Vacuum polarization

In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic field. It is also sometimes referred to as the self energy of the gauge boson (photon).

The effects of vacuum polarization were first observed experimentally prior to 1947 before they were theoretically calculated (by Hans Bethe on the return train ride from the Shelter Island Conference to Cornell) after developments in radar equipment for World War II resulted in higher accuracy for measuring the energy levels of the hydrogen atom (the Lamb shift) and the anomalous magnetic dipole moment of the electron (corresponding to the deviation from the Dirac equation predicted value of 2 of the spectroscopic electron g-factor value), measured by I.I. Rabi.

The effects of vacuum polarization have been routinely observed experimentally since then as very well understood background effects. Vacuum polarization referred to below as the one loop contribution occurs with leptons (electron-positron pairs) or quarks, the former (leptons) first observed in 1940s but also recently observed in 1997 using the TRISTAN particle accelerator in Japan,^[1] the latter (quarks) along with multiple quark-gluon loop contributions from the early 1970s to mid-1990s using the VEPP-2M particle accelerator at the Budker Institute of Nuclear Physics in Siberia in Russia and many other accelerator laboratories worldwide.^[2]

Explanation

According to quantum field theory, the vacuum between interacting particles is not simply empty space. Rather, it contains short-lived "virtual" particle–antiparticle pairs (leptons or quarks and gluons) which are created out of the vacuum in amounts of energy constrained in time by the energy-time version of the Heisenberg uncertainty principle. After the constrained time, having duration inversely correlated with the amount of energy of the fluctuation, the virtual particles annihilate each other.

These particle–antiparticle pairs carry various kinds of charges, such as color charge if they are subject to QCD such as quarks or gluons, or the more familiar electromagnetic charge if they are electrically charged leptons or quarks, the most familiar charged lepton being the electron and since it is the lightest in mass, the most numerous due to the energy-time uncertainty principle as mentioned above; e.g., virtual electron–positron pairs. Such charged pairs act as an electric dipole. In the presence of an electric field, e.g., the electromagnetic field around an electron, these particle–antiparticle pairs reposition themselves, thus partially counteracting the field (a partial screening effect, a dielectric effect). The field therefore will be weaker than would be expected if the vacuum were completely empty. This reorientation of the short-lived particle-antiparticle pairs is referred to as vacuum polarization.

The one-loop contribution of a fermion–antifermion pair to the vacuum polarization is represented by the following diagram:

Electric and magnetic fields

Extremely strong electric and magnetic fields cause an excitation of electron-positron pairs. Maxwell's equations are the classical limit of the quantum electrodynamics which cannot be described by any classical theory. A point charge must be modified at extremely small distances less than the reduced Compton wavelength ${\displaystyle {\lambda _{c}}}$ (${\displaystyle \ ={\frac {\hbar }{mc}}}$). To lowest order in the fine-structure constant, $\alpha$, the QED result for the electrostatic potential of a point charge is:^[3]

${\displaystyle \phi (r)={\frac {q}{4\pi \epsilon _{0}r}}\times \left\{{\begin{array}{ll}1-{\frac {2\alpha }{3\pi }}\ln({\frac {r}{\lambda _{c}}})&{\frac {r}{\lambda _{c}}}\ll 1\\1+{\frac {\alpha }{4{\sqrt {\pi }}}}({\frac {r}{\lambda _{c}}})^{-3/2}e^{-2r/\lambda _{c}}&{\frac {r}{\lambda _{c}}}\gg 1\\\end{array}}\right.}$

This can be understood as a screening of a point charge by a medium with a dielectric permittivity, which is why the term vacuum polarization is used. When observed from distances further than $\lambda _{c}$, the charge is renormalized to the finite value $q$.

The effects of vacuum polarization become significant when the external field approaches:

${\displaystyle E_{c}={\frac {mc^{2}}{e\lambda _{c}}}=1.32\times 10^{18}{\text{ V/m}}}$

${\displaystyle B_{c}={\frac {mc}{e\lambda _{c}}}=4.41\times 10^{9}{\text{ T .}}}$

These effects break the linearity of Maxwell's equations and therefore break the superposition principle. The QED result for slowly varying fields can be written in non-linear relations for the vacuum. To lowest order $\alpha$, virtual pair production generates a vacuum polarization and magnetization given by:

${\displaystyle {\vec {P}}={\frac {2\epsilon _{0}\alpha }{E_{c}^{2}}}{\bigg (}2(E^{2}-c^{2}B^{2}){\vec {E}}+7c^{2}({\vec {E}}\cdot {\vec {B}}){\vec {B}}{\bigg )}}$

${\displaystyle {\vec {M}}=-{\frac {2\alpha }{\mu _{0}E_{c}^{2}}}{\bigg (}2(E^{2}-c^{2}B^{2}){\vec {E}}+7c^{2}({\vec {E}}\cdot {\vec {B}}){\vec {B}}{\bigg )}}$ .

As of 2012, this polarization and magnetization has not been directly measured.

Vacuum polarization tensor

The vacuum polarization is quantified by the vacuum polarization tensor Π^μν(p) which describes the dielectric effect as a function of the four-momentum p carried by the photon. Thus the vacuum polarization depends on the momentum transfer, or in other words, the dielectric constant is scale dependent. In particular, for electromagnetism we can write the fine structure constant as an effective momentum-transfer-dependent quantity; to first order in the corrections, we have

$\alpha _{{\text{eff}}}(p^{2})={\frac {\alpha }{1-[\Pi _{2}(p^{2})-\Pi _{2}(0)]}}$

where Π^μν(p) = (p² g^μν - p^μp^ν) Π(p²) and the subscript 2 denotes the leading order-e² correction. The tensor structure of Π^μν(p) is fixed by the Ward identity.

Note

Vacuum polarization affecting spin interactions has also been reported based on experimental data and also treated theoretically in QCD, as for example in considering the hadron spin structure.

See also

• Renormalization
• QED vacuum
• QCD vacuum
• Virtual particles

References

1. Levine, I.; TOPAZ Collaboration (1997). "Measurement of the Electromagnetic Coupling at Large Momentum Transfer". Physical Review Letters. 78 (3): 424–427. Bibcode:1997PhRvL..78..424L. doi:10.1103/PhysRevLett.78.424.
2. Brown, Douglas H; Worstell, William A (1996). "The Lowest Order Hadronic Contribution to the Muon g − 2 Value with Systematic Error Correlations". Physical Review D. 54 (5): 3237–3249. arXiv:hep-ph/9607319. Bibcode:1996PhRvD..54.3237B. doi:10.1103/PhysRevD.54.3237.
3. Berestetskii, V.B; Lifshitz; Pitaevskii (1980). "Sec 114". Quantum Electrodynamics (2nd ed.). Oxford: Pergamon.

Further reading

• For a derivation of the vacuum polarization in QED, see section 7.5 of M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.