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Quantum Field Theory


Axion Axion was postulated to explain why CP violation is not observed in strong interaction, although it should be according to the Standard Model. The CP violation in SM arises from a certain non-zero parameter related to the QCD vacuum (see instanton). It was shown in 1997 that the parameter could be driven to zero by a Higgs field. A side effect of the transformation is the introduction of a new particle - the axion. If axions do exist, they each would have a mass of around 10-5 ev, but there could be so many of them in the

Figure 10 Axion Detection
[view large image]

Universe, that they contribute a large proportion of the overall mass in the form of dark matter. In the inflationary scenario, axions induce isocurvature fluctuations whereby the system remains homogeneous and constrain the allowed inflationary scale.

A report in March, 2006 claimed the detection of axions in laboratory. However, the results contradict other observations and do not fit with constraints deduced from astrophysics. Since axions interact only weakly with baryonic matter, they would be able to stream out from the central cores of stars into space virtually unobstructed, carrying energy away with them and cooling the stellar cores. This cooling is more effective the heavier the axions are, and if each axion had a mass greater than 0.01 ev this would affect the appearance of stars and the way supernovas exploded. Further experiments such as the one shown in Figure 10 will confirm (or refute) the discovery.

Axion is also proposed to explain the 20 mins cycles of X-ray flares coming from the center of the Milky Way. In the 1990s, computer simulations of clouds of dark matter made of axions showed that giant bubbles of these particles would burst out from the clouds. These axion bubbles would expand and contract with a period of 20 mins - matching the period of infrared and X-ray flares from Sagittarius A*. The model relies on a controversial version of gravity, which proposes that gravity starts to repel as the gravitational field gets stronger. Confirmation by observation would prove the existence of axion and would also raise question about Einstein's General Relativity.

CP Violation The following is an attempt to clarify the rationale for the introduction of this hypothetical particle and the consequence via a simplified mathematical formulation. The problem starts from the QCD Lagrangian density LQCD as shown in a "beyond the Standard Model" form below in Eq.(1). The first term is for the quarks, second term for qluons, and the third term is responsible for the expected CP violation similar to that in weak interaction. However, no strong (interaction) CP violation is observed in the very precise "neutron electric dipole moment experiment" (Figure 11). Therefore the effect must be very small or zero. One of the solutions is to add an additional field called axion with the label to cancel out the CP violation term as shown in Eq.(2).

Figure 11 CP Violation [view large image]

cosmic Time Scale It turns out that the cancellation process is not simply inserting a term into the formula. The axion field evolved in the very early cosmic history starting soon after the inflation period, it gradually generates an effect to cancel out the CP violation, and eventually acquires a steady mass. The sequence can be divided into three stages as shown in Figure 16,b, and described in some details below. But before proceeding further, some definition of terms and time/temperature scales are essential. The cosmic time scale is shown in Figure 12. The relevant period would be from about 10-37 sec in the inflation era to 10-5 sec at QCD phase transition when the quarks and gluons in the quark-gluon plasma combined to form hadrons.

Figure 12 Cosmic Time Scale

The relationship between time t and temperature T is established through a chain such as :
t energy density T (via the Equation of State) t ~ 10-6/T2 (in Gev) within the range of interest.

Now the various definitions related to the axion :
Scalar Field Potential

Figure 13 Scalar Field Potential [view large image]

BTW, the Veff here represents only that part of Figure 13,b within the interval "v" (also see Figure 13a).
See "Theory of Cosmic Inflation" for derivation of the equation of motion for the field.

(Stage 1) Initial Conditions : At the end of the inflation period about 10-30 sec after the Big Bang, a phase transition occurred as the temperature dropped below 1013 Gev. The change converted the inflaton field energy to various kinds of elementary particles including the Higgs which appeared at about 250 Gev and endows mass to the other entities. The axion would be one of those emerging from the hot plasma. Its initial properties are deduced from the following considerations. The very first item has to be the Hubble parameter H which is just the inverse of the age (of the universe) 1/t during the radiation era. In comparison to the the initial mass of axion below 10 -20 Gev (or in comparable unit of ~ 1024 sec-1), H is certainly the dominating factor before and soon after the inflation era of 1032 sec-1). According to the solution of Eq.(4), a ~ 0 e-3Ht. It sets the stage for the later development. Initially,
  • The axion phase angle ~ 0, while the CP violation one ~ /(3)1/2. The value of is obtained by assuming axion to be the dark matter with density parameter DM = 0.23 at the current epoch (see Figure 15).
  • The pre-inflation Veff would be symmetrical as shown in Figure 13,a.
  • Scalar Field Potential Phase Angle
  • By statistical fluctuation, a patch of the space entered into a state of false vacuum converting Veff to the form in Figure 13,b. The initial field should be sitting on top and will roll down as the configuration is unstable.
  • The initial number density Na = 0 as there was no axion at the beginning (Figure 16,c).
  • Figure 14 Types of Solution [view large image]

    Figure 15 Phase Angle
    [view large image]

    Axion Evolution (Stage 2) Axion Evolution : At this stage the axion field and phase angle continued on their downward slide while the mass and density moved up. A critical point was reached when the damping term 3H of the cosmic expansion is equal to the oscillating mass term in Eq.(4). Thereafter, the variables acquired an oscillating component in a state called damped-oscillation and eventually settled down into their final values. For example, the phase angle would oscillate as a = 0 ei(mat). Converting back from natural unit, the oscillating period would be about 10-30 sec while 0 ~ 0 at the current epoch (Figures 13a and 16,c).

    Figure 16 Axion Evolution [view large image]

    (Stage 3) Axion as Dark Matter : The effective potential is often said to be tilted in attaining a mass. Such effect is actually mathematical rather than physical because shifting the origin of the potential Veff by an amount f to the minimum (the true vacuum), a mass term would appear in the Lagrangian density. The topological susceptibility as a function of T from 2 to 0.1 Gev has been constructed by QCD lattice computation in 2016 (Figure 17,a). This numerial function enables the evaluation of Veff(a), which in turn
    Axion Now Axion as Dark Matter allows the computation of the axion field , a, and finally the evolution of the mass ma(T) up to the current epoch (Figure 17b). The red line shows the range of possible axion mass : 5x10-5 eV < ma < 1.5x10-3 eV, and phase angle = /f = 2.155 for making the CP violation disappeared. The number density Na ~ 14x1012/m3 would be in agreement with DM = 0.23 if ma = 10-4 eV (Figure 16,c).

    Figure 17 Evaluation of Axion Parameters [view large image]

    Figure 18 Dark Matter

    Figure 18 lists the various dark matter candidates including the axion and other oddballs. Except the regular neutrinos (not the sterile type), most of them has to be discovered. See more about "Dark Matter".

    See "Calculation of Axion Mass Based on Lattice QCD" for the latest effort to estimate the axion mass.

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