Vacuum Polarization and Running Coupling

1. The Vacuum Polarization Tensor

The one-loop correction to the photon propagator comes from the Feynman diagram where a photon creates a virtual electron-positron pair, which then annihilates back into a photon.

A photon with momentum $q$ creates a virtual electron-positron pair. The electron and positron, with momenta $k$ and $q-k$, propagate in a loop before annihilating back into a photon.

Using the Feynman rules for QED, we can write down the expression for this diagram. It is conventionally written as $i{\Pi }^{\mu \nu }(q)$, where ${\Pi }^{\mu \nu }(q)$ is the vacuum polarization tensor. The integral is:

Let's break this down:

• The initial (-1) comes from the fermion loop.
• The two (-ieγ) factors are the vertices.
• The two fractions are the fermion propagators.
• The Tr[...] is a trace over the gamma matrix indices, as the loop connects back to itself.

By power-counting in $k$, the integral looks like $\int d^{4}k\frac{k^2}{k^4}$, which is logarithmically divergent. This is precisely the kind of UV divergence that dimensional regularization is designed to handle.

2. Calculating the Loop Integral

The calculation follows the exact procedure we learned in the last lesson, but with the added complexity of the gamma matrix trace.

Let's summarize the key results of the calculation. After carrying out the steps of Feynman parametrization, momentum shifting, evaluating the trace, and performing the dimensionally regularized integral, the vacuum polarization tensor is found to have the form:

This tensor structure is extremely important. You can check that it satisfies $q_{\mu }{\Pi }^{\mu \nu }(q)=0$, which is a requirement of gauge invariance (the Ward identity). The fact that dimensional regularization automatically preserves this structure is its greatest strength.

The scalar function $\Pi (q^2)$ contains the result of our loop integral. Setting $d=4-ϵ$, it can be expanded for small $ϵ$. The result from the reading (adjusted for the standard fine-structure constant $\alpha =e^2/4\pi$) is:

Just as in our last lesson, this expression has two distinct parts:

1. A divergent part proportional to $2/ϵ$.
2. A finite part that depends on the external momentum $q^2$, the electron mass $m$, and the arbitrary mass scale $\mu$ we introduced for dimensional regularization.

3. Renormalization of the Electric Charge

What does this loop correction do? It modifies the photon propagator. In a scattering process, the total contribution from the photon is not just the bare propagator, but a sum over all possible loop insertions.

The full, or "dressed," photon propagator (double wavy line) is a geometric series of the bare propagator (single wavy line) and the vacuum polarization loop $i{\Pi }^{\mu \nu }$.

This geometric series sums to:

The interaction term in the Lagrangian for two currents $J^{\mu }$ exchanging a photon is $e^2{J}^{\mu }{D}_{\mu \nu }{J}^{\nu }$. The effect of the loop is to modify the effective coupling:

We now see the problem: $\Pi (q^2)$ is infinite. The solution lies in realizing that the parameter $e$ in our original Lagrangian is just a "bare" parameter, which is not directly observable. We must define a physical, renormalized charge, $e_R$, in terms of the bare charge $e_0$ and the divergence.

We do this by adding a counterterm to the Lagrangian:

The full photon kinetic term is now $-\frac{1}{4}(1+\delta Z_3)F_{\mu \nu }{F}^{\mu \nu }$. This $\delta Z_3$ is chosen to precisely cancel the divergent part of $\Pi (q^2)$. In the widely used MS-bar ($\overline{\text{MS}}$) scheme, we choose $\delta Z_3$ to cancel the $2/ϵ$ pole along with the associated constant factors $\ln (4\pi )-{\gamma }_E$.

After this cancellation, the modification to the coupling is purely finite and momentum-dependent. The physical, renormalized charge $e_R$ is what we measure in low-energy experiments (e.g., Thomson scattering), which corresponds to the limit $q^2\to 0$. The relationship between the bare charge and the renormalized charge is:

All infinities are absorbed into the unobservable bare charge $e_0$.

4. The Running of the Fine-Structure Constant

After renormalization, we are left with a finite, momentum-dependent correction. The effective fine-structure constant, $\alpha (q^2)=e(q^2{)}^2/4\pi$, is now a function of the momentum transfer $q^2$:

where $\alpha (0)\approx 1/137$ is the value we all know and love, and ${\Pi }_{finite}$ is the finite part of our calculation. For large $|q^2|\gg m^2$, the logarithmic term dominates, and we find the famous result:

This equation tells us that the strength of the electromagnetic interaction increases at high energies (large $|q^2|$) or, equivalently, at short distances.

This is not just a theoretical curiosity; it has been precisely measured at particle colliders.

The Beta Function

The running of the coupling is elegantly described by the beta function, which measures how the coupling changes with the energy scale $\mu$.

From our result for $\Pi (q^2)$, we can extract the one-loop beta function for QED:

Since $\beta (e)>0$, the coupling increases with the energy scale $\mu$. This confirms our finding from the formula for $\alpha (q^2)$. A theory with a positive beta function, like QED, is said to be "trivial" because the coupling appears to hit an infinite value at some extremely high energy (the Landau pole). This suggests that QED by itself is not a complete theory at all energy scales.

Conclusion

Today, we have achieved a major milestone. By applying dimensional regularization to the vacuum polarization loop, we have not only rendered a divergent calculation finite but also uncovered a deep physical truth about the nature of the quantum vacuum.

Key Takeaways:

• The one-loop correction to the photon propagator, vacuum polarization ${\Pi }^{\mu \nu }(q)$, is logarithmically divergent.
• Dimensional regularization preserves gauge invariance, ensuring that ${\Pi }^{\mu \nu }(q)$ has the required transverse structure $(q^2{g}^{\mu \nu }-q^{\mu }{q}^{\nu })\Pi (q^2)$.
• The $1/ϵ$ divergence is absorbed into the bare electric charge $e_0$ by introducing a counterterm $\delta Z_3$. This process, renormalization, leaves us with a finite, physical, renormalized charge $e_R$.
• The finite part of the calculation results in a momentum-dependent effective charge, leading to the running of the fine-structure constant $\alpha (q^2)$.
• The strength of QED increases at high energies (short distances), a phenomenon described by a positive beta function and confirmed by experiment.

Preview of the Next Lesson:
Vacuum polarization was just one of the three "primitive" divergences in QED. In the next lesson, we will turn our attention to the others: the electron self-energy and the vertex correction. We will see how renormalization works for these diagrams, leading to the concepts of mass renormalization and wavefunction renormalization, and discover more remarkable results like the electron's anomalous magnetic moment.