## Quantum Field Theory

### Lamb Shift The energy levels calculated from the Dirac equation with the electron in a Coulomb potential is given by: where Z is the number of positive charges, m is the mass of the electron, and is the fine structure constant. The formula predicts that the energy levels of the hydrogen atom should depend only on the principal quantum number n and the total angular momentum j = L + S, so that the energy levels of the 2S1/2 (n=2, j=1/2) and 2P1/2 (n=2, j=1/2) states should be degenerate (the same). In 1947, Lamb and Retherford demonstrated that this is not the case. The 2P1/2 and 2S1/2 energy levels are separated by more than 1000 MHz (see Figure 03a).

#### Figure 03a Lamb Shift [view large image]

This small shift in energy levels is caused by various high-order radiative corrections as mentioned earlier. The regular Coulomb potential is now supplemented by an effective potential V(eff) (up to the 2nd order): • All the terms in V(eff) come from the radiative corrections.
• The first and second terms arise from the vertex and self-energy corrections. Since min represents the photon mass, which is equal to zero, it introduces a divergence into the formulation. However, it is eventually cancelled by the infrared correction and disappears completely in the level shift calculation. These two terms contribute the majority of the Lamb shift with a correction of 1011.41 MHz.
• The third term is related to the vacuum polarization. It actually brings down the Lamb shift by -27.13 MHz.
• The last two terms come from the effect of anomalous magnetic moment interaction or spin-orbit interaction. These two terms together create a Lamb shift of 67.82 MHz.
• Those terms with the (3)(x) is applicable only to the 2S1/2 state, which has a high probability to be near the center of the Coulomb potential. On the other hand, the term with the quantum number L vanishes for the 2S1/2 state since L = 0 in this case.
By taking the average of V(eff) with the corresponding hydrogen wave functions, it yields the Lamb shift in the form The total shift amounts to 1052.10 MHz. When other smaller contributions such as the 4th order radiative corrections and the mass corrections are included into the calculation, the final theoretical value is 1057.864 0.014 MHz. Comparing with the experimental value of 1057.862 0.020 MHz, it provides an excellent indicator of the basic correctness of QED. Actually, it is instrumental in restoring confidence to QED from the dubious methodology of renormalization.

The discoverer of Lamb Shift - Willis Eugene Lamb, Jr. passed away on May 15, 2008. Lamb was born in Los Angeles on 12 July 1913. He studied at the University of California, Berkeley, where he obtained his first degree (in chemistry) in 1934 and his PhD (in physics, under the supervision of Robert Oppenheimer) in 1938. He had taken up posts at various institutions, and became professor of physics and optical sciences at the University of Arizona since 1974. His effort on discovering and measuring the Lamb shift earned him the Nobel Prize of Physics in 1955.

More than sixty years after the celebrated QED triumph, precision optical spectroscopy of H atoms and the corresponding calculation have improved tremendously and reach a point where the proton size is the limiting factor when comparing experiment with theory. A much better way to determine the proton radius is possible by measuring the Lamb shift in muonic hydrogen (atom formed by proton and negative muon). Since the Bohr radius is proportional to 1/m and the muon is about 200 times heavier than the electron, the wave function of its S state  overlapped more with the proton charge cloud and shifted more accordingly. By summing the contributions from : fine structure, hyperfine splittings, radiative corrections, recoil and proton size, ..., the total computed 2S1/2F=1 - 2P3/2F=2 energy difference (the dominate transition) is : E = 209.9779(49) - 5.2262(rp)2 + 0.0347(rp)3
where rp is the root-mean-square charge radius of the proton, numbers in parenthesis indicate uncertainty, and the energy is expressed in unit of

#### Figure 03c Laser Resonance [view large image]

mev = 10-3ev. The first term in the formula is dominated by vacuum polarization, which causes the 2S states to be more tightly bound than the 2P states (Figure 03b,c).
Muonic hydrogen atoms are produced in a cavity filled with H2 gas and bombarded by a beam of low energy muons. Most of them are in highly excited states with n ~ 14 initially. About 99% quickly de-excite to the 1S ground state (blue in Figure 03b,a), but ~ 1% populate the longer living 2S state with a lifetime of about 1 microsecond (red in Figure 03b,a). Tuneable laser pulses with a frequency of around 5 THz are used to illuminate the cavity for inducing transitions from the 2S to the 2P state within the narrow time window (green in Figure 03b,b,c). By recording the 2 kev K X-ray events (red in Figure 03b,b), a resonance curve is obtained with a peak at 49.88188(76) THz (Figure 03c) corresponding to E = 206.2949(32) mev or rp = 0.84184(36)(56) fm (1 fm = 10-13cm) with numbers in parentheses indicating experimental and computational uncertainties respectively. This new value of rp is significantly different from the previous measurements from H atom spectroscopy (CODATA), and from electron-proton scattering experiment (Figure 03c). Much more works are needed before we can find out the cause of the discrepancy. A report in 2013 indicates that the proton size as measured from the muonic hydrogen atoms is indeed smaller (see Fiigure 03d). The accuracy of the new data is two times better than the previous attempt. Various new forces are proposed to resolve the discrepancy. They are constrained by the requirements that it should not deviated too much from the Standard Model, especially the property of neutrinos cannot be altered beyond the current observation. Such constraints can be bypassed with the "dark photon" (associate with dark matter) or gravity leaking to another dimension. All the remedies look rather contrived. The "tidal force" of the electron or muon on the proton has also been measured to be too small to explain the

#### Figure 03d Proton Size Discrepancy [view large image]

discrepancy. It seems that the inherent difficulty of defining the proton size is at the root of the problem as explained below. The traditional assumption for proton mass distribution is about 75% concentrated in a central core (size rp) with the other 25% lying outside in the halo up to 1.4 fm. An August 2010 calculation suggests that the discrepancy between the original proton size and the latest (2010)

#### Figure 03e Latest Proton Size [view large image]

experimental data can be reconciled if the halo band extends 4.7 times as far as the previous definition (Figure 03e).

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