The Quantum Field Theory of the Fundamental Interactions

A Theory Beyond the Standard Model

to explain the origin of the fundamental fermions.

This theory, currently being researched by Nova Software, Inc., has the goal of explaining how all the known fundamental fermions* (and only those fermions) arise from the three known fundamental particle interactions. The Standard Model assumes the existence of the observed fundamental particles, and then is extremely successful in predicting the physics that ensues. Its theoretical framework, based on quantum field theory (QFT) and the local gauge invariance principle, does not, however, constrain the Universe to that observed set of fermions and their eigenvalues. Many others are possible within that framework. It's only experiment that picks out the particular set used in the Standard Model.

Theories "Beyond the Standard Model" often are based on additional interactions (forces) and/or fields (particles) beyond those currently part of the Standard Model, while preserving the QFT & gauge invariance framework. Some mechanism is typically included to explain why the added element(s) are not already readily apparent to us. That mechanism must be overcome in experiments to confirm predictions of the "new physics." This theory is different: It assumes that the interactions and particle spectrum of our Universe are only those already known. There is new physics waiting to be found, but not via the presently explored types of extensions. There is some evidence for this point of view from the fact that none of the many experimental searches based on the ideas to date has conclusively indicated new physics beyond the Standard Model. A lesson from history may be the unsuccessful search for the Ether, Newton's absolute frame, for many years. The answer to today's conundrum may likewise come from changes to one or more fundamental hypotheses.

In that spirit this theory goes beyond the Standard Model in a different direction by postulating a revised form of quantum field theory and local gauge invariance. It adds a hypothesis that enhances the role of the internal symmetry groups smooth manifolds and their U(1) X SU(2) X SU(3) product to the foundations of space-time. The observed sets of the fundamental fermions* and bosons** emerge as derived objects in a natural way from these hypotheses and their mathematical consequences. Those results then help to insure that the theory will be able to match the Standard Model's successes while providing a deeper understanding of how its fermions originate, and possibly provide new, heretofore unsuspected, testable results.

In its current form, in addition to QFT and the Standard Model, the Interaction Theory draws most heavily on mathematical results from the theories of topological groups and manifolds, homotopy theory, Lie groups and Lie algebras and differentiable manifolds. When complete the theory will be posted on a public pre-print server, published in an established physics theory journal, and made available by free download here to enable and encourage other researchers to further investigate its potential.

* Fermions: There are 12 distinct fermions (fields) based on mass alone. When their quantum numbers are taken into account there are 24 fundamental fermions with the 12 distinct masses. These 24 consist of 8 different sets of various strong, E&M, weak and global quantum numbers. These 8 sets are then replicated twice more at higher masses, for some as yet unknown reason outside the Standard Model. These three otherwise identical mass levels, the generations or families of the fundamental fermions and anti-fermions, may be assigned additional pairs of generation "quantum numbers", such as +-1, +-2, +-3, to distinguish them. The ordinary matter in the Universe consists entirely of generation +1 fermions, without -1 anti-fermions. This absence of anti-fermions is also unexplained in the Standard Model, which is nearly symmetric between the two types of fermions, and thus favors an approximately equal distribution of the two. The higher generation fermions and lower anti-fermions are short-lived, decaying ultimately into generation 1 fermions, so do not contribute significantly to the matter in the Universe.

Chirality is a binary property that can only be defined for fermions and anti-fermions. It is closely related to the field's behavior under transformations between right and left handed coordinate systems, i.e. to parity changes. It is not even an approximately conserved quantum number for Standard Model fermions with mass, but is nonetheless crucial for understanding the weak interaction. It splits the 24 fermion fields into 48 left and right chirality fields that must be considered separately when specifying which field particpates in which interaction, and into 96 fields when fermion--anti-fermion duality is accounted for. The left-chirality fermion fields and right-chirality anti-fermions are the only ones participating in the weak interaction. That then results in 36 (quarks) of those 96 chiral fermions and anti-fermions responding to all three interactions, while 42 respond to two interactions (36 quarks to strong + EM, & 6 charged leptons to weak + EM ), 12 respond to only one (left-chirality neutrinos and right-chirality anti-neutrinos only to the weak interaction; right-chirality charged leptons and left-chirality charged anti-leptons only to the EM interaction), and 6 have no known interaction at all (Right-chirality neutrinos and left-chirality anti-neutrinos). These 6 might be considered artifacts only needed to balance the theoretical books of the Standard Model, without physical significance, as there is no practical way to confirm their existence. Their gravitational interaction, if they were to have a typical lepton mass, would be far too weak to detect. Gravity is approximately 10^-32 weaker than the weak interaction, as a pure number, without any unit of measure.

This property of the chirality preference of the weak interaction is allowed by the underlying formalisms of the Standard Model, QFT and gauge theory, but not required. Consequently it is not an a priori theoretical prediction of the Standard Model, but rather another feature injected by hand to match observations.

Results from neutrino oscillation experiments show that a neutrino created as a member of one generation may later interact as a member of a different generation. This clearly shows that generation number, specifically lepton family number, is not a conserved quantum number for neutrinos although it is conserved for the charged leptons. One way this can happen is for the neutrino to have a non-zero rest mass and its free-particle mass eigenstate to be a linear superposition of its generation eigenstates. Such a situation is summarized in the PMNS matrix supplement to the Standard Model, the analog for neutrinos of the CKM matrix for quarks. The most common theoretical framework used to estimate neutrino mass is some version of the general see-saw mechanism. As they generally require additional ultra-heavy, unconfirmed types of neutrinos such additional frameworks are not considered part of the Standard Model in this investigation.

The 100% chiral selectivity of the weak interaction causes it to not conserve parity 100%. Some unknown aspect of the weak interaction causes it to also violate CP conservation at the ~4% level in certain higher generation processes. This result is embodied in the CKM matrix supplement to the Standard Model. Both fermion and anti-fermion chiralities participate in the strong and EM interactions equally, so these two interactions exhibit no chiral, no parity, preference. The Standard Model also allows for the strong interaction to violate CP conservation up to 100%, but it does no do so. Attempts, both theoretical and experimental, to remedy this disagreement have not been successful. It remains an open question, unresolved within the Standard Model. CPT is conserved by all the interactions, universally.

The fundamental fermions all have spin 1/2, anti-fermions spin -1/2. All have a non-zero mass. These mass values cover a wide range of 1:10^10 or greater. A particle and its anti-particle have the same mass including sign: Gravity does not distinguish between the two types. Charged lepton generation masses approximately satisfy the empirical Koide mass formula. Quark generation masses depart from it significantly. Neutrino mass measuremens have only established upper limits to date and are continuing.
** Bosons: There are 4 fundamental vector bosons (spin 1) for the three interactions based on mass alone. When their quantum numbers are taken into account there are 12: 8 vector bosons (gluons) mediate the strong interaction, 1 (photon) mediates the electromagnetic interaction, and 3 (W+- and Z0) mediate the weak interactions of the left-chiral fermions and right-chiral anti-fermions. The W- and W+ are each other's anti-particle. The Z0 boson, of zero hypercharge and being the zero isospin member of a weak isospin triplet, is its own anti-particle. The gluons are self-dual fields, each its own anti-particle. They carry only linearly independent combinations of the three color and three anti-color eigenvalues. The photon has no quantum numbers so is also a self-dual field and its own anti-particle. Gluons and photons mediate their interactions for both fermions and anti-fermions, of both chiralities. In addition the charged weak vector bosons, W- and W+, participate in the electromagnetic interaction by virtue of their unit electric charges, although they are not its primary carriers. This two-interaction property makes them unique among the vector bosons.

The other fundamental boson is the scalar Higgs boson (spin 0) which does not mediate any one of the three fundamental interactions and is of a vastly different character than the vector bosons of the fundamental interactions. The Higgs we observe today might best be called the reduced Higgs, without any electric charge or other quantum numbers, and thus its own anti-particle. It's a single particle remnant of the expanded Higgs particle, which was a weak isospin doublet and anti-doublet that existed only in the ultra-high energy fundamental particle plasma of the first moments of the tightly compressed Universe. When the Standard Model and data from cosmology and astronomy is projected back to those moments they indicate that only the Higgs had rest mass. All the other particles, both fermions and bosons were constrained to be massles by their fundamental interactions, which are all require the local gauge principle. This principle only gives correct results if the fields are massless. As the Universe rapidly expanded from its ultra-hot and compressed state the plasma cooled, and the average energy of the Higgs field eventually fell to the current value of 246 Gev and stabilized there. This stabilization was possible due to the specific fourth power functional form of the potential energy formed from the expanded Higgs field in the Standard Model. The reduced Higgs field is unique among the fundamental particles in having that substantial non-zero average value (energy) throughout the Universe.

Due to this decline in average energy the expanded Higgs field underwent a quasi-phase transition (known as spontaneous symmetry breaking or hiding) that left only the single, self-dual, reduced Higgs field with 0 weak isospin. At the same time the four initial, massless, weak hypercharge and weak vector boson fields mixed (transformed) into the three curently observed weak interaction vector boson fields with rest mass, and into the single massles photon field and electromagnetic interaction that we see today. This mixing is modeled as a unitary transformation of the boson fields to a new basis in the Hilbert space of states, a rotation of approximately 29 degrees in that space. At the same time the weak hypercharge values of the fermions, all of which interact with the weak hypercharge field, became quantized in integer increments of +-1/6. The Standard Model does not predict these values, so the correct weak hypercharge values are added to the fermion fields by hand to agree with their observed electric charges and the Standard Model formula: Y = Q - T, in which Y = weak hypercharge, Q = electric charge, T = weak isospin projection.

Before the mixing there were 12 physical degrees of freedom among the five bosons: Two possible spin polarizations of each for the four massless vector bosons, and four arbitrary real values for the isospin doublet complex-valued expanded Higgs field. After mixing there are still 12 physical degrees of freedom: Three possible spin polarizations for the three weak vector bosons now that they have mass, two for the massless vector photon, and one arbitrary real-valued component of the reduced scalar Higgs field. Thus the Standard Model portrays the three weak vector bosons each gaining mass and one more physical degree of freedom (spin orientation) at the expense of the expanded Higgs field phase-transitioning to the reduced Higgs and loosing three degrees of freedom.

Fundamental fermion masses are not allowed (i.e. give incorrect results) in the Standard Model interaction terms due to the local gauge invariance principle governing those interactions. To account for the fact that the electrically-charged fundamental fermions do have mass the Standard Model assumes an additional simple direct contact (Yukawa) interaction between the expanded Higgs field and a chirality mix of the fermion fields. After the Higgs phase transition these terms result in mass-like Dirac terms proportional to each electically charged fermion field and to the reduced Higg's stable vacuum expectation value. These terms in the Lagrangian are taken to imply that the electrically-charged fundamental fermions feel a drag from the ubiquitious Higgs field due to this constant interaction. This drag effectively slows their motion, making it appear that each has some non-zero rest mass: They no longer are able to travel at the speed of light. The values of these masses are all proportional to the reduced Higg's stable vacuum expectation value and a different arbitrary constant for each. Thus the actual mass values are not predicted by the Standard Model, but must be inserted by hand into numerical calculations.

The gluon field does not have any weak chiral isospin interaction nor weak hypercharge, so could not interact directly with the expanded Higgs field. The photon field likewise has neither interaction, and only arose later as a consequence of the Higgs phase transition and boson field mixing. Consequently neither the gluon nor photon acquired mass from the Higgs phase transition, so they are the only fundamental particles that are massless and can travel at the speed of light. The Standard Model, however, does not allow for free gluons in today's Universe, so only the photon can actually be observed traveling at the speed of light.
Many other fundamental particles with a wide variety of properties have been proposed and have been or are being actively searched for. The above are the only ones unequivocally confirmed by the Particle Data Group thus far and thus the only subjects for the Standard Model and Interaction theory.

Summmary for a general audience: The current physics theory of the known fundamental particles, the Standard Model, explains extremely well how they interact via their strong, electromagnetic and weak interactions and the Higgs field. It does not, however, explain how the particular combination of observed fundamental particles and the pattern of their interactions, and only that combination and pattern, occurs. These facts are inserted into the theory "by hand" from what we see, rather than being derived from fundamental hypotheses. The Quantum Field Theory of the Fundamental Interactions is being researched at Nova Software, Inc., to address this question, to see if there is a theory that can successfully explain those facts in more fundamental terms. If successful the theory will be made available in the usual physics locations for such work, and at this site in the hope that others will find it a worthwhile subject for further study.