Anomaly cancellation in the Standard model requires $B-L$ to be constant, which is done using perturbative diagrammatic expansion. Secondly, baryon number is conserved as an $U(1)$ global field symmetry in the standard model Lagrangian. The question is:

  • Why/how does then baryon number violation take place in the non-perturbative regime of the electroweak theory?

  • And are there any Goldstone modes generated corresponding to the violation of the Baryon number $U(1)$ symmetry in the Lagrangian as well as for the violation of the lepton number symmetry?


1 Answer 1


In general the theory with nontrivial chiral structure isn't invariant under chiral transformations of the fermion field; corresponding phenomena is called anomaly, and it leads to nonconservation of corresponding current. Formally it is related to the fact that there is no way to define bot gauge invarian and chiral invariant regularization for such class of diagrams. Particularly, if we have $U(1)$ global nonchiral symmetry for theory with fermions, but fermions interact through chiral theory, then this $U(1)$ symmetry becomes broken. For example, in SM the baryon number is defined through corresponding global nonchiral symmetry, and to break it we need to look for chiral gauge symmetry group. SM local symmetry group is based on $SU_{c}(3)\times SU_{L}(2)\times U_{Y}(1)$, from which the only chiral group is $SU_{L}(2)$. So that the only anomaly is $U(1)SU_{L}(2)^2$: we have that $$ \tag 1 \partial_{\mu}J^{\mu}_{B} = \frac{3g_{EW}}{16\pi^2}F_{a}^{SU(2)}\tilde{F}_{a}^{SU(2)} $$ Absolutely analogical thing is for lepton current: $$ \tag 2 \sum_{l}\partial_{\mu}J^{\mu}_{l} = \frac{3g_{EW}}{16\pi^2}F_{a}^{SU(2)}\tilde{F}_{a}^{SU(2)} $$ Why this nonconservation is nonperturbative? The reason is that $F\tilde{F}$ can be expressed as full derivative, $F\tilde{F} = \partial K$, and this means that the contribution of corresponding correlator in Feynman diagrams is exactly zero in all orders (look here for details). But in fact such correlator is nonzero due to nontrivial topology of the $SU(2)$ group. Corresponding comfigrations for which this term is not zero called (in electroweak theory) instanton-like configurations (pure instantons are forbidden). The other fact (it is more important here) which relates anomalous current nonconservation to nonperturbative effects is that the anomaly equations $(1),(2)$ are one-loop exact: we need to compute only triangle diagram for getting exact in all orders Eqs. $(1),(2)$. Corresponding theorem was proved by Adler and Bardeen.

Which Goldstone modes you discuss? There can't be spontaneous breaking of symmetries such as baryonic and lepronic symmetry in SM. This is (again) nonperturbative result which was proved by Witten. The breaking of baryon and lepton numbers symmetries is explicit, not spontaneous.

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    $\begingroup$ In fact $U(1)_Y$ is also chiral, insofar as it acts differently on left- and right-handed fields. The divergence of the baryon current should thus look more like that in physics.stackexchange.com/questions/300050/…. I believe this distinction ends up being irrelevant on account that there are no non-trivial $U(1)$ fibre bundles over $S^4$. $\endgroup$
    – gj255
    Commented Mar 15, 2019 at 17:00

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