Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Technical question about loop corrections

  1. May 28, 2015 #1
    Does anyone know a simple explanation for the following statement:

    Gauge invariance ⇒ $Πμνϒϒ(0) = ΠμνϒZ(0) = 0$

    Where ΠVV' is the V to V' one loop correction, ϒ is the photon field and Z is the Z-boson field. The argument of Π is the incoming momentum q2 = 0
  2. jcsd
  3. May 29, 2015 #2


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    Your photon propagator is
    $$D_{\mu \nu}(q)=-\frac{1}{q^2-q^2 \Pi(q^2)+\mathrm{i} 0^+}[g_{\mu \nu}-q_{\mu} q_{\nu}] + A(q^2) q^{\mu} q^{\nu},$$
    where ##A(q^2)## is a gauge-dependent non-interacting piece, which doesn't enter any physical result.

    Now the photon has strictly 0 mass. Together with the Ward-Takahashi identity of the photon polarization tensor, which makes it purely 4-transverse, this implies that
    $$\Pi_{\mu \nu}=q^2 \Pi(q^2) (g_{\mu \nu}-q_{\mu} q_{\nu}).$$
    ##\Pi## is a logarithmically divergent scalar. Now to make the residuum of the photon propgator 1 at ##q^2=0##, you impose the renormalization condition
    The same argument holds for the ##\gamma##-Z mixing piece too.

    Note that the above renormalization condition is dangerous with regard to infrared divergences, which must be resummed. For this purpose it's better to choose the renormalization point in the space-like.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Technical question loop Date
I X-ray dose question Wednesday at 3:54 PM
I X-ray tube to gamma tube question Tuesday at 3:51 AM
I Technicality with Noether's theorem Apr 22, 2017
A technical question -- derivation of an equation in a theoretical paper on dark matter Jan 22, 2016
Majorana particle technical? Jun 3, 2013