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A classical electron moving around a nucleus in a circular orbit has an orbital angular momentum, L=m_{e}vr, and a magnetic dipole moment, = -evr/2, where e, m_{e}, 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)m_{e}R^{2}, and a magnetic dipole moment, = -(3/10)eR^{2}, 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
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## Figure 07 Classical g-ratio [view large image] |
of the magnetic moment to the angular momentum. In both cases one finds _{cl} = /L= -e/(2m_{e}). 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 = e^{2}/(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.
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## Figure 08a Quantum Description |
## Figure 08b Vertex Cor-rection [view large image] |

g - 2 = 2 ( /2 - 0.328 (/2)

In calculating the effects of the cloud of virtual particles, we need to include not just the effects of virtual photons and virtual electron-positron pairs, but also virtual quarks, virtual Higgs particles, and, in fact, all the particles of the Standard Model. It turns out, though, that because of the larger muon mass, any such heavy particles would affects the muon magnetic moment more than the electron magnetic moment. The muon g - 2 has been calculated with the Standard Model to a precision of 0.6 ppm (parts per million). The calculated value with the combined effect is g - 2 = 0.0023318360. A remarkable fact is that the muon g - 2 factor not only can be predicted to high precision, but also measured to equally high precision. The measurements at Brookhaven National Laboratory in 2001/2004 (Figure 09) yields an average value of g - 2 = 0.0023318416. Thus, the comparison of measurement and theory provides a sensitive test of the Standard Model. If there is physics not included in the current theory, and such new physics is of a nature that will

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 (from SM theoretical calculation); _{}(SM) = 11659180x10^{-11}_{} a_{}(BNL) - a_{}(SM) = 28x10^{-11}. |

- After a lapse of 20 years, the Muon g-2 collaboration at the Fermi National Accelerator Laboratory (FNAL) finally reports the latest measurements of updated muon g-2 on 7 April, 2021 (see "Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm"). The new value of a
- In the Standard Model of elementary particles (SM), it is the quantum fields that are ubiquitous in the formalation. As shown in Figure 11, the various quantum fields permeate throughout the Universe at ground state with virtual particles popping out incessantly according to the Uncertainty Principle. The particle becomes real when enough excitation energy is injected into the field. It is the virtual particles from these quantum fields that contribute to the anomalous magnetic dipole moment of muon (actually it is the correction to the charge or/and mass). The second order corrections from the various boson fields as shown in Figure 10,a include :
- The vertex correction (to the electric charge) in QED (Quantum ElectroDynamics) is the dominant contribution to :

a(QED) = /2 = 0.0011617,

where = 1/137 is the fine structure constant. The original derivation was published 74 years ago in 1947 by J. Schwinger, Phys. Rev. 73, 416. The divergences in the expression for very high and very low 4-momentum |k^{2}| of the virtual particles have been removed by mathematical tricks such as the "Regularization Schemes". The a(QED) = 0.00116584719 in Figure 10,a should be more accurate by including more virtual quantum fields such as the leptons. - The vertex correction in EW (Electro-Weak Interaction) could be large since
_{EW}= 1/30, i.e., about 4.5 times larger than

. But the over all value is very small with a(EW) = 0.00000000153 because it is short range with_{EW}= 0 for distance large than 10^{-17}cm from the particle, and the virtual particles in this case are the Z, W, and Higgs bosons which have mass ~ 100 Gev, i.e., more 100 times that of the proton; while the virtual particle (the photon) for QED has no rest mass. - Contribution from hadron introduces some uncertainty because the calculation cannot empoly perturbation method. The a(Had) in Figure 10,a is estimated from the R-ratio methodology which asserts that cross-section in hadronic process is proportional to the similar QED process. The ratio is obtained by experimental measurements. Theoretical values are computed by "Lattice Theory". The numerical calculations consume enormous amount of computer time, but the results improve the agreement between a
_{}(SM) and a_{}(Exp). Figure 12 shows the effect of the 2 different methods on the need for "new physics". Further theoretical and - The experimental work involves simpler mathematics. The efforts are devoted to keep the magnetic field uniform, the alignment of the muon beams, ... (see "How does Muon g-2 work?"). The followings provides a little bit more details on the basic of experimental measurement :

- Frequency of Larmor Precession - When a muon with spin S = /2 is placed in a magnetic field B,
it experiences a torque which can be expressed in the form :

= SB,

for which the direction of spin is assumed to be perpendicular to B and = ge/2m is the gyromagnetic ratio. The torque is also related to S by another formula (see Figure 13,a):

= S (d/dt),

combining these 2 formulas yields :

_{S}= 2/T = B = (ge/2m)B,

where T is the period of the precession. - Cyclotron Frequency - The cause of this effect is different from the case above. It is derived from the balance between the Lorentz and centripetal forces in the form (see Figure 13,b) :

_{C}= (e/m)B. - Anomalous Frequency - These 2 kinds of frequencies are the same in quantum theory with g = 2. However, when virtual particles are included in quantum field theory with g > 2, the difference becomes the anomalous frequency :
_{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

- Frequency of Larmor Precession - When a muon with spin S = /2 is placed in a magnetic field B,
it experiences a torque which can be expressed in the form :
- In the experiment, million muons with prefect alignment in spin direction are injected into the storage ring. The number would be gradually reduced as the muons decay into positrons and neutrinos with half-life
_{a}= 1.56_{}s. However, the detectors lining along the inside of the storage ring will experience cycle of maximum and minimum as the positrons point toward and then away from them with frequency_{a}. The result is recorded in a graph such as the one in Figure 15. The corresponding formula is :

N(t) = N_{0}e^{-(t/a)}[1 + cos(_{a}t +_{})].

A DIY evaluation of a = (g-2)/2 is straight forward by reading off the period T _{a}= 4.4x10^{-6}sec from Figure 15. It follows :

_{a}= 2/T_{a}= 1.43x10^{6}sec^{-1}.

The rest of the input parameters includes B = 1.45 T = 1.45 kg/sec^{2}-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. - As for new particles, the constituent for dark matter is still elusive after extensive searchings since 1980s. The LHC has failed to discover new particles including those SUSY superpartners so far. Another possibility is the 12 very heavy gauge bosons ( > 10
^{15}Gev) from "Grand Unified Theory (GUT)", and the "Axion" etc. Anyway, even if the g-2 experiment has confirmed the requirement of "new particle", physicists would still have a hard time to find such particle to fit into the grand scheme of things. - Since the introduction of the Standard Model (SM) in early 1970s, it is found to be incomplete. It is "guilty" of at least 10 counts for failing to explain :

- The cosmological constant or vacuum energy.
- Dark energy.
- The inflaton in the first fraction of a second of the Big Bang.
- Matter-antimatter asymmetry.
- Cold dark matter.
- The form of the Higgs field.
- Hierarchy problem - huge Higgs boson mass implies huge mass for all elementary particles.
- Gravity.
- Masses of the quarks and leptons.
- Three generations of elementary particles.

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

_{}- Another approach is to keep the notion of composite "Point" particles with consecutive layers :
- A "point" is always Lorentz invariant.
- A "point" avoids the invention of certain shape.
- All quantum systems from molecule, to atom, to nuclei are composite system with the assumption of "point" constituents. The size of each one is getting smaller while the energy scale becomes bigger according to such sequence (see Figure 18).
- The elementary particles in SM could be similar with the next level at even smaller size and requires higher energy (beyond the current capacity by LHC) to probe.
- As mentioned in the "Introduction" 20 years ago, systems in the universe are somewhat similar to the Russian Doll.

It fails to provide further insight into the more "fundamental structure" in "elementary particles" such as electron and quarks.#### Figure 18 GUT

[view large image]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 10 ^{32}years. The theory is abandoned for failure to detect the decay after years of experiments since 1980s.

there is a difference between theory and observations (both from BNL and FNAL, now the experimental value is averaged to a). 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.
_{}(Exp) = 11659206x10^{-11} | |

## 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". |

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## 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 g factor of a bound-electron is modified by the properties of the nucleus such as the nuclear mass, polarizability and the charge radius. However, by studying the differences of the g-factor (g) between isotopes, many common QED contributions cancel out owing to the identical electron configuration, making it possible to resolve the intricate effects stemming from the nuclear differences.

As shown by the table in Figure 19, the values of g for ^{20}Ne^{9+} and ^{22}Ne^{9+} 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
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## Figure 19 g2 Bound-state [view large image] |
## Figure 20 g Experiment |
charge-radius difference (finite nuclear size, FNS) of the isotopes d< r ^{2} >^{1/2} = 0.0530(34)fm. |

- The experiment to measure g (Figure 20):
- Electrons of the neon-20 and neon-22 isotopes were removed sequentially, so that only one bound electron remained in each ion.
- They were prepared in a Penning trap, so that their moments were aligned with the magnetic field. They were then transferred to the precision trap (PT), and placed in the same orbit, thus sampling the same magnetic field.
- A pulse of microwave radiation was applied to rotate the moments into the plane perpendicular to the field. Because the two isotopes have different numbers of neutrons in their nuclei, they rotate with slightly different Larmor frequency
_{L}so that the moments pointing at different directions. - A second pulse rotated them back to the plane of the field, but preserving their direction relative to each other. The image in

Figure 20 happens to shows that they point in opposite direction. In general, the relative direction varies depending on the time when the second pulse is applied. It has been shown that the probability for the various relative direction is given by the formula :

P = A cos[2(_{L})t + ] + 0.5 ---------- (1)

where A is the modulation amplitude, is the differential phase of the spins,_{L}encodes the difference of the Larmor frequencies which ultimately determines

g = (4/B)(m_{e}/e)_{L}---------- (2)

where B is the magnetic field, (m_{e}/e) the mass/charge ratio of the electron. - Thus, from counting the number (of relative direction) at different time t by the analysis trap (AT), a cosine curve would emerge as shown by the insert at the lower left corner of Figure 20, from which the value of
_{L}= 758 Hz can be extracted, and thus g via Eq.(2). For example, with A = 0.26 and = 53^{o}(as shown by the insert), P ~ 0.65 at t = 0 ms, and P ~ 0.24 at t = 0.5 ms.

The g can be applied to set constraint on the parameters for the new-physics (NP) search via the hypothetical relaxion (see "Relaxion Review"). The scheme is similar to the introduction of the as yet discovered axion by adding a boson to the Lagrangian density (to resolve the problem). In this case, it is the "Hierarchy Problem with Higgs". It is suggested that the relaxion couples to the Higgs boson to endow it with heavy mass, and as the early Universe expanded, the expansion would have weakened this coupling, thereby 'relaxing' the Higgs mass down to its currently observed value. Anyway, g is related to the coupling constant y

## Figure 21 Relaxion Constraint [view large image] |

For example, g = 13.475x10^{-9} for Ne^{9+} isotopes as shown by the table in Figure 19; the rest of the parameters are : A = 20, Z = 10, = 1/137, m_{e} = 510 kev; thus, Z = 0.073 << 1, and ~ 1. Figure 22 is a DIY plot with the relaxion coupling y_{e}y_{n} (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 m_{e} ~ 5x10^{2} kev. |

See "Particle physics isn’t going to die — even if the LHC finds no new particles" on the disperate search for New Physics (NP).

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