Understanding 1PI Vertex and Self-Energy Conventions in Quantum Field Theory

[geoduck]

I assume that if your theory has an interaction term written as L_I= ±λ*fields, where λ is the coupling, that the 1PI vertex is defined as:

$$i(\pm \Gamma)=i(\pm \lambda)+loops$$

That way to tree level, Γ is λ, and not -λ. The exception seems to be the self-energy. A mass term has -m²*field², so this suggests that you should write -i Π whenever you insert a self energy. However, it seems textbooks write +iΠ for self-energy insertions. This leads to propagators that look like:

$$\frac{i}{p^2-m^2+\Pi(p^2)}$$

where Π has the opposite sign of m².

Is it conventional to define the self-energy with +i instead of -i, so that in the propagator it has the opposite sign of the mass? Why is this so?

Also, is the self-energy even a 1PI vertex? Don't you have to include the tree-level term? So really:

$$i\Gamma^{(2)}=-i(p^2-m^2+\Pi(p^2) )$$

Or should it be defined:

$$-i\Gamma^{(2)}=-i(p^2-m^2+\Pi(p^2) )$$

It's not clear how to logically define the sign of Γ⁽²⁾. Should it be the same sign as m², or Π(p²)?

 

[vanhees71]

No, usually you identify -\mathrm{i} \Pi for a truncated 1PI self-energy diagram, then you get
\mathrm{i} G(p)=\frac{\mathrm{i}}{p^2-m^2-\Pi}.
The real part of the self-energy is then a correction to m^2 as it should be. Note that I use the west-coast convenction of the metric, \eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1).

It's different for massive vector bosons. There you have
\mathcal{L}_{\text{free}}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} +\frac{1}{2} m^2 A_{\mu} A^{\mu}.
This change in sign of the mass term is due to the west-coast convention of the metric, which is "mostly negative". Then the self-energy (or polarization) tensor as represented by a corresponding truncated 1PI Feynman diagram is +\mathrm{i} \Pi^{\mu \nu}, so that you get again a contribution to the mass with the correct sign.

Of course, there are as many conventions as textbooks, and it's a pain in the a... to sort these signs out when comparing a calculation in one convention with one done in another :-(.

 

[geoduck]

In mark sredinicki's textbook, he has the self energy opposite the sign of the mass in the propagator. He uses the east coast metric however. But I don't see why using the east coast metric (-+++) should lead you to want to define the self energy as having opposite sign to the mass.

I checked cheng and li's book, and they use West coast metric, and they have self energy same sign as mass, but they don't define interaction 1pI's with a prefactor of 'i' at all, so they have $$ \Gamma^4=-i lambda+loops $$.

 

[vanhees71]

As I said, there are as many conventions as books.