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Higgs Field, the Eternal and Ubiquitous (2021 Edition)


Contents

Spontaneous Symmetry Breaking (SSB) and Phase Transition
Inflaton and Higgs as Metastable States
A Novel Higgs Field
(Modified) Quantum Field History
Quantum Field History à la Gauge Theory

Spontaneous Symmetry Breaking (SSB) and Phase Transition

Symmetry Breaking The existence of asymmetric solutions to a symmetric theory is common to many branches of physics. The reason lies in the fact that the symmetric state is not the state of minimum energy, i.e., the ground state, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken. Figure 01 shows that the initial position of the marble on top of the bump is symmetric but not in a

Figure 01 Symmetry Breaking

state of minimum energy. A small perturbation will cause the rotational symmetry to be broken and the system to assume a stable state configuration. When the symmetry of a physical system is broken in this way, it is often referred to as "spontaneous symmetry breaking (SSB)".

As shown in Figure 01, energy is released when the system transits from the symmetric phase to the lower energy Higgs phase in the
Phase Transition Standard Model process of SSB. Such process is similar to the phase transition (see Figure 02,a) between the gas-liquid-solid phases, in which energy is released to the environment and the rotational symmetry changes to translational with the change of gas/liquid phase to solid. By such analogy, the SSB of electro-weak at 10-12 sec after the Big Bang can be considered as the transition of massless particles moving widely at the speed of light to motion constrained by having mass (see Figure 03 of the Standard Model in symmetric and Higgs phases).

Figure 02 Phase Transition

Figure 03 Standard Model

Figure 02,b portrays another interpretation of phase transition via lowering the potential energy V by some kind of catalyzing agent.

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Inflaton and Higgs as Metastable States

Since the confirmation of the Higgs mass at 125 Gev in 2012, there are more researches on the link between the Higgs field and cosmic inflation. It is remarkable that the Planck data on CMB are indeed compatible with treating the Higgs (in particle physics) as the inflaton (in cosmology), provided the field is non-minimally coupled to gravity (see a short discussion in Inflation and the Higgs particle, 2014).

By tweaking the energy curve as shown in Figure 04,a; a novel idea in 2017 (See "The Higgs Bang") merges the inflaton and the Higgs fields into one scalar field, the inflaton and Higgs just correspond to different phase of the universe. In addition, the scheme requires an extra scalar field to make it consistent. It is known as the dilaton (~ cosmological energy density, i.e., the dark energy). Thus, this new theory resolves three outstanding paradoxes in one shoot (1. endowing mass, 2. unifying inflaton, 3. adding drak energy). Such "Inflaton-Higgs-Dilaton" model can be tested by a particular twist in the polarization of the CMBR. The presence of a dilaton field can also be detected by certain gravitational wave imprint on CMBR. This kind of data may be available by 2020.
Higgs as Inflaton The obvious omission in this model is the absence of the reheating process which is responsible for the generation of matter at the end of inflation (see "Inflation Era"). According to another model developed in 2014 (see "Inflation, LHC and the Higgs boson"), it is indeed possible to produce a potential V with an inflationary plateau at large values of the scalar field, and two minima (Figure 04,b).

Figure 04 Higgs as Inflaton
[view large image]

A much simplified version of the mathematic derivation is sketched below to illustrate the theoretical formulation.

The Lagrangian (density) for gravity in General Relativity has the form : = (-g)1/2R ---------- (1)
where R is the scalar curvature and g is the determinant formed from the space-time metric tensor gik.




Higgs, Time Variation
    Figure 05 portrays the time evolution of the inflaton/Higgs field with the potential given by Eq.(3b). It is computed by a constant value of the self-coupling constant . It is also possible to get some idea about each stage of the evolution by inspecting dv/d in Figure 04,b :
  1. For dv/d > 0 at large value of , it is in flation era.
  2. For dv/d = 0 at the 1st minimum, has reached a true vacuum (Figure 05).
  3. For dv/d < 0, it involves reheating and ends up in another false vacuum state.
  4. For dv/d > 0 after the 2nd false vacuum, the is making the transition to the final Higgs phase. This is the electro-weak SSB.
  5. For dv/d = 0 at the 2nd minimum, this is the SM domain as of today.

Figure 05 Higgs/Inflaton, Evolution with Time

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A Novel Higgs Field

§§§ This is just a mathematical model.

There is an alternate way to associate the Higgs before and during cosmic inflation. The formulation starts from a very small universe at Planck scale with Mass/Energy M = MPL and size r0 = LPL. Accordingly, its energy density PL ~ 10114 erg/cm3. Such entity is at the realm of quantum gravity. Unfortunately, there is no such universally accepted theory for now. One of the ad hoc solutions is to quantize the Friedmann Equation for cosmic expansion.

The classical Friedmann Equation for the evolution of scale factor R is in the form : (dR/dt)2 - 2GM / [R(r0)3] = -kc2 ---------- (5a),
where k is the curvature of the universe, it is actually a form of energy curled up in space.

For the Planck scale model, it can be re-written (in term of the Planck time tPL) as : (dR/dt)2 - 2 / [R(tPL)2] = -kc2 ---------- (5b).

In first quantization to endow wave property to a particle the linear momentum px and position x has to satisfy the commutative relation :
Energy Level and Cosmic Expansion

Figure 06 H Atom Energy Level & Cosmic Radius

Now the hydrogenic wave function n(R) can be identified to the Higgs field . Inflation corresponds to the transition from the level of n 1 to the continuum (n = ). Energy should be infused to pump the transition as shown in Figure 07. In addition, can be generalized to be the unified field in the GUT Era (Figure 08).
After inflation, the dynamic equation reverts back to Eq.(5a) with k = 0. The gravity couples only to mass/energy, detached from the Higgs field.
Cosmic Energy Evolution of Fields Effort to formulate theory in time or size smaller than the Planck scale (~10-44 sec, Figure 08) is doomed to fail as any system below such limit violates the uncertainty principle of t tPL and x LPL (for the corresponding Planck mass/energy) rendering it meaningless (see "What Is The Smallest Possible Distance In The Universe?").

Figure 07 Cosmic Energy Density [view large image]

Figure 08 Q Field Evolution

Evolution of Entropy Entropy is defined as the degree of randomness, which can be expressed alternatively as the degree of freedom in a system, i.e., the number of different parameters or arrangements needed to specify completely the state of a system. The evolution of entropy in the universe can be separated into four phases as described in Figure 08a, which shows lot of (???) for the very low entropy at the beginning of the universe. It has been a puzzling problem since a hot Big Bang should involve a lot of entropy (randomness/disorder). The mathematics in the above quantum cosmological model is specified by only "one" parameter, i.e., the Planck length LPL, which is the absolute minimum entropy for any system. Thus, low entropy came naturally in the model resolving the paradox.

Figure 08a Evolution of Entropy [view large image]

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(Modified) Quantum Field History

According to the the above scenario of "quantum cosmology", the quantum field history has to be modified such that the Big Bang occurred amid quantum uncertainty with no definite time and size. It is only at the Planck scale that a concrete event occurred.
Quantum Field History The universe starts with gravity (which goes on by itself) and a quantum wave function that is identified as the Higgs field. This Higgs field went on to undergo many phase changes and finally settled down with 3 distinct quantum fields plus the Higgs field itself as shown in Figure 09. Such phenomena is similar to the phase transition of water (see Figure 02), which can exist in the form of gas, liquid, and solid in macroscopic states (similar to the strong, weak, and em quantum fields in Figure 09).

Figure 09 Quantum Field History (modified) [view large image]


NB : The gravitational field is a classical field that does not follow the rules for the quantum fields. It only coupled to the Higgs field at the very early cosmic time in the form of quantum gravity. It detaches from the Higgs after inflation when the size of the universe is beyond the quantum domain.
    The role of the Higgs field in the different phases is discussed briefly in the followings :

  1. Planck Scale (the Constants) - Soon after Max Planck introduced the Planck constant h in 1899 to account for the spectrum of blackbody radiation, he realized that the only way to construct a unit of length out of h = 6.625x10-27 erg-sec, the velocity of light c = 3x1010 cm/sec, and the gravitational constant G = 6.67x10-8 cm3/sec2-gm, is LPL = (Gh/c3)1/2. This is the Planck length.
    Subsequently, it transpires that the realm of Planck scale would involve entities with size of the order 10-33 cm and associated with high energy phenomena at 1019 Gev (see list below). Such environment is considered to occur at or near the moment of Big Bang - a most favorable theory for the creation of this universe. Here's the Planck scale :
    Quantum Field History
    • Planck Length - LPL = GMPL/c2 = (G/c3)1/2 = 1.62x10-33 cm.
    • Planck Time - tPL = LPL/c = (G/c5)1/2 = 5.39x10-44 sec.
    • Planck Mass - MPL = (c/G)1/2 = 2.17x10-5 gm.
    • Planck Energy - EPL = MPLc2 = (c5/G)1/2 = 1.22x1019 Gev = 1.95x1016 erg.
    • Planck Temperature - TPL = EPL/kB = (c5/GkB2)1/2 = 1.42x1032 oK.
    • Planck Energy Density - PL = (c5/G2) ~ 10114 ergs/cm3.
      Note that at speed of light c, there is enough time tPL for signal to travel the length of LPL, i.e., no problem with homogeneous/isotropic of the universe.

    Figure 09b Convergence of Quantum Gravity [view large image]

    Figure 09b shows all kinds of objects in this world with different size and mass. Ultimately, they all emerge from a tiny speck at the Planck scale and via the action of Quantum Gravity (see the corresponding Table).

  2. Planck Scale (the Theory) - Theoretically, the moment of Big Bang should be related to General Relativity, specifically, the Friedmann Equation in Standard Cosmology. This equation has to be quantized if the initial entity is in Planck scale as discussed in "Novel Higgs Field". It turns out that the wave function is similar to the Hydrogen atom's with the scale factor R as independent variable. This wave function is identified to the Higgs field . The transition of from the state of huge curvature k to k = 0 represents the period of inflation at the end of which it approaches an oscillatory form ~ Ae-i(t-kx), which is the solution for the "Klein-Gordon Equation of the Scalar Field". At this stage, the Higgs field could be the only quantum field around. It used to be called "Grand Unification" by assuming that all the quantum fields are in existence all the time but merged together at this point.
    BTW, the "flatness" problem is resolved by the transition of curvature k ~ 1/LPL to k 0 in the quantum cosmological model.

  3. Quark-Gluon Plasma
  4. Reheating and "Electroweak Unification" - During inflation, the environment lost a lot of energy in the process. Reheating is another process whereby the energy of the environment is replenished by energy from the Higgs field in the form of quarks and gluons (the Quark-Gluon Plasma QGP, see Figure 10). Separation of the quark fields (of strong interaction) at this period represents a form of phase transition. The re-shuffling also moves the Higgs field to a false vacuum state (mathematically by adding the potential V() to the Klein-Gordon Equation, see Eq.(3b)).
  5. Figure 10 Quark-Gluon Plasma [view large image]

  6. Electroweak Spontaneous Symmetry Breaking (SSB) - At 10-12 sec after BB (Figure 09), the Higgs field decays from the false vacuum to the true vacuum precipitating another phase transition. According to the Standard Model (SM, Figure 11), all the quantum fields already were in existence at the state of false vacuum. However, the quantum fields were not interacting with the
    SM, Parameters Mass, Origin of Higgs and hence the quantum particles (excited from the fields) are massless moving at the velocity of light (Figure 12). Most of them acquired mass only after SSB. The event separated the electroweak field into the electromagnetic and weak; it also made the left/right-handed versions of the quarks and leptons acting differently in weak interaction.

    Figure 11 SM, Parameters

    Figure 12 Mass, Origin of
    [view large image]


  7. Stability of the Universe
  8. By taking into account the higher term of the SM (Standard Model) perturbation series, the vacuum potential of the Higgs field V is in the form :

  9. Figure 13 Stability of the Universe [view large image]

    and the sum is over all SM particles acquiring a Higgs-dependent mass Mi. The precise form of V1 is not important in the present context, it just shows that the Higgs potential also depends on the particles it acts upon. Furthermore, only the heaviest top quark in the sum is retained in the following consideration.
    As shown in Figure 13,a and the above formula, the vacuum potential V depends on the mass of the Higgs m and also the mass of the top quark mt. Computation of V with the various values of m and mt yields different forms, which can be stable, meta-stable, or runaway (see Figure 13,a). Figure 13,b plots the result revealing our universe is possibly in an meta-stable state sitting on the verge of catastrophic decay into the true vacuum. The revelation produced sensational news headline around the globe one morning in 2013 (soon after the discovery of Higgs in July, 2012). A second wave of doomsday prophecy occurred in 2014 with Hawking's blessing on the revelation.
See a slightly different version in "Evolution of Quantum Fields" and one more variation below.

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Quantum Field History à la Gauge Theory

Gauge Transformation In Gauge theory, the Largangian or field equation is invariant under the gauge transformation :
' = ei(at) ,
where a is the phase angle associates with the "axis", t the generator in the form of nxn matrix, n the dimension of the Gauge space, the number of phase angles is determined by the formula Na = n2 - 1 (except for n = 1, Na = 1, see a little bit more in "Unitary Groups").
Figure 14 is an example of simplest Gauge transformation with n = 1. It shows that the SO(2) rotational transformation of 2-component field (interpreted as two +/- charge states) is equivalent to a single complex field in U(1)= eiI (I = 1) Gauge group. In general SO(n+1) ~ SU(n), where "S" means that the transformation is "Special" with |det(U)| = +1, which implies continuous rotations without reflection.

Figure 14 [view large image]
Gauge Transformation

This is "Global Gauge Transformation" applicable to all space. It is the spatial dependence "Local Gauge Transformation" that generate the Gauge bosons responsible for interaction of particles.

Formulation of the Standard Model of Elementary Particle (SM) is entirely based on the concept of Gauge invariance of the theory both global and local with the U(1) gauge transformation for electromagnetism, SU(2) for leptons, and SU(3) for quarks.
    The followings trace the variations of the Gauge environment at different phase of the Quantum Field Evolution :

  1. Planck Scale Era :
  2. Grand Unified Theory (GUT) :
  3. Spontaneous + Electroweak, Symmetry Breaking (SSB + ESB) :
  4. Spontaneous + Chiral, Symmetry Breaking (SSB + CSB) :
Finally, here's an analogy with the Crude Oil Distillation which may provide a better understanding of the quantum fields evolution :
Crude Oil Distillation The main way of processing crude oil is to separate it using fractional distillation. Oil is made up of many different hydrocarbons of different boiling points (Quantum Fields). The first step in distillation is to boil the oil to a very high temperature (Big Bang), usually around 400oC. When the solution boils, it forms vapors entering the fractional distillation column (GUT), which has many trays to collect the liquid (Quantum Fields). As the height in the column increases, the temperature gets cooler, so as the vapors rise, they cool. Since every hydrocarbon chain has a different boiling point, as a chain reaches a height where the temperature is lower than the boiling point, it will cool into a liquid and be collected by a tray (Symmetry Breaking Processes). This method allows for all parts of the oil to be separated and collected so that they can be further processed and refined (Standard Model). A very good animation of the distillation process can be found in Figure 20.

Figure 20 Q Fields Evolve and Crude Oil Distillation

According to this scenario, all the quantum fields were unified (mixed together) in the beginning. Each one emerges from the whole and manifests its characteristics only at suitable temperature by the process of Symmetry Breaking. The Higgs field is eternal and ubiquitous (Figure 09).