Yang-Mills Gauge Theory with Higgs Mechanism
Locally gauging the global rigid internal symmetry Lie group G makes a big qualitative difference. Note, there is NO gravity as yet in this theory. It is done on globally flat Minkowski space-time, yet rest masses of all the fundamental particles, i.e. leptons, unconfined quarks, W-bosons emerge from the vacuum coherent ODLRO local order parameter in the U(1)xSU(2) electro-weak force sector. There is NO need for the Haisch-Puthoff transverse virtual photon acceleration drag in the semi-classical SED picture, and no need for James Woodward's "Mach's principle" model. The rest mass of the Higgs particle(s) themselves is irrelevant to the formulae for the rest masses of the leptons and quarks. To say that the high rest mass(s) of the Higgs shows that standard model is wrong is not correct. However, there is a contradiction with the GR equivalence principle of the inseparabilty of gravity and inertia that I think I have corrected by arguing that the curvature of space-time is emergent from a charge neutral Goldstone phase of the electro-weak force.
To review, we have the false vacuum disordered continuous internal symmetry Lie group G, with Lie algebra T^a in a nxn matrix representation. The range of a is not necessarily the same as n, except in the "adjoint representation" where
(T^a)bc = -if^abc the structure constants of the Lie algebra of G
The unbroken ordered vacuum phase has an invariant subgroup H of G such that the H transformations leave the order parameters invariant. This implies that the Lie sub-algebra of H annihilates the order parameter, that are eigenvectors of the sub-algebra charges with zero eigenvalue. That is, the order parameters are "charge neutral" relative to the residual internal symmetry group H of the more ordered COHERENT vacuum phase. The generators of G not in H are called the "broken generators". We use the Mexican Hat quartic potential for the order parameters because the associated quantum operator field theory for small vibrations in Higgs amplitudes and Goldstone phases are then renormalizable. The mass matrix of the global gauge theory is not of direct physical interest. Now locally gauge G, this introduces compensating Cartan 1-form Yang-Mills potentials (in most general non-Abelian G)
A = Audx^u = Au^aT^adx^u
where T^a is the nxn matrix representation of the elements of the G Lie algebra.
Define the internal symmetry gauge covariant partial derivatives in flat Minkowski space-time as
Du = ,u - ieAu^aTa
for a real Lie group representation of n REAL SCALAR FIELDS T^a replaced by iQ^a, where Q^a are real antisymmetric nxn matrices. Let D(g) be a nxn matrix representation of g in G. Then
The scalar field c- number local order parameters transform as
(0|phi(x)i|0) --> D(g)ij(0|phi(x)j|0)
i = 1 ... n
Note n = 2 is a U(1) complex scalar field here. The phij are real functions of space-time in global Minkowski metric.
The internal symmetry Yang-Mills gauge potentials transform INHOMOGENEOUSLY (like Levi-Civita connection under Diff(4)) as
Au --> D(g)AuD(g)^-1 + ie^-1D(g)^-1D(g),u
For the internal symmetry gauge covariant partial derivatives
Du(0|phi|0) --> D(g)Du(0|phi|0)
And for the Yang-Mills field tensors under G (internal curvature)
Fuv --> D(g)FuvD(g)^-1
Fuv^a = A^av,u - A^au,v + ef^abcAu^bAv^c
Fuv = Fuv^aTa
The Lagrangian Density is
L = (1/2)DuphiiD^u(phi^i) - V(phii) - (1/4)F^auvF^auv
Make small vibrations about the VEVs (0|phi(x)i|0)
When we quantize these small vibrations into quantum harmonic oscillators, the theory will be renormalizable.
phi(q-number)j = (0|phi(x)j0) (c-number local order parameter) + phi'(x) q-number operator (boson CR)
The resulting Lagrangian density for the q-number excited states above the vacuum is
L' = (1/2)(phi'(x)^j)^,u(phi'(x)j),u - (1/2)mij^2phi'(x)^iphi'(x)^j - (1/4)(Fauv(x)F^a^u^v(x)) + (1/2)Mab^2A^au(x)A^b^u(x) + Lint
Note there will be THREE kinds of INDEPENDENT MASS MATRICES
I. mij for the HIGGS BOSONS of the UNBROKEN GENERATORS of ordered H.
II. Mab for the MASSIVE GAUGE BOSONS OF THE BROKEN GENERATORS OF G ---> H
III. Yukawa interactions for the Leptons & Quarks with
Fermion(lepto-quark) MASS MATRIX
mij = (Gamma)j(0|phi|0)j
*Eric Davis and James Woodward have confounded I with III in their Red Herring objections.
DEFINE A GAUGE CONSTRAINT for the BROKEN GENERATORS o unorderedf G i.e. T^a NOT in the ordered subgroup H in which the BROKEN VEVS vanish. These broken generators WOULD have had gapless Goldstone phase modes if there were no local gauge fields.
The sum over ij in the mij^2 mass matrix for the scalar fields does not include the broken generators, but only goes over the H invariant subgroup generators in the selected gauge constraint.
mij^2 (over H only) = Functional Second Derivative of V relative to phi(x)i & phi(x)j at (0|phi(x)|0) vacuum in G/H.
*These mij describe the REST MASSES of the HIGGS BOSONS on mass-shell, the number of them depends on the dimension of H.
The gauge force boson MASS matrix is
Mab^2 = e^2(T^aT^b)ij(0|phi(x)^i|0)(0|phi(x)^j)
WHERE THESE T^a are ONLY FOR THE BROKEN GENERATORS OUTSIDE OF H!
That is, the MEISSNER EFFECT of MASSIVE GAUGE BOSONS ONLY APPLIES TO THE BROKEN GENERATORS OF THE UNORDERED LIE GROUP G. For example, W+,W-,W0 is from SU(3)weak that is completely broken. Only U(1)hypercharge is unbroken with a massless gauge boson.
"We see that the vector fields associated with the broken generators acquire non-zero masses, while the gauge fields of the unbroken subgroup H remain massless. The WOULD-BE Goldstone bosons have DISAPPEARED, and the corresponding degrees of freedom have been absorbed as additional spin states of the massive vector fields.
Fermions (leptons & quarks) can be EASILY INCORPORATED .. by adding appropriate kinetic and interaction terms to the Lagrangian ...
psi are the fermion leptons & quarks with their OWN independent COUPLINGS Gammai
* That is, the rest masses of the leptons & quarks are INDEPENDENT of the Higgs boson rest masses!
N = number of generators for UNORDERED FALSE VACUUM internal symmetry Lie group G
N - K = number of UNBROKEN GENERATORS of ORDERED INVARIANT SUBGROUP H of G
i.e. K = number of BROKEN GENERATORS
n = Dimension of REAL REPRESENTATION of G.
e.g. n = 2 is a U(1) single complex scalar field over Minkowski space-time.
There are then
n - K MASSIVE HIGGS BOSONS that are now being looked for.
K massive gauge bosons ---> K = 3 in standard mode,l i.e. only W+, W- & W0
Massless gauge bosons N - K
Note, if we start from
G = U(1)hyperchargeSU(2)weak
N = 4
K = 3
N - K = 1
U(1)xSU(2) is n = 6 real scalar fields 2 for U(1) and 4 for SU(2)
This means 3 massive Higgs bosons in this simplest version of standard model.