplitude ag given by Eq.A23. 0 b  pGT 00  L2 uu L1          3.6. Decay Rate and Asymmetry 00 b mGT 0   uu L2    L1  Coefficients A and B        bF 0 00 bF 3  uuuu  L1 L1 L2 L2  3.6.1. Decay Rate As in Eq.7.5.1 of [3], the decay rate is (52) 2  final states TS dfi (57) 2 1 F 0 bb F 0 2 IIEN fgndr 00 (53) 1  cF 0 where Td is a long decay time. The subscript “final states” refers to four final lepton spin states and all possible 8E   II 3 bb  n fg 00 1  c (54) momenta of the proton, electron and antineutrino, like F 3 F 3 2 F 3 b 00  IIa fgg 00 that in Eq.7.5.2 of [3]. The decay rates are

2  S   F    1  fiF     e  3 3 3 (58)  9 p e KdKdKd      2  T spinse  spins 2 GT    d  S   fiGT   The square of Sfi contains squares of the  functions in   ikkk     ki  cos,expsin  Eq.52, which are “linearized” by Eq.7.5.6 of [3] type of    21 n  3  formula. Following the common approach, integration e  1 ee 2  KiKKK ee  i e ,expsin (59) over the recoil momenta Kp in Eq.58 is carried out first.  KK cos  By Eq.47, this gives Kp  K  Ke, where the superscript e3 ee () has been dropped. Introduce These and Eq.52 are inserted into the decay rate expression Eq.58 to produce   ,, kkkKK  321 ,

F   g 4  m  I F      2 2 EEEEKdKKdK 1  e   (60)   8192 NM 245 ee   ep  n  E   GT   drW  e  IGT  

 I F   iF      sin sin dddd    (61)   eee   spinse spins    IGT   iGT  

 spinse spins F  F  F  F   iiiii F  (62)

 spinse spins GT  GT  GT  GT   iiiii GT  (63) where the both arrows denote the spin directions, separated by the commas in Eq.7.4.19 of [3], of the electron and the antineutrino and

2  K  K bbi 1  e3   bbk e 1  k F  F 0 F 3     F 0 F 3  3   mE ee   mE ee

2  K  K bbi 1  e3  1  bbk e k F  F 0 F 3   3 F 0 F 3      mE ee   mE ee

2  K  K bbi 1  e3  1  bbk e k F  F 0 F 3   3 F 0 F 3      mE ee   mE ee 2 (64)  K  K bbi 1  e3   bbk e 1  k F  F 0 F 3     F 0 F 3  3   mE ee   mE ee

2 2  K  K  bi 1  e3   ,1 ik  b e k GT   pGT   3 GT  mGT     mE ee   mE ee

2 2  K  K i b 1  e3  ,  bik e 1  k GT   pGT     GT  mGT  3   mE ee   mE ee

2 2 K  K  i  b e   ,1  bik 1  e3 k GT   pGT  3 GT   mGT      mE ee   mE ee  (65) 2 2 K  K   bi e  k , i  b 1  e3  1  k GT   pGT   GT   mGT   3  mE ee   mE ee 

Let   mmEE pnpnm  Mev2933.1 (66)     1 g 4 1  2  22 bb 2   F     F   EPdE  F 0 F 3  3 24  eFe 2 GT  GT 128 NM bGT Eq.61 can now be evaluated using Eqs.64-65 and     drW   Eq.59. Carrying out the angular integrations, it is found (69) 22 2 that the cross terms in Eq.64 drop out and one finds  eF   EEmEEP emeee (70)  I   2  22 bb 2  16E  F    4 2  F 0 F 3  e (67) where me  Ee    me and PF (Ee) is the conventional I b2  GT    GT   mE ee Fermi electron energy spectrum. The half life of the neutron to be compared to the kn- which is independent of the antineutrino energy E = own exp = 885.7 sec. is |K|. Carrying out the K integration in Eq.60 using h 1 Eq.66, one gets  2log  2log (71) th 2   I  2  I  GTF 2 F  F     ep   EEEEKdK n    Eem    IGT    IGT   where h is the Planck constant. (68) 3.6.2. Asymmetry Coefficients A and B where 0  K    Ee. Inserting Eqs.67-68 into Eq.60, The asymmetry coefficients A and B are obtained from changing the variable dKeKe to dEeEe and noting Eqs. Eq.61, just like Eq.67, but without carrying out integra- 53-55 leads to tion over  and e, respectively. One finds,

  II  b  8E  K   F  GT    d 4sin  2  FGT   e 1  A e cos    II  ee b   e   F  GT    FGT    mE ee  Ee  2 2 2 2 2 2 FGT FGT  F 0 22 F 3  pGT FGT  F 0 22 F 3  bbbbbbbbb mGT (72) 2 2   4 FF  30   FGTpGT    4 FF  30   bbbbAbbbbA FGTmGT 

The antineutrino moves in the opposite direction rela- both Ke and K are now on equal footing in these both tive to that of the neutrino so that K is to be replaced by expressions and Eq.61 can be written in the form K in Eq.47, hence also Eq.52. With this replacement,   II  b  8E  F  GT    d 4sin  2  FGT   e  B cos1   II   b    F  GT    FGT    mE ee (73) 2 2   4 FF  30  GT FGT   4 FF  30  GT bbbbBbbbbB FGT

3.6.3. Comparison with Data If the symmetric quark postulate [H15] mentioned The expressions in Eqs.71-73 have been evaluated using beneath Eq.9.3.7 of [3] is used, Eq.A9 can be inserted Eqs.53-56, Eq.39 and the normalized radial wave func- into Eqs.A7-A8 and Eq.A6 is no longer needed. The tions g00(r) and f00(r) for the neutron associated with so- expressions Eqs.71-73 remain valid if cF0 and cF3 in me of the confinement cases in Table 1. The results are Eq.56 are replaced by c´ and c´ , summarized in Table 2 below. F0 F3   These results have been derived starting from Eqs.A7-  FF 00 F 3  cccc F 3 14/9,2/ (74) A8, as was mentioned in Subsection 2.2.1, and using Eq.A6 which stems from putting the quartet wave func- The corresponding results are similarly summarized tions Eq.9.2.8 of [3] entering Eqs.A7-A8 to zero. inside parentheses in Table 2.

Table 2. Values of the calculated decay rate th(log2), A and B asymmetry coefficients given by Eqs.71-73 and GT/F by Eq.69 are presented for a number of confinement strengths db2 given in Table 1. The normalization constant b in Eqs.53-56 are chosen such that in one case the A coefficient agrees with data [9] and in the other case the B coefficient agrees with data. The integrals appearing in Eqs.53-56, given by Eq.39, depend upon the approximate, normalized radial wave functions for the neutron g00(r) and f00(r) obtained in Subsection 2.2 and shown in Figure 2 for the db2  0.3202 case. The corresponding results stemming from symmetric 22 2 quark postulate using Eq.A69 and Eq.74 are given inside parentheses. AV22bb F30  GTp b F  4.85.5 for all these cases.

PDF data [9] A  0.1173 B  0.9807 exp  885.7 sec

d12 b A B GT/F th(log2)(sec)

0.1621 3.436 0.0595 0.9807 0.854 73.5 2.245 0.1174 0.9984 1.26 38.0 (2.928 0.0130 0.9808 0.938 56.1)

(2.203 0.1172 1.0000 1.27 36.6) 0.1641 3.472 0.0000 0.9806 0.962 91.1 2.639 0.1172 0.9996 1.26 59.9 (3.049 0.0381 0.9807 1.04 73.0) (2.571 0.1171 0.9965 1.26 56.7) 0.1670 3.413 0.0580 0.9807 1.08 109.2 2.982 0.1172 0.9937 1.25 89.16 (3.081 0.0883 0.9807 1.15 91.60) (2.897 0.1172 0.9879 1.24 83.73) 0.3202 9.585 0.1173 0.9781 1.21 1036 9.374 0.1265 0.9807 1.24 1001 (8.815 0.1173 0.9732 1.20 872) (8.347 0.1422 0.9807 1.28 805) 0.3622 12.41 0.1173 0.9679 1.19 1956 11.25 0.1578 0.9807 1.32 1684 (11.30 0.1173 0.9631 1.18 1614) (10.05 0.1708 0.9807 1.36 1359) 0.4042 17.40 0.1173 0.9571 1.16 4345 14.66 0.1861 0.9807 1.40 3349 (15.55 0.1174 0.9526 1.15 3455) (13.03 0.1969 0.9807 1.43 2671)

1.2

g 00(r) 0.8

0.6 -f00( r) y(1) 0.4 y(4)

0.2

r(1/Gev) 0 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25 5.75 6.25 6.75 7.25 7.75 8.25 8.75 9.25 -0.2

Figure 2. Normalized neutron radial wave functions f00(r) and g00(r) in (A19) for the db2  0.3202 case in Table 1. r is the quark-diquark distance.

Subsection 2.2 shows that the approximate solutions the chosen confinement strength db2  0.3202 is con- to the baryon spectra problem are those having confine- sistent with the approximate solution to the baryon spec- ment strength constant db2 values in the range around tra problem given in Subsection 2.2. Secondly, this ap- 0.32 to 0.45. Table 2 shows that neutron wave func- pro- ximate solution also leads to B coefficient in agree- tions associated with the db2  0.3202 case leads to half ment with data [9]. life  (log2) which are about 15%(12%) off from data th 3.6.4. Detachment of Weak and Electromagnetic [9]. The associated A and B asymmetry coefficients for Couplings this d  0.3202 case are also rather close to data. b2  / and  / in Table 2 have not been measured. These are obtained by adjusting the only free parameter GT F A V  / is the ratio between the decay rates stemming or chosen normalization constant  such that the calcu- A V b from the axial vector or tensor part and the vector part in lated A coefficient coincides with data. The so predicted the decay amplitude Eq.38. Both these ratios are > 1 and B coefficients deviate only 0.3% (0.8%) from data [9], shows that the axial vector or tensor part of the ampli- close to experimental error. If  is adjusted such that the b tude is greater than that of the vector part. They behave calculated B coefficient coincides with data, the so pre- qualitatively in a similar way as does the conventional dicted A coefficient deviates from data [9] by 8% (21%). (g /g )2  1.611 [9], which lies between  / and For d  0. 3622, the predicted half life time are too A V GT F b2  / but cannot be related to them. long and the both the predicted A and B coefficients are A V In the present theory, there is only one weak coupling futher way from data. For d  0. 3202, it was pointed b2 constant g in Eqs.19-21 identified or associated with g out beneath Figure 1 that no satisfactory convergent V in the literature [9]. Gauge transformations in Eqs.19-21 solutions were found in the range –0.23  d  –0.27. b2 does not allow that the axial vector or tensor field is as- For d  0.1670, the predicted half life time are too b2 sociated with a different coupling constant g . That   short and the both the predicted A and B coefficients are A A  is due to that the normalization type of constant  in also futher way from data, particularly for the A coeffi- V b Eq.41, the only free parameter in this article, is con- cient. Thus, agreement of predictions of the half life nected to  but not to  and  is rather large in Table  (log2) and A or B asymmetry coefficients with data [9] A V b th 2. is best for d  0.3202 and deteriorates considerably b2 Even this g or g will drop out in the decay rate. Eq- for larger or smaller d values. This supports the results V b2 uation Eq.69 shows that the magnitude of neutron decay of Subsection 2.2 that confinement strength constant d b2 rate is proportional to (g/M )4. Now, M 2 itself is pro- lies in the range 0.32 to 0.45. W W portional to g2 according to Eq.40 so that In conclusion, the present treatment leads to two app- 2 roximate predictions, bearing in mind that they are based 4   g 2    upon the approximate results of Subsection 2.2. Possible 32GF  (75) M 4  dx0   source of the approximations there has been conjectured W    in Subsection 2.3. Firstly, the predicted half life time for

Copyright © 2010 SciRes. OPEN ACCESS 944 F. C. Hoh / Natural Science 2 (2010) 929-947 is independent of g. Here, GF is the Fermi constant of rious observable baryon phenomena is irretrievably lost. Eq.7.4.29 of [3]. Since g=e/sinW, where W is the However, the point particle approximation of a baryon Weinberg angle and –e the electron charge, Eq.75 is can be valid in certain low energy interactions. For inst- independent of e. ance, the strong charge of a baryon may be considered to Because W is not a basic constant in the present the- be concentrated at a point source in pion-nucleon scat- ory but can be derived as in Eq.7.2.3 and Eq.7.2.12 of [3] tering. Analogously, in Rutherford scattering, the proton or Eq.3.3 and Eq.3.12 of [5], the more genuine weak can be regarded as having a point charge. In these cases, coupling Eq.75 is detached from the much stronger Eq.76 can be applied and J is conserved. electromagnetic coupling characterized by e, just like In weak interactions, however, the gauge boson inter- such a detachment found in the meson case in Subsec- acts differently with the differently flavored quarks, bro- tion 7.5.3 of [3]. Nature is too economical to deal out adly speaking. Therefore, reduction of the baryon to a two fundamental constnts g and e that are so close to point particle cannot be made. This is evident in the intr- each other. Instead, the strength of weak interactions is oduction of a gauge fields in Eqs.12-13 of [4] or 2 ba  characterized by the dimensionless constant FW = 1.737 Eq.7.1.4 of [3], where  operating in the relative  1013 in Eq.7.5. 22a of [3] which is much smaller than space of the diquark and quark cannot be neglected in the corresponding  = 1/137 for electromagnetic inter- the ensuing calculations. Thus, the angular momentum J actions. of Eq.76 is not applicable to the baryon wave functions On the contrary, based upon an anlysis of some vector in Eqs.A1-A2, which depend upon both the laboratory meson decay rates, the strong coupling s and electro- coordinates X and the relative space coordinates x. magnetic coupling   1/137 are unified into one single Therefore, J needs not be conserved in neutron decay. “electrostrong” coupling via the hypothesis Eq.9.2 of [6] This is supported by noting the following. 4 or Eq.8.3.11 of [3], s   or s  0.2923. In the four possile spin combinations for the nucleons That the Fermi constant GF in Eq.75 is a ratio betw- in Eq.37, the total spin of the lepton pair is fixed, being een a large volume and a long relative time indicates that zero for the Fermi decays and unity for the Gamow-Te- the weak interaction is related to the large scale, long ti- ller decays. However, Eqs.62-65 show that this total spin me or low energy aspects of physics. can also be unity for the Fermi decays and zero for the Gamow-Teller decays in violation of angular momentum 3.7. Possible Nonconservation of Angular conservation. Thus, a qualitative prediction is that an- Momentum gular momentum is not conserved in neutron decay and 3.7.1. Theoretical Background of Possible J hence in weak interactions in general. Nonconservation 3.7.2. Experimental Tests of J Conservation The angular momentum J of an observable, spin ½ point Involving Nucleons particle in conventional form J in Eq.76 is conserved in muon decay which involves    four spin ½ point particles. XiJ   i   (76)  X    As was mentioned in Subsection 3.1, Eq.15 makes use of conservation of angular momentum. When combined is a constant of motion, hence a conserved quantity. Th- with Coulomb correction functions, its predictions are erefore, the total angular momentum of an ensemble of relatively consistent with nuclear -decay and free neu- such particles is also conserved. tron decay data. Therefore, there is a prevalent view that However, a baryon is neither a point particle nor a det- angular momentum is conserved in such decays. Arguments achable ensemble of such particles. Therefore, conserva- aginst this view has been given in Subsection 3.1. The con- tion of J in Eq.76 cannot be applied to a baryon without clusion given at the end of Subsection 2.7.1 thus invali- reservation. J conservation also does not apply to the qu- dates Eq.15 by Jackson et al. [11] and hence also the in- arks that constitute the baryon because a quark is not ob- terpretations of the experimental results that ensue from it. servable in the sense implied by Eq.76. For a slowly Already in their classical paper of 1956, Lee and Yang moving doublet baryon, however, Subsection 10.2.4 of [14] remarked that in weak interaction experiments up to [3] shows that Eq.10.2.1 of [3] can be reverted to a Dirac that time, the baryon number, electric charge, energy and equation when the diquark coordinate xI merges into the momentum are conserved. Conservation of angular mo- quark coordinate xII. This merger reduces the present mnetum J and parity P as well as invariances under ch- nonlocal description Eqs.A1-A2 to a local one so that arge conjugation C and time reversal T had however not Eq.76 is applicable and J is conserved. been established. Nonconservation of P and C was soon By reducing a baryon with extension into a point part- discovered and P violation has been extensively meas- icle, a great amount of underlying physics leading to va- ured [15]. A small violation of T has also been detected

Copyright © 2010 SciRes. OPEN ACCESS F. C. Hoh / Natural Science 2 (2010) 929-947 945 and subjected to many experimental investigations [12] thout Higgs- in the scalar strong interaction hadron the- and [16]. ory. Natural Science, 2(4), 398-401. http://www.scirp. In contrast, no experiment dedicated to test conserva- org/journal/ns [5] Hoh, F.C. (2010) Scalar strong interaction hadron the- tion of J in nuclear -decay or free neutron decay has ory(ssi)--kaon and pion decay. In: Columbus, F., Ed., been performed to my knowledge. In fact, no experiment Hadrons: Properties, Interactions, and Production, Nova exists that directly distinguishes Fermi from Gamow- Science Publishers. Teller transitions in free neutron decay without making [6] Hoh, F.C. (1999) Normalization in the spinor strong in- use of Eq.14. teraction theory and strong decay of vector meson VPP. Therefore, specific tests on J conservation in such de- International Journal of Theoretical Physics, 38(10), cays seem to be called for. Such a test is however not 2617-2645. [7] Hoh, F.C. (2007) Epistemological and historical implica- strictly a test of the present theory which holds for free tions for elementary particle physics. International Jour- neutron decay only. As was indicated below Eq.15, the nal of Theoretical Physics, 46(2), 269-299. unknown effects of internucleon interactions intervene [8] Hoh, F. C. (1996) Meson classification and spectra in the the theory and eventual experimental results. To this end, spinor strong interaction theory. Journal of Physics, G22, decays of free neutrons are needed and will give results 85-98. of more fundamental importance. That this has not been [9] Amsler, C., et al. (2008) Particle data group. Physics Letters, B667, 1. done is due to the great technical difficulty of such an [10] Källén, G. (1964) Elementary particle physics. Addi- experiment. son-Wesley, Reading. In view of the wealth of raw data available on coinci- [11] Jackson, J.D., Treiman, S.B. and Wyld, H.W., Jr. (1957) dental experiments with free, polarized neutrons, it may Possible tests of time reversal invariance in  decay. be possible to obtain some indication as to whether J is Physical Review, 106(3), 517-521. conserved in free neutron decay. No analysis along this [12] Lising, L.J., et al. (2000) New Limit on the D coefficient in polarized neutron decay. Physical Review, C62, line has been carried out to my knowledge. Important 055501. clues may be obtained by reviewing existing data. [13] Laporte, O. and Uhlenbeck, G.E. (1931) Applications of spinor analysis to the Maxwell and Dirac equations. REFERENCES Physical Review, 37(11), 1380-1397. [14] Lee, T.D. and Yang, C.N. (1956) Question of parity con- [1] Hoh, F.C. (1993) Spinor strong interaction model for servation in weak interactions. Physical Review, 104(1), meson spectra. International Journal of Theoretical 254-258. Physics, 32(7), 1111-1133. [15] Schreckenbach, K., et al. (1995) A new measurement of [2] Hoh, F.C. (1994) Spinor strong interaction model for the beta emission asymmetry in the free decay of polar- baryon spectra. International Journal of Theoretical ized neutrons. Physics Letters, B349(4), 427-432. Physics, 33(12), 2325-2349. [16] Sromicki, J., et al. (1996) Study of time reversal viola- [3] Hoh, F.C. (2010) Scalar strong interaction hadron the- tion on beta decay of polarized Li. Physical Review, C53, ory, Nova Science Publishers. 932-955. [4] Hoh, F.C. (2010) Gauge boson mass generation—wi-

Appendix Rules for manipulating the spinor indices are given in Appendix B of [3]. Here, an upper index 1(2) can be This appendix provides the basic equations underlying lowered into a index 2(1 and a  sign) and vice versa the present work given in [2] and Chapters 9 and 10 of according to Eq.B5b of [3]. For doublets, the e, f and d [3]. The equations of motion for the baryons of interest indices in Eqs.A1-A2 are raised and lowered. Multiply hd here are given by Eq.2.9 and Eq.8.5 of [2] or Eq.9.3.16 the so-modified Eq.A1 and Eq.A2 by  ge 2/ and  2/ , and Eq.9.3.19 of [3], respectively, and apply Eq.A6. Eqs.A1-A2 are now reduced to ab gh f    xx, IIIIefbh III ba  ef he  1 (A1) I II  I 2  fhb  , xx III  3 ag (A7) iM  xx,, xx 3 a b b IIIe  III  IIIbb  0 ,, xxxxMi III

1 eck  de ck    cbI heII  khI 2  , xx III Ibc Ihk II e xx I, II (A8) 3 (A2)  ,   , xxxxMi 3 d IIIbb 0b III iMbbIII   xx,,   xx III bh where a subscript 0 has been added to  and  on the

 right. 1 6 f hb   , IIIbIIII  s   ,  fIII III  ccxxxxgxx .., 4 hb If the symmetric quark postulate Section 4 of [2] below (9.3.7) of [3] is used, Eq.10.0.7 of [3] (A3)     ,       ekceck  (A9) 1  0  fhbfhb 0 3  mmmM 3 (A4) b 8 qsp can be inserted into Eqs.A7-A8 which now take a sim- pler form. where x and x are the diquark and quark coordinates, I II The quark coordinates are transformed into a labora- respectively.  and  are baryon wave functions. The tory coordinate X and a relative coordinate x according spinor are defined in Eqs.C1-C3 of [3], to Section 5 of [2] or Eq.3.1.3a, Eq.3.1.10a and Eq.3.5.6   abab  0  ab  XXX , ab    ab    ab  of [3], X 0 X X 1 ,  xxxxxX (A10) (A5) 2 III III the m’s are the quark masses obtained from meson spec- As has been mentioned at the end of Section 3.1 of [3],  tra in Table 1 of [8] or Table 5.2 of [3] and are mu = the relative coordinates x  x is a “hidden variable” not 0.6592, md = mu + 0.00215 and ms = 0.7431 in unit of directly observable. Gev. b is the scalar quark-diquark interaction potential Consider solutions of the separable form Eq.5.1 of [2] 6 and the strong quark-quark interaction constant gs /4 can or Eq.10.1.1 of [3], be absorbed into the amplitudes of  and .  e  , xx    e  exp XiKx  ,    (A11) The six component wave functions  and  can each ca III ca  c be decomposed into a doublet part  and b and a where K = (E0, K) is the four momentum of the baryon. quartet part. Since we are only concerned with the dou- The rest frame rest (K = 0) doublet equations in the in blet or spin ½ baryons Eq.9.3.7b of [3], the quartet relative space are obtained from Eqs.A7-A8 using Eqs. baryon wave function components are put to zero. From A10-A11 and read Eq.5.4 of [2] or Eq.10.2.1 of [3], Eq.2.6 of [2] or Eq.9.2.8 of [3], one finds that  ba  ba 2  Ei 2/   E0 4/     x 1 , 0b (A12) b 2  bb11  2  2 3 a  bb   0 xxMi  a 1 aa11  2  2   2 c 2    Ei 2/    E 4/    x cb cb 0 0 (A13) 12 3    , 12 223 11 1 1  bb  0b xxMi ,  12 , (A6) 12 12322  2 2 Similarly, Eq.A3 with Eq.A6, Eq.A10 and Eq.A11 0 leads to Eq.5.5 of [2], 11 2 22   1  4 12 212 11 1 1  a  , b x  Re 0a   0 xx . (A14) 23 3  12 112 22 2 2 23 , The doublet wave functions 0 and 0 include a noma- 11 2  22  1 0 lization type of factor 1/cb factor according to Eq.A23

1 which vanishes for a free baryon. In this case, the right  10     0 xx  side of Eq.A14 drops out and Eqs.A12-13 are linear, 2 1  0 x  0  0 rifrg  ,cos (A19) which is necessary for wave packet building. Equation 4 Eq.A14 has now the solution Eq.10.2.2a [3] dropping 2       xx the nonlinear terms there; 20 0 Substituting Eqs.A16-A17 and Eqs.A18-A19 into db 2 b r bb 10  b2 ,  xrrdrdd (A15) Eqs.A12-A13 yields, respectively, Eq.10.2.12 of [3] and r Eq.6.9 of [2] or Eq.10.2.12 (with f0  f0,) of [3], where the four db’s are unknown integration constants. E 3 E The doublet wave functions in the relative space are  0 3 0   b bd rM   00 rg  8 2  entirely analogous to those of the hydrogen atom and are 2 expanded into spherical harmonics according to Eq.6.3  E0   2    0    0 rf  0 (with gl  g ) of [2] or Eq.10.2.3a of [3]. These rela-  4   rr  tions give for orbital quantum number l  0 and azi- 3 muthal quantum number m  ½ Eq.10.3.8 of [3] which  E0 3 E0   b bd rM   01 rf consists of two equivalent solutions,  8 2  (A20) 1 1 E 2  m  ,  1 x   rifrg  ,cos  0  0  0 0    1  0 rg  0 2 4  4  r 1     0 xx 10 (A16) A with16 0  0 rfrf (A21) 2 1  0  0 rifx i ,expsin The doublet  and  wave functions in the rest frame 4 have been normalized in Subsection 10.3 of [3]. The 2 20    0 xx resulting normalization integral given by Eq.10.3.9b of [3] reads 1 1 1 m  ,   rifx expsin  i  , 2 0 0  E 2 2 4 2 0 2 2 N dr  2 drr  0 0   0 rgrfrg 1  4 10    0 xx (A22) (A17) 2 2  1 rf  2 rf 2 0 2 0   0 x 0  0 rifrg  ,cos r  4  2 ag    0 xx 20 0 , 0 rgrf  00 00 rgrf ,, cb 1 1 1 m  ,  0 x 0  0 rifrg  ,cos E 2 4 g  ,10 a 2  0d (A23) 00 g 2 N 1  dr0      xx 10 0 (A18) dr0  NN dr A18in 0 rf ,with 2 1  0  0 rifx i ,expsin rg replaced ,by rgrf 4 0 00 00 2 according to Eq.10.3.14 of [3]. cb is a large normaliza- 20   0 xx tion volume for the doublet baryon.

1 1 1 m  ,  0  0 rifx expsin  i  , 2 4