Mass Generation via the Phase Transition of the Higgs Field
Abstract
:1. Introduction
2. Derivation of the Higgs Potential from Catastrophe Theory
2.1. Method 1: Relying on Catastrophe Theory and Stable Isolated States
2.2. Method 2: Implementing a Shortcut
2.2.1. Utilizing a Familiarity Heuristic
2.2.2. Looking Back to Landau’s Theory of Phase Transitions
3. Discussion of Phase Transitions
3.1. The Higgs Phase Transition
3.2. The Maxwell Convention and Chemical Reactions
3.3. Overcoming the Energy Barrier
- (a)
- If the Gibbs free energy gained by the parts of the system is not sufficient to push any part up to at least point P (or B), then the perturbed system remains in the neighborhood of point I.
- (b)
- (c)
3.4. Star-Forming Phase Transitions
3.5. Peculiar -Transitions
4. Conclusions
- (a)
- Figure 1 shows the potential functions for Landau’s phenomenological theory of second-order phase transitions [1], including that ascribed to the Higgs field. As the control parameter is increased (to simulate time evolution), the potential develops two features that render it unphysical: two symmetric stable global minima appear on either side of the local maximum at , and they both continue to move away from in time.Thus, assuming an initial state at , as usual [4,5,6], the phase transition does not produce a unique or a universal final state. But these properties are required for the Higgs field at present [9,10,11,12,13,20]; and uniqueness of the final state is required for many phase transitions in solids and fluids [8,29,30,31,32,34,35,36,45,54].
- (b)
- Figure 2 shows how to get around the problems highlighted above. The potential functions of the cusp catastrophe [4] are all shifted so that one minimum is always at (representing the initial state) and another minimum is constrained to always be at , making this (final) state universal [20]. The linear term of the cusp catastrophe (Equation (4)) precludes the appearance of another minimum at before the second-order critical point at is reached; and when such a minimum finally appears for , it is incapable of influencing the dynamics of the phase transition that took place spontaneously already for .The maximum that appears for represents a free-energy barrier between the two isolated stable states. External perturbations may drive a system from to , if they supply the requisite energy to overcome the intervening barrier (a first-order phase transition [2,3,30,31]), otherwise the system remains oscillating about . As the control parameter decreases toward (Figure 2), the barrier becomes shorter (just as in catalyzed chemical reactions [39,40,41,42,43]), and a second-order phase transition appears at , where the barrier disappears [29,32,45]. We believe that such a transition occurred in the massless Higgs field when it acquired its uniquely positive universal VEV [16,17,20] because we cannot imagine vacuum perturbations strong enough to overcome the energy barrier in the interval .Before the phase transition occurs, the Higgs field is induced to executing small amplitude oscillations about the minimum at that represents the equilibrium VEV of the massless state. Such oscillations generate evanescent particles with both positive and negative masses that do not survive long into the future. The observed particles of our times were all assigned masses after the Higgs field had settled to its universal VEV of 246.22 GeV [20,64].
- (c)
- Figure 3 shows the evolutionary path in the control plane of the cusp catastrophe. The path remains within the fold lines of the separatrix at all times (even for ), and exhibits a Maxwell critical point [37,38] for and a second-order critical point for .As the green curve in Figure 2 shows, a first-order stable minimum at becomes available for , but it is not necessarily accessible to a system located at via a first-order phase transition due to the intervening energy barrier. For the Higgs field, such a continuous line of first-order phase transitions was observed in the past (before the discovery of the Higgs boson and the measurement of its mass; see Refs. [16,17,46] and references therein); although the lattice simulations were using Landau’s even-symmetric potential and a Higgs mass of no more than GeV. For higher Higgs masses in non-perturbative simulations, the second-order critical point was replaced by a smooth crossover to the final massive state [16,17,46]. These doubtful results must have their origin in the unphysical potential used, and new simulations are needed to revisit the true nature of the Higgs phase transition.
- (d)
- Figure 4 shows a schematic illustration of the various aspects of first-order phase transitions capable of overcoming the intervening energy barrier [3]. Basically, there are two separate evolutionary modes depending on the amount of energy deposited by acting external perturbations over time: (i) a strongly perturbed system (point J in Figure 4) is not impeded by the barrier any longer and makes the transition to the final equilibrium state S on a dynamical time [30,31,44]; and (ii) a system oscillating about point I, and perturbed gradually upward to point P (or B), gains enough energy to jump over the top of the barrier B and down to the final equilibrium state S [28,29,30].
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. The Potentials of Higher-Order Catastrophes
Appendix A.1. Swallowtail Potentials
Appendix A.2. Butterfly and Triple-Point Potentials
1 | In contrast, Landau [1] was not thinking about isospin or null quantities when he formulated his theory. To him, symmetries were visible in the arrangement of atoms in a crystal or in the (mis)alignment of magnetic moments in magnetic materials. |
2 | In all fairness to Landau [1], Thom’s catastrophe theory [4] did not exist in Landau’s time, so he did not know that his Taylor expansion of the potential was not formally correct near the degenerate critical point. In fact, he was apparently lucky to get the rest of the perturbation () right when he correctly eliminated the cubic term (), albeit based on an inconclusive argument (that, for , the curve of phase transitions degenerates to a single point in the plane, where P is pressure); the counterargument is that functions and may have the same zeroes [5] and/or that . |
3 | |
4 | |
5 | The point or is where the evolutionary path crosses the B-axis in the control parameter plane (see Figure 3). |
6 | Recall that catastrophe theory is applicable only to gradient systems [6], so it does not account for time, and qualitative conventions have been invented to describe actual time evolution before and after a phase transition (or “catastrophe”). |
7 | Thus, not all butterfly phase-transition paths are covered in the present investigation. |
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Christodoulou, D.M.; Kazanas, D. Mass Generation via the Phase Transition of the Higgs Field. Axioms 2023, 12, 1093. https://doi.org/10.3390/axioms12121093
Christodoulou DM, Kazanas D. Mass Generation via the Phase Transition of the Higgs Field. Axioms. 2023; 12(12):1093. https://doi.org/10.3390/axioms12121093
Chicago/Turabian StyleChristodoulou, Dimitris M., and Demosthenes Kazanas. 2023. "Mass Generation via the Phase Transition of the Higgs Field" Axioms 12, no. 12: 1093. https://doi.org/10.3390/axioms12121093
APA StyleChristodoulou, D. M., & Kazanas, D. (2023). Mass Generation via the Phase Transition of the Higgs Field. Axioms, 12(12), 1093. https://doi.org/10.3390/axioms12121093