Table of Contents:

- Higgs Scalar Mass is about 146 GeV
- hep-ph/9512438
- ALL PAPERS
- hep-ph/9501252
- hep-ph/9501252and hep-th/9403007
- hep-th/9402003
- hep-th/9302030
- HISTORY of the BINOMIAL TRIANGLE

Before I read quant-ph/9806009,

by Guang-jiong Ni of the Department of Physics, Fudan University, Shanghai 200433, P. R. China, and the related paper hep-ph/9801264 by Guang-jiong Ni, Sen-yue Lou, Wen-fa Lu, and Ji-feng Yang,

I did not correctly understand the Higgs
mechanism. Therefore, in my earlier papers I had wrongly stated
that the D4-D5-E6 model gives a Higgs scalar mass of about 260 GeV
and a Higgs scalar field vacuum expectation value of about 732 GeV.
**I now see that my earlier values were wrong, and that the correct
values under the D4-D5-E6 model are a Higgs scalar mass of about 146
GeV and a Higgs scalar field vacuum expectation value of about 252
GeV**. The fault was not with the D4-D5-E6 model itself, but with
my incorrect understanding of it with respect to the Higgs
mechanism.

My paper hep-ph/9512438, on the Rb and Rc Crises, should be disregarded because the experimental results on which it was based have been superseded. Here is a brief discussion of the matter.

In ALL PAPERS, I have considered the Standard Model gauge group to be G=SU(3)xSU(2)xU(1), and have not considered the distinction between it and the quotients U(3)xSU(2)=G/Z3 , SU(3)xU(2)=G/Z2, and S(U(3)xU(2))=G/Z5.

Similarly, I have not considered the distinction between groups SU(3)xU(1) and U(3)=(SU(3)xU(1))/Z3 or

SU(2)xU(1) and U(2)=(SU(2)xU(1))/Z2 or

SU(4)xU(1) and U(4)=(SU(4)xU(1))/Z4.

For the purpose of studying color, weak, and electromagnetic gauge bosons as Lie algebra infinitesimal generators, my errors should probably have no serious consequences.

Similarly, for the purpose of studying conformal gravity of SU(4) = Spin(6) or SU(2,2) = Spin(2,4), and its relationship to the Higgs mechanism, my errors should probably have no serious consequences.

What I probably should have done in all cases is to use the smallest group (which roughly amounts to specifying which of the diagonal elements of the SU(n) corresponds to the U(1)). For details, see Group Structure of Gauge Theories, by L. O'Raifeartaigh, Cambridge (1986).

The coset spaces used in calculating the force strength constants should remain the same, as each of the forces considered alone would be represented by Spin(5), SU(3), SU(2), or U(1), respectively. In other words, the discrete groups Zn would not change the geometric structures used to calculate the force strength constants.

In hep-ph/9501252 - Gravity and the Standard Model with 130 GeV Truth Quark from D4-D5-E6 Model using 3x3 Octonion Matrices, and in earlier papers, I have not made a proper distinction between the 4-dimensional associative spacetime and the 4-dimensional coassociative internal symmetry space that come from the original nonassociative octonionic 8-dimensional spacetime.

The two 4-dimensional spaces both have quaternionic structure, so it was easy to fail to make the proper distinction. It is also easy to correct the error, by just reading "internal symmetry space" instead of "spacetime" every time context shows it should be done.

The most important place in which this confusion occurs is in discussion of how gauge bosons "see" "spacetime" (should be "internal symmetry space") when determining the volumes of compact symmetric spaces to be used in calculating force strength constants.

In hep-ph/9501252 - Gravity and the Standard Model with 130 GeV Truth Quark from D4-D5-E6 Model using 3x3 Octonion Matrices, on page 99,

and

In hep-th/9403007 - Higgs and Fermions in D4-D5-E6 Model based on Cl(0,8) Clifford Algebra, on page 14,

I wrote v = 517.798 GeV

instead of v / sqrt(2) = 517.977 GeV.

Elsewhere in these papers (p. 89 and p. 8, respectively) I had calculated that v = 732.53 GeV.

hep-th/9402003 - SU(3)xSU(2)xU(1), Higgs, and Gravity from Spin(0,8) Clifford Algebra CL(0,8):

- on page 8, Cl(0,6) should be R(8) instead of R(16) in a number of places
- a typographical error of only $S_{8+}$ or $S_{8-}$ instead of $S_{8\pm}$. The correct 8-dim Lagrangian is: $$ \int_{V_{8}} F_{8} \wedge \star F_{8} + \partial_{8}^{2} \Phi_{8} \star \partial_{8}^{2} \Phi_{8} + \overline{S_{8\pm}} \not \! \partial_{8} S_{8\pm} + GF + GH $$

- erroneously used $\times$ instead of $\oplus$ for the fermion spinor space. The correct full fermion space of first generation particles and antiparticles is $S_{8+} \oplus S_{8\pm} = ({\bf{R}}P^{1} \times S^{7}) \oplus ({\bf{R}}P^{1} \times S^{7})$. It is the Silov boundary of the 32(real)-dimensional bounded complex domain corresponding to the $Type V$ HJTS $E_{6}/(Spin(10) \times U(1)$
- erroneously listed $SU(3)/SU(2) \times U(1)$, instead of
$Spin(5)/SU(2) \times U(1) = Spin(5)/Spin(3) \times U(1)$, as the
$Type IV_{3}$ HJTS corresponding to the 6(real)-dimensional
bounded complex domain on whose Silov boundary the gauge group
$SU(2)$ naturally acts. The corrected table is: The $Q$ and $D$
manifolds for the gauge groups of the four forces are:
\[ \begin{array}{|c|c|c|c|c|} \hline Gauge & Hermitian & Type & m & Q \\ Group & Symmetric & of & & \\ & Space & D & & \\ \hline & & & & \\ Spin(5) & Spin(7) \over {Spin(5) \times U(1)} & IV_{5} &4 & {\bf R}P^1 \times S^4 \\ & & & & \\ SU(3) & SU(4) \over {SU(3) \times U(1)} & B^6 \: (ball) &4 & S^5 \\ & & & & \\ SU(2) & Spin(5) \over {SU(2) \times U(1)} & IV_{3} & 2 & {\bf R}P^1 \times S^2 \\ & & & & \\ U(1) & - & - & 1 & - \\ & & & & \\ \hline \end{array} \]

HISTORY of the BINOMIAL TRIANGLE:

The Southern Song Chinese mathematician Yang Hui (1261 AD) is given credit in my home page for discovering the Binomial Triangle known to Europeans as Pascal's Triangle. However, I have now (20 Jun 94) seen references to earlier discovery of the Binomial Triangle:

- 1100-1109 AD: Jia Xien states the method of forming the Pascal triangle; the triangle was probably known before his time. (from The Timetables of Science, A. Hellemans and B. Bunch, Simon and Schuster (1988))
- ca. 1100-1180 AD: As-Samaw'al ibn-Yahya al-Maghribi wrote The Luminous Book on Arithmetic, which contained an illustration of the Binomial Triangle up to order 11. He was the son of a Moroccan Rabbi. He moved to Baghdad and coverted to Islam. He did not claim to have discovered the Binomial Triangle, but to have gotten it from Al-Karaji, who in turn may have gotten it from another source. (from From One to Zero, G. Ifrah, Penguin (1985) English translation of 1981 French book)
- ca. 300-200 BC: The Jaina mathematician Halayudha has been given credit for discovering the Binomial Triangle by John McLeish (The Story of Numbers, Fawcett Columbine (1991)). McLeish says that it should be called the Meru-Prastera rule. Ernest G. McClain (The Myth of Invariance, Nicolas-Hays (1976)) says that Halayudha was actually only explaining what Pingala had said about 200 BC, that the mathematical Binomial Triangle represented Mount Meru, or Meru Prastara, the Holy Mountain, and could be used to describe the number of forms of long or short syllables that can be formed from a given number of syllables. It seems likely to me that the Binomial Triangle is actually very much older than about 200 BC, and that our discovery of written records of it only (as of now) goes back that far.

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