Feynman rules for scalar QED

Last updated Dec 29, 2024

• Onepass

The scalar QED Lagrangian can be written out as $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\phi^*(\Box+m^2)\phi - ieA_\mu[\phi^*(\partial_\mu\phi)-(\partial_\mu\phi^*)\phi] + e^2A^2_\mu|\phi|^2$$ From this we can get the Feynman rules:

The scalar propagator is the Feynman propagator as derived before from the kinetic term of the scalar field.

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• Note $\phi$ and $\phi^$ cannot be disentangled, the propagator is therefore defined as $\braket{0|\phi^(x)\phi(0)|0}.$

Photon propagator:

The photon propagator was derived before in $\xi$ gauge.

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• Interactions have the form $$-ieA_\mu[\phi^*(\partial_\mu\phi) - (\partial_\mu\phi^*)\phi]$$
• The derivatives give a factor of $\pm i p_\mu$ since you can pull it into the quantization expression of the fields $$\partial_\mu \phi(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}\partial_\mu(a_pe^{-ipx}+b^\dagger_pe^{ipx})$$
• Each feynman rule gets another factor of $i$ from the expansion of $e^{i\mathcal{L}_\text{int}}$ leading to an overall $i^2=-1$.
• Since both $\phi$ and $\phi^*$ have a term corresponding to the creation/annihilation of a particle/antiparticle we get in total four combinations, each with two contributions from both terms of the interaction respectively
  1. Annihilate $e^-$ create $e^-$: Particle scattering: Pasted image 20230908171948.png Here the $-p^1_\mu$ comes from $\phi^(\partial_\mu \phi)$ since the $\phi$ annihilates the $e^-$, while the $-\phi(\partial_\mu\phi^)$ gives a $-(+p^2_\mu)$ because the $e^-$ is created by the $\phi^*$.
  2. Annihilate $e^+$ create $e^+$: Antiparticle Scattering: $ie(p^1_\mu+p^2_\mu)$ Pasted image 20230908172035.png Here the $\phi^(\partial_\mu \phi)$ creates the $e^+$ while the $\phi(\partial_\mu \phi^)$ annihilates the $e^+$.
  3. Annihilate $e^-$ annihilate $e^+$: Pair annihilation: Pasted image 20230908172125.png
  4. Create $e^-$ create $e^+$: Pair creation: Pasted image 20230908172211.png

This leads to the sum or difference from the two different momenta $p_1$ and $p_2$. The arrows give the Momentum Flow and Particle Flow:

• Momentum flow conventionally goes from left to right
• For antiparticles the Particle flow points opposing to the momentum flow
• Particle flow is always a connected path
• For closed paths particle flow direction is arbitrary
• Since momentum flow is always forward in time and particle flow for antiparticles is opposed to momentum flow, one can interpret antiparticles as going backwards in time
• The Feynman rules generalizing the above diagrams is to have the vertex factor $-iep_\mu$ with a minus sign for momenta of anti-particles

Feynman-Stückelberg Interpretation

Antiparticles travel backwards in time, given that their particle flow is always directed into the past

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• The four point vertex comes from the $$\mathcal{L}_\text{int}=e^2A^\mu A_\mu|\phi|^2$$
• This term comes from gauge invariance which forces the $|D_\mu\phi|^2$ term.
• The factor of $2$ in the Feynman rule is a symmetry factor for the two $A_\mu$ fields (either can create/annihilate the incoming/outgoing photon, so two possibilites $\rightarrow$ two terms).

• For external scalar fields: Everything can be expressed in terms of correlation functions resp. propagators, so we dont need any external states
• For external photons: Since Photon states $\ket{k,\epsilon}$ have added polarization, LSZ reduction formula is modified by adding a factor $\epsilon_\mu$ for every external state:
  • $\epsilon_\mu$ for incoming particles
  • $\epsilon^*_\mu$ for outgoing particles

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• The Ward identity holds in scalar QED, which can be seen, when computing matrix elements and replacing external polarizations e.g. $\epsilon^\nu_3$ with its momentum $p^\nu_3$ which gives zero for the matrix element.
• This has to be the case since these parts of the polarization $\epsilon || p$ are unphysical.
• Note that this is only the case for final matrix elements (all diagrams contributing have to be summed), individual diagrams dont neccessarily vanish

• Gauge invariance which is in this case the independence of phyiscal results of $\xi$ from the photon propagator holds as well.
• One can derive diagrammatic proofs for this
• Gauge invariance and the Ward identity beeing fulfilled have a similar structure:
  • In $\xi$ gauge the photon propagator contains a term $\propto \xi p_\mu p_\nu$.
  • We can write the final matrix element as $\mathcal{M}=M_{\mu\nu}\Pi_{\mu\nu}$.
  • For Gauge invariance to hold we need $\xi M_{\mu\nu}p^\mu p^\nu =0$
  • For the Ward identity we need $M_\mu p^\mu=0$, if $\mathcal{M}=M_\mu\epsilon^\mu$.

• Using the property of lorentz invariance of a matrix element $\mathcal{M}$ in scalar QED one can directly derive charge conservation by using that there are lorentz transformations for which $\epsilon_\mu \to \epsilon_\mu + q_\mu$ for fixed momentum (excatly the little group).
• This is a general result for massless spin 1 theories.
• The proof does not require any applications of gauge invarience or a Lagrangian, but only lorentz invariance of the final matrix element and the transformation of the polarizations
• Similar proofs can be done for a massless spin 2 particle, however one finds that:

There are no interacting theories of massless particles with spin greater than 2