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Lets assume some theory which concludes sets of generations of fermions (lets call them $A$ and $B$). Fermions $A$ have some gauge group $G_{A}$ (for example, SM), while fermions $B$ are charged under other 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 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 $G_{A}$.

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

  1. As for me, by this naive integration (without intoduction some non-invariant terms) we make theory anomalous; but if mother theory is gauge invariant, in an effective theory gauge invariance must hold too. Is it true?

  2. 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 correlator $\langle \bar{\psi}_{B}\gamma_{5}\psi_{B} \rangle$. To contract gauge-variant part of $(1)$ we have to introduce 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 it true?

    1. 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 it true?

    2. Finally, are Wess-Zumino terms the only terms which break unitarity in an effective action $\Gamma$ before introducing the counterterm? Or this is completely wrong statement?

Some prehistory

The questions have arisen after reading of article, in which there is an exlpanation how nontrivial anomaly cancellation in fundamental theory provides effects of non-decoupling of massive fermions. As example there is toy-model with two sets of chiral fermions which lagrangian has $U_{X}(1), U_{Y}(1)$ symmetries; they then are integrated out; after that unsuppressed by fermions masses effective operators arise; look at Eq.(7). I want to know about the nature of these terms; 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 detailed discussion about 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 case is following. 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 lepton fermions $A$ anomaly is cancelled by fermions $B$ anomaly. Let's then integrate $B$ fermions out. 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 Wess-Zumino term, $\Gamma_{WZ}[U, A_{L},A_{R}]$, where $A_{L/R}$ denotes gauge field 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 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 set of background vector fields who correspond to 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 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).

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  • $\begingroup$ Comment to the question (v9): Consider adding references to make the question more accessible to the reader and focus the answers. $\endgroup$ – Qmechanic May 11 '15 at 10:54

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