The sign of the QED vertex in Feynman rules

Question

I am confused with the sign of the QED vertex. It seems that in different books, the signs are different.

1. In Chapter 7 of Griffiths' book, the vertex is of the form $-iq\gamma^\mu$ (the particle (electron) has a charge $q = -e$, so the vertex is $ie\gamma^\mu$)

And if I search online for the "QED vertex", many of the pages and notes state that the vertex is $-iq\gamma^\mu$.

2. But in Peskin and Schroeder's book, equation (A.6), the vertex is $iq\gamma^\mu$ ($q = -e$ for particle (electron))

3. Similarly, in Schwartz's book, equation (13.22), it seems that the vertex is $iq\gamma^\mu$ ($q = -e$ for particle (electron))

I am confused about the sign, which one is correct? $iq\gamma^\mu$ or $-iq\gamma^\mu$?

Answer 1

TL;DR: Different authors have different conventions. E.g. the covariant derivative is $D_{\mu}=d_{\mu}\pm iqA_{\mu}$. Nevertheless, everybody seems to agree that the Dirac Lagrangian density is ${\cal L}=i\bar{\psi}\gamma^{\mu}D_{\mu}\psi+\ldots$, so the interaction term becomes ${\cal L}_{\rm int}=\mp q\bar{\psi}\gamma^{\mu}A_{\mu}\psi$, and hence the Feynman vertex rule is $\mp iq\gamma^{\mu}$, respectively, i.e. it comes down to a sign ambiguity.

For a correlation function with an odd number of external photon legs, this yields an absolute sign ambiguity (but no relative sign ambiguities), and hence no physical consequences for probabilities if we flip the overall sign of all the charges $q\to -q$.

Concerning the references listed by OP it goes as follows:

1. D. Griffiths, Introduction to elementary particles, has

   • Minkowski sign convention $(+,-,-,-)$ (p. 85);

   • charge sign convention $e=|e|$ (p.395);

   • covariant derivative $D_{\mu}=d_{\mu}+iqA_{\mu}$ (p.349,352,365);

   • interaction term $-q\bar{\psi}\gamma^{\mu}A_{\mu}\psi$ (p.348);

   • and Feynman vertex rule $-iq\gamma^{\mu}$ (p.230).

2. M.E. Peskin and D.V Schroeder, An Introduction to QFT, has

   • Minkowski sign convention $(+,-,-,-)$ (p. xix);

   • charge sign convention $e=-|e|$ (p.809);

   • covariant derivative $D_{\mu}=d_{\mu}+iqA_{\mu}$ (p.303);

   • interaction term $-q\bar{\psi}\gamma^{\mu}A_{\mu}\psi$ (p.303);

   • and Feynman vertex rule $-iq\gamma^{\mu}$ (p.303,802).

3. M.D. Schwartz, QFT & the standard model, has

   • Minkowski sign convention $(+,-,-,-)$ (p.12,817);

   • charge sign convention $e=|e|$ (p.818);

   • covariant derivative $D_{\mu}=d_{\mu}-iqA_{\mu}$ (p.140,224,818);

   • interaction term $q\bar{\psi}\gamma^{\mu}A_{\mu}\psi$ (p.225);

   • and Feynman vertex rule $iq\gamma^{\mu}$ (p.226).

For more on conventions in QED, see e.g. this Phys.SE post.

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$^1$ We work in units with $c=1=\hbar$.