This statement may be well-known. For many massless theories, these tadpole diagram graphs vanish in dimensional regularization (by dimensional analysis and the absence of any inherent mass scale in the loop integral).

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How can one show this is QFT statement is true(tadpole diagram graphs vanish) regardless the coupling, say by Yukawa-Higgs via $\lambda \bar{\Psi} \phi \Psi$ or by gauge field $g \bar{\Psi} \gamma^\mu A_\mu \Psi$, etc? i.e. does not depend on $\lambda,g$ or an energy cutoff $\Lambda$?

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    $\begingroup$ if there is a soliton background, then it shall not be zero due to Jackiw-Rebbi or Goldstone-Wilczek effect. $\endgroup$
    – wonderich
    Jun 5, 2014 at 21:51

1 Answer 1


If there is a soliton background (of dashed lines), then this tadpole diagram with extended legs like

enter image description here

shall not be zero due to Jackiw-Rebbi or Goldstone-Wilczek effect; the soliton background (dashed lines) will induce nontrivial (fractionalized) induced quantum numbers (or simply the charge or vacuum expectation value) for the scalar bosons (solid line); even if the original scalar bosons(solid line) are massless.

For example. see the bosonic anomaly in arxiv-1403.5256 - Bosonic Anomaly, Induced Fractional Quantum Number and Degenerate Zero Modes.

Interestingly, such an effect takes place on the surface of Symmetry Protected Topological(SPT) States as noticed in this Ref. And see Reference therein, e.g. about J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981) and R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).


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