Let us assume some theory which consists of sets of generations of fermions (let us call them $A$ and $B$). Fermions $A$ have some gauge group $G_{A}$ (for example, SM), while fermions $B$ are charged under another group $G_{B}$ as well as under some subgroup of $G_{A}$ (for example, $SU(2)\times U_{Y}(1)$). Group $G_{B}$ (for example, $U_{B}(1)$) is spontaneously broken by vacuum averaged values $v_{B}$ due to a Higgs-like mechanism (for case of $G_{B}$ isomorphism to $U_{B}(1)$ - with Higgs-like field $h_{B} = v_{B}e^{i\theta_{B}}$).

Theories of $A$ and $B$ fermions are anomaly-free separately, while the mixed anomalies are cancelled by fixing charges of $B$ fermions under $G_{B}$ and $G_{A}$.

Let us then assume that all $B$ fermions are very massive; we thus can integrate them out for getting an effective action $\Gamma$ that contains an interaction between $G_{A}$ and $G_{B}$ gauge fields. I have a few questions about anomalies in $\Gamma$.

- It seems to me that by this naive integration (without introduction of some non-invariant terms) we make the theory anomalous; but if the mother theory is gauge invariant, gauge invariance must hold in the effective theory too. Is this true?
- An effective action, which is derived after integrating out massive fermions, consists of Wess-Zumino terms $$ \tag 1 \Gamma_{WZ} = c_{1} \int d^{4}x \theta_{B}F_{A}^{\mu\nu}\tilde{F}^{A}_{\mu \nu} + c_{2}\int d^{4}x \theta_{B}F_{A}^{\mu\nu}\tilde{F}^{B}_{\mu \nu}, \quad \tilde {F}_{\mu \nu} = \epsilon_{\mu \nu \alpha \beta}F^{\alpha \beta}, $$ which are connected with mixed anomalies and arise from triangular diagrams in the correlator $\langle \bar{\psi}_{B}\gamma_{5}\psi_{B} \rangle$. To contract the gauge-variant part of $(1)$ we have to introduce a generalized Chern-Simons counterterm $$ \tag 2 \Delta \Gamma_{CS} = c_{3}\int d^{4}x \epsilon^{\mu \nu \alpha \beta}A_{\mu}\left(B_{\nu}\partial_{\alpha}A_{\beta} + \frac{1}{3}e_{A}\epsilon_{abc}A^{a}_{\nu}A^{b}_{\alpha}A^{c}_{\beta}\right), $$ which arises from process $B_{\mu} \to A^{a}_{\mu}A^{b}_{\nu}$ in one-loop approximation. Is this true?
- The coefficients in front of Wess-Zumino terms are determined uniquely (are regularization independent), which is connected with the fact that they collect all anomomaly effects in theory. Is this true?
- Finally, are Wess-Zumino terms the only terms that break unitarity in an effective action $\Gamma$ before introducing the counterterm? Or is this a completely wrong statement?

**Some prehistory**

The questions have arisen after reading of article, in which there is an explanation how nontrivial anomaly cancellation in a fundamental theory provides effects of non-decoupling of massive fermions. As example there is the toy-model with two sets of chiral fermions whose Lagrangian has $U_{X}(1), U_{Y}(1)$ symmetries; they then are integrated out; after that effective operators unsuppressed by fermions masses arise; look at Eq.(7). I want to know about the nature of these terms; the first two terms I identify as Wess-Zumino terms; they are regularization independent, as is claimed in an article. The last term arises, if I understand correctly, as counterterm which arises for the process $A_{1} \to A_{2},A_{2}$.

But in some articles (for example, Preskill's Gauge anomalies) Wess-Zumino terms $(1)$ as well as counterterm $(2)$ are interpreted as counterterms which initially aren't included in an effective action (look at page 25 for a detailed discussion of contraction of gauge-variant terms in an effective action of $SU(2)\times U(1)$ theory). So there is a bad mix in my head about anomalies contraction in an effective field theory.

**An edit**

It seems that the following is the case. Let's temporarily turn off $G_{B}$ interactions. This provides that both $A, B$ fermions interact only with $G_{A}$ fields. Then there is the fact that the lepton fermions $A$ anomaly is cancelled by the fermions $B$ anomaly. Let's then integrate $B$ fermions out. The resulting effective field theory must be anomaly-free, so that it must contain some term which is changed as well as $B$ fermions part of mother action under gauge transformation. This term is called the Wess-Zumino term, $\Gamma_{WZ}[U, A_{L},A_{R}]$, where $A_{L/R}$ denotes a gauge field that interacts with left or right $B$ fermions $\frac{1 \mp \gamma_{5}}{2}\psi_{B}$ (for example, $A_{L} = \gamma + Z$, $A_{R} = \gamma$). By denoting the action which consists of $A$ fermions as $S_{A}$ which has anomaly, $$ \delta S_{A} = \Gamma_{anomaly}, $$ we have $$ \delta_{anomalous} \left( \Gamma_{WZ} + S_{A}\right) = -\Gamma_{anomaly} + \Gamma_{anomaly} = 0 $$ Let's then introduce $G_{B}$ interactions. It seems that only mixed anomalies cancellation is interesting. So maybe it is convenient to introduce $\Gamma_{WZ}[U, A_{L} + b, A_{R} + b]$ term, where $b$ corresponds to a set of *background* vector fields corresponding to the adjoint representation of $G_{B}$. Then $$ \delta_{anomalous} \left( \Gamma_{WZ} + S_{A}\right) = -\Gamma_{anomaly} + \Gamma_{anomaly} + \Gamma_{anomaly}[\varphi , b, A_{L}, A_{R}] \neq 0 $$ We need to introduce a counterterm $\Gamma_{ct}[b, A_{L}, A_{R}, \varphi]$ (if it exists) which has variation equal to $\Gamma_{anomaly}[b, A_{L}, A_{R}]$. It is possible, and the sum of $\Gamma_{WZ} + \Gamma_{ct}$ contains new interactions of type $$ b \wedge Z \wedge \partial Z,\quad b \wedge Z \wedge F^{\gamma},\quad b \wedge Z \wedge \partial b $$ (this is important for the first linked article).

This post imported from StackExchange Physics at 2015-05-11 19:55 (UTC), posted by SE-user Name YYY