- Mass Ratios
**Regularization-Renormalization Method of Ni, Lou, Lu, and Yang**- D4-D5-E6-E7-E7-E8 VoDou Physics Lagrangian
- Weak Force
- Higgs, Truth Quark, and Weak Bosons
- Experimental Results
- The Standard Model hierarchy, fine-tuning, and negativity of the Higgs mass squared, by Marko B. Popovic - hep-ph/0204345
- References

Here is how the mass ratios work:

- the electron / Truth quark mass ratio is fixed, as are the mass ratios of all fundamental fermions;
- the Truth - antiTruth quark pair mass corresponds to the sum of the masses of the weak bosons W+, W-, and W0;
- the W0 / Z0 mass ratio is fixed, and the sum of the masses of the weak bosons W+, W-, and Z0 corresponds to the Vacuum Expectation Value of the Higgs scalar field that transforms the W0 into the Z0;
- the Planck mass is a condensation of fermion-antifermion pairs, somewhat like a pion condensate;
- the SU(5) LeptoQuark X-bosons have a Vacuum Expectation Value corresponding to a decoherence / condensate below the Planck mass.

It is interesting that

which is very close to

the ratio of the geometric part of the Weak Force Strength to the Electromagnetic Fine Structure Constant is 0.253477 / ( 1 / 137.03608 ) = 34.7355.

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-E7-E7-E8
VoDou Physics 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-E7-E8 VoDou Physics model are a Higgs
scalar mass of about 146 GeV and a a Higgs scalar field vacuum
expectation value of about 252 GeV**. The fault was not with the
D4-D5-E6-E7-E8 VoDou Physics model itself, but with my incorrect
understanding of it with respect to the Higgs mechanism.

**Guang-jiong Ni, Sen-yue Lou, Wen-fa Lu, and Ji-feng Yang**,
in hep-ph/9801264,
used a new Regularization-Renormalization (R-R) method to calculate
the Higgs mass in the Standard Model to be about 140 GeV. Guang-jiong
Ni has further described the R-R method in quant-ph/9806009.

When the R-R method of Ni is applied to the D4-D5-E6-E7-E8 VoDou Physics model, it is seen that, at tree level,

** **

**To see how the R-R method of Ni, Lou, Lu, and Yang works**,
consider Integrals over 4-dim SpaceTime or its dual 4-dim Momentum
Space of factors (ignoring factors like 2 and pi) like

for which the Integral I diverges as the Integral of dK^4 / K^4

or as definite integral from 0 to R of dK / K in the limit of R going to infinity, which is a logarithmic divergence of the most weak type. Other Integrals encountered in physics give linear, quadratic, and other higher divergences.

Their new R-R method deals with the divergence (for example, of the Integral I) by:

1 - taking the partial derivative with respect to M^2 of the divergent Integral of

to get the **convergent** Integral d I / d M^2 of

whose value is (ignoring factors like 2 and pi) d I / d M^2 = 1 / M^2.

2 - after working with the convergent Integral, then going back to the original Integral by Integrating the convergent expression d I / d M^2 = 1 / M^2 with respect to M^2, getting

where C1 = - ln M1^2 is a Constant of Integration and M1 has the dimension of mass.

3 - Use the Chain Approximation to derive a renormalized mass MR = M + dM.

then dM = ( Alpha / 4 Pi ) M ( 5 - 3 ln M^2 / M1^2 )

where Alpha = e^2 / 4 Pi is the Electromagnetic Fine Structure Constant.

4 - Fix the constant M1 so that the parameter M in the original Lagrangian is interpreted as the observed renormalized mass MR. This cannot be done by Perturbative Quantum Field Theory, so M1 should be fixed by experiment. The condition dM = 0 gives

5 - When the particle is off the mass-shell so that p^2 =/= M^2, combine the above with other Feynman Diagram Integrals of Quantum ElectroDynamics to calculate such things as the Lamb shift of the 2S(1/2) state with respect to the 2P(1/2) state in Hydrogen, which can then be understood as a mass modification of the electron bound in the Hydrogen atom.

As Ni says in quant-ph/9806009, "... by means of perturbative QFT, we can evaluate the mass modification but never the mass generation. ... we re replace the divergence by arbitrary constant M1, then the latter is fixed by the mass M measured in the experiment. ... the crucial point lies in the fact that we should act from the beginning, act before the counterterm in introduced, act until the bottom is reached. That is, to take the derivative of integral I with respect to M^2 (or to a parameter Sigma added by hand, say M^2 -- goes to -- M^2 + Sigma) enough times until it becomes convergent, then perform the same times of integration with respect to M^2 ( Sigma then setting Sigma -- goes to -- 0 again) for going back to I. Now instead of divergence,we obtain some arbitrary constant Ci ... to be fixed ... Each Ci has its unique meaning and role ..."

** **

**The difference between the D4-D5-E6-E7-E8 VoDou Physics model
Higgs mass of 146 GeV and the Ni, Lou, Lu, and Yang calculated value
of 138 GeV** is due to these facts:

- the D4-D5-E6-E7 model value of 145.789 GeV (about 146 GeV) is a tree level value, while Ni, Lou, Lu, and Yang calculate the Higgs mass beyond tree level using a nonperturbative Gaussian Effective Potential (GEP) method;
- Ni, Lou, Lu, and Yang use as input parameters:
- the particle masses mW+ = mW- = 80.359 GeV and mZ0 = 91.1884 GeV,
- the Weinberg angle sin^2(thetaW) = s2w = 0.2317, and
- the Electromagnetic Fine Structure Constant at the Z0-mass energy level alphaE(mZ0) = 1 / 128.89;

- the D4-D5-E6-E7 model uses as input parameter the vacuum expectation value of the Higgs scalar field v = 252.514 GeV, which is based on the D4-D5-E6-E7 model identity v = mW+ + mW- + mZ0, and which leads to the tree-level particle masses mW+ = mW- = 80.326 GeV and mZ0 = 91.862 GeV as well as the other calculated particle masses and force strength constants of the D4-D5-E6-E7 model;
- Ni, Lou, Lu, and Yang use a T-quark mass of 175 GeV, giving them a larger 1-loop level fermion contribution than would be the case for the D4-D5-E6-E7 model which has a T-quark mass of 130 GeV.

where d is the 8-dim covariant derivative P is the scalar field F is the Spin(8) curvature S' and S are half-spinor fermion spaces D is the 8-dim Dirac operator GF is the gauge-fixing term GG is the ghost term

After dimensional reduction to 4-dim SpaceTime, the scalar Fh8 /\ *Fh8 term becomes the Integral over 4-dim Spacetime of

where cross-terms are eliminated by antisymmetry of the wedge /\ product and 4 denotes 4-dim SpaceTime and I denotes 4-dim Internal Symmetry Space

**The first term** is the integral over 4-dim SpaceTime
of

Since they are both SU(2) gauge group terms, this term merges into the SU(2) weak force term that is the integral over 4-dim SpaceTime of Fw /\ *Fw (where w denotes Weak Force).

**The third term** is the integral over 4-dim SpaceTime of the
integral over 4-dim Internal Symmetry Space of

The third term after integration over the 4-dim Internal Symmetry Space, produces, by a process similar to the Mayer Mechanism developed by Meinhard Mayer, terms of the form

where L is the Lambda term, P is the Phi scalar complex doublet term, and M is the Mu term in the wrong-sign Lamba Phi^4 theory potential term, which describes the Higgs Mechanism. The M and L are written above in the notation used by Kane and Barger and Phillips. Ni, and Ni, Lou, Lu, and Yang, use a different notation

so that the L that I use (following Kane and Barger and Phillips) is different from the Ln of Ni, and Ni, Lou, Lu, and Yang, and the P that I use is different from Pn, and the 2 M^2 that I use is ( 1 / 2 ) Sigma.

** **

**Proposition 11.4 of chapter II of ****volume
1 of Kobayashi and Nomizu** states that

where P takes values in the SU(2) Lie algebra. If the action of the Hodge dual * on P is such that *P = -P and *[P,P] = [P,P], then

If integration of P over the Internal Symmetry Space gives P = (P+, P0), where P+ and P0 are the two components of the complex doublet scalar field, then

which is the Higgs Mechanism potential term.

The value of the fundamental mass scale vacuum expectation value v of the Higgs scalar field is set in the D4-D5-E6-E7 model as the sum of the physical masses of the weak bosons, W+, W-, and Z0, whose tree-level masses will be 80.326 GeV, 80.326 GeV, and 91.862 GeV, respectively, and so that the electron mass will be 0.5110 MeV.

The resulting equations, in my notation (and that of Kane and Barger and Phillips), are:

In their notation, Ni, Lou, Lu, and Yang have 2M^2 = (1/2) Sigma and P^2 = 6 Sigma / Ln, and for the tree-level value of the Higgs scalar particle mass Mh they have Mh^2 / Pn^2 = Ln / 3.

By combining the non-perturbative Gaussian Effective Potential (GEP) approach with their Regularization-Renormalization (R-R) method, Ni, Lou, Lu, and Yang find that:

Mh and Pn are the two fundamental mass scales of the Higgs mechanism, and

the fundamental Higgs scalar field mass scale Pn of Ni, Lou, Lu, and Yang is equivalent to the vacuum expectation value v of the Higgs scalar field in my notation and that of Kane and Barger and Phillips, and

Ln (and the corresponding L) can not only be interpreted as the Higgs scalar field self-coupling constant, but also can be interpreted as determining the invariant ratio between the mass squares of the Higgs mechanism fundamental mass scales, Mh^2 and Pn^2 = v^2. Since the tree-level value of Ln is Ln = 1, and since Ln / 3 = Mh^2 / Pn^2 = Mh^2 / v^2 = 2 L, the tree-level value of L is L = Ln / 6 = 1 / 6, so that, at tree-level

so that it is equal to the sum of the physical masses of the weak bosons, W+, W-, and Z0, whose tree-level masses will be 80.326 GeV, 80.326 GeV, and 91.862 GeV, respectively, and

so that the electron mass will be 0.5110 MeV.

Then, the tree-level mass Mh of the Higgs scalar particle is given by

The Critical Mass (or Energy, or Temperature) Mssb for restoration of the Spontaneous Symmetry Breaking (SSB) symmetry, which is Mssb = Mh sqrt(12/Ln), so that, for the tree-level value Ln = 1,

The High-Energy Singularity of the Higgs Mechanism model, Msing, beyond which the Higgs field vanishes, and the Maximum Energy Scale, Mmax, can be calculated in the Higgs Mechanism model. The fact that the Higgs field vanishes above Msing and Mmax may justify regarding the Higgs Mechanism model as a low energy effective theory, just as the D4-D5-E6-E7-E8 VoDou Physics model is fundamentally a low (with respect to the Planck energy) energy effective theory. The values calculated by Ni, Lou, Lu, and Yang are

The Planck energy is

The Complex Doublet P = ( P+, P0) = (1/sqrt(2)) ( P1 + iP2, P3 + iP4 ) = (1/sqrt(2)) ( 0, v + H ), so that

where v is the vacuum expectation value and H is the real surviving Higgs field.

The value of the fundamental mass scale vacuum expectation value v of the Higgs scalar field is in the D4-D5-E6-E7 physics model set to be 252.514 GeV so that the electron mass will turn out to be 0.5110 MeV.

Now, to interpret the term

in terms of v and H, note that L = M^2 / v^2 and that P = (1/sqrt(2)) ( v + H ), so that

= (1/16) ((M^2 / v^2) ( v + H )^4 - (1/8) M^2 ( v + H )^2 =

= (1/4) M^2 H^2 - (1/16) M^2 v^2 ( 1 - 4 H^3 / v^3 - H^4 / v^4 )

Disregarding some terms in v and H,

The second term after integration over the 4-dim Internal Symmetry Space, produces, by a process similar to the Mayer Mechanism, terms of the form

where P is the Phi scalar complex doublet term and d is the covariant derivative.

** **

**Proposition 11.4 of chapter II of ****volume
1 of Kobayashi and Nomizu** states that

where P(X) takes values in the SU(2) Lie algebra. If the X component of Fh4I(X,Y) is in the surviving 4-dim SpaceTime and the Y component of Fh4I(X,Y) is in the 4-dim Internal Symmetry Space, then the Lie bracket product [X,Y] = 0 so that P([X,Y]) = 0 and therefore

Integration over Internal Symmetry Space of (1/2) dx P(Y) gives (1/2) dx P, where now P denotes the scalar Higgs field and dx denotes covariant derivative in the X direction.

Taking into account the Complex Doublet structure of P, the second term is the Integral over 4-dim SpaceTime of

= (1/4) (1/2) d ( v + H ) /\ *d ( v + H ) = (1/8) dH dH + (some terms in v and H)

Disregarding some terms in v and H,

Combining the second and third terms, since the first term is merged into the weak force part of the Lagrangian:

In the D4-D5-E6-E7-E8 VoDou Physics model, the geometric part of the weak force strength, and the geometric weak charge that is its square root, are:

In more customary particle physics notation, such as that found in Kane, there are two weak charges, g1 and g2, such that their squares are weak force strengths. In the D4-D5-E6-E7 model, the geometric weak charge is the average of the customary two weak charges g1 and g2:

Combining some aspects of the D4-D5-E6-E7 model and some aspects of the customary picture gives tree-level estimate results that are off by a few percent. Estimated Weak Boson masses are approximately

Mz = sqrt( g1^2 + g2^2 ) v / 2 = 93.98 GeV

Some other relations given by Kane, and the results of using in them some D4-D5-E6-E7 model values, and the estimates immediately above, are

from the Weinberg angle ThetaW: g2 = e / sin(ThetaW) = 0.6247

g1 = e / cos(ThetaW) = 0.3463

the Fermi constant, GF = sqrt(2) g2^2 / 8 Mw^2 = 1.11 x 10^(-5) GeV^(-2)

For comparison, the D4-D5-E6-E7 model value of the Fermi constant is

where

= sqrt(80.326^2 + 80.326^2 + 92.862^2) = 146.09298 GeV.

Roughly,

The tree level mass of a Higgs scalar, about 146 GeV, is somewhat higher than, but roughly similar to, the tree level Truth quark mass of about 130 GeV.

the sum of the tree level masses mW+ + mW- + mZ0 of the 3 weak bosons W+, W-, and Z0, that is, the physical weak bosons below the Higgs mass scale, is the fundamental energy level vacuum expectation value v of the Higgs scalar field. To give the tree-level particle masses mW+ = mW- = 80.326 GeV and mZ0 = 91.862 GeV (as well as the other calculated particle masses and force strength constants of the D4-D5-E6-E7 model), P is set equal to 252.514 GeV. The D4-D5-E6-E7 model assumed value v = about 80+80+92 = 252 GeV.

**The Higgs Vacuum Expectation Value of 252 GeV is roughly the
****Compressibility of the
Aether**** and the ****Superposition
Separation of an entire single Tubulin in the Brain****,**
and **is close to the tree level mass of the ****Truth
Quark**** T-T(bar) Meson of about 260 GeV.** It is also of
the same order of magnitude as the geometric mean (about 650 GeV) of
the Planck Energy (10^19 GeV) and the
Hydrogen Lamb Shift Energy (4.3
x 10^(-5) eV = 4.3 x 10^(-14) GeV (see Weinberg, The Quantum
Theory of Fields, Vol. I, p. 593).

Mw, the square root of the sum of the squares of the tree level masses of the 3 weak bosons W+, W-, and Z0, that is, the physical weak bosons below the Higgs mass scale, is 146.09298 GeV, which is very close to the mass of the tree level Higgs scalar mass of 145.789 GeV. The tree level mass of a pair of Higgs scalars, about 292 GeV, is somewhat higher than, but roughly similar to, the fundamental energy level vacuum expectation value v of the Higgs scalar field, about 252 GeV, and the truth quark T-T(bar) meson mass of about 260 GeV.

Each quark in the T-Tbar can decay very rapidly to W + b + X (where X is just an indication for other miscellaneous stuff).

Note that , roughly,

while

Also,

In hep-ph/0102137,
12 Februay 2001, G. Degrassi says: "... The last months of the year
**2000** ... seems a good moment to try to review what we (do not)
know about the **Higgs**.

- On one side we have the impressive amount of data collected at
LEP, SLC, and the Tevatron that allow to probe
**the quantum structure of the Standard Model**(SM), thereby**providing indirect information about the Higgs mass**. - On the other we have the result of the
**direct searches performed at LEP**with the excess of events reported in the combined analysis of the four collaborations.

... Given these two pieces of information it is often said that
one of the greatest achievement of LEP has been to have pin down the
**Higgs mass between 113** (from the 95% C.L. lower bound of the
direct searches) **and 170 GeV** (from the 95% C.L. upper bound of
the global fit to electroweak data). ...

Formulae ...[ that are used for indirect ] Higgs mass inference from precision measurements ... following a Bayesian approach ... to construct f( mH | ind. ), the p.d.f. of the Higgs mass conditioned by this indirect information under the assumption of the validity of the S.M.... are very accurate for 75 < MH < 350 GeV with the other parameters in the ranges 170 < Mt < 181 GeV ... 0.113 > ALPHAs(MZ) > 0.123 ... The experimental inputs I use to construct f( mH | ind. ) are: s2eff = 0.23146 +/- 0.00017, MW = 80.419 +/- 0.038 GeV, ... Mt = 174.3 +/- 5.9 GeV, ALPHAs(MZ) = 0.119 +/- 0.003. ... Table 1 summarizes the results ...

... As expected, the inclusion of the direct search information drifts the p.d.f. towards higher values of MH by cutting regions below 110 GeV. ...".

In my opinion, the fact that the indirect prediction of a 90 to 105 GeV Higgs mass is shown to be wrong by failure to detect a Higgs with a direct search up to 110 GeV indicates that either

- some of the input parameters ( particularly Mt = 174.3 +/- 5.9 GeV and ALPHAs(MZ) = 0.119 +/- 0.003 ) may be incorrect, or
- the details of the Standard Model ElectroWeak structure are poorly modelled by the calculation techniques used, or
- both the input parameters and the calculation techniques may be flawed.

In the paper hep-ph/0204345

Marko B. Popovic says:

"...

The range of Higgs masses in the vicinity of 140 GeV satisfies all theoretical requirements for healthy physics up to the Plank scale ... the theoretical and logical necessity for the existence of ... Supersymmetry, technicolor, extra dimensions and other exotic scenarios around the TeV scale ... is strongly questioned here ... experimental data and theoretical knowledge strongly point towards the SM as the best candidate for the physics that will be observed in near future experiments. ...

... The vacuum energy problem is ... cited as a serious obstacle to the SM. However, the vacuum energy problem only guides toward a conflict between the classical general theory of relativity and its, yet to be understood, "quantum-compatible" version.....

.... The well-known non-zero vacuum expectation value (vev) of the scalar Higgs field, vEW = 2.462 x 10^2 GeV, sets the scale of the electroweak interactions. Another well-known energy scale is the Planck scale, /\planck = 1.2 x 10^19 GeV, probably setting the scale of the unified theory incorporating gravity. Obviously, many orders of magnitude separate these numbers. Traditionally, this fact is considered a serious obstacle for the SM. The problem termed hierarchy problem expresses doubts that the SM alone can provide a good physical description over such a broad range of energies. ...

... Only when united with the fine-tuning problem can the hierarchy look persuasive as an obstacle to the SM at high energies.... To characterize the fine-tuning problem it is useful to introduce the dimensionless mass parameter ... mu = mH(/\)^2 / /\^2 . The parameter mH(/\) represents a renormalized Higgs mass at the cut-off energy scale /\ ...

... Figure 1 ...

... Illustration of the fine-tuning problem in traditional sense (top sketch) and in the more realistic SM case (bottom sketch). The purpose of these drawings is to show the qualitative distinctions between the functional forms of the two cases; by no means should the two plots be quantitatively compared or separately analyzed. The gauge and Yukawa couplings are labeled with gi and gf respectively while mu_a and mu_b identify the two possible evolutions of dimensionless mass parameter. ...

... from fine-tuning ... in traditional sense ... it may be concluded that the SM can be a good description only up to very small energies, maybe ten or so times larger than a physical Higgs mass! ...

... However, there is no fundamental physical or logical basis for the claim that the loop corrections need to be smaller than the tree level value!... the logic behind fine-tuning problem ... in traditional sense ... fails. And the reason is rather simple - the coupling constants g^2 and lambda run logarithmically, and moreover, the parameter mu intersects with g^2 . Therefore, the more the theory is finely tuned, the smaller a hierarchy (see Fig. 1.b). And the fine-tuning is shown to be benign. ...

... The simple truth is that the SM already gives the best explanation! ...

... one should be ambitious and try to explain the hierarchy up to the Planck (or GUT) scale with the SM structure alone. ...

... In Fig. 4 the dimensionless mass parameter mu is shown over the range, 140-230 GeV, of Higgs masses. Due to the stability and perturbativity constraints, only the 140-170 GeV curves are accepted up to the Planck scale. 5 An exciting feature is easily observed; the mu's in the vicinity of 140 GeV are negative over a whole range of energies, and they very slowly approach zero in the high energy limit. The negativity of dimensionless parameter mu guarantees the good behavior of the scalar propagator at high energies. In other words, the negativity of the mass squared suggests that the mass parameter in the propagator at high energies should be set to zero and that the theory is surely not breaking down below the Planck scale.

The stability curve roughly traces the energy scale at which quartic coupling lambda goes to zero (more pedantically the scale at which deeper minima form) as a function of physical Higgs mass. Therefore, in Fig. ... 4 it is seen that the lambda's for Higgses in the vicinity of 140 GeV are positive while mu's are negative over the whole energy range. Therefore, the minimum of the effective potential can not be formed in this range. ... That is the SM explanation of the fictitious hierarchy problem! ...

... It should be noted that at the Planckian energies both dimensionless quantities characterizing scalar potential, i.e. lambda and mu, happens to be suspiciously close to zero! ...

... Interestingly, the Higgs masses in the vicinity of 140 GeV may happen to be those preferred by the fine-tuning quantities presented in Figs. 2 and 3 as well. ...

... mu for Higgses in the vicinty of 140 GeV runs almost logarithmically (i.e. as expected) all the way up to the Planck scale. The slowly varying parameter alpha in Fig. 3 is tightly confined to the region near the value of 0.05.

...

Conclusion.Why does experimental data and theoretical knowledge strongly point towards the SM as the best candidate for the physics that will be observed in near future experiments? The answer is drawn from the following facts:

- -The SM is the correct "theory" at low energies.
- -The hierarchy and fine-tuning are not serious problems of the SM.
- -The Planck scale, as an important gravity related dimensional quantity, may be addressed by the current theoretical framework. The range of Higgs masses in the vicinity of 140 GeV satisfies all theoretical requirements for healthy physics up to the Plank scale:

- stability,
- perturbativity and
- negativity of the dimensionless mass parameter mu.
- -The range of Higgs masses in the vicinity of 140 GeV falls inside 1 sigma of the ... electroweak precision data (ewpd) ... analysis preferred range (or 2 sigma with Z -> bbar b forward backward asymmetries excluded.
- -Current new physics models at the 1 TeV scale are neither minimal nor required by some burning theoretical necessity. ...
... If Higgs happens to [weigh] around 140 GeV ... Nature has indeed chosen the large SM high-energy desert in front of us ...".

Such a desert is consistent with the possibility that the
Truth Quark, through its strong interaction with Higgs Vacua, may
have two excited energy levels at 225 GeV and 173 GeV, above a
**ground state at 130 GeV**. The 173 GeV excited state may exist
due to appearance of a Planck-energy vaccum with < phi_vac2 > =
10^19 GeV in addition to the low-energy Standard Model vacuum with
< phi_vac1 > = 252 GeV.

Weinberg, The Quantum Theory of Fields (2 Vols.), Cambridge 1995,1996.

Barger and Phillips, Collider Physics, updated edition, Addison Wesley 1997.

Gockeler and Schucker, Differential Geometry, Gauge Theories, and Gravity , Cambridge 1987.

Kane, Modern Elementary Particle Physics, updated edition, Addison Wesley 1993.

Kobayashi and Nomizu, Foundations of Differential Geometry, vol. 1, John Wiley 1963.

Kobayashi and Nomizu, Foundations of Differential Geometry, vol. 2, John Wiley 1969.

Mayer, Hadronic Journal 4 (1981) 108-152, and also articles in New Developments in Mathematical Physics, 20th Universitatswochen fur Kernphysik in Schladming in February 1981 (ed. by Mitter and Pittner), Springer-Verlag 1981, which articles are:

- A Brief Introduction to the Geometry of Gauge Fields (written with Trautman);
- The Geometry of Symmetry Breaking in Gauge Theories;
- Geometric Aspects of Quantized Gauge Theories.

Ni, To Enjoy the Morning Flower in the Evening - What does the Appearance of Infinity in Physics Imply?, quant-ph/9806009.

Ni, Lou, Lu, and Yang, hep-ph/9801264.

Particle Properties from the Particle Data Group at LBL

......