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A classical electron moving around a nucleus in a circular orbit has an orbital angular momentum, L=mevr, and a magnetic dipole moment, = -evr/2, where e, me, v, and r are the electron´s charge, mass, velocity, and radius, respectively. A classical electron of homogeneous mass and charge density rotating about a symmetry axis has an angular momentum, L=(3/5)meR2, and a magnetic dipole moment, = -(3/10)eR2, where R and are the electron´s classical radius and rotating frequency, respectively. The classical gyromagnetic ratio of an orbiting or a spinning electron is defined as the ratio | |
Figure 07 Classical g-ratio [view large image] |
of the magnetic moment to the angular momentum. In both cases one finds cl = /L= -e/(2me). The minus sign indicates that is in the opposite direction to L (see Figure 07). |
If the vertex correction as shown in Figure 08b is taken into account, then g = 2 ( 1 + /2), where = e2/(4c) ~ 1/137.036 is the fine structure constant giving g - 2 = 0.002322814. The extra term arises from the electron self-interaction, in which it emits and reabsorbs a virtual photon, making a loop in the Feynman diagram as shown in Figure 08b. The same process also applies to the muon. | ||
Figure 08a Quantum Description |
Figure 08b Vertex Cor-rection [view large image] |
affect the muon's spin, then the measurement would differ from the theory. This is what appears to have been observed, although there are several interpretations of the result that must be considered. One of the missing pieces in the theoretical calculation is the exotic particles predicted by the theory of supersymmetry. Although these particles are rare and unstable their mere existence in the vacuum would modify observable quantities such as the muon magnetic moment. | |
Figure 09 g-2 Experiment [view large image] | The modern version prefers to use a = (g - 2)/2 with the subscript to denote muon, a(BNL) = 11659208x10-11 (from BNL Muon Experiment), a(SM) = 11659180x10-11 (from SM theoretical calculation); a(BNL) - a(SM) = 28x10-11. |
there is a difference between theory and observations (both from BNL and FNAL, now the experimental value is averaged to a(Exp) = 11659206x10-11). Although the discrepancy seems to be rather small to the laymen, it is a big deal for theoretical physicists. For it reiterates the necessity to search for a new theory to resolve the "tension". Anyway, the confidence level of such discrepancy is claimed to be at 4.2 sigma, i.e., a near certainty but not enough to qualify as a discovery at 5 sigma, i.e., 1-in-3.5 million chance of a fluke. | |
Figure 10 Muon g-2, 2021 Update [view large image] |
Figure 10 summarizes the theoretical, experimental processes and the prospect for new physics. The followings provides a little bit more details : |
Figure 11 Quantum Fields |
Figure 12 Muon g-2, QCD | experimental works on muon g-2 are still in progress. So the dis-agressment between theory and experiment is not a "done deal". See "Why You Should Doubt ‘New Physics’ From The Latest Muon g-2 Results". |
a = S - C =
a(e/m)B,
where a = (g-2)/2. The a can be measured as the spin direction is off a little bit from the orbital path. When the muon decays into positron, such product moves in the direction of the original muon spin and can be captured by the 24 detectors lining inside the storage ring (see Figure 14). | ||
Figure 13 Frequency, 3 Kinds [view large image] |
Figure 14 g-2 Experiment |
A DIY evaluation of a = (g-2)/2 is straight forward by reading off the period Ta = 4.4x10-6 sec from Figure 15. It follows : a = 2/Ta = 1.43x106 sec-1. The rest of the input parameters includes B = 1.45 T = 1.45 kg/sec2-A, and (m/e) = 1.176x10-9 kg/sec-A, finally | |
Figure 15 g-2 Calculation (DIY) [view large image] |
a = (g-2)/2 = (a/B)(m/e) ~ 0.00116. |
| |
Figure 16 The 24 Quantum Fields |
Figure 16 lists the 24 elementary particles (or quantum fields) in the Standard Model (should add the Higgs hoson discovered in 2012). |
The Superstring Theory (Figure 17) became the most studied since SM. It posits that the fundamental element is the one dimensional string open or closed. This particular choice is to satisfy with the demand of Lorentz invariance (i.e., the string formalism is unchanged under the Lorentz transformation, other shapes won't do), which guarantees all inertial observers using the same form of the theory. The theory requires the adoption of the not yet proven supersymmetry, adding extra-dimenstions, ... (see a list of other problems). It becomes obsolete over the years as there is no test to support the claims. | |
Figure 17 Superstring Theory |
Figure 18 GUT |
The Grand Unified Theory (GUT) just adds more gauge bosons to the list of elementary particles. The problem is the prediction of proton decay with half life of about 1032 years. The theory is abandoned for failure to detect the decay after years of experiments since 1980s. |
As shown by the table in Figure 19, the values of g for 20Ne9+ and 22Ne9+ have been calculated individually, from which the difference g = 13.474x10-9 is obtained. The experimental value by measuring the g difference of 2 isotopes directly obtains g = 13.47524x10-9 - an improvement of accuracy about 2 orders of magnitude limited in precision solely by the uncertainty of the | ||
Figure 19 g2 Bound-state [view large image] |
Figure 20 g Experiment |
charge-radius difference (finite nuclear size, FNS) of the isotopes d< r2 >1/2 = 0.0530(34)fm. |
Figure 21 Relaxion Constraint [view large image] |
For example, g = 13.475x10-9 for Ne9+ isotopes as shown by the table in Figure 19; the rest of the parameters are : A = 20, Z = 10, = 1/137, me = 510 kev; thus, Z = 0.073 << 1, and ~ 1. Figure 22 is a DIY plot with the relaxion coupling yeyn (in log scale) against m (in unit of kev) according to the formula shown below. Since the most advanced Large Hadron collider (LHC) failed to detect any sign of relaxion up to energy of 7 Tev and according to various predictions on its coupling strength (Figure 21), the theoretical curve in Figure 22 indicates that the coupling/mass combo of relaxion would be weaker (under the curve, should it be a real thing). | |
Figure 22 DIY Relexion |
In comparison, the electromagnetic coupling = 1/137 = 7.3x10-3, while me ~ 5x102 kev. |